# Meyers exponent rules the first-order approach to second-order elliptic boundary value problems

Pascal Auscher, Tim Böhnlein, Moritz Egert

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Pascal Auscher, Tim Böhnlein, Moritz Egert. Meyers exponent rules the first-order approach to second-order elliptic boundary value problems. 2025. hal-05012448

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Distributed under a Creative Commons Attribution 4.0 International License
# MEYERS EXPONENT RULES THE FIRST-ORDER APPROACH TO SECOND-ORDER ELLIPTIC BOUNDARY VALUE PROBLEMS

PASCAL AUSCHER, TIM BÖHNLEIN, AND MORITZ EGERT

**ABSTRACT.** The first-order approach to boundary value problems for second-order elliptic equations in divergence form with transversally independent complex coefficients in the upper half-space rewrites the equation algebraically as a first-order system, much like how harmonic functions in the plane relate to the Cauchy–Riemann system in complex analysis. It hinges on global $L^p$-bounds for some $p > 2$ for the resolvent of a perturbed Dirac-type operator acting on the boundary. At the same time, gradients of local weak solutions to such equations exhibit higher integrability for some $p > 2$, expressed in terms of weak reverse Hölder estimates. We show that the optimal exponents for both properties coincide. Our proof relies on a simple but seemingly overlooked connection with operator-valued Fourier multipliers in the tangential direction.

## 1. INTRODUCTION

We work in Euclidean space $\mathbb{R}^{1+n} := \{(t, x) : t \in \mathbb{R}, x \in \mathbb{R}^n\}$ and consider pure second-order operators in divergence form

$$ (1.1) \qquad \mathcal{L}U := -\operatorname{div}_{t,x}(A\nabla_{t,x}U) $$

with $t$-independent coefficients $A = A(x)$ that are bounded, complex-valued, measurable and strongly elliptic (see Section 3 for precise definitions). Historically, these operators arise as pullbacks of the Laplacian from the domain above a Lipschitz graph. Since the pioneering work of Dahlberg [15], harmonic analysts have shown a particular interest in a priori estimates, uniqueness, and solvability of boundary value problems for the equation $\mathcal{L}U = 0$ in the upper half-space $\mathbb{R}_+^{1+n}$ with prescribed data of Dirichlet- and Neumann-type in $L^p$-spaces.

When the coefficients of $\mathcal{L}$ are real, such questions can efficiently be studied through the associated elliptic measure and layer potentials [20, 21]. These tools are not available for complex coefficients since they rely on the maximum principle and pointwise estimates of weak solutions in style of the celebrated DeGiorgi–Nash–Moser theory.

An alternative measure of regularity for weak solutions, independent of whether the coefficients are real or complex, is the self-improving property of weak reverse Hölder estimates for the gradient of local $\mathcal{L}$-harmonic functions, first established by Meyers [25]. This property states that there exists $p > 2$ and a constant $C \ge 0$ (both depending only on ellipticity

<footnote>Date: March 31, 2025.</footnote>

<footnote n="2020">2020 Mathematics Subject Classification. Primary: 35J25, 47F10. Secondary: 35B65, 46E35.</footnote>

<footnote>Key words and phrases. Elliptic boundary value problems, a priori estimates, Meyers exponent, reverse Hölder estimates, Hodge projector, Cauchy–Riemann systems, operator-valued Fourier multipliers.</footnote>
and dimension) such that for every axes-parallel cube $Q \subseteq \mathbb{R}^{1+n}$ and every weak solution to $\mathcal{L}U = 0$ in $2Q$ we have the weak reverse Hölder estimate

$$ \left( \int_Q |\nabla_{t,x} U|^p d(t,x) \right)^{\frac{1}{p}} \le C \left( \int_{2Q} |\nabla_{t,x} U|^2 d(t,x) \right)^{\frac{1}{2}}. $$

We denote the supremum over all such $p > 2$ by $m_+(\mathcal{L})$ and refer to it as the **Meyers exponent** of $\mathcal{L}$. The original proof relies on real-variable methods, whose underlying ideas originate from Gehring's self-improvement property for conformal mappings [18]. Notably, it implies that weak solutions are Hölder continuous when $n = 1$. However, no direct link between the Meyers exponent and a priori estimates or solvability of boundary value problems is known so far.

