# Abstract Besov-Hardy-Sobolev spaces and elliptic boundary value problems with complex bounded measurable coefficients

Alex Amenta, Pascal Auscher

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Alex Amenta, Pascal Auscher. Abstract Besov-Hardy-Sobolev spaces and elliptic boundary value problems with complex bounded measurable coefficients. 2017, CRM Monographs. hal-01345585v2

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Distributed under a Creative Commons Attribution 4.0 International License
Abstract Besov-Hardy-Sobolev spaces and elliptic
boundary value problems with complex bounded
measurable coefficients

Alex Amenta

Pascal Auscher

DELFT INSTITUTE OF APPLIED MATHEMATICS, DELFT UNIVERSITY OF TECHNOLOGY, P.O. BOX 5031, 2628 CD DELFT, THE NETHERLANDS

E-mail address: amenta@fastmail.fm

LABORATOIRE DE MATHÉMATIQUES D'ORSAY, UNIV. PARIS-SUD, CNRS,
UNIVERSITÉ PARIS-SACLAY, 91405 ORSAY, FRANCE

E-mail address: pascal.auscher@math.u-psud.fr
2010 Mathematics Subject Classification. Primary 35J25, 42B35, 47A60; Secondary 35J57, 35J46, 35J47, 42B25, 42B30, 42B37, 47D06

*Key words and phrases.* Second-order elliptic systems, boundary value problems, tent spaces, Besov–Hardy–Sobolev spaces, bisectorial operators, functional calculus, off-diagonal estimates, interpolation, layer potentials

**ABSTRACT.** In this monograph our main goal is to study the well-posedness of boundary value problems of Dirichlet and Neumann type for elliptic systems div $A\nabla u = 0$ on the upper half-space with coefficients independent of the transversal variable, and with boundary data in fractional Hardy–Sobolev and Besov spaces. Our approach uses minimal assumptions on the coefficients $A$, and in particular does not require De Giorgi–Nash–Moser estimates. Our results are completely new for the Hardy–Sobolev case, and in the Besov case they extend corresponding results recently obtained by Barton and Mayboroda.

First we develop a theory of Besov–Hardy–Sobolev spaces adapted to operators which are bisectorial on $L^2$, with bounded $H^\infty$ functional calculus on their ranges, and which satisfy $L^2$ off-diagonal estimates. In particular, this theory applies to perturbed Dirac operators $DB$. We then prove that for a nontrivial range of exponents (the identification region) the Besov–Hardy–Sobolev spaces adapted to $DB$ are equal to those adapted to the unperturbed Dirac operator $D$ (which correspond to classical Besov–Hardy–Sobolev spaces).

Our main result is the classification of solutions to the elliptic system div $A\nabla u = 0$ within a certain classification region of exponents, defined in terms of the identification regions for certain perturbed Dirac operators associated with $A$. More precisely, we show that if the conormal gradient of a solution belongs to a weighted tent space (or one of their real interpolants) with exponent in the classification region, and in addition vanishes at infinity in a certain sense, then it has a trace in a Hardy–Sobolev (or Besov) space, and can be represented as a semigroup evolution of this trace in the transversal direction. As a particular corollary, any such solution can be represented in terms of an abstract layer potential operator.

Within the classification region, we show that well-posedness of one of the boundary value problems under consideration is equivalent to a certain boundary projection being an isomorphism. We derive various consequences of this characterisation, including interpolation, extrapolation, and duality properties of well-posedness, as well as some results on stability of well-posedness under coefficient perturbation. These consequences are illustrated in various situations, including in particular that of the Regularity problem for real equations.
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# Contents

<table><tbody><tr><td>Chapter 1. Introduction</td><td>1</td></tr><tr><td>1.1. Introduction and context</td><td>1</td></tr><tr><td>1.2. Summary of the monograph</td><td>15</td></tr><tr><td>1.3. Notation</td><td>17</td></tr><tr><td>Acknowledgements</td><td>18</td></tr><tr><td>Chapter 2. Function Space Preliminaries</td><td>19</td></tr><tr><td>2.1. Exponents</td><td>19</td></tr><tr><td>2.2. Tent spaces</td><td>21</td></tr><tr><td>2.3. Z-spaces</td><td>24</td></tr><tr><td>2.4. Unification: tent spaces, Z-spaces, and slice spaces</td><td>31</td></tr><tr><td>2.5. Homogeneous smoothness spaces</td><td>36</td></tr><tr><td>2.6. Appendix: Factorisation and interpolation of Z-spaces</td><td>40</td></tr><tr><td>2.7. Appendix: Table of exponents and function spaces</td><td>44</td></tr><tr><td>Chapter 3. Operator Theoretic Preliminaries</td><td>45</td></tr><tr><td>3.1. Bisectorial operators and holomorphic functional calculus</td><td>45</td></tr><tr><td>3.2. Off-diagonal estimates and the Standard Assumptions</td><td>49</td></tr><tr><td>3.3. Integral operators on tent spaces</td><td>51</td></tr><tr><td>3.4. Extension and contraction operators</td><td>55</td></tr><tr><td>Chapter 4. Adapted Besov-Hardy-Sobolev Spaces</td><td>59</td></tr><tr><td>4.1. Initial definitions, equivalent norms, and duality</td><td>59</td></tr><tr><td>4.2. Mapping properties of the holomorphic functional calculus</td><td>63</td></tr><tr><td>4.3. Completions, interpolation, and inclusions</td><td>64</td></tr><tr><td>4.4. The Cauchy operator on general adapted spaces</td><td>69</td></tr><tr><td>Chapter 5. Spaces Adapted to Perturbed Dirac Operators</td><td>73</td></tr><tr><td>5.1. D-adapted spaces</td><td>73</td></tr><tr><td>5.2. Similarity of functional calculus: going between DB and BD</td><td>75</td></tr><tr><td>5.3. Inclusions and identifications of DB- and D-adapted spaces</td><td>77</td></tr><tr><td>5.4. Projections and BD-adapted spaces</td><td>81</td></tr><tr><td>5.5. The Cauchy operator on DB-adapted spaces</td><td>83</td></tr><tr><td>5.6. Appendix: Technicalities involving extensions and projections</td><td>89</td></tr><tr><td>Chapter 6. Classification of Solutions to Cauchy-Riemann Systems and Elliptic Equations</td><td>93</td></tr><tr><td>6.1. Basic properties of solutions</td><td>93</td></tr><tr><td>6.2. Decay of solutions at infinity</td><td>94</td></tr><tr><td>6.3. Classification and representation theorems</td><td>96</td></tr><tr><td>6.4. Proofs of classification theorems</td><td>99</td></tr></tbody></table>
<table><tr><td>6.5. Interpolation of solution spaces</td><td>116</td></tr><tr><td>6.6. Boundary behaviour of solutions</td><td>118</td></tr><tr><td>Chapter 7. Applications to Boundary Value Problems</td><td>125</td></tr><tr><td>7.1. Characterisation of well-posedness and corollaries</td><td>125</td></tr><tr><td>7.2. Regions of well-posedness for certain classes of coefficients</td><td>130</td></tr><tr><td>7.3. Stability of well-posedness under perturbation of coefficients</td><td>135</td></tr><tr><td>7.4. The method of layer potentials</td><td>139</td></tr><tr><td>Bibliography</td><td>145</td></tr><tr><td>Index</td><td>149</td></tr></table>
CHAPTER 1

# Introduction

## 1.1. Introduction and context

In this monograph we investigate well-posedness of boundary value problems associated with divergence-form elliptic equations

$$ (1.1) \qquad L_A u := \operatorname{div} A \nabla u = 0, $$

where the unknown $u$ is a $\mathbb{C}^m$-valued function on the upper half-space

$$ \mathbb{R}_+^{1+n} := \{(t, x) \in \mathbb{R}^{1+n} : t > 0\}. $$

We work in ambient dimension $1+n \ge 2$, with $m \ge 1$. The equation (1.1) may be considered as system of $m$ scalar equations. The value of $m$ is quite irrelevant to everything that we do, so for simplicity the reader may assume $m = 1$ throughout.

We use the so-called ‘first order approach’, following previous work by the second author with Hofmann, McIntosh, Mourgoglou, Rosén (Axelsson), and Stahlhut [6, 8, 7, 5, 17, 13, 15, 16, 19]. In these articles, boundary value problems are considered with (boundary) data of regularity 0 (i.e. data in Lebesgue spaces $L^p$ and Hardy spaces $H^p$) or of regularity $-1$ (i.e. in Sobolev spaces $W^p_{-1}$, Hölder spaces $\dot{A}^\alpha_{-1}$, or in $B\dot{M}O_{-1}$). Here we address problems with fractional regularity data: that is, data in Hardy–Sobolev spaces $\dot{H}^p_\theta$ or Besov spaces $\dot{B}^{p,p}_\theta$ with $\theta \in (-1, 0)$. These were considered in the previous articles only when $p = 2$.

Boundary value problems for the equation $L_A u = 0$ with data in Besov spaces have recently been studied by Barton and Mayboroda [22]. Under the additional assumptions that $m = 1$ and that solutions to (1.1) satisfy De Giorgi–Nash–Moser estimates (see (7.2)), they establish various well-posedness results via the method of layer potentials. One novelty of their approach is that they can also consider inhomogeneous problems $L_A u = f$, which we do not address. For homogeneous problems, however, our results are more general. The first order approach requires neither the existence of fundamental solutions (which is implied by De Giorgi–Nash–Moser estimates when $m = 1$, and which is crucial in setting up the method of layer potentials) nor the validity of trace theorems (which hold for Besov spaces, but not for Hardy–Sobolev spaces).

Our approach is based on an abstract framework of adapted Besov–Hardy–Sobolev spaces. For applications to boundary value problems with data of regularity 0 and $-1$, a theory of adapted Hardy spaces had been sufficiently developed by the second author and Stahlhut [19]. We extend this theory by exploiting properties of weighted tent spaces $T^p_\theta$ and their real interpolants, the $Z$-spaces $Z^p_\theta$.

### 1.1.1. The elliptic equation.
Consider again the elliptic equation (1.1). The gradient operator $\nabla$ maps $\mathbb{C}^m$-valued functions in $1+n$ variables $f$ to $\mathbb{C}^{m(1+n)}$-valued functions $\nabla f$ by writing $f = (f^j)_{j=1}^m$ as an $m$-tuple of $\mathbb{C}$-valued functions
and acting as the usual gradient operator on each component $f^j$. The divergence operator div is similarly defined in terms of the usual divergence operator, sending $\mathbb{C}^{m(1+n)}$-valued functions to $\mathbb{C}^m$-valued functions. These differential operators are interpreted in the distributional sense. Vectors $v \in \mathbb{C}^{m(1+n)}$ are split into transversal and tangential parts $v = (v_\perp, v_\parallel)$ according to the decomposition

$$ (1.2) \qquad \mathbb{C}^{m(1+n)} = \mathbb{C}^m \oplus \mathbb{C}^{mn}, $$

and likewise $\mathbb{C}^{m(1+n)}$-valued functions $f$ can be split into transversal and tangential parts $f = (f_\perp, f_\parallel)$, valued in $\mathbb{C}^m$ and $\mathbb{C}^{mn}$ respectively. We write $\nabla_\parallel$ and $\text{div}_\parallel$ for the corresponding tangential restrictions of $\nabla$ and $\text{div}$.

Throughout the monograph we assume that the coefficient matrix

$$ A \in L^\infty(\mathbb{R}_+^{n+1} : \mathcal{L}(\mathbb{C}^{m(1+n)})) $$

is bounded, measurable, and t-independent, meaning that $A(t, x) = A(x)$ for almost every $(t, x) \in \mathbb{R}_+^{1+n}$. Thus we may consider $A$ as an element of $L^\infty(\mathbb{R}^n : \mathcal{L}(\mathbb{C}^{m(1+n)}))$. We also assume that $A$ is strictly accretive on curl-free vector fields, meaning that there exists $\kappa > 0$ such that

$$ (1.3) \qquad \text{Re} \int_{\mathbb{R}^n} (A(x)f(x), f(x)) \, dx \geq \kappa \|f\|_2^2 $$

for all $f \in L^2(\mathbb{R}^n : \mathbb{C}^{m(1+n)})$ such that $\text{curl}_\parallel(f_\parallel) = 0$. The round bracket in the integrand above is the usual Hermitean inner product on $\mathbb{C}^{m(1+n)}$. By $\text{curl}_\parallel(f_\parallel) = 0$ we mean that

$$ \partial_j f_k = \partial_k f_j \quad (1 \le j, k \le n, \ j \neq k), $$

where $\partial_j$ is the distributional partial derivative in the j-th coordinate direction of $\mathbb{R}^n$, acting componentwise on $\mathbb{C}^m$-valued functions. The strict accretivity condition (1.3) is weaker than the usual notion of pointwise strict accretivity

$$ \text{Re}(A(x)v, v) \geq \kappa|v|^2 \quad (v \in \mathbb{C}^{m(1+n)}, \ x \in \mathbb{R}^n) $$

unless $m=1$ and $A$ is real, in which case these two notions are equivalent [8, §2].

As we have not assumed any regularity of $A$, we must consider weak solutions to (1.1): we say that a function $u \in W_{1,\text{loc}}^2(\mathbb{R}^n : \mathbb{C}^m)$ solves (1.1) if for all $\varphi \in C_0^\infty(\mathbb{R}_+^{1+n} : \mathbb{C}^m)$ we have

$$ \iint_{\mathbb{R}_+^{1+n}} (A(x)\nabla u(t, x), \nabla \varphi(t, x)) \, dx \, dt = 0. $$

### 1.1.2. Formulation of boundary value problems.
Various boundary value problems for the equation $L_A u = 0$ have been studied. Here we reformulate some of these problems, and introduce some new ones. In Definition 7.1 we concisely synthesise these problems into two categories, each parametrised by a set of exponents and two (related) families of spaces of boundary data.

First, for $1 < p < \infty$, following the groundbreaking work of Dahlberg on the Laplace equation on Lipschitz domains [29], we formulate the $L^p$-Dirichlet problem for $L_A$, denoted by $(D_H)_{0,A}^p$:

$$ (D_H)_{0,A}^p : \left\{ \begin{array}{ll} L_A u = 0 & \text{in } \mathbb{R}_+^{1+n}, \\ \|N_* u\|_{L^p} \lesssim \|f\|_{L^p}, & \\ \lim_{t \to 0} u(t, \cdot) = f \in L^p(\mathbb{R}^n : \mathbb{C}^m). & \end{array} \right. $$

This should be read:
for all $f \in L^p(\mathbb{R}^n : \mathbb{C}^m)$,
there exists $u \in W_{1,\text{loc}}^2(\mathbb{R}_+^{1+n} : \mathbb{C}^m)$ solving $L_A u = 0$,
with $\|N_*u\|_p \lesssim \|f\|_p$ (the interior estimate),
and $u(t, \cdot) \to f$ in $L^p$ as $t \to 0$ (the boundary condition).

Here $N_*$ is the non-tangential maximal function

$$N_* u(x) := \sup_{\substack{(t,y) \in \mathbb{R}_+^{1+n} \\ y \in B(x,t)}} |u(t,y)|.$$

We say that the problem $(D_H)_{0,A}^p$ is well-posed if for all $f \in L^p(\mathbb{R}^n : \mathbb{C}^m)$ there exists a unique $u$ satisfying these conditions.<fnref n="1" />

Well-posedness is defined analogously for all boundary value problems that we consider: for all boundary data in the specified function space, there must exist a unique solution to (1.1)—up to an additive constant, for Regularity and Neumann problems—which satisfies the conditions of the boundary value problem. Of course, a boundary value problem may or may not be well-posed.

For $n/(n+1) < p < \infty$, we formulate the $H^p$-Regularity problem for $L_A$:

$$(R_H)_{0,A}^p : \left\{ \begin{array}{ll} L_A u = 0 & \text{in } \mathbb{R}_+^{1+n}, \\ \| \tilde{N}_*(\nabla u) \|_{L^p} \lesssim \| \nabla \|f\|_{H^p}, \\ \lim_{t \to 0} \nabla \|u(t, \cdot)\| = \nabla \|f\| \in H^p(\mathbb{R}^n : \mathbb{C}^{mn}), \end{array} \right.$$

where $\tilde{N}_*u$ is the modified non-tangential maximal function

$$(1.4) \quad \tilde{N}_* u(x) := \sup_{\substack{(t,y) \in \mathbb{R}_+^{1+n} \\ y \in B(x,t)}} \left( \iint_{\substack{t/2 < \tau < 2t \\ \xi \in B(y,t)}} |u(\tau, \xi)|^2 d\tau \, d\xi \right)^{1/2} \quad (x \in \mathbb{R}^n),$$

and where $H^p(\mathbb{R}^n : \mathbb{C}^{mn})$ is the $(\mathbb{C}^{mn}$-valued) real Hardy space, which may be identified with $L^p(\mathbb{R}^n : \mathbb{C}^{mn})$ when $p > 1$. The restriction on $p$ arises because a gradient $\nabla \|f\|$ need not be in $H^p$ for $p \le n/(n+1)$, as it will only have one vanishing moment.

**Remark 1.1.** If $f$ is a distribution with $\nabla \|f\| \in H^p(\mathbb{R}^n : \mathbb{C}^{mn})$, then $f$ may be identified with an element of $\dot{H}_1^p(\mathbb{R}^n : \mathbb{C}^m)$—the $\mathbb{C}^m$-valued homogeneous Hardy–Sobolev space of order 1, defined in Section 2.5—and the boundary condition $\lim_{t \to 0} \nabla \|u(t, \cdot)\| = \nabla \|f\| \in H^p(\mathbb{R}^n : \mathbb{C}^{mn})$ is equivalent to the condition

$$\lim_{t \to 0} u(t, \cdot) = f \in \dot{H}_1^p(\mathbb{R}^n : \mathbb{C}^m).$$

Therefore, by considering functions $u$ rather than tangential gradients $\nabla \|u\|$, the $H^p$-Regularity problem can be seen as a kind of $\dot{H}_1^p$-Dirichlet problem. Conversely, by considering tangential gradients $\nabla \|u\|$, the $L^p$-Dirichlet problem $(D_H)_{0,A}^p$ can be seen as a kind of $\dot{H}_1^p$-Regularity problem. For technical reasons we generally consider Regularity problems rather than Dirichlet problems.

<footnote n="1">¹Use of the non-tangential maximal function $N_*$ is appropriate for $(D_H)_{0,A}^p$ provided that solutions to $L_A u = 0$ have pointwise values. Otherwise, it is better to use the modified non-tangential maximal function $\tilde{N}_*$ defined in (1.4). Also, the boundary condition $\lim_{t \to 0} u(t, \cdot) = f$ is traditionally imposed as non-tangential convergence almost everywhere rather than $L^p$ convergence; we shall return to this point later. For now, observe that there is a priori no relation between the interior estimate and the boundary condition, as there are no trace theorems for the space $\{u : \|N_*u\|_p < \infty\}$.</footnote>
For $n/(n+1) < p < \infty$, we also formulate the $H^p$-Neumann problem for $L_A$:

$$ (N_H)_{0,A}^p : \left\{ \begin{array}{ll} L_A u = 0 & \text{in } \mathbb{R}_+^{1+n}, \\ \|\tilde{N}_*(\nabla u)\|_{L^p} \lesssim \|\partial_{\nu_A} f\|_{H^p}, \\ \lim_{t \to 0} \partial_{\nu_A} u(t, \cdot) = \partial_{\nu_A} f \in H^p(\mathbb{R}^n : \mathbb{C}^m). \end{array} \right. $$

The A-conormal derivative $\partial_{\nu_A}$ of $u$ is defined by

$$ (1.5) \qquad \partial_{\nu_A} u(t, \cdot) = e_0 \cdot A \nabla u(\cdot, t), $$

where $e_0 = (1, 0, \dots, 0)$ is the normal vector to $\partial \mathbb{R}_+^{1+n}$ 'pointing in the t-direction'.<fnref n="2" />

The boundary value problems $(D_H)_{0,A}^p$, $(R_H)_{0,A}^p$, and $(N_H)_{0,A}^p$ are all problems of order zero: in each of these problems boundary data are assumed to be in the Lebesgue space $L^p$ or the Hardy space $H^p$. One can also formulate Regularity and Neumann problems of order $-1$. For $1 < p < \infty$, the $\dot{H}_{-1}^p$-Regularity problem, which is similar to the $L^p$-Dirichlet problem but with a different interior estimate<fnref n="3" /> and a decay condition at infinity (see Remark 1.1), is

$$ (R_H)_{-1,A}^p : \left\{ \begin{array}{ll} L_A u = 0 & \text{in } \mathbb{R}_+^{1+n}, \\ \|\nabla u\|_{T_{-1}^p} \lesssim \|\nabla \|f\|_{\dot{H}_{-1}^p}, \\ \lim_{t \to \infty} \nabla \|u(t, \cdot)\| = 0 & \text{in } \mathcal{Z}'(\mathbb{R}^n : \mathbb{C}^{mn}), \\ \lim_{t \to 0} \nabla \|u(t, \cdot)\| = \|f\|_{\dot{H}_{-1}^p} \in \mathbb{R}^n : \mathbb{C}^{mn}). \end{array} \right. $$

Here $\mathcal{Z}'(\mathbb{R}^n : \mathbb{C}^{mn})$ is the space of $\mathbb{C}^{mn}$-valued tempered distributions modulo polynomials, in which all homogeneous Hardy-Sobolev and Besov spaces are embedded. The space $T_{-1}^p$ is a particular instance of a weighted tent space: for $0 < p < \infty$, $\theta \in \mathbb{R}$, and $f: \mathbb{R}_+^{1+n} \to \mathbb{C}^N$ measurable with $N \in \mathbb{N}_+$ fixed, we have

$$ \|f\|_{T_{\theta}^{p}} := \left( \int_{\mathbb{R}^n} \left( \int_0^{\infty} \int_{B(x,t)} |t^{-\theta} f(t, y)|^2 \frac{dy dt}{t^{n+1}} \right)^{p/2} dx \right)^{1/p} . $$

We can also formulate boundary value problems with data in BMO-type spaces and homogeneous Hölder spaces $\dot{\Lambda}_\alpha$, thus increase the range of exponents to '$p \ge \infty$. For $0 < \alpha < 1$ we define

$$ (R_H)_{-1,A}^{(\infty,\alpha)} : \left\{ \begin{array}{ll} L_A u = 0 & \text{in } \mathbb{R}_+^{1+n}, \\ \|\nabla u\|_{T_{-1,\alpha}^{\infty}} \lesssim \|\nabla \|f\|_{\dot{\Lambda}_{\alpha-1}^{\infty}}, \\ \lim_{t \to \infty} \nabla \|u(t, \cdot)\| = 0 & \text{in } \mathcal{Z}'(\mathbb{R}^n : \mathbb{C}^{mn}), \\ \lim_{t \to 0} \nabla \|u(t, \cdot)\| = \|f\|_{\dot{\Lambda}_{\alpha-1}^{\infty}} \in \mathbb{R}^n : \mathbb{C}^{mn}). \end{array} \right. $$

and furthermore, with $\alpha = 0$,

$$ (R_H)_{-1,A}^{(\infty,0)} : \left\{ \begin{array}{ll} L_A u = 0 & \text{in } \mathbb{R}_+^{1+n}, \\ \|\nabla u\|_{T_{-1,0}^{\infty}} \lesssim \|\nabla \|f\|_{\dot{\text{BMO}}_{-1}}, \\ \lim_{t \to \infty} \nabla \|u(t, \cdot)\| = 0 & \text{in } \mathcal{Z}'(\mathbb{R}^n : \mathbb{C}^{mn}), \\ \lim_{t \to 0} \nabla \|u(t, \cdot)\| = \|f\|_{\dot{\text{BMO}}_{-1}} \in \mathbb{R}^n : \mathbb{C}^{mn}). \end{array} \right. $$

<footnote n="2">²The Regularity and Neumann problems for Laplace's equation on domains do not require use of $\tilde{N}_*$; in this case $N_*$ works fine (see Dahlberg and Kenig [30] and Brown [24]). The modified non-tangential maximal function $\tilde{N}_*$ is needed for more general operators $L_A$. This modification was introduced by Kenig and Pipher [60], who obtained the first results in this direction.</footnote>

<footnote n="3">³It is known that $\|\tilde{N}_*u\|_{L^p} \lesssim \|\nabla u\|_{T_{-1}^p}$ in the range of $p$ that we shall deal with (see [19]), so the two problems are related. The converse inequality is not known except for the case of real equations [42, Theorem 1.7].</footnote>
The spaces $\dot{\text{BMO}}_{-1}$ and $\dot{\Lambda}_{\alpha-1}$ are most conveniently represented as the homogeneous Triebel-Lizorkin space $\dot{F}_{-1}^{\infty,2}$ and Besov spaces $\dot{B}_{\alpha-1}^{\infty,\infty}$ respectively, as their negative orders prevent traditional characterisations in terms of smoothness. In these problems the limit in the boundary condition is imposed in the weak-star topology, which is possible since $\dot{\Lambda}_{\alpha-1}$ and $\dot{\text{BMO}}_{-1}$ are the Banach duals of $\dot{H}_1^{n/(n+\alpha)}$ and $\dot{H}_1^1$ respectively. The spaces $T_{-1;\alpha}^{\infty}$ are again instances of weighted tent spaces: for $\theta \in \mathbb{R}$ and $\alpha \ge 0$, and for measurable $f: \mathbb{R}_+^{1+n} \to \mathbb{C}^N$, we have the Carleson-type norm

$$ (1.6) \qquad \|f\|_{T_{\theta;\alpha}^{\infty}} := \sup_{B \subset \mathbb{R}^n} \frac{1}{r_B^{\alpha}} \left( \frac{1}{r_B^n} \int_0^{r_B} \int_{B(c_B, r_B-t)} |t^{-\theta} f(t, y)|^2 dy \frac{dt}{t} \right)^{1/2} $$

where the supremum is taken over all open balls $B = B(c_B, r_B) \subset \mathbb{R}^n$.

For $p$ and $\alpha$ as above, we can define order $-1$ Neumann problems $(N_H)^{p}_{-1,A}$, $(N_H)^{(\infty,\alpha)}_{-1,A}$, and $(N_H)^{(\infty,0)}_{-1,A}$ in the same way, with tangential gradients $\nabla_\| \| \cdot \|$ replaced by $A$-conormal derivatives $\partial_{\nu_A}$ in the boundary condition (keeping $\nabla_\| \| \cdot \|$ in the decay condition at infinity).

Note that in the 'order $-1$' problems above, we impose a tent space estimate on $\nabla u$ rather than a nontangential maximal function estimate. We also impose a decay condition on the tangential gradient $\nabla_\| u$ at infinity. For $p$ sufficiently small the decay condition is implied by the other conditions, and if $L_A$ satisfies a De Giorgi-Nash-Moser condition (see (7.2)) then it is implied for all $p < \infty$, and also for some range of $\alpha \ge 0$. (see Lemma 6.4).

**Remark 1.2.** We have not imposed any non-tangential convergence of solutions in the problems above. This is because the classification theorems of the second author and Mourgoglou, in particular [16, Corollaries 1.2 and 1.4], automatically yield almost everywhere (a.e.) non-tangential convergence of Whitney averages (of either the solution or its conormal gradient, whichever is relevant). When $L_A$ satisfies a De Giorgi-Nash-Moser condition this can be improved to a.e. non-tangential convergence without Whitney averages.

Let us summarise the problems we have introduced so far. There are Dirichlet problems of order 0 and 1 (interpreting the $H^p$-Regularity problem as a $\dot{H}_1^p$-Dirichlet problem), Regularity problems of order 0 and $-1$, and Neumann problems of order 0 and $-1$.

In their recent memoir [22], Barton and Mayboroda consider problems of *intermediate order*. They formulate Dirichlet problems of order $\theta \in (0,1)$ and Neumann problems of order $\theta \in (-1,0)$ as follows. For $0 < \theta < 1$ and $n/(n+\theta) < p \le \infty$,

$$ (D_B)^p_{\theta,A} : \left\{ \begin{array}{ll} L_A u = 0 & \text{in } \mathbb{R}_+^{1+n}, \\ \|\nabla u\|_{L(p,\theta,2)} \lesssim \|f\|_{\dot{B}_\theta^{p,p}}, & \\ \text{Tr } u = f \in \dot{B}_\theta^{p,p}(\mathbb{R}^n : \mathbb{C}^m) & \end{array} \right. $$

and

$$ (N_B)^p_{\theta-1,A} : \left\{ \begin{array}{ll} L_A u = 0 & \text{in } \mathbb{R}_+^{1+n}, \\ \|\nabla u\|_{L(p,\theta,2)} \lesssim \|\partial_{\nu_A} f\|_{\dot{B}_{\theta-1}^{p,p}}, & \\ \partial_{\nu_A} u|_{\partial \mathbb{R}_+^{1+n}} = \partial_{\nu_A} f \in \dot{B}_{\theta-1}^{p,p}(\mathbb{R}^n : \mathbb{C}^m). & \end{array} \right. $$
The spaces $L(p, \theta, 2)$ are defined by the norms

$$ \|F\|_{L(p,\theta,2)} := \left( \iint_{\mathbb{R}_+^{1+n}} \left( \iint_{t/2 < \tau < 2t} |\tau|^{1-\theta} F(\tau, \xi)^2 d\xi d\tau \right)^{p/2} dx \frac{dt}{t} \right)^{1/p} $$

with the usual modification when $p = \infty$. We refer to these spaces as Z-spaces starting from Section 2.3 (the letter L already being overused), with an indexing convention such that $Z_{\theta}^{p} = L(p, \theta + 1, 2)$. The boundary condition for $(D_B)_{\theta,A}^{p}$ is phrased in terms of the trace operator, which Barton and Mayboroda show to be bounded from $\{u : \nabla u \in L(p, \theta, 2)\}$ to $\dot{B}_{\theta}^{p,p}$ when $p > n/(n + \theta)$ [22, Theorem 3.9]. A similar argument is used to define the boundary conormal derivative $\partial_{\nu_A} u|_{\partial \mathbb{R}_+^{1+n}}$.

**Remark 1.3.** We warn the reader that our indexing convention for boundary value problems is different to that in [22]: we index our problems by the order of the function space in which the boundary data is assumed to lie. Thus Barton and Mayboroda refer to $(N_B)_{\theta-1,A}^p$ as $(N)_{\theta,A}^p$.

As we stated earlier, for technical reasons we consider Regularity problems rather than Dirichlet problems. Thus for $-1 < \theta < 0$ and $n/(n + \theta + 1) < p \le \infty$ we define

$$ (R_B)_{\theta,A}^p : \left\{ \begin{array}{ll} L_A u = 0 & \text{in } \mathbb{R}_+^{1+n}, \\ \|\nabla u\|_{Z_{\theta}^p} \lesssim \|\nabla \|f\|_{\dot{B}_{\theta}^{p,p}}, & \lim_{t \to \infty} \nabla \|u(t, \cdot)\| = 0 \quad \text{in } \mathcal{Z}'(\mathbb{R}^n : \mathbb{C}^{mn}), \\ \lim_{t \to 0} \nabla \|u(t, \cdot)\| = \|\nabla \|f\| \in \dot{B}_{\theta}^{p,p}(\mathbb{R}^n : \mathbb{C}^{mn}) & \end{array} \right. $$

and

$$ (N_B)_{\theta,A}^p : \left\{ \begin{array}{ll} L_A u = 0 & \text{in } \mathbb{R}_+^{1+n}, \\ \|\nabla u\|_{Z_{\theta}^p} \lesssim \|\partial_{\nu_A} f\|_{\dot{B}_{\theta}^{p,p}}, & \lim_{t \to \infty} \nabla \|u(t, \cdot)\| = 0 \quad \text{in } \mathcal{Z}'(\mathbb{R}^n : \mathbb{C}^{mn}), \\ \lim_{t \to 0} \partial_{\nu_A} u(t, \cdot) = \partial_{\nu_A} f \in \dot{B}_{\theta}^{p,p}(\mathbb{R}^n : \mathbb{C}^m), & \end{array} \right. $$

replacing the trace conditions with limiting conditions for consistency with the 'endpoint order' problems that we have already defined, writing $Z_{\theta}^{p}$ instead of $L(p, \theta + 1, 2)$, and including a decay condition at infinity. As before, when $p = \infty$ we impose the boundary condition in the weak-star topology, using that $\dot{B}_{\theta}^{\infty, \infty}$ is the dual of $\dot{B}_{-\theta}^{1,1}$.