In order to treat boundary value problems for operators with complex coefficients, the first author, together with Rosén (Axelsson) and McIntosh, introduced a ‘first-order approach’ in [3], which rewrites the equation $\mathcal{L}U = 0$ algebraically as a first-order system $\partial_t F + DBF = 0$, much like how harmonic functions in the plane relate to the Cauchy–Riemann system in complex analysis. Here, $D$ is a Dirac operator and the perturbation $B$ arises from the coefficients $A$ through an explicit, yet intricate transformation on elliptic matrices. This approach applies to all equations as above and leads to remarkable results that all, in one way or the other, impose an $L^p$-type topology on the data and the solution $F$ when $p$ belongs to a specific range of exponents $[2, p_+(DB))$. Let us mention the explicit construction of solutions through a $DB$-semigroup [1, 9], representation and uniqueness of solutions to boundary value problems [6, 7], bounds for layer potentials beyond singular integral operators [6, 26], well-posedness of Neumann problems for equations in block form [5, 7], and identification of Hardy spaces [1, 5, 9]. The limiting exponent $p_+(DB)$ was introduced in [9] as the upper endpoint of the maximal interval of exponents $p$ around 2 for which $DB$ is bisectorial in $L^p(\mathbb{R}^n)$, that is, $DB$ satisfies a specific type of resolvent estimates along the imaginary axis.

A natural and frequently asked question is how this exponent $p_+(DB)$ relates to the underlying second-order equation. Our main result offers a concise answer.

**Theorem 1.1.** *The exponent $p_+(DB)$ from the first-order approach and the Meyers exponent $m_+(\mathcal{L})$ coincide.*

We highlight again that the two exponents have fundamentally different natures. The Meyers exponent $m_+(\mathcal{L})$ primarily captures local properties of weak solutions in the interior, uniformly across all scales, whereas $p_+(DB)$ relates to a global estimate at the boundary.

Theorem 1.1 confirms that results obtained via self-improvement properties for second-order equations and extrapolation in the first-order approach are compatible in terms of admissible exponents. In particular, this shows that $p_+(DB)$ is not merely an artifact of the first-order approach, which constructs specific global solutions; rather, it genuinely reflects interior behavior of all local weak solutions in the second-order setting. This may be surprising.
1.1. **Principal ideas for the proof of Theorem 1.1.** We successively replace the weak reverse Hölder estimates with other $L^p$-estimates that hold for the same range of exponents $p$, with the possible exception of the endpoints.

The **first idea** is to 'globalize' by replacing the set of reverse Hölder estimates (one for each cube) with a global estimate for one single operator.

Multiple choices for the global operator are possible. A particularly easy proof can be given for the Hodge projector $\nabla_{t,x} \mathcal{L}^{-1} \text{div}_{t,x}$ by using an extrapolation technique due to Shen [27]. The general theory of $L^p$-families associated with divergence-form operators from the textbook [5] allows us to replace the Hodge projector further with $s \nabla_{t,x}(1 + s^2 \mathcal{L})^{-N}$ for some large $N$, as long as the $L^p$-bound remains uniform with respect to $s > 0$. These higher-order resolvents offer decay properties that are needed for technical reasons only.

The **second idea** is to use a seemingly overlooked connection with operator-valued Fourier multipliers through the partial Fourier transform in the $t$-variable already at the level of second-order theory, hence not involving the first-order approach.

Indeed, partial Fourier transform turns the homogeneous gradient $\nabla_{t,x}$ in $\mathbb{R}^{1+n}$ into a family $S_\tau := [i\tau, \nabla_x]^\top$ of $\tau$-dependent inhomogeneous gradients in $\mathbb{R}^n$. Since the coefficients of $\mathcal{L}$ are independent of $t$, this operator corresponds to the family $L_\tau = S_\tau^* A S_\tau$ of inhomogeneous second-order operators. Hence, the global operator from above can be written as an operator-valued Fourier multiplier

$$s \nabla_{t,x} (1 + s^2 \mathcal{L})^{-N} = \mathcal{F}_t^{-1} (s S_\tau (1 + s^2 L_\tau)^{-N}) \mathcal{F}_t.$$