**Remark 1.4.** If we omit the decay condition, the Regularity problem $(R_B)_{\theta,A}^p$ is equivalent to the Dirichlet problem $(D_B)_{\theta+1,A}^p$ defined above by an argument similar to that of Remark 1.1, and the Neumann problem $(N_B)_{\theta,A}^p$ is simply a rewriting of the previously-defined Neumann problem. As stated earlier, the decay condition is often redundant. Furthermore, the trace theorem of Barton and Mayboroda [22, Theorem 3.9] implies that the boundary limiting condition and the trace condition are equivalent given that $\nabla u \in Z_{\theta}^p$.

The Besov spaces $\dot{B}_{\theta}^{p,p}$ with $\theta \in (-1, 0)$ are not the only function spaces situated between $H_0^p$ and $\dot{H}_{-1}^p$. One of our new contributions to this problem is that we also consider the Hardy-Sobolev spaces $\dot{H}_\theta^p$ with $\theta \in (-1, 0)$. These are defined in Section 2.5; they may be identified with the homogeneous Triebel-Lizorkin spaces $\dot{F}_{\theta}^{p,2}$, whereas the Besov spaces $\dot{B}_{\theta}^{p,p}$ may be identified with $\dot{F}_{\theta}^{p,p}$ when $p < \infty$.
We use Hardy–Sobolev spaces to formulate the following Regularity and Neumann problems, with $-1 < \theta < 0$ and $n/(n + \theta + 1) < p < \infty$:

$$ (R_H)^p_{\theta,A} : \left\{ \begin{array}{ll} L_A u = 0 & \text{in } \mathbb{R}_+^{1+n}, \\ \|\nabla u\|_{T_\theta^p} \lesssim \|\nabla_\| f\|_{\dot{H}_\theta^p}, \\ \lim_{t \to \infty} \nabla_\| u(t, \cdot) = 0 & \text{in } \mathcal{Z}'(\mathbb{R}^n : \mathbb{C}^{mn}), \\ \lim_{t \to 0} \nabla_\| u(t, \cdot) = \nabla_\| f \in \dot{H}_\theta^p(\mathbb{R}^n : \mathbb{C}^{mn}) \end{array} \right. $$

and

$$ (N_H)^p_{\theta,A} : \left\{ \begin{array}{ll} L_A u = 0 & \text{in } \mathbb{R}_+^{1+n}, \\ \|\nabla u\|_{T_\theta^p} \lesssim \|\partial_{\nu_A} f\|_{\dot{H}_\theta^p}, \\ \lim_{t \to \infty} \nabla_\| u(t, \cdot) = 0 & \text{in } \mathcal{Z}'(\mathbb{R}^n : \mathbb{C}^{mn}), \\ \lim_{t \to 0} \partial_{\nu_A} u(t, \cdot) = \partial_{\nu_A} f \in \dot{H}_\theta^p(\mathbb{R}^n : \mathbb{C}^m). \end{array} \right. $$

Recall that the $T_\theta^p$ (quasi-)norm is defined in (1.1.2). Furthermore, for $-1 < \theta < 0$ we formulate ‘endpoint’ problems $(R_H)^p_{\theta,A}$ and $(N_H)^p_{\theta,A}$ by replacing $\dot{H}_\theta^p$ with the homogeneous BMO-Sobolev space $BMO_\theta$, which may be identified with the homogeneous Triebel–Lizorkin space $\dot{B}_\theta^{\infty,2}$, and replacing $T_\theta^p$ with the weighted tent space $T_{\theta;0}^\infty$ (defined in (1.6)). In this case the boundary condition is imposed in the weak-star topology, using that $BMO_\theta$ is the dual of $\dot{H}_\theta^1$. Unlike the function space $\{u : \nabla u \in Z_\theta^p\}$, there is no trace theorem for the function space $\{u : \nabla u \in T_\theta^p\}$.

Let us briefly summarise the Regularity and Neumann problems that we have introduced.

* At order 0 we have problems $(R_H)^p_{0,A}$ and $(N_H)^p_{0,A}$ for $n/(n+1) < p < \infty$, with boundary data in $H^p$ and a modified non-tangential maximal estimate on the interior.

* At order $-1$ we have $(R_H)^p_{-1,A}$ and $(N_H)^p_{-1,A}$ for $1 < p < \infty$, with boundary data in $\dot{H}_{-1}^p$ and a $T_{-1}^p$ interior estimate. Furthermore, for $0 \le \alpha < 1$, we have $(R_H)^{(
abla, \alpha)}_{-1,A}$ and $(N_H)^{(
abla, \alpha)}_{-1,A}$ with boundary data in $\dot{A}_{\alpha-1}$ (or $B\dot{M}O_{-1}$ when $\alpha = 0$) and a $T_{-1;\alpha}^\infty$ interior estimate.

* In between, i.e. for order $\theta \in (-1, 0)$, and with $n/(n + \theta + 1) < p \le \infty$, we have $(R_B)^p_{\theta,A}$ and $(N_B)^p_{\theta,A}$ with boundary data in $\dot{B}_\theta^{p,p}$, and $(R_H)^p_{\theta,A}$ and $(N_H)^p_{\theta,A}$ with boundary data in $\dot{H}_\theta^p$ ($B\dot{M}O_\theta$ when $p = \infty$). Here the interior estimates are in $Z_\theta^p$ and $T_\theta^p$ ($T_{\theta;0}^\infty$ when $p = \infty$) respectively.

For all problems of negative order we also impose a decay condition on $\nabla_\| u(t, \cdot)$ as $t \to \infty$ in the space $\mathcal{Z}'$ of tempered distributions modulo polynomials, which is redundant in many cases (see Lemma 6.4). Note that for $\theta \in (-1, 0)$, the problems $(R_H)^2_{\theta,A}$ and $(R_B)^2_{\theta,A}$ (and likewise for Neumann problems) coincide, since $\dot{H}_\theta^2 = \dot{B}_{\theta,2}^{2,2}$ and $Z_\theta^2 = T_\theta^2$.

Since we assume no regularity of our coefficients, and since we consider weak solutions without further regularity properties, the only Regularity and Neumann problems which are meaningful are those of order between $-1$ and $0$. Thus, within this context, this list of boundary value problems is essentially complete.

### 1.1.3. Perturbed Dirac operators and Cauchy–Riemann systems.
Let $D$ denote the differential operator on $\mathbb{C}^{m(1+n)}$-valued functions given by

$$ D := \begin{bmatrix} 0 & \mathrm{div}_\| \\ -\nabla_\| & 0 \end{bmatrix} $$
with respect to the transversal/tangential splitting (1.2) of $\mathbb{C}^{m(1+n)}$. We refer to $D$ as a Dirac operator, because $D^2$ acts as the Laplacian $\Delta$ on the range of $D$. When $B \in L^\infty(\mathbb{R}^n : \mathcal{L}(\mathbb{C}^{m(1+n)}))$ is a coefficient matrix satisfying the same assumptions as $A$, we refer to the composition $DB$ as a perturbed Dirac operator.

The Cauchy-Riemann system associated with $DB$ is the first-order partial differential system

$$ (\mathrm{CR})_{DB} : \left\{ \begin{array}{l} \partial_t F + DBF = 0 \\ \mathrm{curl}_{\|} F_{\|} = 0 \end{array} \right. \quad \text{in } \mathbb{R}_+^{1+n} $$

interpreted in the weak $(L_{\mathrm{loc}}^2)$ sense: that is, we say that $F \in L_{\mathrm{loc}}^2(\mathbb{R}_+^{1+n} : \mathbb{C}^{m(1+n)})$ solves $(\mathrm{CR})_{DB}$ if for all test functions $\varphi \in C_c^\infty(\mathbb{R}_+^{1+n} : \mathbb{C}^{m(1+n)})$

$$ \iint_{\mathbb{R}_+^{1+n}} (F(t,x), \partial_t \varphi(t,x)) \, dx \, dt = \iint_{\mathbb{R}_+^{1+n}} (F(t,x), B^*(x)D\varphi(t,x)) \, dx \, dt, $$

and for all $\psi \in C_c^\infty(\mathbb{R}_+^{1+n} : \mathbb{C}^m)$ and $1 \le j, k \le n, j \ne k$,

$$ \iint_{\mathbb{R}_+^{1+n}} (F_k(t,x), \partial_j \psi(t,x)) \, dx \, dt = - \iint_{\mathbb{R}_+^{1+n}} (F_j(t,x), \partial_k \psi(t,x)) \, dx \, dt. $$

The condition $\mathrm{curl}_{\|} F_{\|} = 0$ in $(\mathrm{CR})_{DB}$ is preserved for all limits, so the Cauchy-Riemann system may be considered as an evolution equation in a restricted space involving a differential structure.

The first-order approach to boundary value problems for elliptic equations $L_{Au} = 0$ exploits a correspondence between these elliptic equations and Cauchy-Riemann systems $(\mathrm{CR})_{DB}$. Recall that $A \in L^\infty(\mathbb{R}^n : \mathcal{L}(\mathbb{C}^{m(1+n)}))$. Write $A$ in matrix form with respect to the transversal/tangential splitting (1.2) of $\mathbb{C}^{m(1+n)}$ as

$$ (1.7) \qquad A = \begin{bmatrix} A_{\perp\perp} & A_{\perp\|} \\ A_{\|\perp} & A_{\|\|} \end{bmatrix}, $$

and using this representation of $A$ define auxiliary matrices

$$ \bar{A} := \begin{bmatrix} A_{\perp\perp} & A_{\perp\|} \\ 0 & I \end{bmatrix} \quad \text{and} \quad \underline{A} := \begin{bmatrix} I & 0 \\ A_{\|\perp} & A_{\|\|} \end{bmatrix} $$

in $L^\infty(\mathbb{R}^n : \mathcal{L}(\mathbb{C}^{m(1+n)}))$. Strict accretivity of $A$ implies that $A_{\perp\perp}$ is invertible in $L^\infty(\mathbb{R}^n : \mathcal{L}(\mathbb{C}^m))$, and so $\bar{A}$ is invertible in $L^\infty(\mathbb{R}^n : \mathcal{L}(\mathbb{C}^{m(1+n)}))$. Thus we may define

$$ \hat{A} := \underline{A}\bar{A}^{-1}. $$

The transformed coefficient matrix $\hat{A}$ satisfies the same assumptions as $A$, and $\hat{A} = A$ [8, Proposition 3.2].

The correspondence between elliptic equations $L_{Au} = 0$ and Cauchy-Riemann systems $(\mathrm{CR})_{DB}$ is given by the following theorem. See [8, §3], [5, Proposition 4.1], [68, §2], and [16, Lemma 7.1] for proofs and discussions.

**Theorem 1.5 (Auscher-Axelsson-McIntosh).** Let $A$ be as above, and let $B = \hat{A}$. If $u$ solves $L_{Au} = 0$, then the $A$-conormal gradient $\nabla_{Au}$ solves the Cauchy-Riemann system $(\mathrm{CR})_{DB}$. Conversely, if $F$ solves $(\mathrm{CR})_{DB}$, then there exists a function $u$, unique up to an additive constant, such that $L_{Au} = 0$ and $F = \nabla_{Au}$.
The A-conormal gradient $\nabla_A u$ of a function $u: \mathbb{R}_+^{1+n} \to \mathbb{C}^m$ is defined by

$$ (1.8) \qquad \nabla_A u = \begin{bmatrix} \partial_{\nu_A} u \\ \nabla_{\|} u \end{bmatrix}, $$

where the A-conormal derivative $\partial_{\nu_A}$ is defined in (1.5). The components of $\nabla_A u$ appear in the boundary conditions of the Regularity and Neumann problems; hence our preference for Regularity problems over Dirichlet problems.

Thus in our consideration of elliptic equations we may focus instead on Cauchy–Riemann systems. The principal advantage of Cauchy–Riemann systems over el- liptic equations is that the Cauchy equation $\partial_t F + DBF = 0$ can be solved by semigroup methods. We sketch how this is done, following [8] and [5].

Consider $D$ as an unbounded operator on $L^2 := L^2(\mathbb{R}^n : \mathbb{C}^{m(1+n)})$ with natural domain, and consider $B$ as a multiplication operator on $L^2$. Then we have the following theorem [8, Proposition 3.3 and Theorem 3.4]. This is a very deep result: it is part of the framework developed by Axelsson, Keith, and McIntosh [21], which encompasses the solution of the Kato square root problem [10].

**Theorem 1.6 (Axelsson–Keith–McIntosh).** The perturbed Dirac operator $DB$ is bisectorial and has bounded $H^\infty$ functional calculus on the closure $\overline{\mathcal{R}(D)} = \overline{\mathcal{R}(DB)}$ of its range.

These notions are properly discussed in Section 3.1. Using the direct sum decomposition

$$ L^2 = \mathcal{N}(DB) \oplus \overline{\mathcal{R}(DB)} $$

(which follows from bisectoriality of $DB$) and the bounded $H^\infty$ functional calculus of $DB$ on $\overline{\mathcal{R}(DB)}$, we obtain a decomposition

$$ (1.9) \qquad L^2 = \mathcal{N}(DB) \oplus \overline{\mathcal{R}(DB)}^+ \oplus \overline{\mathcal{R}(DB)}^- $$

The positive and negative spectral subspaces $\overline{\mathcal{R}(DB)}^\pm$ are the images of $\overline{\mathcal{R}(DB)}$ under the projections $\chi^\pm(DB)$, which are defined via the functions $\chi^\pm: \mathbb{C} \setminus i\mathbb{R} \to \{0, 1\}$ given by

$$ \chi^\pm(z) := \mathbf{1}_{z \pm \mathrm{Re}(z) > 0}. $$

These are the characteristic functions of the right and left half-plane, restricted to $\mathbb{C} \setminus i\mathbb{R}$. They are bounded and holomorphic on every bisector, so they fall within the scope of the $H^\infty$ functional calculus.

By way of the $H^\infty$ functional calculus, we may define a generalised Cauchy operator $C_{DB}^+ : \overline{\mathcal{R}(DB)} \to L^\infty(\mathbb{R}_+ : \overline{\mathcal{R}(DB)})$ by

$$ (C_{DB}^+ f)(t) = e^{-tDB} \chi^+(DB)f, $$

corresponding to the family of functions $(z \mapsto e^{-tz}\chi^+(z))$, which are bounded and holomorphic on every bisector. When restricted to the positive spectral subspace $\overline{\mathcal{R}(DB)}^+$, the Cauchy operator $C_{DB}^+$ acts as a strongly continuous semigroup. The Cauchy operator is used in the following classification theorem, which is a combi- nation of parts of [8, Theorem 2.3] and [5, Corollary 8.4].

**Theorem 1.7 (Auscher–Axelsson–McIntosh).** If $f \in \overline{\mathcal{R}(DB)}^+$, then the Cauchy extension $C_{DB}^+ f$ solves (CR)$_{DB}$, with

$$ \|\tilde{N}_*(C_{DB}^+ f)\|_2 \approx \|f\|_2 \quad \text{and} \quad \lim_{t \to 0} (C_{DB}^+ f)(t, \cdot) = f \quad \text{in } L^2. $$
Conversely, if $F$ solves $(CR)_{DB}$ and $\tilde{N}_*(F) \in L^2$, then $F = C_{DB}^+f$ for a unique $f \in \overline{\mathcal{R}(DB)}^+$.

Combining this with Theorem 1.5 yields a characterisation of well-posedness of the boundary value problems $(R_H)_{0,A}^2$ and $(N_H)_{0,A}^2$. Consider the $L^2$-Regularity problem $(R_H)_{0,A}^2$ and let $B = \hat{A}$. A function $u$ solves $L_A u = 0$ with $\tilde{N}_*(\nabla u) \in L^2(\nabla u)$ and $\nabla_A u$ are interchangeable in this assumption) if and only if $\nabla_A u = C_{DB}^+g$ for some $g \in \overline{\mathcal{R}(DB)}^+$, hence $\nabla_{\|}u(t, \cdot) = (C_{DB}^+g)(t)_{\|}$ and $\lim_{t \to 0} \nabla_{\|}u(t, \cdot) = g_{\|}$. Therefore $(R_H)_{0,A}^2$ is well-posed if and only if $g \mapsto g_{\|}$ is an isomorphism from $\overline{\mathcal{R}(DB)}^+$ to $L^2(\mathbb{R}^n : \mathbb{C}^{mn}) \cap \mathcal{N}(\mathrm{curl}_{\|})$. By the same argument, $(N_H)_{0,A}^2$ is well-posed if and only if $g \mapsto g_{\perp}$ is an isomorphism from $\overline{\mathcal{R}(DB)}^+$ to $L^2(\mathbb{R}^n : \mathbb{C}^m)$.

By characterising solutions to $(CR)_{DB}$ within various function spaces, we show that well-posedness of corresponding Regularity and Neumann problems is equivalent to the transversal and tangential projections being isomorphisms between certain ‘boundary function spaces’. In this section we only described how to handle boundary value problems of order 0 with $L^2$ boundary data. We need to extend this technique to boundary value problems of more general order, and beyond $L^2$. In the argument we just described, abstract semigroup theory (accessed via holomorphic functional calculus) did a lot of the work for us. However, once we go beyond $L^2$, we cannot rely on abstract semigroup theory on general Banach spaces. This would only classify solutions $F$ to $(CR)_{DB}$ such that $F(t)$ is in a fixed function space for all $t > 0$, ruling out consideration of many tent spaces and $Z$-spaces. Furthermore, abstract semigroup methods do not always allow us to move from initial value problems for $(CR)_{DB}$ to boundary value problems for $L_A u = 0$. On top of these defects we also need to consider quasi-Banach spaces, for which no semigroup theory seems to be available. This makes things difficult.

**1.1.4. Adapted function spaces.** ‘Adapted’ Hardy spaces $H_L^p$ with respect to which some operator $L$ has good properties (such as bounded $H^\infty$ functional calculus), have been developed in various contexts. For example, Hardy spaces of differential forms on Riemannian manifolds were constructed by the second author with McIntosh and Russ [14] (these are adapted to the Hodge-Dirac operator $d+d^*$ on the de Rham complex); Hardy spaces adapted to non-negative self-adjoint operators satisfying Davies–Gaffney estimates on spaces of homogeneous type were studied by Hofmann, Lu, Mitrea, Mitrea, and Yan [44] (generalising the aforementioned example); Hardy spaces adapted to divergence-form elliptic operators on $\mathbb{R}^n$ were developed by Hofmann and Mayboroda [45] and also McIntosh [46]. Some further developments can be found, for example, in the work of Hytönen, van Neerven, and Portal [51], Jiang and Yang [52], Anh and Li [4], and Duong and Li [32]. This is a very small sample of the work that has been done.

Hardy spaces $H_{DB}^p$ and Sobolev spaces $W_{-1, DB}^p$ adapted to perturbed Dirac operators $DB$ were introduced by the second author and Stahlhut [19] (See also Stahlhut’s thesis [71], and for a different approach see Frey, McIntosh, and Portal [35]). These spaces (defined along with more general spaces in Chapter 4) consist of $\mathbb{C}^{m(1+n)}$-valued functions (at least formally); the simplest case is

$$H_D^2 = H_{DB}^2 = \overline{\mathcal{R}(DB)} = \overline{\mathcal{R}(D)} \subset L^2(\mathbb{R}^n : \mathbb{C}^{m(1+n)}).$$
The bounded $H^\infty$ functional calculus of $DB$ on $H_{DB}^2$ extends to $H_{DB}^p$ and $W_{-1, DB}^p$, yielding spectral decompositions

$$H_{DB}^p = H_{DB}^{p,+} \oplus H_{DB}^{p,-}, \quad W_{-1, DB}^p = W_{-1, DB}^{p,+} \oplus W_{-1, DB}^{p,-}$$

analogous to (1.9). Furthermore, the Cauchy operator $C_{DB}^+$ on $\overline{\mathbb{R}(DB)}$ extends to operators on $H_{DB}^p$ and $W_{-1, DB}^p$, both of which we denote by $C_{DB}^+$.

The main application of these spaces, which incorporates results from work of the second author with both Stahlhut [19] and Mourgoglou [16], is a classification of solutions to the Cauchy–Riemann system (CR)$_{DB}$ with $L^p$-type interior estimates, for $p$ such that certain $DB$-adapted spaces may be identified with $D$-adapted spaces. For simplicity we only state results for $1 < p < \infty$ in this introduction; corresponding results for $p \le 1$ and $p \ge \infty$ (i.e. for boundary data in BMO-type and Hölder spaces) are also available, but these may not be stated so simply in terms of Hölder conjugate exponents.

**Theorem 1.8 (Auscher–Mourgoglou–Stahlhut).** Let $1 < p < \infty$.

(i) Assume $H_{DB}^p \simeq H_D^p$. If $f \in H_{DB}^{p,+}$, then $C_{DB}^+f$ solves (CR)$_{DB}$, with

$$\| \tilde{N}_*(C_{DB}^+f) \|_p \simeq \|f\|_{H^p} \quad \text{and} \quad \lim_{t \to 0} C_{DB}^+f(t) = f \quad \text{in } H^p.$$

Conversely, if $F$ solves (CR)$_{DB}$ and $\tilde{N}_*F \in L^p$, then $F = C_{DB}^+f$ for a unique $f \in H_{DB}^{p,+}$.

(ii) Assume $H_{DB}^{p'} \simeq H_D^{p'}$. If $f \in W_{-1, DB}^{p,+}$, then $C_{DB}^+f$ solves (CR)$_{DB}$, with

$$\| C_{DB}^+ f \|_{T_{-1}^p} \simeq \| f \|_{\dot{W}_{-1}^p} \quad \text{and} \quad \lim_{t \to 0} C_{DB}^+ f(t) = f \quad \text{in } \dot{W}_{-1}^p.$$

Conversely, if $F \in T_{-1}^p$ solves (CR)$_{DB}$ and $\lim_{t \to \infty} F(t) \| = 0$ in $Z'(\mathbb{R}^n)$, then $F = C_{DB}^+f$ for a unique $f \in W_{-1, DB}^{p,+}$.

Furthermore, it is shown that for every $B$ there exists an open interval containing 2, which we denote $I_0(H, DB)$, such that $H_{DB}^p \simeq H_D^p$ for all $p \in I_0(H, DB)$ [19, Theorem 5.1]. Thus there is a nontrivial range of exponents for which Theorem 1.8 applies.

**Remark 1.9.** The assumption in part (ii) of the theorem is given in terms of the conjugate exponent $p'$ and the adjoint coefficients $B^*$. As part of our theory we show—as a corollary of what will eventually be termed ‘$\heartsuit$-duality’—that this is equivalent to $W_{-1, DB}^p \simeq W_{-1, D}^p$, thus unifying the assumptions of parts (i) and (ii).

As we described in the case where $p = 2$, Theorem 1.8 implies a characterisation of well-posedness of various Regularity and Neumann problems, both of order 0 and order $-1$, in terms of certain transversal and tangential projections being isomorphisms. We will not explicitly state this characterisation now; instead, we state our extension of this result in Theorem 1.11.

The technical heart of this monograph is an extension of Theorem 1.8 to fractional order $\theta \in (-1, 0)$, incorporating both Hardy–Sobolev spaces and Besov spaces. To this end, we introduce Hardy–Sobolev spaces $H_{\theta, L}^p$ and Besov spaces $B_{\theta, L}^p$ adapted to operators $L$ satisfying ‘Standard Assumptions’, which are satisfied
in particular by the perturbed Dirac operators DB and BD. We define extension operators

$$ (\mathbb{Q}_{\varphi,L} f)(t) = \varphi(tL)f \quad (t > 0, f \in \overline{\mathbb{R}(L)}) $$

for appropriate holomorphic functions $\varphi$, and the adapted Hardy–Sobolev and Besov norms are then, roughly speaking, defined by

$$ \|f\|_{\mathbf{H}_{\theta,L}^p} := \|\mathbb{Q}_{\varphi,L} f\|_{T_{\theta}^p}, \quad \|f\|_{\mathbf{B}_{\theta,L}^p} := \|\mathbb{Q}_{\varphi,L} f\|_{Z_{\theta}^p}. $$

These definitions are reminiscent of the $\varphi$-transform characterisations of Triebel–Lizorkin and Besov spaces due to Frazier and Jawerth [34] (the letter $\varphi$ has a different meaning there), with functional calculus and tent/Z-spaces replacing discretised Littlewood–Paley decompositions and sequence spaces respectively.

Chapters 3 to 5 are occupied with the construction of a sufficiently rich general theory of adapted Besov–Hardy–Sobolev spaces. The theory is relatively straightforward once enough preliminaries have been collected, but getting to this stage takes some time. We emphasise in particular the amount of work needed to quantify independence on $\varphi$ of the spaces $\mathbf{H}_{\theta,L}^p$ and $\mathbf{B}_{\theta,L}^p$ (essentially all of Sections 3.3 and 3.4) and the care which must be taken in discussing completions (Section 4.3), which is necessary to discuss and fully exploit interpolation.

**1.1.5. Classification of solutions to CR systems, and applications to well-posedness.** Our main theorem is the following classification of solutions to the Cauchy–Riemann system (CR)$_{DB}$. In this statement we restrict ourselves to $1 < p < \infty$. As with our statement of Theorem 1.8, this is a simplification of the full result (Theorems 6.8 and 6.9): our theorem also allows for $p \le 1$ and $p \ge \infty$, but the corresponding results are better stated in terms of the ‘exponent notation’ that we introduce in Section 2.1.

**Theorem 1.10.** Let $-1 < \theta < 0$ and $1 < p < \infty$.

(i) Suppose that $\mathbf{H}_{\theta, DB}^p \simeq \mathbf{H}_{\theta, D}^p$. If $f \in \mathbf{H}_{\theta, DB}^{p, +}$, then $\mathbf{C}_{DB}^+ f$ solves (CR)$_{DB}$, with

$$ \|\mathbf{C}_{DB}^+ f\|_{T_{\theta}^p} \simeq \|f\|_{\dot{H}_{\theta}^p} \quad \text{and} \quad \lim_{t \to 0} \mathbf{C}_{DB}^+ f(t) = f \quad \text{in } \dot{H}_{\theta}^p, $$

and furthermore $\lim_{t \to \infty} \mathbf{C}_{DB}^+ f(t) \| = 0$ in $\mathcal{Z}'(\mathbb{R}^n)$. Conversely, if $F \in T_{\theta}^p$ solves (CR)$_{DB}$ and $\lim_{t \to \infty} F(t) \| = 0$ in $\mathcal{Z}'(\mathbb{R}^n)$, then $F = \mathbf{C}_{DB}^+ f$ for a unique $f \in \mathbf{H}_{\theta, DB}^{p, +}$.

(ii) Suppose that $\mathbf{B}_{\theta, DB}^p \simeq \mathbf{B}_{\theta, D}^p$. If $f \in \mathbf{B}_{\theta, DB}^{p, +}$, then $\mathbf{C}_{DB}^+ f$ solves (CR)$_{DB}$, with

$$ \|\mathbf{C}_{DB}^+ f\|_{Z_{\theta}^p} \simeq \|f\|_{\dot{B}_{\theta}^{p,p}} \quad \text{and} \quad \lim_{t \to 0} \mathbf{C}_{DB}^+ f(t) = f \quad \text{in } \dot{B}_{\theta}^{p,p}, $$

and furthermore $\lim_{t \to \infty} \mathbf{C}_{DB}^+ f(t) \| = 0$ in $\mathcal{Z}'(\mathbb{R}^n)$. Conversely, if $F \in Z_{\theta}^p$ solves (CR)$_{DB}$ and $\lim_{t \to \infty} F(t) \| = 0$ in $\mathcal{Z}'(\mathbb{R}^n)$, then $F = \mathbf{C}_{DB}^+ f$ for a unique $f \in \mathbf{B}_{\theta, DB}^{p, +}$.

Parts (i) and (ii) of this theorem are essentially identical, the only modifications being the replacement of (adapted) Hardy–Sobolev spaces with (adapted) Besov spaces, and of tent spaces with Z-spaces. In fact, our arguments apply equally well to both parts, so we prove them simultaneously. Although the theorem can be thought of as ‘intermediate to’ Theorem 1.8, it does not follow by any interpolation argument. It is proven similarly, but the underlying techniques must be generalised, and this takes a considerable amount of work. Neither direction is easy, but
the ‘converse’ direction—finding a ‘trace’ $f$ given a solution $F$—is certainly more
difficult.

A consequence of this theorem (and of the abstract theory that we construct) is that certain solution spaces for the equation $L_A u = 0$ form interpolation scales (Section 6.5). This provides another viewpoint on the structure of the solution spaces, although it is weaker than the explicit description that we obtain. Another consequence is a representation theorem for solutions $u$ of $L_A u = 0$, rather than for their conormal gradients $\nabla_A u$ (which solve the Cauchy-Riemann system (CR)$_{DB}$). This result is much more complicated to state; we refer the reader to Theorem 6.13.

Identifying regions of exponents where Theorem 1.10 applies is important. These are regions in which we can identify $DB$-adapted spaces with $D$-adapted spaces, so we call them *identification regions*. Starting from information on the intervals $I_0(\mathbf{H}, DB), I_0(\mathbf{H}, DB^*) \ni 2$ (as given by the second author and Stahlhut), a procedure of ‘$\mathcal{H}$-duality’ and interpolation yields non-trivial regions of exponents $(p, \theta)$ for which Theorem 1.10 applies (Section 5.3).

With Theorem 1.10 as a springboard, we extend the characterisation of well-
posedness of Regularity and Neumann problems—described for $p = 2$ after the
statement of Theorem 1.5 and then extended to $p \neq 2$ and $\theta \in \{-1, 0\}$ by the second
author with Mourgoglou and Stahlhut—as follows. For all exponents $(p, \theta)$, we show
that $\mathbf{H}_{\theta, D}^p$ is equal to the set of those $f \in \dot{H}_\theta^p(\mathbb{R}^n : \mathbb{C}^{m(1+n)})$ with $\mathrm{curl}_\parallel f_\parallel = 0$.
Let $N_\perp$ and $N_\parallel$ denote the projections from $\mathbf{H}_{\theta, D}^p$ onto $\mathbf{H}_{\theta, \perp}^p := \dot{H}_\theta^p(\mathbb{R}^n : \mathbb{C}^m)$ and
$\mathbf{H}_{\theta, \parallel}^p := \dot{H}_\theta^p(\mathbb{R}^n : \mathbb{C}^{mn}) \cap \mathcal{N}(\mathrm{curl}_\parallel)$ respectively. For $(p, \theta)$ as in Theorem 1.10 we
have an identification of $\mathbf{H}_{\theta, DB}^{p,+}$ as a subset of $\mathbf{H}_{\theta, D}^p$, and so we can use $N_\perp$ and $N_\parallel$
to define

$$
N_{H,DB,\parallel}^{(p,\theta)} : \mathbf{H}_{\theta,DB}^{p,+} \to \mathbf{H}_{\theta,\parallel}^{p} \quad \text{and} \quad N_{H,DB,\perp}^{(p,\theta)} : \mathbf{H}_{\theta,DB}^{p,+} \to \mathbf{H}_{\theta,\perp}^{p}.
$$

Corresponding definitions of $N_{B,DB,\parallel}^{(p,\theta)}$ and $N_{B,DB,\perp}^{(p,\theta)}$ are also made for Besov spaces.
It is important to understand that we can define these restrictions on $DB$-adapted
spaces only after we have identified them with $D$-adapted spaces, on which the
projections are initially defined (the only exception to this is when $B$ has a block
structure). These operators carry the well-posedness of Regularity and Neumann
problems, as shown by the following theorem. Again, this is a simplification of the
full result (Theorem 7.5). The $\theta \in \{-1,0\}$ endpoints follow from Theorem 1.8.

**Theorem 1.11.** Let $B = \hat{A}$, $-1 \le \theta \le 0$, and $1 < p < \infty$. Suppose that $\mathbf{H}_{\theta, DB}^p \simeq \mathbf{H}_{\theta, D}^p$. Then $(R_H)_{\theta, A}^{p_0}$ (resp. $(N_H)_{\theta, A}^{p_0}$) is well-posed if and only if $N_{H,DB,\parallel}^{(p,\theta)}$ (resp. $N_{H,DB,\perp}^{(p,\theta)}$) is an isomorphism. The same results hold mutatis mutandis for problems with Besov boundary data.

The notion of well-posedness can be refined when considering boundary value
problems with different exponents. Consider $-1 \le \theta_0, \theta_1 \le 0$ and $1 < p_0, p_1 < \infty$.
We say that the problems $(R_H)_{\theta_0, A}^{p_0}$ and $(R_H)_{\theta_1, A}^{p_1}$ are mutually well-posed if they
are both well-posed, and if for all $\nabla_{\parallel} f \in \dot{H}_{\theta_0}^{p_0} \cap \dot{H}_{\theta_1}^{p_1}$ the unique solutions to $(R_H)_{\theta_0, A}^{p_0}$
and $(R_H)_{\theta_1, A}^{p_1}$ with boundary data $\nabla_{\parallel} f$ are equal. This definition simply extends to
all the boundary value problems that we consider. Two well-posed boundary value
problems need not be mutually well-posed: this phenomenon was first observed by
Axelsson [20]. The concept of mutual well-posedness extends the notion of compat-
ible well-posedness introduced by Barton and Mayboroda [22, §2.4]. More precisely,
mutual well-posedness defines an equivalence relation on the set of exponents $(p, \theta)$ for which a problem (e.g. $(R_H)^{p}_{\theta, A}$) is well-posed, and compatible well-posedness corresponds to the equivalence class of the 'energy exponent' $(2, -1/2)$. The problems $(R_H)^2_{-1/2, A}$ and $(N_H)^2_{-1/2, A}$ are always well-posed [13, Theorems 3.2 and 3.3].

By Theorem 1.11, $(R_H)^{p_0}_{\theta_0, A}$ and $(R_H)^{p_1}_{\theta_1, A}$ are mutually well-posed if and only if $N^{(p_0, \theta_0)}_{H, DB, \|}$ and $N^{(p_1, \theta_1)}_{H, DB, \|}$ are isomorphisms whose inverses are equal on the intersection $\mathbf{H}^{p_0}_{\theta_0, \|} \cap \mathbf{H}^{p_1}_{\theta_1, \|}$ (and likewise for Neumann problems, and with Besov boundary data). This allows us to interpolate mutual well-posedness as a straightforward corollary of Theorem 1.11. Furthermore, by using real interpolation instead of complex interpolation, we can deduce mutual well-posedness of boundary value problems with Besov boundary data from that of those with Hardy-Sobolev boundary data. The following theorem makes this precise. The full result is Theorem 7.7.

**Theorem 1.12.** Suppose $-1 \le \theta_0, \theta_1 \le 0$, $1 < p_0, p_1 < \infty$, and $\alpha \in (0, 1)$, and let

$$ \frac{1}{p} = \frac{1 - \alpha}{p_0} + \frac{\alpha}{p_1} \quad \text{and} \quad \theta = (1 - \alpha)\theta_0 + \alpha\theta_1. $$

(i) If $\mathbf{H}^{p_j}_{\theta_j, DB} \simeq \mathbf{H}^{p_j}_{\theta_j, D}$ for $j \in \{0, 1\}$, and if $(R_H)^{p_0}_{\theta_0, A}$ and $(R_H)^{p_1}_{\theta_1, A}$ are mutually well-posed, then $(R_H)^p_{\theta_0, A}$ is mutually well-posed with both $(R_H)^{p_0}_{\theta_0, A}$ and $(R_H)^{p_1}_{\theta_1, A}$; and furthermore if $\theta_0 \ne \theta_1$ then $(R_B)^p_{\theta_0, A}$ is mutually well-posed with both $(R_H)^{p_0}_{\theta_0, A}$ and $(R_H)^{p_1}_{\theta_1, A}$.

(ii) If $\mathbf{B}^{p_j}_{\theta_j, DB} \simeq \mathbf{B}^{p_j}_{\theta_j, D}$ for $j \in \{0, 1\}$, and if $(R_B)^{p_0}_{\theta_0, A}$ and $(R_B)^{p_1}_{\theta_1, A}$ are mutually well-posed, then $(R_B)^p_{\theta_0, A}$ is mutually well-posed with both $(R_B)^{p_0}_{\theta_0, A}$ and $(R_B)^{p_1}_{\theta_1, A}$.

*Corresponding results are also true for Neumann problems.*

Since invertibility is stable in complex interpolation scales, well-posedness of our boundary value problems is also stable in a way that preserves mutuality. This is made precise in the following simplification of Theorem 7.8.

**Theorem 1.13.** Let $-1 < \theta < 0$ and $1 < p < \infty$, and suppose that $\mathbf{H}^{p}_{\theta, DB} = \mathbf{H}^{p}_{\theta, D}$. Suppose also that $(R_H)^{p}_{\theta_0, A}$ is well-posed. Then $(R_H)^{p_1}_{\theta_1, A}$ and $(R_H)^{p}_{\theta_1, A}$ are mutually well-posed for all $(p_1, \theta_1)$ in some neighbourhood of $(p, \theta)$. Similar results hold for Neumann problems, and for problems with Besov boundary data.

This theorem does not apply when $\theta \in \{-1, 0\}$. However, the result is true if $\theta \in \{-1, 0\}$ and $\theta_1 = \theta$, with $p_1$ in a neighbourhood of $p$. This is implicit in the work of the second author with Mourgoglou and Stahlhut, and the proof of Theorem 1.13 still applies in this case.

Finally, we have a duality result for well-posedness (this is a simplification of Theorem 7.11).

**Theorem 1.14.** Let $-1 \le \theta_0, \theta_1 \le 0$ and $1 < p_0, p_1 < \infty$, and suppose that $\mathbf{H}^{p_0}_{\theta_0, DB} \simeq \mathbf{H}^{p_0}_{\theta_0, D}$ and $\mathbf{H}^{p_1}_{\theta_1, DB} \simeq \mathbf{H}^{p_1}_{\theta_1, D}$. Then $(R_H)^{p_0}_{\theta_0, A}$ and $(R_H)^{p_1}_{\theta_1, A}$ are mutually well-posed if and only if $(R_H)^{p_0}_{-\theta_0-1, A^*}$ and $(R_H)^{p_1}_{-\theta_1-1, A^*}$ are mutually well-posed. Similar results hold for Neumann problems and with Besov spaces.

We may take $(p_0, \theta_0) = (p_1, \theta_1)$ in this result. The mapping $(p, \theta) \mapsto (p', -\theta - 1)$ is the reflection about the point $(1/2, -1/2)$ in the $(1/p, \theta)$-plane, which corresponds
to the aforementioned ‘energy exponent’. This reflection describes what we call ‘$\mathcal{H}$-duality of exponents’.

These theorems can be used to derive new well-posedness results for Regularity problems $(R_H)_{\theta,A}^p$ with fractional order $\theta \in (-1, 0)$, and also to derive known results for $(R_B)_{\theta,A}^p$ which were recently obtained by different methods by Barton and Mayboroda [22] under the De Giorgi-Nash-Moser assumption. For details see Section 7.2.

**Remark 1.15.** All of our results can be reformulated with the lower half-space replacing the upper half-space. The main difference is that in this case positive spectral subspaces must be replaced with negative spectral subspaces.

## 1.2. Summary of the monograph

In Chapter 2 we discuss two types of function spaces. First, the ‘ambient spaces’: tent spaces, $Z$-spaces, and slice spaces. Many of the results here are new, or have not been used in this context. We then consider the ‘smoothness spaces’: Hardy–Sobolev spaces, Besov spaces, and so on. After a quick review of these spaces, we characterise them in terms of tent spaces and $Z$-spaces (Theorem 2.57). We also introduce a new system of notation for exponents: these are written as boldface letters, typically $\mathbf{p}$ and $\mathbf{q}$, and encode both integrability and regularity information. This is not strictly necessary, but it truly cleans up the exposition of later parts of the monograph and makes the flow of ideas more transparent.

In Chapter 3 we discuss basic operator theoretic notions. The operators that we use in applications (i.e. the perturbed Dirac operators $DB$ and $BD$) are bi-sectorial, with bounded $H^\infty$ functional calculi on their ranges, and satisfy certain off-diagonal estimates. Most of our abstract theory applies to any operator $A$ satisfying these ‘Standard Assumptions’, so we work with such operators until we are forced to use more specific properties of perturbed Dirac operators. We establish the boundedness of certain integral operators between tent spaces and $Z$-spaces. Particular examples of these operators are given in terms of ‘extension’ and ‘contraction’ operators $Q_{\varphi,A}$ and $S_{\psi,A}$, which we discuss. This chapter culminates in Theorem 3.19, which quantifies when operators of the form $Q_{\psi,A}\eta(A)S_{\varphi,A}$ are bounded between different tent/$Z$-spaces, where $\eta$ is a holomorphic function (not necessarily bounded) on an appropriate bisector.

In Chapter 4 we consider Besov–Hardy–Sobolev spaces adapted to an operator $A$ satisfying the aforementioned Standard Assumptions. In Section 4.1 we introduce ‘pre’-Besov–Hardy–Sobolev spaces $H_A^p$ and $E_A^p$ and establish their basic properties. Mapping properties of the holomorphic functional calculus between these spaces, including boundedness for $H^\infty$ functions of $A$ and ‘regularity shifting’ estimates for operators such as powers of $A$, are collected in Section 4.2. These all follow from Theorem 3.19. In Section 4.3 we discuss completions. This issue is more subtle than it initially seems. We define ‘canonical completions’ $\psi H_A^p$ and $\psi B_A^p$ in terms of auxiliary functions $\psi$, and show how these can be used to formulate satisfactory duality and interpolation results (Proposition 4.23 and Theorem 4.28). We also introduce ‘inclusion regions’ of exponents $\mathbf{p}$ such that $H_{A_0}^p \hookrightarrow H_{A_1}^p$ (likewise for Besov spaces) for two operators $A_0, A_1$ satisfying the Standard Assumptions with $\bar{R}(A_0) = \bar{R}(A_1)$. An interpolation result for these regions (Theorem 4.32) is proven. Finally, in Section 4.4 we show that the extended Cauchy operator $C_A$ produces
strong solutions of the Cauchy problem for A with initial data in any completion of any adapted pre-Besov-Hardy-Sobolev space, and we also show the quasinorm equivalence

$$ (1.10) \qquad \|f\|_{H_A^p} \approx \|C_A f\|_{T^p} \quad (f \in H_A^{p, \pm}) $$

when $\mathbf{p} = (p, s)$ with $p \le 2$ and $s < 0$, and likewise for Besov spaces and $Z$-spaces (Corollary 4.36).

Up until this point, we work with $\mathbb{C}^N$-valued functions for an arbitrary $N \in \mathbb{N}$, as in this abstract setting we gain nothing from the transversal/tangential structure of $\mathbb{C}^{m(1+n)}$.

In Chapter 5 we consider the case when A is a perturbed Dirac operator of the form $DB$ or $BD$ (and so we finally specialise to $\mathbb{C}^{m(1+n)}$-valued functions). We show that for all exponents $\mathbf{p}$ the spaces $H_D^p$ and $B_D^p$ are equal to projections of classical smoothness spaces intersected with $L^2$ (Theorem 5.3), and so we may take projections of these classical smoothness spaces (without intersecting with $L^2$) as completions. We denote the resulting spaces by $H_D^p$ and $B_D^p$. Then we define ‘identification regions’ $I(H, DB)$ and $I(B, DB)$, consisting of exponents $\mathbf{p}$ for which we can identify $H_D^p$ and $B_D^p$ as completions of $H_{DB}^p$ and $B_{DB}^p$ respectively. These regions turn out to be open (with some minor restrictions; see Theorem 5.18) and stable under interpolation and $\heartsuit$-duality (in a sense which interchanges $B$ and $B^*$; Corollary 5.14). Finally, in Theorem 5.26 we show that for $\mathbf{p} = (p, s) \in I(H, DB)$ with $p > 2$ and $s < 0$ we have boundedness of the Cauchy operator $C_{DB}^+$ from $H_{DB}^p$ to $T^p$, extending the ‘abstract’ estimate (1.10) (and likewise for Besov spaces and $Z$-spaces). This is a long argument which requires various ad-hoc estimates. The result is known to fail for $s = 0$, so it does not follow by interpolation.

In Chapter 6 we turn our attention to differential equations. After presenting some basic properties of gradients of solutions to $L_A u = 0$ (or equivalently solutions of $(CR)_{DB}$) we prove Theorems 6.8 and 6.9, which classify solutions to $(CR)_{DB}$ in tent/$Z$-spaces with a decay condition at infinity (the decay condition is removed for certain exponents in Section 6.2). This leads us to a range of exponents, related to the identification region, called the classification region. The argument is quite long, particularly for exponents $\mathbf{p} = (p, s)$ with $p > 2$, and uses all of the preceding material. We have been (perhaps excessively) pedantic in citing dependence on previous results, so it should be possible to treat certain technical lemmas as ‘black boxes’ in initial readings. Although these results are ‘intermediate to’ Theorem 1.8 and proven by similar arguments, they do not follow by any interpolation procedure. The results must be reproven manually. As a corollary of Theorems 6.8 and 6.9 we prove that certain solution spaces for the equation $L_A u = 0$ are interpolation scales (Theorem 6.41), and that the Whitney averages of such solutions have non-tangential boundary limits (Theorem 6.42).

In Chapter 7 we present applications to boundary value problems. Most of these have already been summarised in the introduction (Subsection 1.1.5). In particular, we derive ranges of well-posedness for Regularity and Neumann problems for various classes of coefficients in Section 7.2. Our results for real coefficient scalar equations with boundary data in Hardy-Sobolev spaces are new. In Section 7.3 we show that well-posedness of a boundary value problem is stable under perturbation of the coefficients, given certain a priori assumptions which are known to hold in some cases. Finally, we show the relationship between our approach and the method
of layer potentials in Section 7.4. For exponents in the classification region, all solutions to boundary value problems with gradients in the corresponding tent/Z-space and with the appropriate decay condition are given by (generalised) layer potentials.

## 1.3. Notation

The following notation will be used throughout the monograph.

We let $\mathbb{N} := \{0, 1, 2, \dots\}$ denote the natural numbers (including 0), and $\mathbb{N}_+ := \{1, 2, \dots\}$ denote the positive natural numbers.

For $a, b \in \mathbb{R}$ and $t > 0$ we write

$$m_a^b(t) := \begin{cases} t^a & (t \le 1) \\ t^{-b} & (t \ge 1). \end{cases}$$

For $0 < p, q \le \infty$, we define the number

$$\delta_{p,q} := \frac{1}{q} - \frac{1}{p},$$

with the interpretation $1/\infty = 0$.

We write the Euclidean distance on $\mathbb{R}^n$ as $d(x, y) = d(y, x) := |x - y|$, the open ball with centre $x \in \mathbb{R}^n$ and radius $r > 0$ by $B(x, r) := \{y \in \mathbb{R}^n : d(x, y) < r\}$, and the (half closed, half open) annulus with centre $x \in \mathbb{R}^n$, inner radius $r_0 > 0$, and outer radius $r_1 > r_0$ by

$$A(x, r_0, r_1) := B(x, r_1) \setminus B(x, r_0) = \{y \in \mathbb{R}^n : r_0 \le d(x, y) < r_1\}.$$

For subsets $E, F \subset \mathbb{R}^n$ we write

$$d(E, F) := \mathrm{dist}(E, F) = \inf\{d(x, y) : x \in E, y \in F\}.$$

We let $L^0(\Omega : E)$ denote the set of strongly measurable functions from a measure space $\Omega$ to a Banach space $E$. As usual, we identify two functions if they agree almost everywhere.

We write quasinorms as either $\|f\|_X$ or $\|f\|_X$ according to typographical need. For two quasinormed spaces $X$ and $Y$, we write $X \hookrightarrow Y$ to mean that $X \subset Y$ (possibly after some identification has been made) and that the identity map is bounded. We write $X \simeq Y$ to mean that $X \hookrightarrow Y$ and $Y \hookrightarrow X$, i.e. to mean that $X$ and $Y$ are isomorphic, and $X = Y$ to mean that the sets $X$ and $Y$ are equal and that the associated quasinorms are equivalent. Often we refer to norms as ‘quasinorms’ even though they are actually norms; for example, we refer to the $L^p$ quasinorm when $p \in (0, \infty]$, even though this is a norm when $p \ge 1$. For a quick introduction to quasi-Banach spaces the reader can consult the early sections of [54]. When necessary, we label dual pairings by the space on the left: for example, by $\langle f, g \rangle_{L^p}$, we mean the usual duality pairing between $L^p$ and $L^{p'}$, with $f \in L^p$ and $g \in L^{p'}$.

When $a$ and $b$ are real numbers, we write $a \lesssim b$ to mean that $a \le Cb$ for some $C > 0$ which is independent of $a$ and $b$, and which may vary from line to line. If $C$ depends on some other quantities $c_1, c_2, \dots$, we write $a \lesssim_{c_1, c_2, \dots} b$. We write $a \simeq b$ to mean that $a \lesssim b$ and $b \lesssim a$. Context prevents us from confusing this meaning of the symbol $\simeq$ and that introduced in the previous paragraph (representing isomorphism of quasinormed spaces).
We use a new and extensive notation for exponents, which appear in boldface
p. This is described in Section 2.1.

## Acknowledgements

The first author acknowledges financial support from the Australian Research
Council Discovery Grant DP120103692, and the VIDI subsidy 639.032.427 of the
Netherlands organisation for Scientific Research (NWO). Both authors were par-
tially supported by the ANR project "Harmonic analysis at its boundaries" ANR-
12-BS01-0013. The majority of this work was completed while the first author was
a doctoral student at the Australian National University and Université Paris-Sud.
Both authors thank Moritz Egert for valuable discussions on and around this topic.
# Function Space Preliminaries

Throughout this chapter we consider $\mathbb{C}^N$-valued functions for some fixed $N \in \mathbb{N}$, but since nothing really changes whether we choose $N = 1$ or $N \neq 1$ (this is justified in Remark 2.14) we do not refer to $\mathbb{C}^N$ in the notation. So we write $L^2(\mathbb{R}^n) = L^2(\mathbb{R}^n : \mathbb{C}^N)$, $T^p(\mathbb{R}^n) = T^p(\mathbb{R}^n : \mathbb{C}^N)$, and so on. For $z \in \mathbb{C}^N$ we write $|z|$ in place of $\|z\|_{\mathbb{C}^N}$.

## 2.1. Exponents

Our theory makes heavy use of relationships between different function spaces, which may be seen as relationships between the exponents used in their parametrisation. The most efficient way to keep track of these relationships, balancing economy of notation and clarity of ideas, is to introduce an enriched notation for exponents right at the beginning, and to work with it consistently.

Our exponent notation depends implicitly on a fixed dimension $n \in \mathbb{N}_+$. The set of exponents is defined to be the disjoint union

$$ \mathbf{E} := \mathbf{E}_{\text{fin}} \sqcup \mathbf{E}_{\infty}, $$

where $\mathbf{E}_{\text{fin}} := \{(p, s) : p \in (0, \infty), s \in \mathbb{R}\}$ and $\mathbf{E}_{\infty} := \{(\infty, s; \alpha) : s \in \mathbb{R}, \alpha \ge 0\}$. We say that an exponent is *finite* if it is in $\mathbf{E}_{\text{fin}}$, and *infinite* if it is in $\mathbf{E}_{\infty}$.

We define two functions $i: \mathbf{E} \to (0, \infty]$, $r: \mathbf{E} \to \mathbb{R}$, representing integrability and a kind of regularity, by

$$ i(p, s) := p, $$

$$ i(\infty, s; \alpha) := \infty, $$

$$ r(p, s) := s, $$

$$ r(\infty, s; \alpha) := s + \alpha. $$

We also define functions $j, \theta: \mathbf{E} \to \mathbb{R}$ by

$$ j(p, s) := 1/p, $$

$$ j(\infty, s; \alpha) := -\alpha/n $$

$$ \theta(p, s) := s, $$

$$ \theta(\infty, s; \alpha) := s. $$

Note that $\mathbf{p}$ is finite if and only if $j(\mathbf{p})$ is positive, and furthermore every exponent $\mathbf{p}$ is uniquely determined by the pair $(j(\mathbf{p}), \theta(\mathbf{p}))$. Thus we may consider the set of exponents as being parametrised by the points in the $(j, \theta)$-plane, as pictured in Figure 1.

For $r \in \mathbb{R}$ and $\mathbf{p} \in \mathbf{E}$, define $\mathbf{p} + r \in \mathbf{E}$ to be the unique exponent satisfying

$$ j(\mathbf{p} + r) = j(\mathbf{p}) \quad \text{and} \quad \theta(\mathbf{p} + r) = \theta(\mathbf{p}) + r. $$

We similarly define $\mathbf{p} - r$.

For every exponent $\mathbf{p}$, we define the dual exponent $\mathbf{p}' \in \mathbf{E}$ to be the unique exponent satisfying $j(\mathbf{p}') + j(\mathbf{p}) = 1$ and $\theta(\mathbf{p}') + \theta(\mathbf{p}) = 0$. For finite exponents we
have

$$ (p, s)' := \begin{cases} (p', -s) & (p > 1) \\ (\infty, -s; n(\frac{1}{p} - 1)) & (p \le 1), \end{cases} $$

where $p'$ is the usual Hölder conjugate of $p$. Clearly $p'' = p$. We also define the $\heartsuit$-dual exponent

$$ \mathbf{p}^{\heartsuit} := \mathbf{p}' - 1, $$

and a quick computation shows that $\mathbf{p}^{\heartsuit} = \mathbf{p}$.

For two exponents $\mathbf{p}, \mathbf{q} \in \mathbf{E}$, we write $\mathbf{p} \leftrightarrow \mathbf{q}$ to mean that

$$ \theta(\mathbf{p}) \geq \theta(\mathbf{q}) \quad \text{and} \quad \theta(\mathbf{q}) - \theta(\mathbf{p}) = n(j(\mathbf{q}) - j(\mathbf{p})). $$

We always have $\mathbf{p} \leftrightarrow \mathbf{p}$. Observe that $\mathbf{p} \leftrightarrow \mathbf{q}$ and $\mathbf{q} \leftrightarrow \mathbf{r}$ implies $\mathbf{p} \leftrightarrow \mathbf{r}$, and $\mathbf{p} \leftrightarrow \mathbf{q}$ if and only if $\mathbf{q}' \leftrightarrow \mathbf{p}'$. This notation reflects embedding properties of function spaces.

For $\eta \in \mathbb{R}$, define $[\mathbf{p}, \mathbf{q}]_{\eta} \in \mathbb{E}$ to be the unique exponent satisfying

$$ j([\mathbf{p}, \mathbf{q}]_{\eta}) = (1 - \eta)j(\mathbf{p}) + \eta j(\mathbf{q}), $$

$$ \theta([\mathbf{p}, \mathbf{q}]_{\eta}) = (1 - \eta)\theta(\mathbf{p}) + \eta\theta(\mathbf{q}). $$

Note that $[\mathbf{p}, \mathbf{q}]_0 = \mathbf{p}$ and $[\mathbf{p}, \mathbf{q}]_1 = \mathbf{q}$. This notation is particularly useful for interpolation results.

**Lemma 2.1.** Suppose $\mathbf{p}$ and $\mathbf{q}$ are exponents with $\mathbf{p} \leftrightarrow \mathbf{q}$. Then $[\mathbf{p}, \mathbf{q}]_{\eta_0} \leftrightarrow [\mathbf{p}, \mathbf{q}]_{\eta_1}$ whenever $\eta_0 \le \eta_1$.

**PROOF.** Write

$$ \begin{align} \theta([\mathbf{p}, \mathbf{q}]_{\eta_1}) - \theta([\mathbf{p}, \mathbf{q}]_{\eta_0}) &= ((1 - \eta_1)\theta(\mathbf{p}) + \eta_1\theta(\mathbf{q})) - ((1 - \eta_0)\theta(\mathbf{p}) + \eta_0\theta(\mathbf{q})) \tag{2.1} \\ &= (\eta_1 - \eta_0)(\theta(\mathbf{q}) - \theta(\mathbf{p})) \tag{2.2} \\ &= n(\eta_1 - \eta_0)(j(\mathbf{q}) - j(\mathbf{p})) \\ &= n(((1 - \eta_1)j(\mathbf{p}) + \eta_1 j(\mathbf{q})) - ((1 - \eta_0)j(\mathbf{p}) + \eta_0 j(\mathbf{q}))) \\ &= n(j([\mathbf{p}, \mathbf{q}]_{\eta_1}) - j([\mathbf{p}, \mathbf{q}]_{\eta_0})). \end{align} $$

Line (2.2) follows from $\mathbf{p} \leftrightarrow \mathbf{q}$. Furthermore, line (2.1), $\eta_1 - \eta_0 \ge 0$, and $\theta(\mathbf{p}) \ge \theta(\mathbf{q})$ imply that $\theta([\mathbf{p}, \mathbf{q}]_{\eta_0}) \ge \theta([\mathbf{p}, \mathbf{q}]_{\eta_1})$. Thus $[\mathbf{p}, \mathbf{q}]_{\eta_0} \leftrightarrow [\mathbf{p}, \mathbf{q}]_{\eta_1}$. $\square$

<figure><img src="image_3.png" /></figure>

A straightforward computation shows the following lemma.

**Lemma 2.2.** Suppose $\mathbf{p}, \mathbf{q} \in \mathbb{E}$ and $\eta_0, \eta_1, \lambda \in \mathbb{R}$. Then

$$ [[\mathbf{p}, \mathbf{q}]_{\eta_0}, [\mathbf{p}, \mathbf{q}]_{\eta_1}]_{\lambda} = [\mathbf{p}, \mathbf{q}]_{(1-\lambda)\eta_0 + \lambda\eta_1}. $$

In particular this implies

$$ \mathbf{p} = [[\mathbf{p}, \mathbf{q}]_{-1}, \mathbf{q}]_{1/2} $$

$$ \mathbf{q} = [\mathbf{p}, [\mathbf{p}, \mathbf{q}]_2]_{1/2}. $$

The most convenient way of visualising exponents is by identifying them with points in the $(j, \theta)$ plane. In Figure 1 we show two exponents $\mathbf{p}$ and $\mathbf{q}$ with $\mathbf{p} \leftrightarrow \mathbf{q}$, their dual exponents, their $\heartsuit$-duals, and various other exponents which may be constructed from them. The operations $\mathbf{p} \leftrightarrow \mathbf{p}'$ and $\mathbf{p} \leftrightarrow \mathbf{p}^{\heartsuit}$ are given by reflection about the marked points at $(1/2, 0)$ and $(1/2, -1/2)$ respectively. The exponent $(1/2, -1/2)$ is special: in Section 7.1 we introduce it as the ‘energy exponent’. Certain boundary value problems associated with this exponent are automatically well-posed. Observe that $\mathbf{p} \leftrightarrow \mathbf{q}$ if and only if the line segment from $(j(\mathbf{p}), \theta(\mathbf{p}))$
<figure><img src="image_4.png" /><figcaption>FIGURE 1. Various exponents in the $(j, \theta)$ plane.</figcaption></figure>

to $(j(\mathbf{q}), \theta(\mathbf{q}))$ is parallel to that from $((n+1)/n, 0)$ to $(1, -1)$, with the same orientation.

In Subsection 2.7 we provide a convenient table detailing the function spaces associated with different exponents.

## 2.2. Tent spaces

The most fundamental function spaces in this monograph are the tent spaces. These were first introduced by Coifman, Meyer, and Stein [26, 27], and they have since proven their worth in harmonic analysis and PDE. The other ‘ambient spaces’ that we use—Z-spaces and slice spaces—are closely related to tent spaces, so a solid knowledge of tent spaces will be useful.

For $x \in \mathbb{R}^n$ we define the cone with vertex $x$ by

$$ \Gamma(x) := \{(t, y) \in \mathbb{R}_+^{1+n} : y \in B(x, t)\}, $$

and for each open ball $B \subset X$ we define the tent with base $B$ by

$$ T(B) := \mathbb{R}_+^{1+n} \setminus \left( \bigcup_{x \notin B} \Gamma(x) \right). $$

Equivalently, $T(B)$ is the set of points $(y, t) \in \mathbb{R}_+^{1+n}$ such that $B(y, t) \subset B$.

The tent space quasinorms are defined in terms of the Lusin operator $\mathcal{A}$ and Carleson operators $C_\alpha$, which are defined as follows: for all $\alpha \ge 0$ and $f \in L^0(\mathbb{R}_+^{1+n})$,

$$ (2.3) \qquad \mathcal{A}f(x) := \left( \iint_{\Gamma(x)} |f(t, y)|^2 \frac{dy dt}{t^{n+1}} \right)^{1/2} \quad (x \in \mathbb{R}^n) $$
and

$$
C_{\alpha}f(x) := \sup_{B \ni x} \frac{1}{r_B^{\alpha}} \left( \frac{1}{r_B^n} \iint_{T(B)} |f(t, y)|^2 dy \frac{dt}{t} \right)^{1/2} \quad (x \in \mathbb{R}^n).
$$

For each $s \in \mathbb{R}$ we define an operator $\kappa^s$ on $L^0(\mathbb{R}_+^{1+n})$ by

$$
(\kappa^s f)(t, x) := t^s f(t, x) \quad ((t, x) \in \mathbb{R}_+^{1+n}).
$$

This operator is used to define ‘weighted’ spaces.

In the following definition, and indeed throughout the whole monograph from
this point, we use the exponent notation from Section 2.1.

**Definition 2.3.** For a finite exponent **p** = (p, s), the tent space T**p** = T**p**(R**n**) is the set

$$
T^{\mathbf{p}} = T_s^{\mathbf{p}} := \{ f \in L^0(\mathbb{R}_+^{1+n}) : \mathcal{A}(\kappa^{-s} f) \in L^p(\mathbb{R}^n) \}
$$

equipped with the quasinorm

$$
\|f\|_{T_s^p} := \|\mathcal{A}(\kappa^{-s} f)\|_{L^p(\mathbb{R}^n)}.
$$

For an infinite exponent **p** = (∞, *s*; *α*) we define T**p** by

$$
T^{\mathbf{p}} = T_{s;\alpha}^{\infty} := \{ f \in L^0(\mathbb{R}_+^{1+n}) : \mathcal{C}_{\alpha}(\kappa^{-s} f) \in L^{\infty}(\mathbb{R}^n) \}
$$

with its natural norm.

**Remark 2.4.** The spaces $T_s^p$ agree with those defined by Hofmann, Mayboroda, and McIntosh [46, §8.3], and with the spaces $T_{2,s}^{p,2}$ of Huang [47]. Our spaces $T_{s;0}^\infty$ agree with Huang’s spaces $T_{2,s}^{\infty,2}$.

All tent spaces are quasi-Banach spaces (Banach when $i(\mathbf{p}) \ge 1$). For all finite exponents $\mathbf{p}$, the subspace $T^{\mathbf{p};c} \subset T^{\mathbf{p}}$ of compactly supported functions is dense in $T^{\mathbf{p}}$, and $L_c^2(\mathbb{R}_+^{1+n})$ is densely contained in $T^{\mathbf{p}}$.

**Definition 2.5.** Let **p** be an exponent with $i(\mathbf{p}) \le 1$, and suppose $B \subset \mathbb{R}^n$ is a ball. We say that a function $a \in L^0(\mathbb{R}_+^{1+n})$ is a $T^{\mathbf{p}}$-atom (associated with B) if $a$ is essentially supported in $T(B)$ and if

$$
\|a\|_{T_s^2} \le |B|^{\delta_{p,2}},
$$

where $\delta_{p,2} = \frac{1}{2} - \frac{1}{p}$ (as defined in Section 1.3).

An atomic decomposition for our tent spaces follows immediately from the
original tent space atomic decomposition theorem [27, Theorem 1c].