The operators $L_\tau$ previously appeared in a second-order reformulation of $L^p$-bisectoriality of $DB$ in [9], and it turns out that the pieces fit together beautifully: The condition on $L_\tau$ that arises from $L^p$-bisectoriality of $DB$ is exactly the one that is needed to verify that the Fourier symbol of $s \nabla_{t,x} (1 + s^2 \mathcal{L})^{-N}$ is an $\mathcal{R}$-Mihlin symbol with values in the bounded operators on $L^p(\mathbb{R}^n)$. Thus, this operator is bounded on $L^p(\mathbb{R}; L^p(\mathbb{R}^n)) \cong L^p(\mathbb{R}^{1+n})$ by virtue of Weis' operator-valued multiplier theorem [30].

The above line of reasoning shows that $p_+(DB)$ is at least as large as $m_+(\mathcal{L})$. For the converse, we reverse the 'globalizing' step in one dimension lower and replace the estimates from $L^p$-bisectoriality of $DB$ with weak reverse Hölder estimates for the gradient of solutions to $L_\tau u = 0$. Care is required, however, as we need to develop the theory uniformly in the Fourier variable $\tau$. It then remains to show that the Meyers exponents for the operators $L_\tau$ cannot exceed that of $\mathcal{L}$. This follows immediately from the observation that if $u$ is a weak solution to $L_\tau u = 0$, then $U(t,x) = e^{i\tau t} u(x)$ defines a solution to $\mathcal{L}U = 0$ in one dimension higher.

1.2. **Structure of the paper and additional findings.** Figure 1 comprises the structure of the proof of our main result. The full equivalence reveals some further characterizations of $p_+(DB)$ that have consequences for the operators $DB$ and $\mathcal{L}$ beyond the scope of Theorem 1.1. They mostly involve further exponents linked to the operators $L_\tau$ which are interesting on their own. We summarize these additional findings in Theorem 10.2 at the end of our paper.
<figure><img src="image_2.png" /><figcaption>FIGURE 1. Roadmap to Theorem 1.1. The exponent $p_+(DB)$ coincides with the Meyers exponent for $\mathcal{L}$.</figcaption></figure>

Sections 2–6 contain preliminary material. The proof of Theorem 1.1 is presented in Sections 7–10. For reasons of homogeneity related to Sobolev embeddings, our argument applies in boundary dimensions $n \ge 2$. In the case $n = 1$, it is well-known from [9] that $p_+(DB) = \infty$ and from [10, App. B] that $m_+(\mathcal{L}) = \infty$. We provide a direct argument for $m_+(\mathcal{L}) = \infty$ in Section 11.

1.3. **Acknowledgment.** The second and third authors acknowledge the support of the CNRS and the Laboratoire de Mathématiques d'Orsay, where this project was partly carried out during two research stays in September 2023 and October 2024. The authors are also grateful to Sebastian Bechtel for highlighting the square function techniques in [13, 24]. A CC-BY 4.0 https://creativecommons.org/licenses/by/4.0/ public copyright license has been applied by the authors to the present document and will be applied to all subsequent versions up to the Author Accepted Manuscript arising from this submission.

## 2. NOTATION

Most of our notation is standard and we make use of the following additional conven-
tions:

* We abbreviate $\mathbb{R}^* := \mathbb{R} \setminus \{0\}$.

* We write $Q(x, r)$ for the open, axes-parallel cube with center $x$ and length $2r$. Likewise, $B(x, r)$ denotes the ball centered at $x$ with radius $r$.

* For suitable exponents $p$, we define conjugate indices $p' := p/(p-1)$ (Hölder), $p^* := pn/(n-p)$ (upper Sobolev) and $p_* := pn/(n+p)$ (lower Sobolev), where the ambient dimension $n$ will be clear from the context.

* We write $X^*$ for the Banach space of bounded and anti-linear functionals from $X$ to $C$.

* Our one-dimensional Fourier transform is

$$ (\mathcal{F}_t f)(\tau) := \frac{1}{\sqrt{2\pi}} \int_{\mathbb{R}} f(t) e^{-it\cdot\tau} \, dt. $$

* For dilations of functions we write $(\delta_t f)(x) := f(tx)$.
* We use the notation $(f)_E := \int_E f \, dx := \frac{1}{|E|} \int_E f \, dx$ for averages.