**Theorem 2.6 (Atomic decomposition).** Let **p** be an exponent with $i(\mathbf{p}) \le 1$. Then a function $f \in L^0(\mathbb{R}_+^{1+n})$ is in $T^{\mathbf{p}}$ if and only if there exists a sequence $(a_k)_{k \in \mathbb{N}}$ of $T^{\mathbf{p}}$-atoms and a sequence $\lambda \in \ell^p(\mathbb{N})$ such that

$$
(2.4) \qquad f = \sum_{k \in \mathbb{N}} \lambda_k a_k
$$

with convergence in $T^{\mathbf{p}}$. Furthermore we have

$$
\|f\|_{T^{\mathbf{p}}} \simeq \inf \|\lambda\|_{\ell^p(\mathbb{N})},
$$

where the infimum is taken over all such decompositions.

The following duality theorem includes all finite exponents; our exponent no-
tation allows us to state the result without needing to separate the cases $i(\mathbf{p}) \le 1$ and $i(\mathbf{p}) > 1$.
**Theorem 2.7 (Duality).** Suppose that **p** is a finite exponent. Then for all $f, g \in L^0(\mathbb{R}_+^{1+n})$ we have

$$ (2.5) \qquad \iint_{\mathbb{R}_+^{1+n}} |(f(t,x), g(t,x))| dx \frac{dt}{t} \lesssim \|f\|_{T^{\mathbf{P}}} \|g\|_{T^{\mathbf{P}'}} $$

and the pairing

$$ (2.6) \qquad \langle f, g \rangle := \iint_{\mathbb{R}_+^{1+n}} (f(t,x), g(t,x)) dx \frac{dt}{t} $$

identifies the Banach space dual of $T^{\mathbf{P}}$ with $T^{\mathbf{P}'}$.

Note in particular that the integral in (2.5) converges absolutely.

**Remark 2.8.** Throughout this monograph we refer to the pairing in (2.6) as the $L^2$ duality pairing.

When **p** is finite and $i(\mathbf{p}) \ge 2$, $T^{\mathbf{P}}$ may also be characterised in terms of the Carleson operator $C_0$. This is a straightforward extension of [27, Theorem 3].

**Theorem 2.9 (Carleson characterisation of $T^{\mathbf{P}}$).** Suppose **p** is a finite exponent with $i(\mathbf{p}) > 2$. Then for all $f \in L^0(\mathbb{R}_+^{1+n})$ we have

$$ \|f\|_{T^{\mathbf{P}}} \simeq \|C_0(\kappa^{-\theta(\mathbf{p})}f)\|_{L^p(\mathbb{R}^n)}. $$

The following ‘change of aperture’ theorem was proven by Coifman, Meyer, and Stein for $T_0^p$ [27, Proposition 4], and the extension to the more general tent spaces here is immediate.

**Theorem 2.10 (Change of aperture).** For $\beta \in (0, \infty)$ and $x \in \mathbb{R}^n$ define

$$ \Gamma_{\beta}(x) := \{(t, y) \in \mathbb{R}_+^{1+n} : y \in B(x, \beta t)\}, $$

and for $f \in L^0(\mathbb{R}_+^{1+n})$ define $A_{\beta}f(x)$ as in (2.3), with $\Gamma_{\beta}(x)$ in place of $\Gamma(x)$. Then for $\beta \in (0, \infty)$ and each finite exponent **p** we have an equivalence of quasinorms

$$ \|f\|_{T^{\mathbf{P}}} \simeq \|A_{\beta}(\kappa^{-\theta(\mathbf{p})}f)\|_{L^{i}(\mathbf{p})(\mathbb{R}^n)}. $$

The following embedding theorem, which can be seen as a tent space analogue of the Hardy–Littlewood–Sobolev embedding theorem, was proven by the first author [3, Theorem 2.19]. The case where **p** and **q** are both infinite is not explicitly shown there, but it follows by the same argument.

**Theorem 2.11 (Embeddings).** Let **p** and **q** be exponents with $\mathbf{p} \mapsto \mathbf{q}$. Then we have the embedding $T^{\mathbf{P}} \hookrightarrow T^{\mathbf{q}}$.

The following complex interpolation theorem was proven by Hofmann, Mayboroda, and McIntosh for finite exponents [46, Lemma 8.23], and the extension to one infinite exponent follows by duality [3, Theorem 2.1].

**Theorem 2.12 (Complex interpolation).** Suppose **p** and **q** are exponents with $j(\mathbf{p}), j(\mathbf{q}) \ge 0$ (with equality for at most one exponent), and let $0 < \eta < 1$. Then

$$ [T^{\mathbf{p}}, T^{\mathbf{q}}]_{\eta} = T^{[\mathbf{p} \cdot \mathbf{q}]_{\eta}}. $$

**Remark 2.13.** Here, and throughout the monograph, by ‘complex interpolation’ we mean the Kalton–Mitrea complex interpolation method, introduced in [56, §3]. This agrees with the usual (Calderón) complex interpolation method on couples of Banach spaces, and is well-defined for all quasi-Banach couples.
**Remark 2.14.** In contrast with the earlier work of the first author [3], we define the operator $\kappa^s$ in terms of powers of $t$ rather than powers of volumes of balls, so our tent spaces $T_s^p(\mathbb{R}^n)$ correspond to his tent spaces $T_{s/n}^{p,2}(\mathbb{R}^n)$. We also use $\mathbb{C}^N$-valued functions instead of $\mathbb{C}$-valued functions. This does not change the validity of previous results, as one can always split $T_s^p(\mathbb{R}^n : \mathbb{C}^N) \simeq \bigoplus_{j=1}^N T_s^p(\mathbb{R}^n : \mathbb{C})$ and apply the results to each summand individually. This reduction would fail if we were to replace $\mathbb{C}^N$ with a general Banach space, but thankfully we have no need for such generality.

## 2.3. Z-spaces

We now introduce a class of function spaces, called Z-spaces, which are related to tent spaces by real interpolation. The Z-spaces play the role for Besov spaces $\dot{B}_{s,p}^{p}$ that the tent spaces play for Hardy-Sobolev spaces $\dot{H}_{s}^{p}$.

**Definition 2.15.** We refer to a pair

$$c = (c_0, c_1) \in (0, \infty) \times (3/2, \infty)$$

as a Whitney parameter. To each Whitney parameter $c$ and each $(t, x) \in \mathbb{R}_+^{1+n}$ we associate the Whitney region

$$\Omega_c(t, x) := (c_1^{-1}t, c_1t) \times B(x, c_0t) \subset \mathbb{R}_+^{1+n},$$

and for $f \in L^0(\mathbb{R}_+^{1+n})$ we define the L$^2$-Whitney averages

$$W_c f(t, x) := \left( \iint_{\Omega_c(t,x)} |f(\tau, \xi)|^2 d\xi d\tau \right)^{1/2} \quad ((t, x) \in \mathbb{R}_+^{1+n}).$$

For an exponent $p$ and a Whitney parameter $c$, and for all $f \in L^0(\mathbb{R}_+^{1+n})$, we define the quasinorm

$$(2.7) \qquad \|f\|_{Z_c^p} := \|W_c(\kappa^{-r(\mathbf{p})}f)\|_{L^{i(\mathbf{p})}(\mathbb{R}_+^{1+n})}$$

and a corresponding function space

$$Z_c^p = Z_c^p(\mathbb{R}^n) := \{f \in L^0(\mathbb{R}_+^{1+n}) : \|f\|_{Z_c^p} < \infty\}.$$

We write $\Omega(t, x) := \Omega_{(1,2)}(t, x)$ when a particular Whitney parameter is not needed.

**Remark 2.16.** The spaces $Z_c^p$ coincide with the spaces $L(i(\mathbf{p}), r(\mathbf{p}) + 1, 2)$ introduced by Barton and Mayboroda [22]. We use these spaces for the same purpose: as an ambient space for the gradient of a solution to an elliptic BVP with boundary data in a Besov space. The connection with tent spaces presented here extends that established by the first author [3].

**Remark 2.17.** The restriction $c_1 > 3/2$ is for technical reasons. The first time that it is actually needed is in our proof of the atomic decomposition theorem. It is possible to allow for $c_1 > 1$ by a straightforward covering argument, but this would take extra work, and $c_1 > 3/2$ is sufficient for our applications.

The following real interpolation theorem appears in [3, Theorem 2.9]. In Theorem 2.30 we extend it to infinite exponents.
**Theorem 2.18 (Real interpolation for tent spaces with finite exponents).** Suppose that $\mathbf{p}$ and $\mathbf{q}$ are finite exponents with $\theta(\mathbf{p}) \neq \theta(\mathbf{q})$, and suppose $0 < \eta < 1$. Then for all Whitney parameters $c$ we have

$$ (T^{\mathbf{p}}, T^{\mathbf{q}})_{\eta, p_{\eta}} = Z_{c}^{[\mathbf{p}, \mathbf{q}]_{\eta}}, $$

where $p_{\eta} = i([\mathbf{p}, \mathbf{q}]_{\eta})$.

Thus for finite $\mathbf{p}$ the Z-spaces $Z_c^{\mathbf{p}}$ are complete and independent of $c$ (up to equivalence of quasinorms); we extend this to infinite exponents later. Hence we write $Z^{\mathbf{p}}$ in place of $Z_c^{\mathbf{p}}$.

We establish further properties of the Z-spaces ‘by hand’ rather than arguing by interpolation, because this yields stronger results. In particular, it yields absolute convergence of $L^2$ duality pairings, while interpolation would only prove this on dense subspaces. This is important in applications. Our main tool is a dyadic characterisation of the $Z^{\mathbf{p}}$-quasinorm, is stated and used by Barton and Mayboroda [22, Proof of Theorem 4.13] but without proof. To establish this characterisation we need some notation and a preliminary counting lemma.

Let $\mathcal{Q}(\mathbb{R}^n)$ be a system of (open) dyadic cubes in $\mathbb{R}^n$. For every cube $Q \in \mathcal{Q}(\mathbb{R}^n)$ and for $k \in \mathbb{Z}$, define the *Whitney cube*

$$ \bar{Q}^k := (2^k \ell(Q), 2^{k+1} \ell(Q)) \times Q, $$

and the *Whitney grid*

$$ \mathcal{G}^k := \{\bar{Q}^k : Q \in \mathcal{Q}(\mathbb{R}^n)\}. $$

For each $k \in \mathbb{Z}$, $\mathcal{G}^k$ is a partition of $\mathbb{R}_+^{1+n}$ up to a set of measure zero.

For each Whitney parameter $c$, each $k \in \mathbb{Z}$, and each Whitney cube $\bar{Q}^k \in \mathcal{G}^k$, we define

$$ \mathcal{G}_c(\bar{Q}^k) := \{\bar{R}^k \in \mathcal{G}^k : \bar{R}^k \cap \Omega_c(t, x) \neq \emptyset \text{ for some } (t, x) \in \bar{Q}^k\}. $$

**Lemma 2.19.** Let $c$ be a Whitney parameter and $k \in \mathbb{N}$. Then for all $\bar{Q}^k \in \mathcal{G}^k$ we have

$$ |\mathcal{G}_c(\bar{Q}^k)| \lesssim_{c,k,n} 1 $$

(where $|\cdot|$ denotes cardinality).

**PROOF.** The condition $\bar{R}^k \cap \Omega_c(t, x) \neq \emptyset$ may be rewritten as

$$ \ell(R) \in [t/2^{k+1}c_1, 2^{-k}c_1t] \text{ and } \mathrm{dist}(R, x) < c_0t. $$

By rescaling and translating, the number of $R \in \mathcal{Q}(\mathbb{R}^n)$ such that this condition is satisfied is equal to the number of $R \in \mathcal{Q}(\mathbb{R}^n)$ such that

$$ \ell(R) \in (1/2^{k+1}c_1, 2^{-k}c_1) \text{ and } \mathrm{dist}(R, 0) < c_0, $$

which is finite and depends only on $c$, $k$, and $n$. $\square$

**Proposition 2.20 (Dyadic characterisation).** Let $\mathbf{p}$ be a finite exponent, $c$ a Whitney parameter, and $k \in \mathbb{Z}$. Then

$$ \|f\|_{Z_c^{\mathbf{p}}} \simeq_{c,k,\mathbf{p}} \left\| \ell(Q)^{-r(\mathbf{p})} [|f|^2]^{1/2}_{\bar{Q}^k} \right\|_{\ell^{\mathcal{P}}(\mathcal{G}^k, \ell(Q)^n)}, $$

$$ \text{where } [|f|^2]^{1/2}_{\bar{Q}^k} = \|f\|_{L^2(\bar{Q}^k, d\tau d\xi/\tau^{1+n})}. $$
PROOF. Write $\mathbf{p} = (p, s)$ and estimate

$$ (2.8) \qquad \|f\|_{Z_c^p}^p = \sum_{\overline{Q}^k \in \mathcal{G}^k} \iint_{\overline{Q}^k} \mathcal{W}_c(\kappa^{-s}f)(t, x)^p \frac{dt}{t} dx \\ \qquad \approx \sum_{\overline{Q}^k \in \mathcal{G}^k} (2^k \ell(Q))^{-ps} \iint_{\overline{Q}^k} \|f\| L^2(\Omega_c(t, x), d\tau d\xi/\tau^{1+n})^p \frac{dt}{t} dx $$

$$ (2.9) \qquad \lesssim \sum_{\overline{Q}^k \in \mathcal{G}^k} (2^k \ell(Q))^{-ps} \iint_{\overline{Q}^k} \sum_{\overline{R}^k \in \mathcal{G}_c(\overline{Q}^k)} \|f\| L^2(\overline{R}^k, d\tau d\xi/\tau^{1+n})^p \frac{dt}{t} dx $$

$$ (2.10) \qquad \approx_{k,p,s} \sum_{\substack{\overline{Q}^k \in \mathcal{G}^k \\ \overline{R}^k \in \mathcal{G}_c(\overline{Q}^k)}} \ell(R)^{n-ps} \|f\| L^2(\overline{R}^k, d\tau d\xi/\tau^{1+n})^p $$

$$ (2.11) \qquad \approx \sum_{\overline{R}^k \in \mathcal{G}^k} \ell(R)^n \left( \ell(R)^{-s} \|f\| L^2(\overline{R}^k, d\tau d\xi/\tau^{1+n}) \right)^p. $$

The equivalence (2.8) comes from the fact that $\tau \approx 2^k \ell(Q)$ when $(\tau, \xi) \in \Omega_c(t, x)$ and $(t, x) \in \overline{Q}^k$. The upper bound (2.9) comes from covering $\Omega_c(t, x)$ with the Whitney cubes $\overline{R}^k \in \mathcal{G}_c(\overline{Q})$, of which there are boundedly many by Lemma 2.19. The equivalence (2.10) comes from noting that $\ell(R) \approx \ell(Q)$ when $\overline{R}^k \in \mathcal{G}_c(\overline{Q}^k)$. Finally, (2.11) follows from the fact that every cube $\overline{R}^k \in \mathcal{G}^k$ appears at least once, and at most a bounded number of times, in the multiset $\{\overline{R}^k \in \mathcal{G}_c(\overline{Q}^k) : \overline{Q}^k \in \mathcal{G}\}$.

To prove the converse statement, we need only prove the converse direction of (2.9). This follows from the existence of a Whitney parameter $\tilde{c}$ such that whenever $\overline{R}^k \in \mathcal{G}_c(\overline{Q}^k)$ and $(t, x) \in \overline{Q}^k$, we have $\overline{R}^k \subset \Omega_{\tilde{c}}(t, x)$. To this end one can take $\tilde{c}_0 = 2(c_0 + 2^{-k}\sqrt{n}(c_1+1))$ and $\tilde{c}_1 = 4c_1$. This, along with the independence of $Z_c^p$ on $c$, completes the proof. $\square$

**Remark 2.21.** The same proof works for infinite exponents once we show that the corresponding Z-space norms are independent of $c$.

The dyadic characterisation of the Z-space quasinorm can be used to prove a duality theorem when $i(\mathbf{p}) > 1$. As with the corresponding result for tent spaces, this is not just an abstract identification of dual spaces (which could be deduced by real interpolation), but also includes absolute convergence of the $L^2$ duality pairing.

**Proposition 2.22 (Duality: reflexive range).** Suppose $i(\mathbf{p}) \in (1, \infty)$. Then for all $f, g \in L^0(\mathbb{R}_+^{1+n})$ we have

$$ (2.12) \qquad \iint_{\mathbb{R}_+^{1+n}} |(f(t, x), g(t, x))| dx \frac{dt}{t} \lesssim \|f\|_{Z^{\mathbf{p}}} \|g\|_{Z^{\mathbf{p}'}}, $$

and the $L^2$ duality pairing identifies the Banach space dual of $Z^{\mathbf{p}}$ with $Z^{\mathbf{p}'}$.

PROOF. Let $\overline{Q}_0 = (1, 2) \times (0, 1)^n$ be equipped with the measure $dx dt/t^{1+n}$, and for each function $f \in L^0(\mathbb{R}_+^{1+n})$ and each cube $\overline{Q} \in \mathcal{G}$, let $f_{\overline{Q}}$ be the function on $\overline{Q}_0$ which is the affine reparametrisation of $\mathbf{1}_{\overline{Q}}f$, so that $[|f|^2]_{\overline{Q}}^{1/2} = \|f_{\overline{Q}}\|_{L^2(\overline{Q}_0)}$. Then by Proposition 2.20, writing $\mathbf{p} = (p, s)$, we have

$$ \|f\|_{Z^{\mathbf{p}}} \approx \|\ell(Q)^{-s} [|f|^2]_{\overline{Q}}^{1/2} \|_{\ell^p(\mathcal{G}, \ell(Q)^n)} $$
$$ \begin{aligned} & \simeq \|\ell(Q)^{-s} f_{\overline{Q}}\|_{\ell^p(\mathcal{G}, \ell(Q)^n; L^2(\overline{Q}_0))} \\ & =: \|f_{\overline{Q}}\|_{\ell_s^p(\mathcal{G}, \ell(Q)^n; L^2(\overline{Q}_0))}. \end{aligned} $$

Evidently the map $f \mapsto (f_{\overline{Q}})_{\overline{Q} \in \mathcal{G}}$ is an isomorphism between $Z^p$ and $\ell_s^p(\mathcal{G}, \ell(Q)^n : L^2(\overline{Q}_0))$. Furthermore, for all $f, g \in L^0(\mathbb{R}_+^{1+n})$ we have

$$ \int_{\mathbb{R}_+^{1+n}} |(f(t, x), g(t, x))| dx \frac{dt}{t} \simeq \sum_{\overline{Q} \in \mathcal{G}} \ell(Q)^n \iint_{\overline{Q}_0} |(f_{\overline{Q}}(t, x), g_{\overline{Q}}(t, x))| \frac{dx dt}{t^{1+n}}, $$

so $f \mapsto (f_{\overline{Q}})_{\overline{Q} \in \mathcal{G}}$ identifies the $L^2(\mathbb{R}_+^{1+n})$ and $\ell^2(\mathcal{G}, \ell(Q)^n : L^2(\overline{Q}_0))$ duality pairings (up to a constant).

Since we have

$$ \sum_{\overline{Q} \in \mathcal{G}} \ell(Q)^n |(f_{\overline{Q}}, g_{\overline{Q}})_{L^2(\overline{Q}_0)}| \lesssim \|f_{\overline{Q}}\|_{\ell_s^p(\mathcal{G}, \ell(Q)^n; L^2(\overline{Q}_0))} \|g_{\overline{Q}}\|_{\ell_{-s}^{p'}(\mathcal{G}, \ell(Q)^n; L^2(\overline{Q}_0))} $$

and since the $\ell^2(\mathcal{G}, \ell(Q)^n : L^2(\overline{Q}_0))$ duality pairing identifies $\ell_s^{p'}(\mathcal{G}, \ell(Q)^n : L^2(\overline{Q}_0))$ as the dual of $\ell_{-s}^p(\mathcal{G}, \ell(Q)^n : L^2(\overline{Q}_0))$, the corresponding results for $Z^p$ follow. $\square$

The dyadic characterisation may also be used to prove an atomic decomposition theorem for Z-spaces.

**Definition 2.23.** Let $\mathbf{p} = (p, s)$ be a finite exponent and $c$ a Whitney parameter. We say that a function $a \in L^0(\mathbb{R}_+^{1+n})$ is a $Z^p$-atom associated with the point $(t, x) \in \mathbb{R}_+^{1+n}$ if $a$ is essentially supported in $\Omega_c(t, x)$ and if

$$ \|\kappa^{-s} a\|_{L^2(\Omega_c(t,x), dx dt/t)} \le t^{n\delta_{p,2}}. $$

(recall that $\delta_{p,2} = \frac{1}{2} - \frac{1}{p}$ is defined in Section 1.3).

**Lemma 2.24.** Let $\mathbf{p}$ be a finite exponent and suppose $a$ is a $Z^p$-atom associated with $(t_0, x_0) \in \mathbb{R}_+^{1+n}$. Then

$$ \|a\|_{Z^p} \lesssim_{c, \mathbf{p}} 1. $$

**PROOF.** A reasonably quick computation shows that

$$ \{(t, x) \in \mathbb{R}_+^{1+n} : \Omega_c(t, x) \cap \Omega_c(t_0, x_0) \neq \emptyset\} \subset \Omega_{\tilde{c}}(t_0, x_0) $$

where $\tilde{c}_0 = c_0(1 + c_1^2)$ and $\tilde{c}_1 = c_1^2$. Hence we can estimate, using the support and size conditions for $a$ and writing $\mathbf{p} = (p, s)$,

$$ \begin{aligned} \|a\|_{Z^p} &\le \left( \iint_{\Omega_{\tilde{c}}(t_0, x_0)} \left( \frac{1}{t^{1+n}} \iint_{\Omega_c(t_0, x_0)} \tau |\tau^{-s} a(\xi, \tau)|^2 d\xi \frac{d\tau}{\tau} \right)^{p/2} dx \frac{dt}{t} \right)^{1/p} \\ &\lesssim_c t_0^{n\delta_{p,2}} \left( \iint_{\Omega_{\tilde{c}}(t_0, x_0)} t_0^{-np/2} dx \frac{dt}{t} \right)^{1/p} \approx_{c,p} t_0^{n\delta_{p,2} - \frac{n}{2} + \frac{n}{p}} = 1 \end{aligned} $$

as required.

$\square$

**Theorem 2.25 (Atomic decomposition of Z-spaces).** Suppose $\mathbf{p} = (p, s)$ with $p \le 1$, and let $c$ be a Whitney parameter. Then a function $f \in L^0(\mathbb{R}_+^{1+n})$ is in $Z^p$ if and only if there exists a sequence $(a_k)_k \in \mathbb{N}$ of $Z^p$-atoms and a sequence $\lambda \in \ell^p(\mathbb{N})$ such that

$$ \sum_{k \in \mathbb{N}} \lambda_k a_k = f $$
with convergence in $Z^P$. Furthermore we have

$$ \|f\|_{Z^P} \simeq \inf \|\lambda\|_{\ell^p(\mathbb{N})}, $$

where the infimum is taken over all such decompositions.

PROOF. Given such a decomposition of $f$, we have

$$ \|f\|_{Z^P}^p = \left\| \sum_{k \in \mathbb{N}} \lambda_k a_k \right\|_{Z^P}^p \lesssim \|\lambda\|_{\ell^p(\mathbb{N})}^p $$

by Lemma 2.24, and so $\|f\|_{Z^P} \lesssim \inf \|\lambda\|_{\ell^p(\mathbb{N})}$. It remains to prove the reverse estimate. For each $k \in \mathbb{Z}$ we can write

$$ (2.13) \qquad f = \sum_{\overline{Q}^k \in \mathcal{G}^k} f_{\overline{Q}^k} $$

where $f_{\overline{Q}^k} = \mathbf{1}_{\overline{Q}^k} f$ (note that this notation differs from that in the proof of Proposition 2.22). By Proposition 2.20, and using dominated convergence, this sum converges in $Z^P$. If $k \ge \log_2(c_0^{-1}\sqrt{n}/3) + 1$ and if $c_1 > 3/2$ (this is the first place where we actually use this assumption), then

$$ \overline{Q}^k \subset \Omega_c(c_Q, t_Q) $$

for all $Q \in \mathcal{Q}(\mathbb{R}^n)$, where $c_Q$ is the center of $Q$ and $t_Q$ is the midpoint of $2^k \ell(Q)$ and $2^{k+1} \ell(Q)$. Therefore, under this condition on $k$, each $f_{\overline{Q}^k}$ satisfies the support condition required of a $Z_c^P$-atom. By Proposition 2.20 the norms

$$ \|\kappa^{-s} f_{\overline{Q}^k}\|_{L^2(\Omega_c(c_Q, t_Q), dx \, dt/t)} $$

are all finite, so we can define

$$ \lambda_{\overline{Q}^k} := t_{\overline{Q}^k}^{-n\delta_{p,2}} \|\kappa^{-s} f_{\overline{Q}^k}\|_{L^2(\Omega_c(c_{\overline{Q}}, t_{\overline{Q}}), dx \, dt/t)} $$

and

$$ a_{\overline{Q}^k} := \begin{cases} \lambda_{\overline{Q}^k}^{-1} f_{\overline{Q}^k} & (f_{\overline{Q}^k} \neq 0) \\ 0 & (f_{\overline{Q}^k} = 0). \end{cases} $$

Each $a_{\overline{Q}^k}$ is a $Z_c^P$-atom and

$$ f = \sum_{\overline{Q}^k \in \mathcal{G}^k} \lambda_{\overline{Q}^k} a_{\overline{Q}^k} $$

with convergence in $Z^P$, and furthermore

$$ \begin{aligned} \|(\lambda_{\overline{Q}^k})\|_{\ell^p(\mathcal{G}^k)} &= \left\| t_{\overline{Q}^k}^{-n\delta_{p,2}} \|\kappa^{-s} f_{\overline{Q}^k}\|_{L^2(\Omega_c(c_{\overline{Q}^k}, t_{\overline{Q}^k}), dx \, dt/t)} \right\|_{\ell^p(\mathcal{G}^k)} \\ &\simeq \|(2^k \ell(Q))^{-n\delta_{p,2}-s} |Q|^{1/2} [|f|^2]_{\overline{Q}^k}^{1/2} \|_{\ell^p(\mathcal{G}^k)} \\ &\simeq \|\ell(Q)^{(n/p)-s} [|f|^2]_{\overline{Q}^k}^{1/2} \|_{\ell^p(\mathcal{G}^k)} \\ &\simeq \|f\|_{Z^P} \end{aligned} $$

again using Proposition 2.20. $\square$
In contrast with tent spaces, it is very easy to construct atomic decompositions of functions $f \in Z^P$: as in the proof of the theorem, simply decompose $f$ via the Whitney grid $G^k$ for sufficiently large $k$. This works for all finite $\mathbf{p}$, even if $i(\mathbf{p}) > 1$. Abstract decompositions are useful in the proof of $Z^P-Z^{P'}$ duality when $i(\mathbf{p}) \le 1$, which we now build towards.

**Lemma 2.26.** For all Whitney parameters $c$ and all $f \in L^0(\mathbb{R}_+^{1+n})$, the function $W_c f$ is lower semicontinuous.

**PROOF.** Fix $M > 0$ and suppose that $W_c f(t, x) > M$. Then there exists a small $\varepsilon > 0$ such that $W_{(c_0-\varepsilon, c_1-\varepsilon)} f(t, x) > M$ also. A short computation shows that if $\tilde{x} \in B(x, \varepsilon t/2)$ and if $|\tilde{t} - t| < (c_1/(c_1 - \varepsilon) - 1)t$, then $\Omega_c(\tilde{t}, \tilde{x})$ contains $\Omega_{(c_0-\varepsilon, c_1-\varepsilon)}(t, x)$, so for all such $(\tilde{t}, \tilde{x})$ we have

$$W_c f(\tilde{t}, \tilde{x}) \ge W_{(c_0-\varepsilon, c_1-\varepsilon)} f(t, x) > M.$$

Therefore the set $\{(t, x) \in \mathbb{R}_+^{1+n} : W_c f(t, x) > M\}$ is open. $\square$

<figure><img src="image_5.png" /></figure>

**Corollary 2.27.** Let $\mathbf{p}$ be an infinite exponent. Then

$$\|f\|_{Z_c^P} = \sup_{(t,x) \in \mathbb{R}_+^{1+n}} W_c(\kappa^{-r(\mathbf{p})} f)(t, x),$$

i.e. the essential supremum in the definition of the $Z_c^P$-norm can be replaced with a supremum.

**PROOF.** Lower semicontinuity of $W_c(\kappa^{-r(\mathbf{p})} f)$ implies that if

$$W_c(\kappa^{-r(\mathbf{p})} f)(t, x) > M$$

for some $M < \infty$ at one point $(t, x)$, then it continues to hold in an open neighborhood of $(t, x)$, and in particular on a set of positive measure. $\square$

<figure><img src="image_6.png" /></figure>

We can finally prove the duality theorem for $Z^P$ with $i(\mathbf{p}) \le 1$. As with the other duality results so far, this includes absolute convergence of the $L^2$ duality pairing.

**Theorem 2.28 (Duality: non-reflexive range).** Suppose $i(\mathbf{p}) \le 1$ and let $c$ be a Whitney parameter. Then for all $f, g \in L^0(\mathbb{R}_+^{1+n})$ we have

$$(2.14) \qquad \iint_{\mathbb{R}_+^{1+n}} |(f(t, x), g(t, x))| dx \frac{dt}{t} \lesssim \|f\|_{Z_c^P} \|g\|_{Z_c^{P'}},$$

and the $L^2$ duality pairing identifies the Banach space dual of $Z_c^P$ with $Z_c^{P'}$.

**PROOF.** Write $\mathbf{p} = (p, s)$, so that $\mathbf{p}' = (\infty, -s, n\delta_{p,1})$. First suppose $a$ is a $Z_c^P$-atom associated with a point $(t_0, x_0) \in \mathbb{R}_+^{1+n}$. Then we have

$$\iint_{\mathbb{R}_+^{1+n}} |(a(t, x), g(t, x))| dx \frac{dt}{t} \le \|\kappa^{-s} a\|_{L^2(\mathbb{R}_+^{1+n}, dx dt/t)} \|\kappa^s g\|_{L^2(\Omega_c(t_0, x_0), dx dt/t)} \\ \lesssim t_0^{n\delta_{p,2}+n\delta_{1,p}+(n/2)} \|\kappa^{s-n\delta_{1,p}} g\|_{L^2(\Omega_c(t_0, x_0), dx dt/t^{1+n})} \\ \le \|g\|_{Z_c^P}$$

by Corollary 2.27. For general $f \in Z^P$, write $f$ as a sum of $Z_c^P$-atoms as in Theorem 2.25, so that

$$\iint_{\mathbb{R}_+^{1+n}} |(f(t, x), g(t, x))| dx \frac{dt}{t} \le \sum_{k \in \mathbb{N}} |\lambda_k| \iint_{\mathbb{R}_+^{1+n}} |(a_k(t, x), g(t, x))| dx \frac{dt}{t}$$
$$ \lesssim \|g\|_{Z_c^{P'}} \|\lambda\|_{\ell^p(\mathbb{N})} $$

since $p \le 1$. Taking the infimum over all atomic decompositions of $f$ proves (2.14).

Now suppose that $\phi \in (Z_c^{P'})'$. By the same technique as in the proof of Proposition 2.22, we find that there exists a sequence $(g_{\overline{Q}}) \in \ell_{-s}^{\infty}(G : L^2(Q_0))$ corresponding to the induced action of $\phi$ on $\ell_s^p(G, \ell(Q)^n : L^2(Q_0))$ (since $\ell^p(\mathbb{N})' = \ell^{\infty}(\mathbb{N})$ for $p \le 1$). Hence there exists a function $G_{\phi} \in L^0(\mathbb{R}_+^{1+n})$ corresponding to the action of $\phi$ on $Z_c^P$. We need to show that $G_{\phi}$ is in $Z_c^{P'}$.

Suppose $(t, x) \in \mathbb{R}_+^{1+n}$. Then we can estimate

$$ \begin{align*} W_c(\kappa^{s-n\delta_{1,p}} G_{\phi})(t,x) &\approx t^{-n\delta_{1,p}} \|G_{\phi}\|_{L_{-s}^2(\Omega_c(t,x), d\xi d\tau/\tau^{1+n})} \\ &= t^{-n\delta_{1,p}} \sup_{\substack{F \in L_s^2(\Omega_c(t,x), d\xi d\tau/\tau^{1+n}) \\ \|F\| \le 1}} |(F, G_{\phi})| \\ &\lesssim t^{-n\delta_{1,p} - (n/2) - n\delta_{p,2}} \|\phi\|_{(Z_s^p)'} \sup_{\substack{F \in L_s^2(\Omega_c(t,x), d\xi d\tau/\tau) \\ \|F\| \le t^{n\delta_{p,2}}}} \|F\|_{Z_s^p} \\ &\le \|\phi\|_{(Z_c^P)'} \end{align*} $$

using $n\delta_{1,p} + (n/2) + n\delta_{p,2} = 0$, the fact that the condition in the final supremum implies that $F$ is a $Z_c^P$-atom, and Lemma 2.24. Therefore we have

$$ \|G_{\phi}\|_{Z_c^{P'}} = \sup_{(t,x) \in \mathbb{R}_+^{1+n}} W_c(\kappa^{s-n\delta_{1,p}} G_{\phi})(t,x) \lesssim \|\phi\|_{(Z_c^P)'} $$

as desired.

□

**Corollary 2.29.** For all infinite exponents $p$ and all Whitney parameters $c$, the $Z_c^P$ norms are mutually equivalent. Hence for all exponents $p$ we write $Z^P$ in place of $Z_c^P$.

Having identified the duals of all $Z^P$ spaces for finite $p$, we can give a full interpolation theorem.

**Theorem 2.30 (Real interpolation of tent spaces: full range).** Suppose that $p$ and $q$ are exponents with $\theta(p) \ne \theta(q)$, and $0 < \eta < 1$. Then

$$ (T^p, T^q)_{\eta, p_\eta} = Z^{[p, q]_\eta}, $$

where $p_\eta = i([p, q]_\eta)$.

**PROOF.** For finite exponents this is Theorem 2.18. If $1 < i(p), i(q) \le \infty$, this follows by writing

$$ (T^p, T^q)_{\eta, p_\eta} = ((T^{p'})', (T^{q'})')_{\eta, p_\eta} = (T^{p'}, T^{q'})'_{\eta, p_\eta} $$

via the duality theorem for real interpolation [23, Theorem 3.7.1], using that $T^{p'} \cap T^{q'}$ is dense in both $T^{p'}$ and $T^{q'}$, and then noting that

$$ p'_{\eta} = i([p, q]_{\eta})' = i([p', q']_{\eta}). $$

The full result follows by Wolff reiteration [80, Theorem 1].

□

As an immediate consequence we can prove a real interpolation theorem for $Z$-spaces. This follows from Theorem 2.30 along with the reiteration theorem for real interpolation [23, Theorem 5.2.4].
**Proposition 2.31 (Real interpolation of Z-spaces).** Let **p** and **q** be exponents which are not both infinite and with $\theta(\mathbf{p}) \neq \theta(\mathbf{q})$, and let $\eta \in (0, 1)$. Then

$$ (Z^{\mathbf{p}}, Z^{\mathbf{q}})_{\eta, p_{\eta}} = Z^{[\mathbf{p}, \mathbf{q}]_{\eta}}, $$

where $p_{\eta} = i([\mathbf{p}, \mathbf{q}]_{\eta})$.

We can also establish a complex interpolation theorem for Z-spaces. Within the Banach range, this has already been done by Barton and Mayboroda [22, Theorem 4.13]. However, in the quasi-Banach range—that is, for exponents **p** with $i(\mathbf{p}) < 1$—one must argue differently. We remind the reader that we perform complex interpolation using the Kalton–Mitrea method from [56, §3], which is well-defined for all quasi-Banach couples.

**Proposition 2.32 (Complex interpolation of Z-spaces).** Let **p** and **q** be exponents which are not both infinite, and let $\eta \in (0, 1)$. Then

$$ [Z^{\mathbf{p}}, Z^{\mathbf{q}}]_{\eta} = Z^{[\mathbf{p}, \mathbf{q}]_{\eta}}. $$

We defer the proof to Section 2.6, as it requires the introduction of spaces $Z_s^{p,q}$ with $q \neq 2$.

**Remark 2.33.** The Z-spaces can be seen as Wiener amalgam spaces associated with the semidirect product $\mathbb{R}_+ \times \mathbb{R}^n$ corresponding to the dilation action of the multiplicative group $\mathbb{R}_+$ on $\mathbb{R}^n$. Topologically $\mathbb{R}_+ \times \mathbb{R}^n = \mathbb{R}_+^{1+n}$, and the group operation is given by $(t, x) \cdot (s, y) := (ts, x+ty)$. Thus many of the properties above can be deduced from properties of abstract Wiener amalgam spaces. For a review of these spaces, see [39] and the references therein. However, if we were to use Wiener amalgam space arguments, we would not obtain absolute convergence of $L^2$ duality pairings (only abstract duality pairings). Furthermore, these arguments would not show the connection with tent spaces. On the other hand, this observation shows a previously unrecognised link between tent spaces and certain Wiener amalgam spaces, which is interesting in its own right, and which merits further investigation.

## 2.4. Unification: tent spaces, Z-spaces, and slice spaces

Tent spaces and Z-spaces share the same fundamental properties. To make this explicit, we write $X$ as a placeholder for either $T$ or $Z$ when a statement holds for both tent spaces and Z-spaces. When considering two different spaces, either of which can be a tent space or a Z-space independently, we use subscripts $X_0$, $X_1$. For example, one can concisely write the conclusions of Theorem 2.30 and Proposition 2.31 as

$$ (X^{\mathbf{p}}, X^{\mathbf{q}})_{\eta, p_{\eta}} = Z^{[\mathbf{p}, \mathbf{q}]_{\eta}}, $$

and the tent space and Z-space duality results can be written extremely concisely as

$$ (X^{\mathbf{p}})' \simeq X^{\mathbf{p}} \quad (\mathbf{p} \text{ finite}). $$

In this section we establish further properties of tent spaces and Z-spaces, including some relations between the two.

First, we point out that for all $s \in \mathbb{R}$ we have $X_s^2 = L_s^2(\mathbb{R}_+^{1+n})$, where

$$ (2.15) \qquad \|f\|_{L_s^2} := \|\kappa^{-s} f\|_{L^2(\mathbb{R}_+^{1+n})}. $$

The following embedding theorem extends Theorem 2.11 not only to Z-spaces, but also to combinations of tent and Z-spaces.
**Theorem 2.34 (Mixed embeddings).** Let $X_0, X_1 \in \{T, Z\}$ and let $\mathbf{p} \hookrightarrow \mathbf{q}$ with $\mathbf{p} \neq \mathbf{q}$. Then we have the embedding

$$ (X_0)^{\mathbf{p}} \hookrightarrow (X_1)^{\mathbf{q}}. $$

**PROOF.** When $X_0 = X_1 = T$, this is Theorem 2.11.

Let $\mathbf{r} = [\mathbf{p}, \mathbf{q}]_2$, so that $\mathbf{p} \hookrightarrow \mathbf{r}$ and $[\mathbf{p}, \mathbf{r}]_{1/2} = \mathbf{q}$ (by Lemmas 2.1 and 2.2). Then we have embeddings $T^{\mathbf{p}} \hookrightarrow T^{\mathbf{p}}$ (trivially) and $T^{\mathbf{p}} \hookrightarrow T^{\mathbf{r}}$ (Theorem 2.11). Hence

$$ T^{\mathbf{p}} \hookrightarrow (T^{\mathbf{p}}, T^{\mathbf{r}})_{1/2, i([\mathbf{p}, \mathbf{r}]_{1/2})} = Z^{[\mathbf{p}, \mathbf{r}]_{1/2}} = Z^{\mathbf{q}} $$

by Theorem 2.30, using that $\mathbf{p} \neq \mathbf{q}$ and $\mathbf{p} \hookrightarrow \mathbf{q}$ imply $\theta(\mathbf{p}) \neq \theta(\mathbf{q})$. Similarly, putting $\mathbf{s} = [\mathbf{p}, \mathbf{q}]_{-1}$, we have $T^{\mathbf{s}} \hookrightarrow T^{\mathbf{q}}$ and $T^{\mathbf{q}} \hookrightarrow T^{\mathbf{q}}$, so

$$ Z^{\mathbf{p}} = (T^{\mathbf{s}}, T^{\mathbf{q}})_{1/2, i([\mathbf{s}, \mathbf{q}]_{1/2})} \hookrightarrow T^{\mathbf{q}}. $$

Finally, putting $\mathbf{t} = [\mathbf{p}, \mathbf{q}]_{1/2}$ and using the previous results, we have

$$ Z^{\mathbf{p}} \hookrightarrow T^{\mathbf{t}} \hookrightarrow Z^{\mathbf{q}}, $$

which completes the proof. □

<figure><img src="image_7.png" /></figure>

We also have convenient mixed embeddings for fixed exponents.

**Lemma 2.35.** If $i(\mathbf{p}) \leq 2$ then

$$ Z^{\mathbf{p}} \hookrightarrow T^{\mathbf{p}}, $$

and if $i(\mathbf{p}) \geq 2$ (and in particular if $\mathbf{p}$ is infinite) then

$$ T^{\mathbf{p}} \hookrightarrow Z^{\mathbf{p}} $$

**PROOF.** The first embedding is proven in [3, Corollary 2.16]. The second then follows by duality. □

<figure><img src="image_8.png" /></figure>

**Proposition 2.36 (Density of intersections).** Let $\mathbf{p}$ and $\mathbf{q}$ be exponents. If $\mathbf{p}$ is finite then $(X_0)^{\mathbf{p}} \cap (X_1)^{\mathbf{q}}$ is dense in $(X_0)^{\mathbf{p}}$. Otherwise, $(X_0)^{\mathbf{p}} \cap (X_1)^{\mathbf{q}}$ is weak-star dense in $(X_0)^{\mathbf{p}}$.

**PROOF.** This follows immediately from the fact that $L_c^2(\mathbb{R}_+^{1+n})$ is (weak-star) dense in $T^{\mathbf{r}}$ for (infinite) exponents $\mathbf{r}$, and likewise in $Z^{\mathbf{r}}$ (this can be proven directly, or by real interpolation, or by the embeddings of Theorem 2.34). □

<figure><img src="image_9.png" /></figure>

For all $r \in \mathbb{R}_+$, define a 'downward shift' operator $S_r$ on $L^0(\mathbb{R}_+^{1+n})$ by

$$ (S_r f)(t, y) := f(t + r, y) $$

for all $f \in L^0(\mathbb{R}_+^{1+n})$. These operators are well-behaved on certain tent spaces and $Z$-spaces, as shown in the following proposition.

**Proposition 2.37 (Uniform boundedness of downward shifts).** Let $\mathbf{p}$ be an exponent.

(i) If $i(\mathbf{p}) \leq 2$ and $\theta(\mathbf{p}) < -1/2$, then the operators $(S_r)_{r \in \mathbb{R}_+}$ are uniformly bounded on $X^{\mathbf{p}}$.

(ii) If $i(\mathbf{p}) \in (2, \infty]$ and $r(\mathbf{p}) < -(n+1)/2$, then the operators $(S_r)_{r \in \mathbb{R}_+}$ are uniformly bounded on $X^{\mathbf{p}}$.
**Remark 2.38.** Note that the assumptions for $i(\mathbf{p}) \le 2$ and $i(\mathbf{p}) > 2$ are quite different: there is a sudden jump in dimensional dependence at $i(\mathbf{p}) > 2$. We do not have a good explanation for this behaviour, and there is no interpolation procedure to obtain stronger results when $2 < i(\mathbf{p}) < \infty$. Note that we can include endpoints when considering tent spaces (i.e. we can include $\theta(\mathbf{p}) = -1/2$ or $r(\mathbf{p}) = -(n+1)/2$ respectively). However, to realise the spaces $Z^{\mathbf{p}}$ as interpolants of tent spaces, we need to interpolate between tent spaces $T^{\mathbf{p}_0}$ and $T^{\mathbf{p}_1}$ with $\theta(\mathbf{p}_0) \neq \theta(\mathbf{p}_1)$, and so the endpoint Z-space results cannot be proven by this argument.

**PROOF OF PROPOSITION 2.37.** It suffices to prove the tent space results, as the Z-space results follow by real interpolation.

First we prove boundedness on tent spaces for $i(\mathbf{p}) \le 1$ and for $i(\mathbf{p}) = 2$; the rest of part (i) follows by complex interpolation. Suppose $\mathbf{p} = (p, s)$ with $p \le 1$ and let $a$ be a $T^{\mathbf{p}}$-atom associated with a ball $B$ of radius $r_B$. Then $S_r a$ is supported on $T(B)$, and we have

$$
\begin{align*}
\|S_r a\|_{T_s^2} &\approx \left( \int_0^{r_B-r} \int_{\mathbb{R}^n} t^{-2s-1} |a(t+r, x)|^2 dx dt \right)^{1/2} \\
&\le \left( \int_0^{r_B-r} \int_{\mathbb{R}^n} (t+r)^{-2s-1} |a(t+r, x)|^2 dx dt \right)^{1/2} \\
&= \left( \int_r^{r_B} \int_{\mathbb{R}^n} |\tau^{-s} a(\tau, x)|^2 dx \frac{d\tau}{\tau} \right)^{1/2} \\
&\le \|a\|_{T_s^2} \\
&\le |B|^{\delta_{p,2}}
\end{align*}
$$

using that $-2s - 1 > 0$. Therefore $S_r a$ is, up to a uniform constant, a $T^{\mathbf{p}}$-atom associated with $B$. Hence if $f = \sum_{k \in \mathbb{N}} \lambda_k a_k$ is an atomic decomposition of $f$ in $T^{\mathbf{p}}$, then $S_r f = \sum_{k \in \mathbb{N}} \lambda_k (S_r a_k)$ is an atomic decomposition of $S_r f$ in $T^{\mathbf{p}}$ up to a uniform constant. Therefore the operators $(S_r)_{r \in \mathbb{R}_+}$ are uniformly bounded on $T^{\mathbf{p}}$. A similar argument (without need of atoms) works for $\mathbf{p} = (2, s)$ provided $s < -1/2$.

Now let $\mathbf{p} = (p, s)$ with $p \in (2, \infty)$ and $s < -(n+1)/2$, and fix $f \in L^0(\mathbb{R}_+^{1+n})$. First we estimate $S_r f$ in $T^{\mathbf{p}}$:

$$
\begin{align*}
\|S_r f\|_{T^{\mathbf{p}}} &= \left( \int_{\mathbb{R}^n} \left( \iint_{\Gamma(x)} t^{-2s-n-1} |f(t+r,y)|^2 dy dt \right)^{p/2} dx \right)^{1/p} \\
&\le \left( \int_{\mathbb{R}^n} \left( \iint_{\Gamma(x)} (t+r)^{-2s-n-1} |f(t+r,y)|^2 dy dt \right)^{p/2} dx \right)^{1/p} \\
&= \left( \int_{\mathbb{R}^n} \left( \iint_{\Gamma(x)+r} \tau^{-2s-n-1} |f(\tau,y)|^2 dy d\tau \right)^{p/2} dx \right)^{1/p} \\
&\le \|f\|_{T^{\mathbf{p}}}
\end{align*}
$$

using that $-2s - n - 1 > 0$ and $\Gamma(x) + r \subset \Gamma(x)$, where $\Gamma(x) + r$ is the 'vertically translated cone'

$$
\Gamma(x) + r := \{(t, y) \in \mathbb{R}_+^{1+n} : (t - r, y) \in \Gamma(x)\}.
$$

This proves part (ii) in the case where $\mathbf{p}$ is finite.
Now suppose $\mathbf{p} = (\infty, s; \alpha)$ with $s+\alpha < -(n+1)/2$, and let $B = B(c, R) \subset \mathbb{R}^n$ be a ball. If $r \le R$, then we can write

$$
\begin{align*}
& R^{-\alpha-n/2} \left( \iint_{T(B)} t^{-2s-1} |f(t+r, y)|^2 dy dt \right)^{1/2} \\
\le{}& R^{-\alpha-n/2} \left( \iint_{T(B)} (t+r)^{-2s-1} |f(t+r, y)|^2 dy dt \right)^{1/2} \\
\lesssim{}& (R+r)^{-\alpha-n/2} \left( \iint_{T(B(c,R+r))} \tau^{-2s-1} |f(\tau, y)|^2 dy d\tau \right)^{1/2} \\
\lesssim{}& \|f\|_{T_s^\infty}
\end{align*}
$$

using that $-2s-1 > 0$. If $r > R$ then instead we write

$$
\begin{align*}
& R^{-\alpha-n/2} \left( \iint_{T(B)} t^{-2s-1} |f(t+r, y)|^2 dy dt \right)^{1/2} \\
&= R^{-\alpha-n/2} \left( \iint_{T(B)+r} \tau^{-2s-1} \left( \frac{\tau-r}{\tau} \right)^{-2s-1} |f(\tau, y)|^2 dy d\tau \right)^{1/2} \\
&\le R^{-\alpha-n/2} \left( \frac{R}{r} \right)^{-s-1/2} \left( \iint_{T(B)+r} \tau^{-2s-1} |f(\tau, y)|^2 dy d\tau \right)^{1/2} \\
&\le \left( \frac{R+r}{R} \right)^{\alpha+n/2} \left( \frac{R}{r} \right)^{-s-1/2} \|f\|_{T_\alpha^\infty} \\
&\lesssim \|f\|_{T_\alpha^\infty}
\end{align*}
$$

using that $s+\alpha \le -(n+1)/2$ in the last line, where $T(B)+r$ is defined analogously to $\Gamma(x)+r$. These estimates imply that $\|S_r f\|_{T_\alpha^\infty} \lesssim \|f\|_{T_\alpha^\infty}$ as desired, completing the proof. $\square$

Now we define the *slice spaces*. These were introduced in connection with tent spaces and boundary value problems by the second author and Mourgoglou [16]. The name comes from the fact that functions in slice spaces are, roughly speaking, horizontal 'slices' of functions in tent or Z-spaces (this is made precise in Proposition 2.40).

**Definition 2.39.** Suppose $\mathbf{p}$ is an exponent and $t > 0$. For $f \in L^0(\mathbb{R}^n)$ we define

$$
\|f\|_{E^{\mathbf{p}}(t)} := t^{-r(\mathbf{p})} \|x \mapsto \|f\|_{L^2(B(x,t), dy/t^n)}\|_{L^i(\mathbf{p})(\mathbb{R}^n)}.
$$

These quasinorms define the *slice spaces*

$$
E_{r(\mathbf{p})}^{i(\mathbf{p})}(t) = E^{\mathbf{p}}(t) = E^{\mathbf{p}}(t)(\mathbb{R}^n) := \{f \in L^0(\mathbb{R}^n) : \|f\|_{E^{\mathbf{p}}(t)} < \infty\}.
$$

For $f \in L^0(\mathbb{R}^n)$, $t > 0$, and $h > 3/2$ (this restriction corresponds to that in the definition of Whitney parameter), define $\iota_{t,h}(f) \in L^0(\mathbb{R}_+^{1+n})$ by setting

$$
\iota_{t,h}(f)(s,x) := f(x)\mathbf{1}_{[t,ht]}(s)
$$

for all $(s, x) \in \mathbb{R}_+^{1+n}$, and for $g \in L^0(\mathbb{R}_+^{1+n})$ define $\pi_t(g) \in L^0(\mathbb{R}^n)$ by

$$
\pi_{t,h}(g)(x) := \int_t^{ht} g(s,x) \frac{ds}{s}.
$$

for all $x \in \mathbb{R}^n$.
**Proposition 2.40.** For all exponents **p**, the operators

$$E^{\mathbf{P}}(t) \xrightarrow{t_{t,h}} X^{\mathbf{P}} \xrightarrow{\pi_{t,h}} E^{\mathbf{P}}(t),$$

are bounded uniformly in t. Furthermore, the compositions of these operators are identity maps.

**PROOF.** The tent space results with $\theta(\mathbf{p}) = 0$ are already stated in [16, §3]; the extension to all tent spaces is simple. Likewise, the composition statement is clear. The proof for Z-spaces is a straightforward (one page) argument that we omit. □

Therefore we can view the spaces $E^{\mathbf{P}}(t)$ as retracts of $X^{\mathbf{P}}$. Consequently, properties of tent spaces and Z-spaces descend to slice spaces.

**Proposition 2.41.** If $0 < t_0, t_1 < \infty$ and **p**, **q** are exponents with $i(\mathbf{p}) = i(\mathbf{q})$, then $E^{\mathbf{P}}(t_0) = E^{\mathbf{q}}(t_1)$ with equivalent quasinorms.

This follows from change of aperture for tent spaces [16, Lemma 3.5]. For $p \in (0, \infty]$ we write $E^p := E^{\mathbf{P}}(1)$ for any **p** with $i(\mathbf{p}) = p$; all $E^{\mathbf{P}}(t)$ quasinorms are equivalent to the $E^p$ quasinorm (but not uniformly in t or **p**).

We have a duality theorem for slice spaces, and of course this includes absolute convergence of the $L^2$ duality pairing (now on $\mathbb{R}^n$ rather than $\mathbb{R}_+^{1+n}$). This is proven in [16, Lemma 3.2].

**Proposition 2.42 (Duality).** Fix $t > 0$ and let **p** be a finite exponent. Then we have

$$(2.16) \qquad \int_{\mathbb{R}^n} |(f(x), g(x))| dx \lesssim \|f\|_{E^{\mathbf{P}}} \|g\|_{E^{\mathbf{P}'}}$$

and the $L^2(\mathbb{R}^n)$ duality pairing identifies the Banach space dual of $E^{\mathbf{P}}$ with $E^{\mathbf{P}'}$.

The tent space and Z-space embedding results also descend to slice spaces, though for slice spaces the 'regularity' parameters are not important.

**Proposition 2.43 (Embeddings).** Suppose $0 < p_0 \le p_1 \le \infty$. Then $E^{p_0} \hookrightarrow E^{p_1}$.

**PROOF.** Fix $\mathbf{p}_0$ and $\mathbf{p}_1$ with $i(\mathbf{p}_0) = p_0$, $i(\mathbf{p}_1) = p_1$, and $\mathbf{p}_0 \hookrightarrow \mathbf{p}_1$. Then we have bounded operators

$$E^{p_0} \xrightarrow{t_{1,2}} X^{\mathbf{P}_0} \hookrightarrow X^{\mathbf{P}_1} \xrightarrow{\pi_{1,2}} E^{p_1}$$

whose composition is the identity map, with the inclusion following from Theorem 2.34. □

Slice spaces contain the Schwartz functions, and are contained in the space of tempered distributions. This is contained in [16, Lemma 3.6] (only the case $p < \infty$ is included there, but the argument extends to $p = \infty$).

**Proposition 2.44.** For all $p \in (0, \infty]$ we have $S \subset E^p \subset S'$.

We also have a straightforward integration by parts formula for functions in slice spaces. This is part of [16, Lemma 3.8].

**Proposition 2.45 (Integration by parts in slice spaces).** Let **p** be a finite exponent and suppose that $\partial$ is a first-order differential operator with constant
coefficients, and let $\partial^*$ be the adjoint operator for the $L^2$ duality pairing on $\mathbb{R}^n$. If $f, \partial f \in E^p$ and $g, \partial^*g \in E^p$, then

$$ \int_{\mathbb{R}^n} (\partial f(x), g(x)) dx = \int_{\mathbb{R}^n} (f(x), \partial^* g(x)) dx. $$

Finally, we have an equivalent dyadic quasinorm for the slice spaces. This follows from the dyadic characterisation of Z-spaces (Proposition 2.20 and the remark following it) and Proposition 2.40.

**Proposition 2.46 (Dyadic characterisation).** For all $p \in (0, \infty]$ and $k \in \mathbb{Z}$ we have

$$ \|f\|_{E^p} \approx \|(f)_{Q \in \mathcal{D}_k} \|_{\ell^p(\mathcal{D}_k)} $$

where $\mathcal{D}_k$ is any grid of dyadic cubes in $\mathbb{R}^n$ with sidelength $2^k$.

**Remark 2.47.** The slice spaces $E^p$ are equal to the Wiener amalgam spaces $W(L^2, L^p)(\mathbb{R}^n)$ when $p \ge 1$ (see [39] and the references therein). Therefore, as with Z-spaces, many properties of slice spaces can be deduced from properties of Wiener amalgam spaces, but this would not emphasise the connection with tent- and Z-spaces.

## 2.5. Homogeneous smoothness spaces

We only give a quick definition of these, and state specific properties that we will need. These definitions are special cases of the Littlewood-Paley definitions of Triebel-Lizorkin and Besov spaces, which we do not need in full generality. For a quick introduction the reader can consult Grafakos [37, Chapter 6]; the less time-pressed reader may consult Triebel's oeuvre (for example [76, §5]).

Let $\mathcal{Z}(\mathbb{R}^n) \subset \mathcal{S}(\mathbb{R}^n)$ be the set of Schwartz functions $f$ such that $D^\alpha f(0) = 0$ for every multi-index $\alpha$, and let $\mathcal{Z}'(\mathbb{R}^n)$ be the topological dual of $\mathcal{Z}(\mathbb{R}^n)$. The space $\mathcal{Z}'(\mathbb{R}^n)$ can be identified with the quotient space $\mathcal{S}'(\mathbb{R}^n)/\mathcal{P}(\mathbb{R}^n)$, where $\mathcal{P}(\mathbb{R}^n)$ is the space of polynomials on $\mathbb{R}^n$. In this section all our spaces will be of objects defined on $\mathbb{R}^n$, so we omit this from our notation, writing $\mathcal{Z}, \mathcal{Z}'$, and so on. We denote the Fourier transform and inverse Fourier transform of a function $\varphi$ by $\hat{\varphi}$ and $\hat{\varphi}'$ respectively.

**Definition 2.48.** Let $\Psi \in \mathcal{S}$ be a radial bump function with

$$ \hat{\Psi} \ge 0, \quad \text{supp} \hat{\Psi} \subset A(0, 6/7, 2), \quad \text{and} \quad \hat{\Psi}|_{A(0, 1, 12/7)} = 1 $$

(of course these precise parameters are not so important), and for $j \in \mathbb{Z}$ let $\Delta_j$ denote the associated Littlewood-Paley operators $\Delta_j(f) := f * (2^{jn}\Psi(2^j \cdot))$.

For $f \in \mathcal{Z}', \alpha \in \mathbb{R}$, and $0 < p < \infty$ define

$$ \|f\|_{\dot{H}_{\alpha}^{p}} := \|\|j \mapsto 2^{j\alpha}(\Delta_j f)(\cdot)\|_{\ell^2(\mathbb{Z})} \|_{L^p}, $$

and for $0 < p \le \infty$ define

$$ \|f\|_{\dot{B}_{\alpha}^{p, p}} := \|\|j \mapsto 2^{j\alpha} \|\Delta_j f(\cdot)\|_{L^p} \|_{\ell^p(\mathbb{Z})}. $$

The homogeneous Hardy-Sobolev spaces $\dot{H}_\alpha^p = \dot{F}_\alpha^{p,2}$ and Besov spaces $\dot{B}_\alpha^{p,p}$ are then the sets of those $f \in \mathcal{Z}'$ for which the corresponding quasinorms are finite.

These quasinorms are independent of the choice of $\Psi$ (up to equivalence), and $\dot{H}_\alpha^p$ and $\dot{B}_\alpha^{p,p}$ are Banach spaces (quasi-Banach when $p < 1$).
The Hardy–Sobolev spaces $\dot{H}_\alpha^p$ may be characterised by Fourier multipliers in tent spaces: this characterisation can be found in [77, page 180].

**Theorem 2.49.** Suppose $p \in (0, \infty)$ and $\alpha \in \mathbb{R}$. Fix $a > n/\min(p, 2)$ and $s_0, s_1 \in \mathbb{R}$ with $s_0 + a < s < s_1$. Fix also two Schwartz functions $h, H \in S$ with

$$ \operatorname{supp} h \subset \{x \in \mathbb{R}^n : |x| \le 2\}, \quad h(x) = 1 \text{ if } |x| \le 1, \\ \operatorname{supp} H \subset \{x \in \mathbb{R}^n : 1/4 \le |x| \le 4\}, \quad H(x) = 1 \text{ if } 1/2 \le |x| \le 2. $$

Let $\psi: \mathbb{R}^n \setminus \{0\} \to \mathbb{C}$ be a measurable function satisfying the conditions

$$ (2.17) \qquad |\psi(x)| > 0 \quad \text{if } 1/2 \le |x| \le 2, $$

$$ (2.18) \qquad \int_{\mathbb{R}^n} \left| \left( \frac{\psi(\cdot)h(\cdot)}{|\cdot|^{s_1}} \right)^{\vee} (y) \right| (1 + |y|)^a dy < \infty, $$

$$ (2.19) \qquad \sup_{m=1,2,...} 2^{-ms_0} \int_{\mathbb{R}^n} |(\psi(2^m \cdot)H(\cdot))^{\vee}(y)| (1 + |y|)^a dy < \infty. $$

Then for all $f \in \mathcal{Z}'$,

$$ \|t \mapsto (\psi(t \cdot) \hat{f})^{\vee}\|_{T_{\alpha}^p} \approx \|f\|_{\dot{H}_{\alpha}^p}. $$

Conditions (2.18) and (2.19) are a bit complicated, but if $\tilde{\psi} \in \mathcal{Z}$ then they are automatically satisfied for any choice of $a, s_0, s_1, h$, and $H$. In the following corollary we rephrase Theorem 2.49 in convolution form with this simplification. This is a consequence of the identity $\varphi * f = (\hat{\varphi}\hat{f})^{\vee}$.

**Corollary 2.50.** Suppose $p \in (0, \infty)$ and $\alpha \in \mathbb{R}$. Let $\varphi \in \mathcal{Z}(\mathbb{R}^n)$ be such that $\hat{\varphi}(\xi) > 0$ if $1/2 \le |\xi| \le 2$. Then for all $f \in \mathcal{Z}'$,

$$ \|t \mapsto t^{-n} \varphi(t^{-1} \cdot) * f\|_{T_{\alpha}^p} \approx \|f\|_{\dot{H}_{\alpha}^p}. $$

We extend this to more general smoothness spaces in Theorem 2.57. Before that, we will go through some other equivalent norms.

For $f \in \mathcal{Z}'$ and $\alpha \in \mathbb{R}$, the Riesz potential $I_\alpha f \in \mathcal{Z}'$, given by

$$ I_{\alpha} f(x) := (|\cdot|^{-\alpha} \hat{f}(\cdot))^{\vee}(x) \quad (x \in \mathbb{R}^n), $$

is well-defined. These can be used to characterise Hardy–Sobolev spaces.

**Theorem 2.51.** Suppose $p \in (0, \infty)$, $\alpha \in \mathbb{R}$, and $f \in \mathcal{Z}'$. Then $f \in \dot{H}_\alpha^p$ if and only if $I_{-\alpha}f \in L^p$, and $f \mapsto \|I_{-\alpha}f\|_{L^p}$ is an equivalent norm on $\dot{H}_\alpha^p$. Furthermore, for all $s \in \mathbb{R}$, $I_\alpha$ is an isomorphism from $\dot{H}_s^p$ to $\dot{H}_{s+\alpha}^p$.

We can also characterise Hardy–Sobolev and Besov spaces by integrals of differences. For all $p \in [1, \infty]$, $g \in L^0$, and $s \in \mathbb{R}$, define

$$ \mathcal{D}_s^p g(x) := \left( \int_{\mathbb{R}^n} \frac{|g(x+y) - g(x)|^p}{|y|^{n+ps}} dy \right)^{1/p} \quad (x \in \mathbb{R}^n). $$

**Theorem 2.52.** Suppose $\alpha \in (0, 1)$ and $p \in (2n/(n+\alpha), \infty)$. Then for all $f \in L^2$,

$$ (2.20) \qquad \|f\|_{\dot{H}_{\alpha}^{p}} \approx \|\mathcal{D}_{\alpha}^{2} f\|_{L^{p}}. $$

**PROOF.** Whenever $f = I_\alpha \varphi$ for some $\varphi \in C_c^\infty$, the estimate (2.20) follows from a lemma of Stein [72, Lemma 1] combined with the Riesz potential characterisation of $\dot{H}_\alpha^p$ (Theorem 2.51). A density argument, using the fact that elements of $\dot{H}_\alpha^p$ may be represented as $L_{\text{loc}}^2$ functions when $\alpha \in (0, 1)$, completes the proof. $\square$
The corresponding characterisation for Besov spaces is in [76, Theorem 5.2.3.2].