* We use $\lesssim$ and $\gtrsim$ to denote inequalities that hold up to multiplicative constants independent of the relevant quantities.

### 3. FUNCTION SPACES AND ELLIPTIC OPERATORS

For purely second-order operators, we typically use homogeneous Sobolev spaces.

**Definition 3.1.** For $p \in (1, \infty)$, we define $\dot{W}^{1,p}(\mathbb{R}^n)$ as the space of all $L_{\text{loc}}^p$-functions $u$ for which $\nabla_x u$ belongs to $L^p(\mathbb{R}^n)^n$ modulo $\mathbb{C}$. We endow this space with the norm

$$ \|u\|_{\dot{W}^{1,p}} := \|\nabla_x u\|_p. $$

We also write $\dot{W}^{-1,p}(\mathbb{R}^n) := \dot{W}^{1,p'}(\mathbb{R}^n)^*.$

The same notation will be used in dimension $n+1$, where we write $\nabla_{t,x}$ for the gradient. Throughout this article, we work under the following assumption on the coefficients.

**Assumption 3.2 (Ellipticity).** The coefficients $A: \mathbb{R}^n \to L^\infty(\mathbb{R}^n; \mathbb{C}^{(1+n)\times(1+n)})$ are strongly elliptic in the sense that there exists $\lambda > 0$ such that

$$ \mathrm{Re}\langle A(x)\xi, \xi \rangle \geq \lambda |\xi|^2 \quad (x \in \mathbb{R}^n, \xi \in \mathbb{C}^{1+n}). $$

On the (partial) Fourier side, we work with inhomogeneous operators where $\nabla_{t,x}$ is replaced by inhomogeneous gradients that scale with the Fourier variable. We introduce the natural Sobolev spaces that capture the scaling.

**Definition 3.3.** Let $\tau \in \mathbb{R}$ and $p \in (1, \infty)$.

(i) We write $S_\tau := [i\tau, \nabla_x]^\top$ and $S_\tau^* := [-i\tau, -\mathrm{div}_x]$.

(ii) The space $W_\tau^{1,p}(\mathbb{R}^n)$ is the usual inhomogeneous Sobolev space but with the $\tau$-adapted norm

$$ \|u\|_{W_\tau^{1,p}} := \|S_\tau u\|_p $$

and $W_\tau^{-1,p}(\mathbb{R}^n) := W_\tau^{1,p'}(\mathbb{R}^n)^*$ is its dual.

**Remark 3.4.** When $\tau = 0$, the space $W_\tau^{1,p}(\mathbb{R}^n)$ is understood as $\dot{W}^{1,p}(\mathbb{R}^n)$ and $S_\tau$ as the usual gradient $\nabla_x$.

Let $\tau \in \mathbb{R}$. In view of the Lax-Milgram lemma, we can define the isomorphism

$$ L_\tau := S_\tau^* A S_\tau : W_\tau^{1,2}(\mathbb{R}^n) \to W_\tau^{-1,2}(\mathbb{R}^n), $$

$$ \langle L_\tau u, v \rangle := \int_{\mathbb{R}^n} A S_\tau u \cdot \overline{S_\tau v} \, dx. $$

The ellipticity assumption ensures that the maximal restriction of $L_\tau$ to $L^2(\mathbb{R}^n)$ is accretive and its negative generates a holomorphic $C_0$-semigroup of contractions in $L^2(\mathbb{R}^n)$. Background material can be found e.g. in [23]. For simplicity, we do not distinguish notationally between $L_\tau$ and this negative generator.
Writing the matrices $A(x)$ according to the $(t, x)$-notation as

$$ (3.1) \qquad A := \begin{bmatrix} a & b \\ c & d \end{bmatrix}, $$

we see that $L_\tau$ is the weak interpretation of the divergence-form operator with lower-order terms

$$ (3.2) \qquad L_{\tau}u = -\operatorname{div}_{x}(d\nabla_{x}u) - i\tau \operatorname{div}_{x}(cu) - i\tau b \nabla_{x}u + \tau^{2}au. $$

The main elliptic operator $\mathcal{L}$ in (1.1) has an identical weak interpretation with $\tau = 0$ in one dimension higher. In particular, it is also m-accretive.