**Theorem 2.53.** Suppose $\alpha \in (0, 1)$ and $p \in [1, \infty)$. Then for all $f \in L^2$,

$$ \|f\|_{\dot{B}_{\alpha}^{p, p}} \approx \|\mathcal{D}_{\alpha}^{p} f\|_{L^{p}}. $$

For $\alpha \in (0, 1)$, the Besov space $\dot{B}_{\alpha}^{\infty, \infty}$ may be identified with the more familiar homogeneous Hölder–Lipschitz space $\dot{\Lambda}_{\alpha}$: this is the space of functions $f$ on $\mathbb{R}^n$ such that

$$ \|f\|_{\dot{\Lambda}_{\alpha}} := \sup_{x,y \in \mathbb{R}^n} \frac{|f(x) - f(y)|}{|x - y|^{\alpha}} < \infty, $$

modulo constants. Such functions are continuous.

**Remark 2.54.** We do not obtain any well-posedness results for Besov spaces $\dot{B}_{\alpha}^{\infty, \infty}$ with $\alpha \notin (0, 1)$, but nonetheless these spaces fit into our abstract framework.

We must also consider the Triebel–Lizorkin spaces $\dot{F}_{\alpha}^{\infty, 2}$ for $\alpha \in \mathbb{R}$, which are the subspaces of $\mathcal{Z}'$ determined by the quasinorms

$$ \|f\|_{\dot{F}_{\alpha}^{\infty, 2}} := \inf \|j \mapsto 2^{j\alpha} |f_j(\cdot)|\|_{\ell^2(\mathcal{Z})}\|_{L^{\infty}}, $$

where the infima are taken over all decompositions

$$ f = \sum_{j \in \mathbb{Z}} \Delta_j f_j \quad \text{in } \mathcal{Z}' $$

with each $f_j \in L^{\infty}$, where $\Delta_j$ are Littlewood–Paley operators as in Definition 2.48. For all $\alpha \in \mathbb{R}$ we refer to these spaces as homogeneous BMO-$Sobolev$ spaces $B\dot{M}O_{\alpha} := \dot{F}_{\alpha}^{\infty, 2}$. When $\alpha \ge 0$, $B\dot{M}O_{\alpha}$ is usually defined as the image of BMO under the Riesz potential $I_{\alpha}$ defined above, as a subspace of $\mathcal{Z}'$, with a corresponding norm; this definition agrees with the Triebel–Lizorkin definition that we just made. Of course $B\dot{M}O_0 = \text{BMO}$. See Strichartz [73] and Triebel [76, §5.1.4] for further information. In particular, we have the following characterisation of $B\dot{M}O_{\alpha}$ for $\alpha \in (0, 1)$ [73, Theorem 3.3].

**Theorem 2.55.** Suppose $\alpha \in (0, 1)$. Then for all $f \in L^2$ we have

$$ \|f\|_{B\dot{M}O_{\alpha}} \approx \sup_{Q} \left( \frac{1}{|Q|} \int_{Q} \int_{Q} \frac{|f(x) - f(y)|^2}{|x - y|^{n+2\alpha}} dy dx \right)^{1/2}, $$

where the supremum may be taken over all cubes or all balls.

We introduce some unconventional but useful notation for these spaces. For a finite exponent $\mathbf{p} = (p, s)$, define

$$ \mathbf{H}^{\mathbf{p}} := \dot{H}_{s}^{p} = \dot{F}_{s}^{p, 2} \quad \text{and} \quad \mathbf{B}^{\mathbf{p}} := \dot{B}_{s}^{p, p}. $$

For $\mathbf{p} = (\infty, s; 0)$, define

$$ \mathbf{H}^{\mathbf{p}} := \dot{F}_{s}^{\infty, 2} = B\dot{M}O_{s} \quad \text{and} \quad \mathbf{B}^{\mathbf{p}} := \dot{B}_{s}^{\infty, \infty}. $$

recalling that $\dot{B}_{s}^{\infty, \infty} = \dot{\Lambda}_{s}$ when $s \in (0, 1)$. Finally, for $\mathbf{p} = (\infty, s; \alpha)$ with $\alpha > 0$, define

$$ \mathbf{H}^{\mathbf{p}} := \mathbf{B}^{\mathbf{p}} := \dot{B}_{s+\alpha}^{\infty, \infty}. $$

As a consequence of these definitions and the various duality identifications for classical smoothness spaces, for all finite exponents $\mathbf{p}$ we have

$$ (\mathbf{X}^{\mathbf{p}})' \simeq \mathbf{X}^{\mathbf{p}}' $$
whenever $\mathbf{X}$ denotes either $\mathbf{H}$ or $\mathbf{B}$.

We also have the following interpolation theorem. This is a combination of standard results (see for example Mendez and Mitrea [65, Theorem 11], Triebel [74, Theorems 8.1.3 and 8.3.3a]), and Bergh and Löfström [23, Theorem 6.4.5]).<fnref n="1" />

**Theorem 2.56.** Let $\mathbf{p}$ and $\mathbf{q}$ be finite exponents, and suppose $\eta \in (0,1)$ and $p_\eta := i([\mathbf{p}, \mathbf{q}]_\eta)$. Then we have

$$[\mathbf{H}^{\mathbf{p}}, \mathbf{H}^{\mathbf{q}}]_{\eta} = \mathbf{H}^{[\mathbf{p}, \mathbf{q}]_{\eta}}$$

and (also allowing infinite exponents)

$$[\mathbf{B}^{\mathbf{p}}, \mathbf{B}^{\mathbf{q}}]_{\eta} = \mathbf{B}^{[\mathbf{p}, \mathbf{q}]_{\eta}} \quad \text{and} \quad (\mathbf{B}^{\mathbf{p}}, \mathbf{B}^{\mathbf{q}})_{\eta, p_{\eta}} = \mathbf{B}^{[\mathbf{p}, \mathbf{q}]_{\eta}}.$$

Furthermore if $\theta(\mathbf{p}) \neq \theta(\mathbf{q})$, then we have

$$(\mathbf{H}^{\mathbf{p}}, \mathbf{H}^{\mathbf{q}})_{\eta, p_{\eta}} = \mathbf{B}^{[\mathbf{p}, \mathbf{q}]_{\eta}}.$$

Combining these duality and interpolation results with Corollary 2.50, and using our unconventional notation, we obtain a typographically slick characterisation of all the smoothness spaces that we consider. This theorem and its proof foreshadow our approach to adapted Besov-Hardy-Sobolev spaces.

**Theorem 2.57.** Let $\mathbf{p}$ be an exponent, and let $\varphi \in \mathcal{Z}$ be such that $\hat{\varphi}(\xi) > 0$ if $1/2 \le |\xi| \le 2$. Then for all $f \in \mathcal{Z}'$,

$$\|t \mapsto t^{-n}\varphi(t^{-1}) * f\|_{X^{\mathbf{p}}} \approx \|f\|_{X^{\mathbf{p}}}$$

where $X = T$ if $\mathbf{X} = \mathbf{H}$, and $X = Z$ if $\mathbf{X} = \mathbf{B}$.

**PROOF.** Let $Q_\varphi$ be the operator $Q_\varphi f(t) := t^{-n}\varphi(t^{-1}) * f$. Corollary 2.50 says that

$$(2.21) \qquad \|Q_\varphi f\|_{T^{\mathbf{p}}} \approx \|f\|_{H^{\mathbf{p}}}$$

when $\mathbf{p}$ is finite, so in this case $Q_\varphi$ is bounded from $\mathbf{H}^{\mathbf{p}}$ to $T^{\mathbf{p}}$. A quick computation shows that the adjoint $(Q_\varphi)^*: T^{\mathbf{p}}' \to \mathbf{H}^{\mathbf{p}}'$ with respect to the $L^2(\mathbb{R}_+^{1+n})$ and $L^2(\mathbb{R}^n)$ duality pairings is given by

$$((Q_\varphi)^* G)(x) = S_{\tilde{\varphi}} G(x) = \int_0^\infty (t^{-n} \tilde{\varphi}(t^{-1}) * G(t, \cdot))(x) \frac{dt}{t} \quad (G \in T^{\mathbf{p}}')$$

where $\tilde{\varphi}(y) := \varphi(-y)$, and where the integral converges in the weak-star topology on $\mathbf{H}^{\mathbf{p}}'$. Since $Q_{\tilde{\varphi}}: \mathbf{H}^{\mathbf{p}} \to T^{\mathbf{p}}$ is bounded for all finite $\mathbf{p}$, we find that $S_\varphi: T^{\mathbf{q}} \to \mathbf{H}^{\mathbf{q}}$ is bounded for all $\mathbf{q}$ with $i(\mathbf{q}) > 1$. One can extend this to all finite $\mathbf{q}$: this is done by Coifman, Meyer, and Stein for $\theta(\mathbf{q}) = 0$ [27, Theorem 6],<fnref n="2" /> and the reduction to this case for general $\mathbf{q}$ is done by writing

$$(-\Delta)^{-s} S_\varphi G = t^{-2s} S_\psi G$$

with $\psi = (-\Delta)^{-s}\varphi$.

<footnote n="1">¹The results cited in [74] and [23] are for inhomogeneous spaces. As always, the same technique proves the result for homogeneous spaces. To obtain the stated results for Besov spaces with $\theta(\mathbf{p}) = \theta(\mathbf{q})$, write $\dot{B}_\theta^{p,p} = \dot{F}_\theta^{p,p}$ and use the interpolation results for Triebel-Lizorkin spaces.</footnote>

<footnote n="2">²This is stated for compactly supported $\varphi$ there, but the argument extends to $\varphi \in \mathcal{Z}$.</footnote>
By interpolation, for all $\mathbf{p}$, $Q_\varphi$ is bounded from $\mathbf{B}^\mathbf{P}$ to $Z^\mathbf{P}$ and $S_\varphi$ is bounded from $Z^\mathbf{P}$ to $\mathbf{B}^\mathbf{P}$. The Calderón reproducing formula says that there exists a function $\psi \in \mathcal{Z}$, also with $\hat{\psi}(\xi) > 0$ when $1/2 \le |\xi| \le 2$, such that

$$S_{\psi}Q_{\varphi}f = f$$

for all $f \in \mathcal{Z}'$, and hence also for all $f \in \mathbf{H}^\mathbf{P}$ or $f \in \mathbf{B}^\mathbf{P}$. It follows that $\mathbf{X}^\mathbf{P}$ may be identified with $Q_\varphi S_\psi X^\mathbf{P}$ for all $\mathbf{p}$, and by the retraction/coretraction interpolation theorem [75, §1.2.4] we get

$$\|Q_{\varphi}f\|_{X^{\mathbf{P}}} \approx \|f\|_{X^{\mathbf{P}}}$$

for all $\mathbf{p}$ and all $f \in \mathcal{Z}'$, which completes the proof. □

**Remark 2.58.** We were a bit sketchy in the above proof, as we will carry out the argument in much greater detail, and in an abstract setting, in the following chapters. See in particular Section 3.4 and Chapter 4.

## 2.6. Appendix: Factorisation and interpolation of Z-spaces

In this section we prove the complex interpolation result for Z-spaces, including the quasi-Banach range (Proposition 2.32). We argue by factorisation, exploiting the following ‘Calderón product formula’, which is a special case of [56, Theorem 3.4].

**Theorem 2.59 (Kalton–Mitrea).** Let $X_0, X_1$ be a pair of quasi-Banach function spaces on $\mathbb{R}_+^{1+n}$. Suppose that both $X_0$ and $X_1$ are A-convex and separable. Then

$$[X_0, X_1]_{\theta} = X_0^{1-\theta} X_1^{\theta},$$

where $X_0^{1-\theta} X_1^{\theta}$ is the quasi-Banach function space consisting of those $h \in L^0(\mathbb{R}_+^{1+n})$ such that the quasinorm

$$\|h\|_{X_0^{1-\theta} X_1^{\theta}} := \inf\{\|f\|_{X_0^{1-\theta}} \|g\|_{X_1^{\theta}} : |h| \le |f|^{1-\theta} |g|^{\theta}, f \in X_0, g \in X_1\}$$

is finite, with the usual convention that the infimum of an empty set is $+\infty$.

A quasi-Banach function space X is A-convex if and only if it is lattice r-convex for some $r > 0$ (see [57, Theorem 2.2] and [58, Theorem 4.4]), which means that for all finite collections $f_1, \dots, f_k \in X$

$$\left\| \left( \sum_{i=1}^{k} |f_i|^r \right)^{1/r} \right\|_{X} \lesssim \left( \sum_{i=1}^{k} \|f_i\|_{X}^{r} \right)^{1/r}.$$

See [56] for further discussion of A-convexity.

We will use the following extension of the Z-space scale.

**Definition 2.60.** For $p, q \in (0, \infty]$ and $s \in \mathbb{R}$, we define $Z_s^{p,q}$ to be the set of all $f \in L^0(\mathbb{R}_+^{1+n})$ such that the quasinorm

$$\|f\|_{Z_s^{p,q}} := \left\| \ell(Q)^{-s} [|f|^q|_Q^{1/q} | \right\|_{\ell^p(\mathcal{G}, \ell(Q)^n)} \\ = \left( \sum_{\bar{Q} \in \mathcal{G}} \ell(Q)^n |\ell(Q)^{-s} [|f|^q|_{\bar{Q}}^{1/q} |^p \right)^{1/p}$$

is finite, with the usual modifications when p or q is infinite.
When $q = 2$, $Z_s^{p,q}$ is just the Z-space $Z^P$ seen through the dyadic characterisation (Proposition 2.20), with $p = (p, s)$ when $p < \infty$ and $p = (\infty, s; 0)$ when $p = \infty$. It is straightforward to prove that $Z_s^{p,q}$ is a quasi-Banach function space (Banach when $p, q \ge 1$), and that $Z_s^{p,q}$ is min$(p, q)$-convex, hence A-convex. Furthermore, $Z_s^{p,q}$ is separable when $p, q < \infty$.

We will prove the following interpolation theorem, from which Proposition 2.32 follows by taking $q_0 = q_1 = 2$.

**Theorem 2.61.** Suppose $s_0, s_1 \in \mathbb{R}$, $p_0, p_1, q_0, q_1 \in (0, \infty)$, and $\theta \in (0, 1)$. Then

$$[Z_{s_0}^{p_0, q_0}, Z_{s_1}^{p_1, q_1}]_{\theta} = Z_{s_{\theta}}^{p_{\theta}, q_{\theta}},$$
$$\text{where } 1/p_{\theta} = (1-\theta)/p_0 + \theta/p_1, \quad 1/q_{\theta} = (1-\theta)/q_0 + \theta/q_1, \text{ and } s_{\theta} = (1-\theta)s_0 + \theta s_1.$$

This was conjectured by Barton and Mayboroda, who proved the result when $p_0, p_1 \ge 1$ and $q_0 = q_1 \ge 1$. [22, Theorem 4.13]. To deduce this from Theorem 2.59 we need to establish multiplication and factorisation results.

**Definition 2.62.** Let $s_0, s_1 \in \mathbb{R}$ and $p_0, p_1, q_0, q_1 \in (0, \infty]$. We write

$$Z_s^{p,q} \leftrightarrow Z_{s_0}^{p_0, q_0} \cdot Z_{s_1}^{p_1, q_1}$$

to mean that the following multiplication and factorisation properties hold:

* if $f \in Z_{s_0}^{p_0, q_0}$ and $g \in Z_{s_1}^{p_1, q_1}$, then $fg \in Z_s^{p,q}$ with

$$(2.22) \qquad \|fg\|_{Z_s^{p,q}} \lesssim \|f\|_{Z_{s_0}^{p_0,q_0}} \|g\|_{Z_{s_1}^{p_1,q_1}},$$

* if $h \in Z_s^{p,q}$ then there exist $F \in Z_{s_0}^{p_0, q_0}$ and $G \in Z_{s_1}^{p_1, q_1}$ such that $h = FG$ and

$$(2.23) \qquad \|F\|_{Z_{s_0}^{p_0, q_0}} \|G\|_{Z_{s_1}^{p_1, q_1}} \lesssim \|h\|_{Z_s^{p,q}}.$$

We abbreviate the multiplication and factorisation properties as

$$Z_s^{p,q} \leftarrow Z_{s_0}^{p_0, q_0} \cdot Z_{s_1}^{p_1, q_1} \quad \text{and} \quad Z_s^{p,q} \rightarrow Z_{s_0}^{p_0, q_0} \cdot Z_{s_1}^{p_1, q_1}$$

respectively. This notation is easily extended to more than two factors, and formal computations involving commutativity and associativity (as in the proof of Proposition 2.66) are valid.

**Remark 2.63.** In all the cases that we cover here, the implicit constants in (2.22) and (2.23) can be taken to be 1. This is because we use the dyadic definition of the $Z_s^{p,q}$-quasinorm, and it need not be true for other equivalent quasinorms.

**Proposition 2.64 (Multiplication).** Suppose $s_0, s_1 \in \mathbb{R}$ and $p_0, p_1, q_0, q_1 \in (0, \infty]$, and let

$$\frac{1}{p} := \frac{1}{p_0} + \frac{1}{p_1}, \quad \frac{1}{q} := \frac{1}{q_0} + \frac{1}{q_1}, \quad s := s_0 + s_1.$$

Then $Z_s^{p,q} \leftarrow Z_{s_0}^{p_0, q_0} \cdot Z_{s_1}^{p_1, q_1}$.

**PROOF.** Let $f \in Z_{s_0}^{p_0, q_0}$ and $g \in Z_{s_1}^{p_1, q_1}$. Using Hölder's inequality twice and the assumptions on the exponents we have

$$\begin{align*}
& \|\ell(Q)^{-s} [|fg|^q|\frac{1}{Q}]^{\frac{1}{q}}\|_{\ell^p(G, \ell(Q)^n)} \\
& \le \|\ell(Q)^{-s_0} [|f|^{q_0}|\frac{1}{Q}\ell(Q)^{-s_1} [|g|^{q_1}|\frac{1}{Q}]^{\frac{1}{q_1}}\|_{\ell^p(G, \ell(Q)^n)} \\
& \le \|\ell(Q)^{-s_0} [|f|^{q_0}|\frac{1}{Q}\|_{\ell^{p_0}(G, \ell(Q)^n)} \|\ell(Q)^{-s_1} [|f|^{q_1}|\frac{1}{Q}]^{\frac{1}{q_1}}\|_{\ell^{p_1}(G, \ell(Q)^n)}
\end{align*}$$
$$= \|f\|_{Z_{s_0}^{p_0, q_0}} \|g\|_{Z_{s_1}^{p_1, q_1}},$$

proving the multiplication property.

<figure><img src="image_10.png" /></figure>

**Lemma 2.65 (Single-exponent factorisation).** Suppose $p_0, p_1 \in (0, \infty]$ and $s_0, s_1 \in \mathbb{R}$, and let

$$\frac{1}{p} := \frac{1}{p_0} + \frac{1}{p_1} \quad \text{and} \quad s := s_0 + s_1.$$

Then

$$Z_s^{p, \infty} \to Z_{s_0}^{p_0, \infty} \cdot Z_{s_1}^{p_1, \infty} \quad \text{and} \quad Z_s^{\infty, p} \to Z_{s_0}^{\infty, p_0} \cdot Z_{s_1}^{\infty, p_1}.$$

**PROOF.** First suppose $f \in Z_s^{p, \infty}$; we may assume without loss of generality that $f$ is nonnegative. Let

$$(2.24) \quad F := \sum_{\overline{Q} \in \mathcal{G}} \overline{1}_{\overline{Q}} \ell(Q)^{-s \frac{p}{p_0} + s_0} f^{p/p_0} \quad \text{and} \quad G := \sum_{\overline{Q} \in \mathcal{G}} \overline{1}_{\overline{Q}} \ell(Q)^{s \frac{p}{p_0} - s_0} f^{1-p/p_0},$$

so that $FG = f$. A straightforward estimate gives

$$\|F\|_{Z_{s_0}^{p_0, \infty}} \le \|f\|_{Z_s^{p, \infty}}^{p/p_0} \quad \text{and} \quad \|G\|_{Z_{s_1}^{p_1, \infty}} \le \|f\|_{Z_s^{p, \infty}}^{p/p_1},$$

and so

$$\|F\|_{Z_{s_0}^{p_0, \infty}} \|G\|_{Z_{s_1}^{p_1, \infty}} \le \|f\|_{Z_s^{p, \infty}}.$$

Now assume $g \in Z_s^{\infty, p}$ is nonnegative and define $F'$ and $G'$ as in (2.24), but with $f$ replaced by $g$. The same argument as before yields

$$\|F'\|_{Z_{s_0}^{\infty, p_0}} \|G'\|_{Z_{s_1}^{\infty, p_1}} \le \|g\|_{Z_s^{\infty, p}}.$$

completing the proof.

<figure><img src="image_11.png" /></figure>

**Proposition 2.66 (Factorisation).** Suppose $s_0, s_1 \in \mathbb{R}$ and $p_0, p_1, q_0, q_1 \in (0, \infty]$, and let

$$\frac{1}{p} := \frac{1}{p_0} + \frac{1}{p_1}, \quad \frac{1}{q} := \frac{1}{q_0} + \frac{1}{q_1}, \quad s := s_0 + s_1.$$

Then

$$Z_s^{p, q} \to Z_{s_0}^{p_0, q_0} \cdot Z_{s_1}^{p_1, q_1}.$$

**PROOF.** It suffices to show that

$$(2.25) \qquad Z_s^{p, q} \to Z_{s_0}^{\infty, q} \cdot Z_{s_1}^{p, \infty},$$

because by Proposition 2.64 and Lemma 2.65 along with (2.25) we have

$$\begin{align*}
Z_s^{p,q} &\to Z_{s/2}^{p,\infty} \cdot Z_{s/2}^{\infty,q} \\
&\to Z_{s_0/2}^{p_0,\infty} \cdot Z_{s_1/2}^{p_1,\infty} \cdot Z_{s_0/2}^{\infty,q_0} \cdot Z_{s_0/2}^{\infty,q_1} \\
&= Z_{s_0/2}^{p_0,\infty} \cdot Z_{s_0/2}^{\infty,q_0} \cdot Z_{s_1/2}^{p_1,\infty} \cdot Z_{s_1/2}^{\infty,q_1} \\
&\to Z_{s_0}^{p_0,q_0} \cdot Z_{s_1}^{p_1,q_1}.
\end{align*}$$

We now prove (2.25). Given $h \in Z_s^{p, q}$, let

$$F := \sum_{\overline{Q} \in \mathcal{G}} \overline{1}_{\overline{Q}} \ell(Q)^{s_0} h / [|h|^q]_{\overline{Q}}^{1/q} \quad \text{and} \quad G := \sum_{\overline{Q} \in \mathcal{G}} \overline{1}_{\overline{Q}} \ell(Q)^{-s_0} [|h|^q]_{\overline{Q}}^{1/q},$$

so that $FG = h$. We immediately have

$$\|F\|_{Z_{s_0}^{\infty, q}} = \sup_{\overline{Q} \in \mathcal{G}} \ell(Q)^{-s_0} \ell(Q)^{s_0} = 1$$
and

$$
\begin{align*}
\|G\|_{Z_{s_1}^{p,\infty}} &= \|\ell(Q)^{-s_1} \ell(Q)^{-s_0} [|h|^q]_{\overline{Q}}^{1/q} \|\ell^p(G, \ell(Q)^n) \\
&= \|\ell(Q)^{-s} [|h|^q]_{\overline{Q}}^{1/q} \|\ell^p(G, \ell(Q)^n) \\
&= \|h\|_{Z_s^{p,q}},
\end{align*}
$$

proving (2.25).

□

Now we can prove Theorem 2.61.

PROOF OF THEOREM 2.61. Since $Z_{s_0}^{p_0, q_0}$ and $Z_{s_1}^{p_1, q_1}$ are A-convex and separa-
ble, by Theorem 2.59 we have

$$
[Z_{s_0}^{p_0, q_0}, Z_{s_1}^{p_1, q_1}]_{\theta} = (Z_{s_0}^{p_0, q_0})^{1-\theta} (Z_{s_1}^{p_1, q_1})^{\theta}.
$$

We will show that

$$
(2.26) \qquad (Z_{s_0}^{p_0, q_0})^{1-\theta} (Z_{s_1}^{p_1, q_1})^{\theta} = Z_{s_0}^{p_0, q_0}.
$$

This follows from Propositions 2.64 and 2.66 in a standard way, but we include the
argument for completeness. Suppose that $h \in L^0(\mathbb{R}_+^{1+n})$ with $|h| \le |f|^{1-\theta}|g|^{\theta}$ for
some $f \in Z_{s_0}^{p_0, q_0}$, $g \in Z_{s_1}^{p_1, q_1}$. Then by Proposition 2.64 we have

$$
\begin{align*}
\|h\|_{Z_{s_\theta}^{p_\theta, q_\theta}} &\le \|f\|_{Z_{s_\theta}^{p_\theta, q_\theta}}^{1-\theta} |g|^{\theta} \|h\|_{Z_{s_\theta}^{p_\theta, q_\theta}} \\
&\le \|f\|_{Z_{s_0(1-\theta)}^{p_0/(1-\theta), q_0/(1-\theta)}}^{1-\theta} \|g\|^{\theta} \|h\|_{Z_{s_1\theta}^{p_1/\theta, q_1/\theta}} \\
&= \|f\|_{Z_{s_0}^{p_0, q_0}}^{1-\theta} \|g\|_{Z_{s_1}^{p_1, q_1}}^{\theta},
\end{align*}
$$

so by taking the infimum over all such $f$ and $g$ we get

$$
\|h\|_{Z_{s_\theta}^{p_\theta, q_\theta}} \le \|h\|_{(Z_{s_0}^{p_0, q_0})^{1-\theta} (Z_{s_1}^{p_1, q_1})^\theta}.
$$

Conversely, suppose that $h \in Z_{s_\theta}^{p_\theta, q_\theta}$. Then by Proposition 2.66 there exist $F \in Z_{s_0(1-\theta)}^{p_0/(1-\theta), q_0/(1-\theta)}$ and $G \in Z_{s_1\theta}^{p_1/\theta, q_1/\theta}$ such that $h = FG$ with

$$
\begin{align*}
\|h\|_{Z_{s_\theta}^{p_\theta, q_\theta}} &\ge \|F\|_{Z_{s_0(1-\theta)}^{p_0/(1-\theta), q_0/(1-\theta)}} \|G\|_{Z_{s_1\theta}^{p_1/\theta, q_1/\theta}} \\
&= \|F|^{1/(1-\theta)}\|_{Z_{s_0}^{p_0, q_0}}^{1-\theta} \|G|^{1/\theta} \|h\|_{Z_{s_1}^{p_1, q_1}}^{\theta}.
\end{align*}
$$

Setting $f = |F|^{1/(1-\theta)}$ and $g = |G|^{1/\theta}$ we see that $f^{1-\theta}g^{\theta} = |h|$, $f \in Z_{s_0}^{p_0, q_0}$, and $g \in Z_{s_1}^{p_1, q_1}$, so we find that

$$
\|h\|_{Z_{s_\theta}^{p_\theta, q_\theta}} \ge \|h\|_{(Z_{s_0}^{p_0, q_0})^{1-\theta} (Z_{s_1}^{p_1, q_1})^{\theta}}.
$$

This completes the proof of (2.26), and hence also that of Theorem 2.61.

□
## 2.7. Appendix: Table of exponents and function spaces

Here we provide a small ‘dictionary’ detailing precisely what is meant by the data $(X^P, X^P)$. Here $\mathbf{p}$ is an exponent, and $(X, X)$ is either $(T, \mathbf{H})$ or $(Z, \mathbf{B})$. The definition of the spaces $T^P$, $Z^P$, $\mathbf{H}^P$, and $\mathbf{B}^P$ change depending on whether $\mathbf{p}$ is finite or infinite, with a special case when $j(\mathbf{p}) = 0$ (i.e. $\mathbf{p} = (\infty, s; 0)$). We advise the reader to keep a copy of this table at hand while reading the monograph. In the table we assume that $\alpha > 0$.

<table><thead><tr><th>(X, X)<br>p</th><th>(T, H)</th><th>(Z, B)</th></tr></thead><tbody><tr><td>(p, s)</td><td>(T<sub>s</sub><sup>p</sup>, ˙H<sub>s</sub><sup>p</sup>)</td><td>(Z<sub>s</sub><sup>p</sup>, ˙B<sub>s</sub><sup>p,p</sup>)</td></tr><tr><td>(∞, s; 0)</td><td>(T<sub>s;0</sub><sup>∞</sup>, B˙MO<sub>s</sub>)</td><td>(Z<sub>s</sub><sup>∞</sup>, ˙B<sub>s</sub><sup>∞,∞</sup>)</td></tr><tr><td>(∞, s; α)</td><td>(T<sub>s;α</sub><sup>∞</sup>, ˙B<sub>s+α</sub><sup>∞,∞</sup>)</td><td>(Z<sub>s+α</sub><sup>∞</sup>, ˙B<sub>s+α</sub><sup>∞,∞</sup>)</td></tr></tbody></table>

In Section 5 we will define $X_D^P$ to be the image of $X^P$ under the projection $\mathbb{P}_D$ onto the closure of the range of the Dirac operator $D$, $\overline{\mathcal{R}(D)}$, which may be characterised by a tangential curl-free condition. Thus $X_D^P$ may be thought of as the set of tangential curl-free functions in $X^P$.
# Operator Theoretic Preliminaries

## 3.1. Bisectorial operators and holomorphic functional calculus

The material of this section is not new, but we present it here to fix notation. Useful standard references are [63, 64, 1, 38], and a particularly nice recent exposi- tion which focuses on bisectorial operators on reflexive Banach spaces is contained in Egert's thesis [33, Chapter 3].

Let $0 < \omega < \pi/2$. The open bisector of angle $\omega$ is the set

$$S_{\omega} := \{z \in \mathbb{C} \setminus \{0\} : |\arg(z)| < \omega \text{ or } |\arg(-z)| < \omega\} \subset \mathbb{C},$$

where the argument $\arg(z)$ takes values in $(-\pi, \pi]$. The closed bisector of angle $\omega$ is the topological closure $\bar{S}_{\omega}$ of $S_{\omega}$ in $\mathbb{C}$.

Throughout this section we write $L^2 = L^2(\mathbb{R}^n)$.

**Definition 3.1.** Let $0 \le \omega < \pi/2$. A closed linear operator $A$ on $L^2$ is called bisectorial of angle $\omega$ if $\sigma(A) \subset \bar{S}_{\omega}$, and if for all $\mu \in (\omega, \pi/2)$ and all $z \in \mathbb{C} \setminus \bar{S}_{\mu}$ we have the resolvent bound

$$(3.1) \qquad \|(z - A)^{-1}\|_{L(L^2)} \lesssim_{\mu} |z|^{-1}.$$

Closedness of $A$ is included in this definition; this is not standard, but it is convenient. Generally the precise angle $\omega$ is not important, so we may simply refer to $A$ as *bisectorial*.

The following proposition is proven in [33, Proposition 3.2.2] (except for the adjoint statement, which is a simple computation).

**Proposition 3.2.** Let $A$ be a bisectorial operator on $L^2$. Then $A$ is densely-defined, and we have a topological (not necessarily orthogonal) splitting

$$(3.2) \qquad L^2 = \mathcal{N}(A) \oplus \overline{\mathcal{R}(A)}.$$

Furthermore, $A^*$ is also bisectorial.

We let $\mathbb{P}_A$ denote the projection onto $\overline{\mathcal{R}(A)}$ along $\mathcal{N}(A)$.

The procedure of constructing an operator $\varphi(A)$ from a given bisectorial operator $A$ and holomorphic function $\varphi$ on an appropriate bisector, known as *holomorphic functional calculus*, plays a central role in our theory. In order to introduce holomorphic functional calculus properly, we must first define some classes of holomorphic functions.

For an angle $\mu \in (0, \pi/2)$, the set of holomorphic functions $\varphi: S_{\mu} \to \mathbb{C}$ is denoted by $H(S_{\mu})$. For $\sigma, \tau \in \mathbb{R}$ and $\varphi \in H(S_{\mu})$ we define

$$\|\varphi\|_{\Psi_{\sigma}^{\tau}(S_{\mu})} = \|z \mapsto \varphi(z)/m_{\sigma}^{\tau}(|z|)\|_{L^{\infty}(S_{\mu})} \\ \text{(the function } m_{\sigma}^{\tau} \text{ is defined in Section 1.3) and}$$

$$\Psi_{\sigma}^{\tau}(S_{\mu}) := \{\varphi \in H(S_{\mu}) : \|\varphi\|_{\Psi_{\sigma}^{\tau}(S_{\mu})} < \infty\}.$$
Each $\Psi_{\sigma}^{\tau}(S_{\mu})$ is a Banach space when normed by $\|\cdot\|_{\Psi_{\sigma}^{\tau}(S_{\mu})}$, consisting of all holomorphic functions on $S_{\mu}$ which decay of order $\sigma$ at 0 and of order $\tau$ at $\infty$, interpreting growth as decay of negative order. An important special case is $\Psi_{0}^{0}(S_{\mu}) = H^{\infty}(S_{\mu})$, the set of bounded holomorphic functions on $S_{\mu}$. We surpress reference to $S_{\mu}$ in this notation when the relevant bisector is clear from context.

The spaces $\Psi_{\sigma}^{\tau}$ are decreasing in $\sigma$ and $\tau$, in the sense that if $\sigma < \sigma'$ and $\tau < \tau'$, then $\Psi_{\sigma'}^{\tau'} \hookrightarrow \Psi_{\sigma}^{\tau}$. For $\sigma, \tau \in \mathbb{R}$ we define the set

$$ \Psi_{\sigma}^{\tau+} := \bigcup_{\tau' > \tau} \Psi_{\sigma}^{\tau'}, $$

and we define the sets $\Psi_{\sigma+}^{\tau}$ and $\Psi_{\sigma+}^{\tau+}$ analogously. The set $\Psi_{+}^{\tau} := \Psi_{0+}^{0+}$ is particularly important: it is the set of holomorphic functions (on the relevant bisector) with polynomial decay of some positive order at 0 and $\infty$. We also define

$$ \Psi_{\sigma}^{\infty} := \bigcap_{\tau} \Psi_{\sigma}^{\tau}, $$

the set of functions with polynomial decay of arbitrarily large order at $\infty$. Similarly we define $\Psi_{\infty}^{\tau}$, $\Psi_{\infty}^{\infty}$, $\Psi_{\sigma+}^{\infty}$, and so on.

There are a few holomorphic functions which we use extensively. First define $\chi^{+}, \chi^{-} \in H^{\infty}(S_{\mu})$ by

$$ (3.3) \qquad \chi^{+}(z) := \mathbf{1}_{z:\mathrm{Re}(z)>0}(z) \quad \text{and} \quad \chi^{-}(z) := \mathbf{1}_{z:\mathrm{Re}(z)<0}(z) \quad (z \in S_{\mu}). $$

These are the indicator functions of the two halves of the bisector $S_{\mu}$. Define also

$$ [z] := \begin{cases} z & (\mathrm{Re}(z) > 0) \\ -z & (\mathrm{Re}(z) < 0) \end{cases} = (\chi^{+}(z) - \chi^{-}(z))z. $$

This lets us define a bounded version of the exponential map,

$$ (3.4) \qquad \mathrm{sgp} := [z \mapsto e^{-[z]}] \in \Psi_{0}^{\infty}. $$

For $\lambda \in \mathbb{R} \setminus \{0\}$ we define the power function

$$ [z \mapsto z^{\lambda}] \in \Psi_{\lambda}^{-\lambda} $$

via a branch cut on the half-line $i(-\infty, 0] \subset \mathbb{C}$.

We say that a function $\varphi \in H(S_{\mu})$ is nondegenerate if it does not vanish on any open subset of $S_{\mu}$. All the holomorphic functions defined above are nondegenerate except for $\chi^{+}$ and $\chi^{-}$.

Let us introduce some useful operations on holomorphic functions. Let $\varphi \in H(S_{\mu})$. There is a natural involution $\varphi \mapsto \tilde{\varphi}$ on $H(S_{\mu})$ defined by

$$ \tilde{\varphi}(z) := \overline{\varphi(\bar{z})} \quad (z \in S_{\mu}). $$

This involution is isometric on $\Psi_{\sigma}^{\tau}$ for all $\sigma, \tau \in \mathbb{R}$. For $t > 0$ we define the dilation $\varphi_t \in H(S_{\mu})$ by

$$ \varphi_t(z) := \varphi(tz). $$

The following lemma is a simple consequence of the above definitions.

**Lemma 3.3.** Let $\sigma \in \mathbb{R}$. Then for all $t > 0$ we have

$$ \|\varphi_t\|_{\Psi_{\sigma}^{-\sigma}} = t^{\sigma} \|\varphi\|_{\Psi_{\sigma}^{-\sigma}}. $$
Fix an angle $\omega \in [0, \pi/2)$ and let $A$ be an $\omega$-bisectorial operator on $L^2$. If $\mu \in (\omega, \pi/2)$ and $\varphi \in \Psi_+^+(S_\mu)$, then we can define an operator $\varphi(A)$ on $L^2$ by

$$ (3.5) \qquad \varphi(A)f := \frac{1}{2\pi i} \int_{\partial S_\nu} \varphi(z)(z-A)^{-1} f \, dz \quad (f \in L^2) $$

for any choice of $\nu \in (\omega, \mu)$, where $\partial S_\nu$ is oriented counterclockwise. The integral (3.5) is well-defined and independent of the choice of $\nu$, and we have

$$ \|\varphi(A)\|_{\mathcal{L}(L^2)} \lesssim_{A, \sigma, \tau, \mu} \|\varphi\|_{\Psi_{\sigma}^{\tau}(S_{\mu})}. $$

The proof is straightforward. The following $*$-homomorphism property holds: for all $\varphi, \psi \in \Psi_+^+$, we have $(\varphi\psi)(A) = \varphi(A)\psi(A)$, and $\psi(A)^* = \tilde{\psi}(A^*)$.

Often it is convenient to assume that the operator $A$ is injective and has dense range. In our applications this does not hold, but the splitting $L^2 = \mathcal{N}(A) \oplus \overline{\mathcal{R}(A)}$ from Proposition 3.2 shows that the restriction $A|_{\overline{\mathcal{R}(A)}}$, acting on the Hilbert space $\overline{\mathcal{R}(A)}$ (with inner product induced by that of $L^2$), is injective and has dense range. One can also show that $A|_{\overline{\mathcal{R}(A)}}$ is bisectorial. Thus we bypass the problem by working on $\overline{\mathcal{R}(A)}$ instead of the whole space $L^2$.

The integral in (3.5) converges whenever $\varphi \in \Psi_+^+$, but if $\varphi \in H^\infty$ is merely bounded, convergence is not guaranteed. Nevertheless, we are able to construct operators $\varphi(A)$ when $\varphi \in H^\infty$ for certain $A$.

**Definition 3.4.** Let $A$ be a bisectorial operator on $L^2$. We say that $A$ has bounded $H^\infty$ functional calculus on $\overline{\mathcal{R}(A)}$ if for all $\varphi \in \Psi_+^+$ and all $f \in \overline{\mathcal{R}(A)}$,

$$ \|\varphi(A)f\|_2 \lesssim \|\varphi\|_{\infty} \|f\|_2. $$

The property of having bounded $H^\infty$ functional calculus on $\overline{\mathcal{R}(A)}$ is equivalent to certain quadratic estimates: this is an important theorem due to McIntosh (see [63, §7 and §8] and the other references at the start of this section).

**Theorem 3.5 (McIntosh).** Let $A$ be a bisectorial operator on $L^2$. Then $A$ has bounded $H^\infty$ functional calculus on $\overline{\mathcal{R}(A)}$ if and only if the quadratic estimate

$$ (3.6) \qquad \|f\|_2 \simeq_{\varphi} \left( \int_0^\infty \|\varphi_t(A)f\|_2^2 \frac{dt}{t} \right)^{1/2} $$

holds for all $f \in \overline{\mathcal{R}(A)}$ and some (equivalently, all) nondegenerate $\varphi \in \Psi_+^+$.

Note that the quadratic estimate (3.6) need not hold for $\varphi \in H^\infty$.

If $A$ has bounded $H^\infty$ functional calculus on $\overline{\mathcal{R}(A)}$, then for all $\varphi \in H^\infty$ we can define a bounded operator $\varphi(A)$ on $\overline{\mathcal{R}(A)}$ by

$$ (3.7) \qquad \varphi(A)f := \lim_{\alpha} \varphi_{\alpha}(A)f \quad (f \in \overline{\mathcal{R}(A)}), $$

where $(\varphi_\alpha)$ is a net in $\Psi_+^+$ which converges to $\varphi$ in $H^\infty$. We then have

$$ \|\varphi(A)\|_{\mathcal{L}(\overline{\mathcal{R}(A)})} \lesssim \|\varphi\|_{H^\infty}, $$

and for $\varphi, \psi \in H^\infty$ we have the homomorphism property $\varphi(A)\psi(A) = (\varphi\psi)(A)$. For further details see [63, 64]. Thus we may define bounded operators $\chi^\pm(A)$ and $e^{-t[A]} = \text{sgp}_t(A)$ (for all $t > 0$) on $\overline{\mathcal{R}(A)}$, using the corresponding $H^\infty$ functions defined in (3.3) and (3.4).
If $\varphi \in \Psi_+^0$, then we can extend $\varphi(A)$ from $\overline{\mathcal{R}(A)}$ to all of $L^2$ by

$$ \varphi(A)f := \varphi(A)\mathbb{P}_A f. $$

The operator $\varphi(A)$ then maps $L^2$ into $\overline{\mathcal{R}(A)}$. We have

$$ \|\varphi(A)\|_{\mathcal{L}(L^2)} \lesssim \|\mathbb{P}_A\|_{\mathcal{L}(L^2)} \|\varphi(A)\|_{\mathcal{L}(\overline{\mathcal{R}(A)})}, $$

and the homomorphism property $\psi(A)\varphi(A) = (\psi\varphi)(A)$ persists for all $\psi \in H^\infty$.

We refer to the operators $\chi^+(A)$ and $\chi^-(A)$ on $\overline{\mathcal{R}(A)}$ as the positive and negative spectral projections associated with $A$. These are complementary projections. We define the positive and negative spectral subspaces as their images

$$ \overline{\mathcal{R}(A)}^{\pm} := \chi^{\pm}(A)\overline{\mathcal{R}(A)}, $$

and we have a topological direct sum decomposition

$$ \overline{\mathcal{R}(A)} = \overline{\mathcal{R}(A)}^{+} \oplus \overline{\mathcal{R}(A)}^{-}. $$

We define the Cauchy operators $C_A: \overline{\mathcal{R}(A)} \to L^\infty(\mathbb{R} \setminus \{0\} : \overline{\mathcal{R}(A)})$ by
$$ (3.8) \quad C_A f(t) := e^{-tA} \chi^{\operatorname{sgn}(t)}(A) f = e^{-|t|\overline{[A]}} \chi^{\operatorname{sgn}(t)}(A) f = \operatorname{sgp}_{|t|}(A) \chi^{\operatorname{sgn}(t)}(A) f, $$

where we write $\chi^{\pm 1}(A) = \chi^{\pm}(A)$. Write $C_A^+$ and $C_A^-$ for the restrictions of $C_A$ to $\mathbb{R}_+$ and $\mathbb{R}_-$ respectively. These are solution operators for the Cauchy problems associated with $A$ on the upper and lower half-spaces [8].

**Proposition 3.6.** Suppose that $A$ has bounded $H^\infty$ functional calculus on $\overline{\mathcal{R}(A)}$. If $f \in \overline{\mathcal{R}(A)}$, then

$$ F := C_A f \in C^\infty(\mathbb{R} \setminus \{0\} : \overline{\mathcal{R}(A)}) $$

solves the Cauchy problem

$$ \partial_t F(t) + AF(t) = 0, \quad \lim_{t \to 0^+} F(t) = \chi^\pm(A)f, \quad \lim_{t \to \pm\infty} F(t) = 0 $$

with limits taken in $\overline{\mathcal{R}(A)}$.

Finally we discuss unbounded operators arising from holomorphic functional calculus, and some situations in which their compositions are bounded. Suppose that $\varphi \in \Psi_+^\tau$ with $\min(\sigma, \tau) \le 0$, so that the integral (3.5) need not be absolutely convergent. We can define an unbounded operator $\varphi(A)$ on $\overline{\mathcal{R}(A)}$ as follows. Fix $\delta > \max(-\sigma, -\tau) \ge 0$ and define $\eta^\delta \in \Psi_+^\delta$ by

$$ \eta^\delta(z) := \left( \frac{z}{(1+z)^2} \right)^\delta. $$

Then $\eta^\delta\varphi \in \Psi_+^+$, so $(\eta^\delta\varphi)(A)$ is defined by (3.5). We also have $\eta^\delta \in \Psi_+^+$, so $\eta^\delta(A)$ is also defined by (3.5), and since $A$ is injective with dense range on $\overline{\mathcal{R}(A)}$, so is $\eta^\delta(A)$. Therefore the unbounded operator $\eta^\delta(A)^{-1}$ is defined, with $\mathcal{D}(\eta^\delta(A)^{-1}) := \overline{\mathcal{R}(\eta^\delta(A))}$. We then define the unbounded operator

$$ (3.9) \qquad \varphi(A) := \eta^\delta(A)^{-1} (\eta^\delta\varphi)(A) $$

with domain

$$ \mathcal{D}(\varphi(A)) := \{f \in \overline{\mathcal{R}(A)} : (\eta^\delta\varphi)(A)f \in \mathcal{D}(\eta^\delta(A)^{-1})\}. $$
The operator $\varphi(A)$ is closed, densely-defined, and independent of the choice of $\delta$. Of course, if $\min(\sigma, \tau) > 0$, then we can take $\delta = 0$ in the definition (3.9) and recover the original definition of $\varphi(A)$ by the Cauchy integral (3.5).

Now suppose $\psi \in \Psi_{\sigma_1}^{\tau_1}$ and $\varphi \in \Psi_{\sigma_2}^{\tau_2}$. Then a quick computation shows that $\varphi(A)\psi(A) \subseteq (\varphi\psi)(A)$. If $\sigma_1 + \sigma_2 > 0$ and $\tau_1 + \tau_2 > 0$, then the operator $(\varphi\psi)(A)$ is bounded and given by (3.5), while the operator $\varphi(A)\psi(A)$ is not a priori given by such a representation. This observation is convenient in what follows.

## 3.2. Off-diagonal estimates and the Standard Assumptions

For $x \in \mathbb{R}$, write $\langle x \rangle := \max\{1, |x|\}$. We continue to write $L^2 = L^2(\mathbb{R}^n)$.

**Definition 3.7.** Suppose $\Omega \subset \mathbb{C} \setminus \{0\}$, and let $(S_z)_{z \in \Omega}$ be a family of operators in $\mathcal{L}(L^2)$. Let $M \ge 0$. We say that $(S_z)$ satisfies off-diagonal estimates of order $M$ if for all Borel subsets $E, F \subset \mathbb{R}^n$, all $z \in \Omega$, and all $f \in L^2$,

$$ (3.10) \qquad \|\mathbf{1}_F S_z (\mathbf{1}_E f)\|_2 \lesssim \left\langle \frac{d(E, F)}{|z|} \right\rangle^{-M} \|\mathbf{1}_E f\|_2. $$

Many families of operators constructed from first-order differential operators (in particular, certain families of resolvents) satisfy off-diagonal estimates of some order. The following theorem shows that certain families constructed via holomorphic functional calculus of a bisectorial operator $A$ satisfy off-diagonal estimates, under the assumption that a certain resolvent family satisfies off-diagonal estimates. This is a slight extension of [71, Proposition 2.7.1].

**Theorem 3.8 (Off-diagonal estimates for families constructed by functional calculus).** Fix $0 \le \omega < \nu < \mu < \pi/2$, $M \ge 0$, and $\sigma, \tau > 0$. Let $A$ be an $\omega$-bisectorial operator on $L^2$ with bounded $H^\infty$ functional calculus on $\overline{\mathcal{R}}(A)$, such that $((I + \lambda A)^{-1})_{\lambda \in \mathbb{C} \setminus S_\nu}$ satisfies off-diagonal estimates of order $M$. Suppose that $(\eta(t))_{t>0}$ is a continuous family of functions in $H^\infty(S_\mu)$ which is uniformly bounded.<fnref n="1" /> If $\psi \in \Psi_\sigma^\tau(S_\mu)$, then the family of operators $(\eta(t)(A)\psi_t(A))_{t>0}$ satisfies off-diagonal estimates of order $\min\{\sigma, M\}$, with constants depending linearly on $\|\psi\|_{\Psi_\sigma^\tau}\|\eta\|$, where $\|\eta\| := \sup_{t>0} \|\eta(t)\|_\infty$, and also depending on $A, M, \sigma$, and $\tau$.

**PROOF.** Fix Borel sets $E, F \subset \mathbb{R}^n$. Because $\psi_t(A)$ maps into $\overline{\mathcal{R}}(A)$ for each $t$, we can apply $\eta(t)(A)$ to $\psi_t(A)(\mathbf{1}_E f)$ for each $t > 0$. We must prove the estimate

$$ \|\mathbf{1}_F \eta(t)(A) \psi_t(A) (\mathbf{1}_E f)\|_2 \lesssim_{A,M,\sigma,\tau} \|\eta\| \|\psi\|_{\Psi_\sigma^\tau(S_\mu)} \left\langle \frac{d(E, F)}{t} \right\rangle^{-\min\{\sigma,M\}} \|\mathbf{1}_E f\|_2 $$

for all $f \in L^2$. Fix $\nu' \in (\nu, \mu)$ throughout the proof.

If $d(E, F) \le t$, then $\langle d(E, F)/t \rangle \simeq 1$, and so we have
(3.11)

$$ \begin{align} \|\mathbf{1}_F \eta(t)(A) \psi_t(A) (\mathbf{1}_E f)\|_2 &\le \int_{\partial S_{\nu'}} |\eta(t)(z)| \|\psi(tz)\| \|\mathbf{1}_F(z-A)^{-1}(\mathbf{1}_E f)\|_2 |dz| \tag{3.12} \\ &\lesssim_A \|\eta\| \|\psi\|_{\Psi_\sigma^\tau} \|\mathbf{1}_E f\|_2 \int_{\partial S_{\nu'}} m_\sigma^\tau(|z|) \frac{|dz|}{|z|} \nonumber \\ &\simeq_{\sigma, \tau, M} \|\eta\| \|\psi\|_{\Psi_\sigma^\tau} \left\langle \frac{d(E, F)}{t} \right\rangle^{-\min\{\sigma, M\}} \|\mathbf{1}_E f\|_2, \nonumber \end{align} $$

<footnote n="1">¹Continuity isn't really needed here. We only assume it to avoid thinking about measurability.</footnote>
using the resolvent bound coming from bisectoriality of A in (3.12).

Now suppose that $d(E, F) > t$. Rearranging (3.11) and using the assumed off-diagonal estimates for $((I + \lambda A)^{-1})_{\lambda \in C \setminus S_\nu}$, we have

$$ \begin{align} & \|\mathbf{1}_F \eta(t)(A) \psi_t(A)(\mathbf{1}_E f)\|_2 \nonumber \\ & \lesssim_A \|\eta\| \|\psi\|_{\Psi_\sigma} \|\mathbf{1}_E f\|_2 \int_{\partial S_\nu'} m_\sigma^\tau(|z|) \left\langle \frac{d(E, F)}{t/|z|} \right\rangle^{-M} \frac{|dz|}{|z|} \nonumber \\ \tag{3.13} & \lesssim \|\eta\| \|\psi\|_{\Psi_\sigma} \|\mathbf{1}_E f\|_2 (\mathbf{I}_0 + \mathbf{I}_\infty), \end{align} $$

where

$$ \mathbf{I}_0 := \int_0^{t/d(E,F)} m_\sigma^\tau(\lambda) \frac{d\lambda}{\lambda} $$

and

$$ \mathbf{I}_\infty := \int_{t/d(E,F)}^\infty m_\sigma^\tau(\lambda) \left(\frac{\lambda}{t} d(E, F)\right)^{-M} \frac{d\lambda}{\lambda}. $$

The integral $\mathbf{I}_0$ is estimated by

$$ (3.14) \qquad \mathbf{I}_0 \le \int_0^{t/d(E,F)} \lambda^\sigma \frac{d\lambda}{\lambda} \approx_\sigma \left(\frac{t}{d(E,F)}\right)^\sigma \lesssim_M \left\langle \frac{d(E,F)}{t} \right\rangle^{-\min(\sigma,M)}. $$

To estimate $\mathbf{I}_\infty$, we use that $t/d(E, F) \le 1$ to write

$$ \mathbf{I}_\infty \approx_M \left\langle \frac{d(E, F)}{t} \right\rangle^{-M} \left( \int_{t/d(E,F)}^1 \lambda^{\sigma-M} \frac{d\lambda}{\lambda} + C(\tau, M) \right) $$

where

$$ C(\tau, M) = \int_1^\infty \lambda^{-\tau-M} \frac{d\lambda}{\lambda}. $$

If $\sigma \le M$, then we have

$$ \int_{t/d(E,F)}^1 \lambda^{\sigma-M} \frac{d\lambda}{\lambda} \lesssim_{\sigma,M} \left(\frac{t}{d(E,F)}\right)^{\sigma-M}, $$

and in this case

$$ \begin{align} \mathbf{I}_\infty &\lesssim_{\sigma,M} \left\langle \frac{d(E,F)}{t} \right\rangle^{-M} \left( \left\langle \frac{d(E,F)}{t} \right\rangle^{M-\sigma} + C(\tau,M) \right) \nonumber \\ \tag{3.15} &\lesssim_{\tau,M} \left\langle \frac{d(E,F)}{t} \right\rangle^{-\min\{\sigma,M\}}. \end{align} $$

Otherwise, we have

$$ \int_{t/d(E,F)}^1 \lambda^{\sigma-M} \frac{d\lambda}{\lambda} \lesssim_{\sigma,M} 1, $$

and this also yields (3.15). Putting the estimates (3.14) and (3.15) into (3.13) completes the proof. $\square$

Off-diagonal estimates can also be used to deduce uniform boundedness and convergence results for families of operators on slice spaces. These propositions were proven by the second author and Mourgoglou [16, §4].
**Proposition 3.9 (Uniform boundedness of families on slice spaces).**
Let $p \in (0, \infty]$. If $(T_s)_{s>0}$ is a family of operators on $L^2$ satisfying off-diagonal estimates of order greater than $n \max(|\delta_{p,2}|, 1/2)$, then $T_s$ extends to a bounded operator on $E_0^p(t)$ uniformly in $0 < s \le t$.

**Proposition 3.10 (Strong convergence in slice spaces).** Let $p \in (0, \infty)$. Suppose $(T_s)_{s>0}$ is a family of operators on $L^2$ satisfying off-diagonal estimates of order greater than $n \max(1/p, 1/2)$, and such that $\lim_{s \to 0} T_s = I$ strongly in $L^2$. Then $\lim_{s \to 0} T_s = I$ strongly in $E^p$.

Throughout the ‘abstract’ part of this work, the following assumptions will be sufficient. They can be a bit of a mouthful if stated in full, so we give them a name.

**Definition 3.11.** We say that an operator $A$ satisfies the Standard Assumptions if

* $A$ is a $\omega$-bisectorial operator on $L^2$ for some $\omega \in [0, \pi/2)$,

* $A$ has bounded $H^\infty$ functional calculus on $\bar{\mathcal{R}}(A)$, and

* for all $\nu \in (\omega, \pi/2)$ the family $((I + \lambda A)^{-1})_{\lambda \in \mathbb{C} \setminus S_\nu}$ satisfies off-diagonal estimates of arbitrarily large order.

The main examples we have in mind are perturbed Dirac operators.

**Theorem 3.12.** Suppose $D$ and $B$ are as in Subsection 1.1.3 of the introduction. Then the perturbed Dirac operators $DB$ and $BD$ satisfy the Standard Assumptions (see Definition 3.11).

This was proven by the second author and Stahlhut ([18, Proposition 2.1] and [19, Lemma 2.3, Propositions 3.1 and 3.2]). The off-diagonal estimates stated there are in a different but equivalent form. We mention once more that establishing bounded $H^\infty$ functional calculus is a deep result due to Axelsson, Keith, and McIntosh [21].

### 3.3. Integral operators on tent spaces

Let $(S_{t,\tau})_{t,\tau>0}$ be a continuous two-parameter family of bounded operators on $L^2 = L^2(\mathbb{R}^n)$, and for all $f \in L_c^2(\mathbb{R}_+ : L^2)$ define $Sf \in L^0(\mathbb{R}_+ : L^2)$ by

$$ (3.16) \qquad Sf(t) := \int_0^\infty S_{t,\tau} f(\tau) \frac{d\tau}{\tau}. $$

Since $f$ is compactly supported in $\mathbb{R}_+$, the Cauchy–Schwarz inequality shows that this integral is absolutely convergent. We write $S \sim (S_{t,\tau})_{t,\tau>0}$ to say that $S$ is given by the kernel $(S_{t,\tau})_{t,\tau>0}$.

We would like to know when $S$ can be extended from $L_c^2(\mathbb{R}_+ : L^2)$ to an operator between various tent spaces and $Z$-spaces. A first step is given by the following Schur-type lemma. Recall that $L_s^2$ is defined in (2.15) and coincides with $X_s^2$.

**Lemma 3.13.** Let $s, \delta \in \mathbb{R}$, and consider $S \sim (S_{t,\tau})_{t,\tau>0}$ on $L^2$ as above. Suppose that there exists $\gamma \in L^1(\mathbb{R}_+ : \mathbb{R})$ (where $\mathbb{R}_+$ is equipped with the Haar measure $dt/t$) such that for all $t, \tau > 0$,

$$ \|\tau^{-\delta} S_{t,\tau}\|_{L(L^2)} \le \gamma(t/\tau)(t/\tau)^{s+\delta}. $$

Then for all $f \in L_c^2(\mathbb{R}_+ : L^2)$,

$$ (3.17) \qquad \|Sf\|_{L_{s+\delta}^2} \lesssim \|\gamma\|_{L^1(\mathbb{R}_+)} \|f\|_{L_s^2}. $$
PROOF. We argue by duality. For all $g \in L^2_{-(s+\delta)}$, using the assumed estimate on the operators $\tau^{-\delta}S_{t,\tau}$, we can estimate

$$ \begin{aligned} |\langle Sf, g \rangle| & \le \int_0^\infty \int_0^\infty \|S_{t,\tau}f(\tau)\|_2 \|g(t)\|_2 \frac{d\tau}{\tau} \frac{dt}{t} \\ & \le \int_0^\infty \int_0^\infty \gamma(t/\tau) \|\tau^{-s}f(\tau)\|_2 \|t^{s+\delta}g(t)\|_2 \frac{d\tau}{\tau} \frac{dt}{t} \\ & \le \left( \int_0^\infty \int_0^\infty \gamma(t/\tau) \frac{dt}{t} \|\tau^{-s}f(\tau)\|_2^2 \frac{d\tau}{\tau} \right)^{1/2} \\ & \qquad \cdot \left( \int_0^\infty \int_0^\infty \gamma(t/\tau) \frac{d\tau}{\tau} \|t^{s+\delta}g(t)\|_2^2 \frac{dt}{t} \right)^{1/2} \\ & \le \|\gamma\|_{L^1(\mathbb{R}_+)} \|f\|_{L^2} \|g\|_{L^2_{-s-\delta}}, \end{aligned} $$

which implies (3.17). □

<figure><img src="image_12.png" /></figure>

For certain kernels $(S_{t,\tau})$, assuming an $L^2_s \to L^2_{s+\delta}$ estimate such as (3.17) and some off-diagonal estimates, we can prove boundedness of $S$ from $T^P$ to $T^{P+\delta}$ for some exponents $p$ with $i(p) \in (0, 1]$. This is a generalisation of an argument of the second author, McIntosh, and Russ [14] (see also [46]).

**Theorem 3.14 (Extrapolation of $L^2$ boundedness to tent spaces).** Let $\mathbf{p} = (p, s)$ be an exponent with $p \le 1$, let $\delta \in \mathbb{R}$, and let $(S_{t,\tau})_{t,\tau>0}$ be a continuous two-parameter family of bounded operators on $L^2$ such that for all $t_0, \tau_0 > 0$ the one-parameter families $(t^{-\delta}S_{t,\tau_0})_{t \in (\tau_0, \infty)}$ and $(\tau^{-\delta}S_{t_0,\tau})_{\tau \in (t_0, \infty)}$ both satisfy off-diagonal estimates of order $M$, with implicit constant $K$ uniform in $\tau_0$ and $t_0$ respectively. Suppose $a, b \in \mathbb{R}$, and let $\mathbf{S} \sim (m_a^b(t/\tau)S_{t,\tau})_{t,\tau>0}$.

If

$$ (3.18) \qquad \|\mathbf{S}f\|_{L^2_{s+\delta}} \lesssim \|f\|_{L^2_s} $$

holds for all $f \in L^2_c(\mathbb{R}_+ : L^2)$, with

$$ (3.19) \qquad -n\delta_{p,2} < b+s<M \quad \text{and} \quad a>s+\delta, $$

then

$$ \|\mathbf{S}f\|_{T^{P+\delta}} \lesssim \|f\|_{T^P} $$

for all $f \in L^2_c(\mathbb{R}_+ : L^2)$, and the implicit constant is a linear combination of $K$ and
$$ \|\mathbf{S}\| := \|\mathbf{S}\|_{L^2_s \to L^2_{s+\delta}}. $$

PROOF. **Step 1: an estimate for compactly supported atoms.** Suppose that $f$ is a compactly supported $T^P$-atom associated with a ball $B = B(c, r) \subset \mathbb{R}^n$ (see Definition 2.5). Then $f \in L^2_c$, and so $\mathbf{S}f$ is defined. We will show that $\mathbf{S}f$ is in $T^{P+\delta} = T_{s+\delta}^p$ with quasinorm bounded independently of $f$. To do this we will exhibit an atomic decomposition of $\mathbf{S}f$, and we will estimate $\|\mathbf{S}f\|_{T^{P+\delta}}$ using the coefficients of this decomposition.

Let $T_1 := T(4B)$ and $T_k := T(2^{k+1}B) \setminus T(2^k B)$ for all integers $k \ge 2$. Then define $F_k := \mathbf{1}_{T_k} \mathbf{S}f$ for all $k \in \mathbb{N}$, so that $\mathbf{S}f = \sum_{k=1}^\infty F_k$ almost everywhere. For each $k \in \mathbb{N}$ the function $F_k$ is supported in a tent, so we can renormalise by writing $F_k = \lambda_k f_k$ for some $\lambda_k \in \mathbb{C}$ and some $T^{P+\delta}$-atom $f_k$. We need only estimate the coefficients $\lambda_k$.
**Estimate for the local part.** For $k = 1$, since $F_1$ is supported in $T(4B)$, we must estimate $\|F_1\|_{L_{s+\delta}^2}$ in terms of $|4B|^{\delta_{p,2}}$. It follows from (3.18) and the fact that $f$ is a $T^p$-atom that

$$ \|F_1\|_{L_{s+\delta}^2} \le \|\mathbf{S}\| |B|^{\delta_{p,2}} \simeq_{n,p} \|\mathbf{S}\| |4B|^{\delta_{p,2}}, $$

and so we can set $\lambda_1 \simeq_{n,p} \|\mathbf{S}\|$.