### 4. TOOLS FROM $L^p - L^q$ OFF-DIAGONAL THEORY

We review some abstract theory on off-diagonal estimates for uniformly bounded families $\mathcal{T} = (T(t))_{t \in I} \subseteq \mathcal{L}(L^2)$ defined on some set $I \subseteq \mathbb{R}^*$. For us, $\mathcal{T}$ will be a semigroup or resolvent family associated with a differential operator or a singleton. Most of the material is taken from [5].

**Definition 4.1.** Let $1 \le p \le q \le \infty$. We say that $\mathcal{T}$

• is $L^p - L^q$-bounded if there is $C > 0$ such that

$$ \|T(t)f\|_q \le C|t|^{\frac{n}{q}-\frac{n}{p}}\|f\|_p $$

for all $t \in I$ and $f \in L^p \cap L^2$.

• satisfies $L^p - L^q$ off-diagonal estimates if there are $C, c > 0$ such that

$$ \|\mathbf{1}_F T(t) \mathbf{1}_E f\|_q \le C|t|^{\frac{n}{q}-\frac{n}{p}} e^{-c \frac{d(E,F)}{|t|}} \|\mathbf{1}_E f\|_p $$

for all $t \in I$, measurable sets $E, F \subseteq \mathbb{R}^n$ and $f \in L^p \cap L^2$.

For $p = q$, we speak of $L^p$-boundedness and $L^p$ off-diagonal estimates, respectively.

These notions interpolate as expected and we cite the general principle in the form that we need later on.

**Lemma 4.2.** Let $1 \le p < r \le \sigma < q \le \infty$. Suppose that $\mathcal{T}$ satisfies $L^2$ off-diagonal estimates and that it is $L^p - L^q$-bounded. Then $\mathcal{T}$

(i) satisfies $L^r - L^\sigma$ off-diagonal estimates and

(ii) is $L^r$-bounded.

**Proof.** Upon modifying $p$ and $q$ slightly, but preserving the relation with $r$ and $\sigma$, we can assume that $\mathcal{T}$ satisfies $L^p - L^q$ off-diagonal estimates, see the interpolation principle in [5, Lem. 4.14]. This implies $L^p$ and $L^q$ off-diagonal estimates [5, Rem. 4.8] and (i) follows again by interpolation. Finally, (i) implies (ii) by taking $\sigma = r$ and $E = F = \mathbb{R}^n$ in the definition of off-diagonal estimates. ■

There is also a useful bootstrapping mechanism [5, Lem. 4.4]. (The reference uses $q = 2$ as reference point but this does not have an impact on the argument.)
**Lemma 4.3.** Let $1 \le p < r \le \sigma < q \le \infty$ and suppose that $\mathcal{T}$ is $L^q$-, $L^p$- and $L^\sigma - L^q$-bounded. Then there exists some $k \in \mathbb{N}$ such that $\mathcal{T}^k = (T(t)^k)_{t \in I}$ is $L^r - L^q$-bounded.

We remark that implicit constants in both results depend only on $p, q, r, \sigma$ and the constants in the assumption [5].

### 5. $\tau$-DEPENDENT HODGE PROJECTORS

We begin with the theory outlined in Figure 1 and introduce a family of $\tau$-dependent Hodge projectors. Again, this comprises the corresponding theory for $\mathcal{L}$ because this operator is of the same type as $L_0$ in one dimension higher.

**Definition 5.1.** Let $\tau \in \mathbb{R}$. The operator

$$S_{\tau} L_{\tau}^{-1} S_{\tau}^{*} = \begin{bmatrix} \tau^2 L_{\tau}^{-1} & -i\tau L_{\tau}^{-1} \operatorname{div}_x \\ -i\tau \nabla_x L_{\tau}^{-1} & -\nabla_x L_{\tau}^{-1} \operatorname{div}_x \end{bmatrix}$$

is called **Hodge projector** associated with $L_\tau$.