**Estimate for the global parts.** Suppose $k \ge 2$. Since $F_k$ is supported in the tent $T(2^{k+1}B)$, we must estimate $\|F_k\|_{L_{s+\delta}^2}$ in terms of $|2^{k+1}B|^{\delta_{p,2}}$. We use Minkowski's integral inequality to estimate

$$ \begin{aligned} \|F_k\|_{L_{s+\delta}^2} &= \left( \int_0^{2^{k+1}r} \left\| t^{-(s+\delta)} \mathbf{1}_{B(c,2^{k+1}r-t)} \int_0^r m_a^b(t/\tau) S_{t,\tau} f(\tau) \frac{d\tau}{\tau} \right\|_2^2 \frac{dt}{t} \right)^{1/2} \\ &\le \left( \int_0^{2^{k+1}r} \left( \int_0^r t^{-(s+\delta)} m_a^b(t/\tau) \|\mathbf{1}_{A(c,2^{k}r-t,2^{k+1}r-t)} S_{t,\tau} f(\tau) \|_2 \frac{d\tau}{\tau} \right)^2 \frac{dt}{t} \right)^{1/2} \\ &\le \int_0^r \left( \int_0^{2^{k+1}r} t^{-2(s+\delta)} m_a^b(t/\tau)^2 \|\mathbf{1}_{A(c,2^{k}r-t,2^{k+1}r-t)} S_{t,\tau} f(\tau) \|_2^2 \frac{dt}{t} \right)^{1/2} \frac{d\tau}{\tau}, \end{aligned} $$

where

$$ A(c, 2^k r - t, 2^{k-1}r - t) = B(c, 2^{k-1}r - t) \setminus B(c, 2^k r - t) $$

as defined in Section 1.3. Note that $f(\tau)$ is supported in $B(c, r - \tau)$. We have

$$ \begin{aligned} d(\operatorname{supp} f(\tau), A(c, 2^k r - t, 2^{k+1} r - t)) &\ge d(B(c, r), \mathbb{R}^n \setminus B(c, 2^k r - t)) \\ &= ((2^k - 1)r - t)_+, \end{aligned} $$

so we split the region of integration $(0, 2^{k+1}r) \times (0, r)$ into three subregions,

$$ R_1 := \{(t, \tau) : t < \tau < r\} $$

$$ R_2 := \left\{ (t, \tau) : \tau < t < \frac{2^k - 1}{2} r \right\} $$

$$ R_3 := \left\{ (t, \tau) : t > \frac{2^k - 1}{2} r \right\}, $$

and denote the corresponding integrals by $\mathbf{I}_1$, $\mathbf{I}_2$, and $\mathbf{I}_3$.

On $R_3$, where $t > \tau$ and where there is no spatial separation, we have

$$ \begin{aligned} \mathbf{I}_3 &\lesssim K \int_0^r \left( \int_{\frac{2^k-1}{2}r}^{2^{k+1}r} t^{-2(s+\delta)} (t/\tau)^{-2b} t^{2\delta} \|f(\tau)\|_2^2 \frac{dt}{t} \right)^{1/2} \frac{d\tau}{\tau} \\ &\lesssim_{b,s} K \int_0^r \|\tau^{-s} f(\tau)\|_2 \tau^{b+s} (2^k r)^{-(b+s)} \frac{d\tau}{\tau} \\ &\le K 2^{-k(b+s)} \left( \int_0^r \|\tau^{-s} f(\tau)\|_2^2 \frac{d\tau}{\tau} \right)^{1/2} \left( \int_0^r (\tau/r)^{2(b+s)} \frac{d\tau}{\tau} \right)^{1/2} \\ &\approx_{b,s} K 2^{-k(b+s)} \|f\|_{L_s^2} \\ &\le K 2^{-k(b+s)} |B|^{\delta_{p,2}} \\ &= K 2^{-k(b+s)} |2^{k+1} B|^{\delta_{p,2}} \left( \frac{|2^{k+1} B|}{|B|} \right)^{-\delta_{p,2}} \end{aligned} \quad (3.20) $$
$$ \approx K 2^{-k(b+s+n\delta_{p,2})} |2^{k+1}B|^{\delta_{p,2}}, $$

where we used $b+s > -n\delta_{p,2} > 0$ in (3.20).

On $R_1$, where $t < \tau$ and where the off-diagonal estimates for $(\tau^{-\delta}S_{t,\tau})_{\tau \in (t,\infty)}$ involve spatial separation,

$$
\begin{aligned}
\mathbf{I}_1 & \lesssim K \int_0^r \left( \int_0^\tau t^{-2(s+\delta)} (t/\tau)^{2a} \left( \left( \frac{(2^k-1)r-t}{\tau} \right)^{-M} \|f(\tau)\|_2 \right)^2 \frac{dt}{t} \right)^{1/2} \tau^\delta \frac{d\tau}{\tau} \\
& \approx_{a,s,\delta} K 2^{-kM} \int_0^r \left( \frac{\tau}{r} \right)^M \|\tau^{-s} f(\tau)\|_2 \frac{d\tau}{\tau} \\
& \le K 2^{-kM} \|f\|_{L_s^2} \left( \int_0^r \left( \frac{\tau}{r} \right)^{2M} \frac{d\tau}{\tau} \right)^{1/2} \\
& \lesssim_M K 2^{-k(M+n\delta_{p,2})} |2^{k+1}B|^{\delta_{p,2}}
\end{aligned}
$$

using that $a > s + \delta$ in the second line, and then arguing as for $\mathbf{I}_3$.

On $R_2$, we have $t > \tau$ and the off-diagonal estimates for $(t^{-\delta}S_{t,\tau})_{t \in (\tau, \infty)}$ again involve spatial separation, and the restrictions on $t$ imply $(2^k - 1)r - t > \frac{2^k - 1}{2}r \gtrsim 2^k r$. Therefore

$$
\begin{aligned}
\mathbf{I}_2 & \lesssim K \int_0^r \left( \int_{\tau}^{\frac{2^k-1}{2}r} t^{-2s} (t/\tau)^{-2b} \left( \frac{(2^k-1)r-t}{t} \right)^{-2M} \|f(\tau)\|_2^2 \frac{dt}{t} \right)^{1/2} \frac{d\tau}{\tau} \\
& \lesssim K \int_0^r \|\tau^{-s} f(\tau)\|_2 \tau^{b+s} \left( \int_{\tau}^{\frac{2^k-1}{2}r} t^{-2(b+s)} \left( \frac{2^k r}{t} \right)^{-2M} \frac{dt}{t} \right)^{1/2} \frac{d\tau}{\tau} \\
& \lesssim_{b,s,M} K 2^{-kM} \int_0^r \|\tau^{-s} f(\tau)\|_2 \tau^{b+s} r^{-M} (2^k r)^{-(b+s-M)} \frac{d\tau}{\tau} \\
& \lesssim_{b,s} K 2^{-k(b+s+n\delta_{p,2})} |2^{k+1}B|^{\delta_{p,2}}
\end{aligned}
$$

using that $M > s + b$ and arguing as before.

Summing up, we have

$$
\begin{aligned}
\|F_k\|_{L_{s+\delta}^2} & \le \mathbf{I}_1 + \mathbf{I}_2 + \mathbf{I}_3 \\
& \lesssim_{a,b,M,p,s,\delta} K(2^{-k(M+n\delta_{p,2})} + 2^{-k(b+s+n\delta_{p,2})}) |2^{k+1}B|^{\delta_{p,2}},
\end{aligned}
$$

so for $k \ge 2$ we can set

$$ \lambda_k \approx K(2^{-k(M+n\delta_{p,2})} + 2^{-k(b+s+n\delta_{p,2})}). $$

This yields

$$
\begin{aligned}
\|(\lambda_k)\|_{\ell_p(\mathbb{N})}^p & \approx \|\mathbf{S}\|^p + K^p \sum_{k=2}^{\infty} (2^{-k(M+n\delta_{p,2})} + 2^{-k(b+s+n\delta_{p,2})})^p \\
& \le \|\mathbf{S}\|^p + K^p \sum_{k=2}^{\infty} 2^{-kp(M+n\delta_{p,2})} + 2^{-kp(b+s+n\delta_{p,2})},
\end{aligned}
$$

which is finite because of the assumption (3.19). The implicit constants do not depend on the atom $f$. Therefore $\mathbf{S}f$ is in $T_{s+\delta}^p$, with quasinorm bounded independently of $f$ and controlled by a linear combination of $\|\mathbf{S}\|$ and $K$.

**Step 2: from compactly supported atoms to $T_s^p \cap L_c^2(\mathbb{R}_+ : L^2)$**. This final part of the argument exactly follows [14, Proof of Theorem 4.9, Step 3]. One must show that every function in $T_s^p \cap L_c^2(\mathbb{R}_+ : L^2)$ may be decomposed into a sum
of compactly supported atoms, and that such decompositions converge in both $T_s^p$ (which is automatic) and in $L_s^2$. We omit further details. $\square$

### 3.4. Extension and contraction operators

Throughout this section we assume that $A$ satisfies the Standard Assumptions (see Definition 3.11).

**Definition 3.15.** For all $\psi \in H^\infty$ define the extension operator

$$Q_{\psi,A}: \overline{R(A)} \to L^\infty(\mathbb{R}_+ : L^2)$$

by $(Q_{\psi,A}f)(t) := \psi_t(A)f$ for $f \in \overline{R(A)}$ and $t \in \mathbb{R}_+$. If in addition $\psi \in \Psi_+^+$, then $Q_{\psi,A}$ is defined on all of $L^2$, and by Theorem 3.5 we have boundedness $Q_{\psi,A}: L^2 \to L^2(\mathbb{R}_+^{1+n}) = L^2(\mathbb{R}_+ : L^2)$ (recall that the measure $dt/t$ is always used on $\mathbb{R}_+$). For $\varphi \in \Psi_+^+$, this allows us to define the contraction operator

$$S_{\varphi,A} := (Q_{\tilde{\varphi},A^*})^*: L^2(\mathbb{R}_+^{1+n}) \to \overline{R(A)}.$$

Note that $Q_{\psi,A} = (S_{\tilde{\psi},A^*})^*$ when $\psi \in \Psi_+^+$.

A quick computation yields an integral representation of $S_{\varphi,A}$.

**Proposition 3.16.** Suppose $\varphi \in \Psi_+^+$. Then for all $f \in L_c^2(\mathbb{R}_+ : L^2)$ we have

$$(3.21) \qquad S_{\varphi,A}f := \int_0^\infty \varphi_t(A)f(t) \frac{dt}{t}.$$

The integral in (3.21) converges absolutely since $f \in L^1(\mathbb{R}_+ : L^2)$. However, if one only assumes $f \in L^2(\mathbb{R}_+ : L^2)$ (dropping the requirement of compact support in $\mathbb{R}_+$), then the integral converges weakly in $L^2$.

Compositions of the form $Q_{\psi,A}\eta(A)S_{\varphi,A}$, with $\eta$ a (possibly unbounded) holomorphic function, are important in what follows, so we begin investigation of their properties. Fix $\delta \in \mathbb{R}$, and suppose $\eta \in \Psi_{-\delta}^\delta$, $\psi \in H^\infty$, and $\varphi \in \Psi_\delta^{-\delta} \cap \Psi_+^+$. Then for all $f \in L_c^2(\mathbb{R}_+ : L^2)$ we have $S_{\varphi,A}f \in D(\eta(A))$ and the integral representation

$$(3.22) \qquad (Q_{\psi,A}\eta(A)S_{\varphi,A}f)(t) = \int_0^\infty (\psi_t(A)\eta(A)\varphi_\tau(A))f(\tau) \frac{d\tau}{\tau}.$$

Therefore we can write

$$Q_{\psi,A}\eta(A)S_{\varphi,A} \sim ((\psi_t\eta\varphi_\tau)(A))_{t,\tau>0}.$$

Our task now is to show when the results of Section 3.3 apply to these operators. We draw some conclusions even when $\eta$ is not bounded, provided that $\psi_t\eta\varphi_\tau \in \Psi_+^+$. This will ultimately lead to Theorem 4.14.

**Lemma 3.17.** Suppose $\sigma + \tau \ge 0$ and $\delta \in \mathbb{R}$. Let $\psi \in \Psi_\sigma^\tau$, $\varphi \in \Psi_{\tau+\delta}^{\sigma-\delta}$, and $\eta \in \Psi_{-\delta}^\delta$, and define the operator

$$(3.23) \qquad \tilde{S}_{t,r} := m_\sigma^{\tau+\delta}(t/r)^{-1}(\psi_t\eta\varphi_r)(A).$$

Then for all $t_0, r_0 > 0$ the operator families $(t^{-\delta}\tilde{S}_{t,r_0})_{t \in (r_0, \infty)}$ and $(r^{-\delta}\tilde{S}_{t_0,r})_{r \in (t_0, \infty)}$ satisfy off-diagonal estimates of order $\sigma + \tau$, uniformly in $r_0$ and $t_0$ respectively. The implicit constants in these off-diagonal estimates depend linearly on $\|\eta\|_{\Psi_{-\delta}^\delta}$.

The proof is a variation of [14, Lemma 3.7].
PROOF. If $t_0 \le r$ we can write

$$ 
\begin{aligned}
r^{-\delta} \tilde{S}_{t_0,r} &= r^{-\delta}(t_0/r)^{-\sigma} [\psi_{t_0}(z)\eta(z)\varphi_r(z)](A) \\
&= [(t_0z)^{-\sigma}\psi_{t_0}(z)\eta^\delta(z)(rz)^{\sigma-\delta}\varphi_\tau(z)](A)
\end{aligned}
$$

where $\eta^\delta \in H^\infty$ is defined by $\eta^\delta(z) := z^\delta \eta(z)$. Note that $\|\eta^\delta\|_\infty = \|\eta\|_{\Psi^\delta_{-\delta}}$. Since $\psi \in \Psi^\tau_\sigma$ and $\sigma + \tau \ge 0$, the function

$$ \gamma(t_0) : z \mapsto (t_0z)^{-\sigma} \psi_{t_0}(z) \eta^\delta(z) $$

is in $H^\infty$ with bound uniform in $t_0$, linear in $\|\eta\|_{\Psi^\delta_{-\delta}}$, and clearly independent of $r$. Furthermore, the function $\theta : z \mapsto z^{\sigma-\delta}\varphi(z)$ is in $\Psi^0_{\sigma+\tau}$, and so we can write

$$ r^{-\delta} \tilde{S}_{t_0,r} = \gamma(t_0)(A)\theta_\tau(A) $$

where $\gamma(t)$ is uniformly in $H^\infty$ and $\theta \in \Psi^0_{\sigma+\tau}$. Theorem 3.8 then implies that the family $(\tilde{S}_{t_0,r})_{r \in (t_0, \infty)}$ satisfies off-diagonal estimates of order $\sigma + \tau$ uniformly in $t_0 > 0$, with implicit constants linear in $\|\eta\|_\infty$.

Likewise, if $r_0 \le t$ we can write

$$ 
\begin{aligned}
t^{-\delta} \tilde{S}_{t,r_0} &= t^{-\delta}(t/r_0)^{\tau+\delta} [\psi_t(z)\eta(z)\varphi_{r_0}(z)](A) \\
&= [(r_0z)^{-(\tau+\delta)}\varphi_{r_0}(z)\eta^\delta(z)(tz)^\tau\psi_t(z)](A)
\end{aligned}
$$

and proceed in the same way, the consequence being that $(t^{-\delta} \tilde{S}_{t,r_0})_{t \in (r_0, \infty)}$ satisfies off-diagonal estimates of order $\sigma + \tau$ uniformly in $\tau_0 > 0$, with implicit constants linear in $\|\eta\|_{\Psi^\delta_{-\delta}}$. $\square$

**Lemma 3.18.** Fix $s, \delta \in \mathbb{R}$. Suppose $\psi \in \Psi_{(s+\delta)_+}^{-(s+\delta)_+}$, $\varphi \in \Psi_{-s+}^{s+}$, and $\eta \in \Psi_{-\delta}^\delta$. Then the operator $S \sim ((\psi_t\eta\varphi_r)(A))_{t,r>0}$ extends to a bounded operator $L_s^2 \to L_{s+\delta}^2$. Moreover, this extension is the weak limit (in $L_{s+\delta}^2$) of the integral (3.22).

PROOF. Fix $\varepsilon > 0$ such that $\psi \in \Psi_{\varepsilon+s}^{\varepsilon-(s+\delta)}$ and $\varphi \in \Psi_{-\varepsilon-s}^{\varepsilon+s}$. First note that $\psi_t\eta\varphi_r \in \Psi_+^+$, so the operators $S_{t,r} := (\psi_t\eta\varphi_r)(A)$ are all bounded and defined by the integral (3.5) on $L^2$. We will make use of Lemma 3.13, so we write $r = \kappa t$ and begin by estimating

$$ 
\begin{aligned}
\|r^{-\delta} S_{t,r}\|_{L(L^2(\mathbb{R}^n))} &\lesssim_{\psi,\varphi} (\kappa t)^{-\delta} \|\eta_{1/t}\|_{\Psi^\delta_{-\delta}} \int_0^\infty m_{\varepsilon+s}^{\varepsilon-s}(t\lambda) m_{\varepsilon-s}^{\varepsilon+s}(\kappa t\lambda) \frac{d\lambda}{\lambda} \\
(3.24) \qquad &\le \kappa^{-\delta} \|\eta\|_{\Psi^\delta_{-\delta}} \int_0^\infty m_{\varepsilon+s}^{\varepsilon-s}(\lambda) m_{\varepsilon-s}^{\varepsilon+s}(\kappa\lambda) \frac{d\lambda}{\lambda}
\end{aligned}
$$

using Lemma 3.3 to eliminate the powers of $t$ in (3.24). If $\kappa \le 1$, we have

$$ 
\begin{aligned}
&\kappa^{-\delta} \int_0^\infty m_{\varepsilon+s}^{\varepsilon-s}(\lambda) m_{\varepsilon-s}^{\varepsilon+s}(\kappa\lambda) \frac{d\lambda}{\lambda} \\
&= \kappa^{-\delta} \left( \kappa^{\varepsilon-s} \int_0^1 \lambda^{2\varepsilon} \frac{d\lambda}{\lambda} + \kappa^{\varepsilon-s} \int_1^{1/\kappa} \frac{d\lambda}{\lambda} + \kappa^{-\varepsilon-s} \int_{1/\kappa}^\infty \lambda^{-2\varepsilon} \frac{d\lambda}{\lambda} \right) \\
&\lesssim \kappa^{\varepsilon-s-\delta} (2 + \log(1/\kappa)).
\end{aligned}
$$

If $\kappa \ge 1$, then by the same argument we have

$$ \kappa^{-\delta} \int_0^\infty m_{\varepsilon+s}^{\varepsilon-s}(\lambda) m_{\varepsilon-s}^{\varepsilon+s}(\kappa\lambda) \frac{d\lambda}{\lambda} \lesssim \kappa^{-\varepsilon-s-\delta} (2 + \log(\kappa)). $$
Since the function

$$ \gamma(\kappa) := \begin{cases} \kappa^{\varepsilon}(2 + \log(1/\kappa)) & (\kappa \le 1) \\ \kappa^{-\varepsilon}(2 + \log(\kappa)) & (\kappa \ge 1) \end{cases} $$

is in $L^1(\mathbb{R}_+)$, Lemma 3.13 completes the proof of boundedness on the dense subspace $L_c^2(\mathbb{R}_+ : L^2)$. The extension to all of $L_s^2$ as a weak integral follows by the comment after Proposition 3.16. □

The following theorem is the basis of Chapter 4. From the viewpoint of applications, the most important part is the decay condition on $\psi$ at 0. We would particularly like to take $\psi = \text{sgp} \in \Psi_0^\infty$ and $\delta = 0$, which is possible provided that $i(\mathbf{p}) \le 2$ and $\theta(\mathbf{p}) < 0$.

**Theorem 3.19 (Boundedness of contraction/extension compositions).** Suppose $\mathbf{p}$ is an exponent, $\delta \in \mathbb{R}$ and $\eta \in \Psi_{-\delta}^\delta$. Suppose that either

* $i(\mathbf{p}) \le 2$ and

$$ (3.25) \quad \begin{array}{l} \psi \in \Psi_{(\theta(\mathbf{p})+\delta)+}^{(-\theta(\mathbf{p})+\delta)+n|\frac{1}{2}-j(\mathbf{p})|+} \cap H^\infty \quad \text{and} \quad \varphi \in \Psi_{(-\theta(\mathbf{p})+n|\frac{1}{2}-j(\mathbf{p})|+)}^{\theta(\mathbf{p})+} \cap \Psi_+^+, \\ \text{or} \end{array} $$

* $i(\mathbf{p}) \ge 2$ and

$$ \psi \in \Psi_{(\theta(\mathbf{p})+n|\frac{1}{2}-j(\mathbf{p})|+)}^{(-\theta(\mathbf{p}))+} \cap \Psi_+^+ \quad \text{and} \quad \varphi \in \Psi_{(-\theta(\mathbf{p})+\delta)+}^{(\theta(\mathbf{p})-\delta+n|\frac{1}{2}-j(\mathbf{p})|+)} \cap \Psi_+^+. $$

then $\mathbb{Q}_{\psi,A}\eta(A)\mathbb{S}_{\varphi,A}$ extends to a bounded operator $X^{\mathbf{P}} \to X^{\mathbf{P}+\delta}$ (by density when $i(\mathbf{p}) \le 2$ and by duality when $i(\mathbf{p}) > 2$), with bounds linear in $\|\eta\|_{\Psi_{-\delta}^\delta}$.

**PROOF.** We only prove the result for tent spaces. The Z-space result can be deduced by real interpolation, or alternatively by the dyadic characterisation of Proposition 2.20. Furthermore, the result for $i(\mathbf{p}) \ge 2$ follows from that for $i(\mathbf{p}) \le 2$ by duality, so we need only consider $i(\mathbf{p}) \le 2$. Note that (3.22) and the assumptions on $\psi$ and $\varphi$ imply that $\mathbb{Q}_{\psi,A}\eta(A)\mathbb{S}_{\varphi,A}$ contains the integral operator with kernel $((\psi_t\eta\varphi_\tau)(A))_{t,\tau>0}$, so it suffices to work with this operator. Furthermore, the assumptions (3.25) and Lemma 3.18 imply that this operator (extended as a weak limit) is bounded from $T_{\theta(\mathbf{p})}^2$ to $T_{\theta(\mathbf{p})+\delta}^2$, which yields the result when $i(\mathbf{p}) = 2$.

**Step 1:** $i(\mathbf{p}) \le 1$. The assumptions (3.25) imply that there exists $\varepsilon > 0$ such that

$$ \psi \in \Psi_{\sigma+\varepsilon}^{\tau+\varepsilon} \quad \text{and} \quad \varphi \in \Psi_{(\tau+\varepsilon)+\delta}^{(\sigma+\varepsilon)-\delta}, $$

where $\sigma := \theta(\mathbf{p})+\delta$ and $\tau := -\theta(\mathbf{p}+\delta)+n|(1/2)-j(\mathbf{p})|$. Therefore by Lemma 3.17, the operator families $(t^{-\delta}\tilde{S}_{t,r_0})_{t \in (r_0, \infty)}$ and $(r^{-\delta}\tilde{S}_{t_0,r})_{r \in (t_0, \infty)}$, where $\tilde{S}_{t,r}$ is defined as in (3.23), satisfy off-diagonal estimates of order $n|(1/2) - j(\mathbf{p})| + 2\varepsilon$. Theorem 3.14 then applies with $a = \sigma + \varepsilon$, $b = \tau + \varepsilon + \delta$, and $M = 2\varepsilon + n|(1/2) - j(\mathbf{p})|$, and we can conclude that $\mathbb{Q}_{\psi,A}\eta(A)\mathbb{S}_{\varphi,A}$ is bounded from $X^{\mathbf{P}}$ to $X^{\mathbf{P}+\delta}$.

**Step 2:** $i(\mathbf{p}) \in (1, 2)$. The following argument originates from Stahlhut's thesis [71, Proof of Lemma 3.2.6, Step 4]. For $\lambda \in \mathbb{C}$, define functions $\psi^\lambda$ and $\varphi^\lambda$ by

$$ \psi^\lambda(z) := \left( \frac{[z]}{1+[z]} \right)^\lambda \psi(z), \quad \varphi^\lambda(z) := \left( \frac{1}{1+[z]} \right)^\lambda \varphi(z). $$

If $\text{Re} \lambda \ge n(1 - j(\mathbf{p}))$, then Step 1 applies with exponent $(1, \theta(\mathbf{p}))$, and we find that $\mathbb{Q}_{\psi^\lambda, A}\eta(A)\mathbb{S}_{\varphi^\lambda, A}$ is bounded from $T_{\theta(\mathbf{p})}^1$ to $T_{\theta(\mathbf{p})+\delta}^1$. Furthermore, if $\text{Re} \lambda \ge$
$-n|(1/2) - j(\mathbf{p})|$, then the discussion of the first paragraph of the proof applies, and we find that $\mathbb{Q}_{\psi^{\lambda},A}\eta(A)\mathbb{S}_{\varphi^{\lambda},A}$ is bounded from $T_{\theta(\mathbf{p})}^2$ to $T_{\theta(\mathbf{p})+\delta}^2$. By Stein interpolation in tent spaces (see [11, Proof of Lemma 3.4]), when $\mathrm{Re}\,\lambda = 0$, we have that $\mathbb{Q}_{\psi^{\lambda},A}\eta(A)\mathbb{S}_{\varphi^{\lambda},A}$ is bounded from $T_{\theta(\mathbf{p})}^p$ to $T_{\theta(\mathbf{p})+\delta}^p$ when $p \in (1, 2)$ and $\theta \in (0, 1)$ satisfy

$$ \frac{1}{p} = (1 - \theta) + \frac{\theta}{2}, \quad 0 = (1 - \theta)(1 - j(\mathbf{p})) + \theta\left(\frac{1}{2} - j(\mathbf{p})\right). $$

This occurs when $p = i(\mathbf{p})$. Applying this with $\lambda = 0$ yields boundedness of $\mathbb{Q}_{\psi,A}\eta(A)\mathbb{S}_{\varphi,A}$ from $T^p$ to $T^{p+\delta}$. $\square$

Finally, we present an abstract Calderón reproducing formula, which is ubiquitous in the study of abstract Hardy spaces. Whenever $\psi \in \Psi_+^+$ and $\varphi \in H^\infty$, we can define a bounded holomorphic function

$$ \Phi_{\psi,\varphi}(z) := \int_{0}^{\infty} \psi_{t}(z)\varphi_{t}(z) \frac{dt}{t}, \quad z \in S_{\mu}. $$

The integral converges absolutely because $\psi\varphi \in \Psi_+^+$. It is not hard to show that

$$ \mathbb{S}_{\psi,A}\mathbb{Q}_{\varphi,A} \subseteq \Phi_{\psi,\varphi}(A) $$

as operators on $\overline{\mathbb{R}}(A)$, with equality when $\varphi \in \Psi_+^+$.

In [19, Proposition 4.2] it is shown that if $\varphi \in H^\infty$ is nondegenerate, then there exists $\psi \in \Psi_\infty^+$ such that $\Phi_{\varphi,\psi} \equiv 1$. This implies the following abstract Calderón reproducing formula.

**Theorem 3.20.** Suppose $\varphi \in H^\infty$ is nondegenerate. Then there exists $\psi \in \Psi_\infty^+$ such that

$$ (3.26) \qquad \mathbb{S}_{\psi,A}\mathbb{Q}_{\varphi,A} \subseteq I_{\overline{\mathbb{R}}(A)} $$

as operators on $\overline{\mathbb{R}}(A)$, with equality when $\varphi \in \Psi_+^+$. Furthermore, if $\varphi \in \Psi_+^+$, then the operator $\mathbb{Q}_{\psi,A}\mathbb{S}_{\varphi,A}$ is a projection from $L^2(\mathbb{R}_+^{1+n})$ onto $\mathbb{Q}_{\psi,A}\overline{\mathbb{R}}(A)$.

We refer to a pair $(\varphi, \psi)$, with $\varphi \in H^\infty$, $\psi \in \Psi_+^+$ and satisfying (3.26), as *Calderón siblings*. Here is a simple example of the use of the abstract Calderón reproducing formula.

**Corollary 3.21.** Suppose $\varphi \in H^\infty$ is nondegenerate. Then the extension operator $\mathbb{Q}_{\varphi,A}: \overline{\mathbb{R}}(A) \to L^\infty(\mathbb{R}_+ : \overline{\mathbb{R}}(A))$ is injective.

**PROOF.** Let $\psi \in \Psi_+^+$ be a Calderón sibling of $\varphi$, and suppose $f \in \overline{\mathbb{R}}(A)$ with $\mathbb{Q}_{\varphi,A}f = 0$. Then $\mathbb{Q}_{\varphi,A}f \in L^2(\mathbb{R}_+ : \overline{\mathbb{R}}(A))$, so $\mathbb{S}_{\psi,A}\mathbb{Q}_{\varphi,A}f$ is defined, and by (3.26) we have

$$ f = \mathbb{S}_{\psi,A}\mathbb{Q}_{\varphi,A}f = 0, $$

proving injectivity of $\mathbb{Q}_{\varphi,A}$. $\square$
# Adapted Besov-Hardy-Sobolev Spaces

Throughout this chapter we fix an operator $A$ satisfying the Standard Assumptions (see Definition 3.11). As in the previous chapter, we implicitly work with $C^N$-valued functions without referencing this in the notation.

## 4.1. Initial definitions, equivalent norms, and duality

The adapted Hardy-Sobolev and Besov spaces are defined, roughly speaking, by measuring extensions by $Q_{\psi,A}$ in tent spaces and $Z$-spaces respectively. This mimics the characterisation of classical Hardy-Sobolev and Besov spaces given in Theorem 2.57. We will soon show that the resulting function spaces are independent of $\psi$, provided that $\psi$ has sufficient decay at 0 and $\infty$.

**Definition 4.1.** Let $\psi \in H^\infty$ and let $p$ be an exponent. We define the sets

$$ \mathbb{H}_{\psi,A}^{p} := \{f \in \overline{\mathbb{R}(A)} : Q_{\psi,A}f \in T^{p} \cap L^{2}(\mathbb{R}_{+}^{1+n})\}, $$

$$ \mathbb{B}_{\psi,A}^{p} := \{f \in \overline{\mathbb{R}(A)} : Q_{\psi,A}f \in Z^{p} \cap L^{2}(\mathbb{R}_{+}^{1+n})\}, $$

equipped with what we will show to be quasinorms (Proposition 4.3)

$$ \|f\|_{\mathbb{H}_{\psi,A}^{p}} := \|Q_{\psi,A}f\|_{T^{p}}, \quad \|f\|_{\mathbb{B}_{\psi,A}^{p}} := \|Q_{\psi,A}f\|_{Z^{p}}. $$

We call these spaces pre-Hardy-Sobolev and pre-Besov spaces adapted to $A$ (respectively), and we call $\psi$ an auxiliary function.

Generally we want to refer to the pre-Hardy-Sobolev and pre-Besov spaces simultaneously. Thus we write

$$ \mathbb{X}_{\psi,A}^{p} := \{f \in \overline{\mathbb{R}(A)} : Q_{\psi,A}f \in X^{p} \cap L^{2}(\mathbb{R}_{+}^{1+n})\}, $$

where the pair $(X, \mathbb{X})$ is either $(T, \mathbb{H})$ or $(Z, \mathbb{B})$, following the convention initiated in Section 2.4, and speak of pre-Besov-Hardy-Sobolev spaces,

**Remark 4.2.** The condition $Q_{\psi,A}f \in L^2(\mathbb{R}_+^{1+n})$ is automatically satisfied when $\psi \in \Psi_+^+$. We impose it so that $S_{\varphi,A}Q_{\psi,A}f$ is defined for all $f$ in $\mathbb{X}_{\psi,A}^p$ when $\psi \in H^\infty \setminus \Psi_+^+$.

**Proposition 4.3.** Let $\psi \in H^\infty$ and let $p$ be an exponent. Then $\|\cdot\|_{\mathbb{X}_{\psi,A}^p}$ is a quasinorm on $\mathbb{X}_{\psi,A}^p$.

**PROOF.** The only quasinorm property which does not follow directly from linearity of $Q_{\psi,A}$ and the corresponding quasinorm properties of $X^p$ is positive definiteness. To show this, suppose $f \in \mathbb{X}_{\psi,A}^p$ and $\|f\|_{\mathbb{X}_{\psi,A}^p} = 0$. Then $Q_{\psi,A}f = 0$ in $X^p$, and hence also in $L^\infty(\mathbb{R}_+ : \overline{\mathbb{R}(A)})$. By injectivity of $Q_{\psi,A}: \overline{\mathbb{R}(A)} \to L^\infty(\mathbb{R}_+ : \overline{\mathbb{R}(A)})$ (Corollary 3.21), we conclude that $f = 0$. $\square$