By the Lax–Milgram lemma, we have for all $u \in W_{\tau}^{1,2}$ and all $\tau \in \mathbb{R}$ that

$$\|L_{\tau}^{-1} u\|_{W_{\tau}^{1,2}} \le \lambda^{-1} \|u\|_{W_{\tau}^{-1,2}}.$$

Since $S_\tau : W_{\tau}^{1,2} \to L^2$ is isometric, the Hodge projector is bounded on $L^2$ with norm $\lambda^{-1}$ independently of $\tau$. Its $L^p$-boundedness can be characterized as follows.

**Proposition 5.2.** Let $\tau \in \mathbb{R}$ and $p \in (1, \infty)$. The following are equivalent:

(i) $S_\tau L_\tau^{-1} S_\tau^*$ is $L^p$-bounded.

(ii) There is a constant $C > 0$ such that

$$\|L_{\tau}^{-1} u\|_{W_{\tau}^{1,p}} \le C \|u\|_{W_{\tau}^{-1,p}} \quad (u \in W_{\tau}^{-1,p} \cap W_{\tau}^{-1,2}).$$

In this case, the bound in (i) and the constant in (ii) can be taken the same up to a factor depending only on $p$ and $n$.

**Proof.** The implication (ii) $\implies$ (i) follows as above since $S_\tau : W_{\tau}^{1,p} \to L^p$ is isometric. The converse for $\tau = 0$ is done in [5, Lem. 13.4] and the argument makes the dependence of constants transparent. It remains to prove (i) $\implies$ (ii) in the inhomogeneous case $\tau \ne 0$, which, in fact, is much easier than the homogeneous counterpart.

To this end, we assume (i) with $L^p$-bound $C'$ and let $u \in W_{\tau}^{-1,p} \cap W_{\tau}^{-1,2}$. Since $S_\tau^* S_\tau = (\tau^2 - \Delta_x)$, we can write $u = S_\tau^* v$ with $v := S_\tau(\tau^2 - \Delta_x)^{-1} u \in L^p \cap L^2$. In this way, we obtain

$$\|L_{\tau}^{-1} u\|_{W_{\tau}^{1,p}} = \|S_{\tau} L_{\tau}^{-1} S_{\tau}^* v\|_{p} \le C' \|v\|_{p}$$

and we need to bound $v$ independently of $\tau$. Through dilations with parameters $\tau, \tau^{-1}$, we find

$$v = S_{\tau}(\tau^2 - \Delta_x)^{-1} u = \tau^{-1} \delta_{\tau} (S_1(1 - \Delta_x)^{-1}) \delta_{\tau^{-1}} u.$$

A composition of three bounded operators acts on the right-hand side: $\tau^{-1} \delta_\tau : L^p \to L^p$ has norm $|\tau|^{-1-n/p}$ by the transformation rule, $S_1(1 - \Delta_x)^{-1} : W^{-1,p} \to L^p$ is bounded
by the lifting properties of the Bessel potentials [19, Sec. 1.3.1], and $\delta_{\tau^{-1}}: W_{\tau}^{-1,p} \to W^{1,p}$ has norm $|\tau|^{1+n/p}$ as the dual of $|\tau|^n \delta_{\tau}: W^{1,p'} \to W_{\tau}^{1,p'}$. Thus, $\|v\|_p \lesssim \|u\|_{W_{\tau}^{-1,p}}$ with an implicit constant independent of $\tau$, and we are done. ■

## 6. THE FIRST-ORDER DIRAC OPERATOR DB

As a final ingredient for the proof of our main result, we introduce the perturbed Dirac operators $DB$ and the key exponent $p_+(DB)$ as in [9]. There is one new result in this section: we characterize $p_+(DB)$ through the Hodge family $(S_{\tau}L_{\tau}^{-1}S_{\tau}^*)_{\tau \in \mathbb{R}^*}$. This necessitates sorting out certain subtleties related to compatible inverses of linear operators.

Recall the block form of the coefficients of $\mathcal{L}$ in (3.1). The ellipticity assumption on $A$ implies that $A$ is invertible in $L^{\infty}$ and we can introduce the following matrix-valued functions:

**Definition 6.1 ([2]).** We let

$$ \underline{A} := \begin{bmatrix} 1 & 0 \\ c & d \end{bmatrix}, \quad \overline{A} := \begin{bmatrix} a & b \\ 0 & 1 \end{bmatrix}, \quad \text{and} \quad B = \underline{A} \overline{A}^{-1}. $$

Next, we define (perturbed) Dirac operators.

**Definition 6.2.** We define the **Dirac operator** $D$ in the distributional sense as

$$ D := \begin{bmatrix} 0 & \operatorname{div}_x \\ -\nabla_x & 0 \end{bmatrix} : L_{\text{loc}}^1(\mathbb{R}^n)^{1+n} \to \mathcal{D}'(\mathbb{R}^n)^{1+n} $$

and for $p \in (1, \infty)$ we denote by $D_p$ its maximal restriction to a linear operator in $L^p$. The **perturbed Dirac operator** is the composition operator $DB$ and its part in $L^p$ is $(DB)_p = D_p B$.

**Remark 6.3.** The operator $D_p$ in $L^p$ is closed as a first-order differential operator with maximal $L^p$-domain. Since $B$ is bounded, also $(DB)_p$ is closed.

We usually drop the subscript $p$ from our notation when $p = 2$. It is shown in [11, Prop. 2.5] that $DB$ is a **bisectorial** operator in $L^2$: There exists some $\mu \in (0, \pi/2)$ such that the spectrum of $DB$ is contained in the closure of the bisector

$$ S_{\mu} := \{z \in \mathbb{C} : |\arg(\pm z)| < \mu\} $$

and for each $\nu \in (\mu, \pi/2)$ there is $C > 0$ such that

$$ \|(1 + zDB)^{-1}\|_{L^2 \to L^2} \le C \quad (z \in \mathbb{C} \setminus \overline{S_{\nu}}). $$

**Theorem 6.4 ([8], [9, Thm. 3.6]).** There is a maximal open interval $I(DB) \subseteq (1, \infty)$ around 2 such that $(DB)_p$ is bisectorial in $L^p$ for all $p \in I(DB)$. Moreover, bisectoriality fails at the endpoints.

This result gives rives to the key exponent $p_+(DB)$.

**Definition 6.5 ([9, Sec. 3.2]).** We denote by $p_+(DB) > 2$ the upper endpoint of $I(DB)$.
**Remark 6.6.** Due to our strong ellipticity assumption, A is invertible in L$^{∞}$. Hence, B is also invertible, and in particular, multiplication by B induces an isomorphism on any L$^{p}$ space. This would not be true under weaker ellipticity conditions as they are typically imposed for elliptic systems and avoids the discussion of L$^{p}$-coercivity, compare with [9, Rem. 3.5].

The following considerations were tacitly used in [9]. We provide the details for the reader's convenience and to clarify the necessity of restricting ourselves to intervals around $p = 2$.

**Lemma 6.7.** Let $p \in (1, \infty)$. Then the following assertions are equivalent.

(i) $((1 + itDB)^{-1})_{t \in \mathbb{R}^*}$ is L$^{p}$-bounded.

(ii) $p \in I(DB)$.

(iii) $p \in I(DB)$ and the resolvents $(1 + it(DB)_p)^{-1}$ and $(1 + it(DB)_2)^{-1}$ agree on L$^p \cap L^2$ for all $t \in \mathbb{R}^*$.

**Proof.** The implication (iii) $\implies$ (i) is obvious but the rest requires some work.

(i) $\implies$ (ii). We first observe that it is enough to conclude that $(DB)_p$ is bisectorial. Indeed, since the assumption (i) holds for $p = 2$ and interpolates with respect to $p$, this would also imply $p \in I(DB)$. To this end, we denote by $R_p(t)$ the unique bounded extension of $(1 + itDB)^{-1}$ to L$^p$ and show that $1 + it(DB)_p$ is bijective with $(1 + it(DB)_p)^{-1} = R_p(t)$.

To prove that $1 + it(DB)_p$ is surjective, we let $f \in L^p$ and pick a sequence $(f_j)_j \subseteq L^p \cap L^2$ with $f_j \to f$ in L$^p$. Then $u_j := (1 + itDB)^{-1}f_j \to R_p(t)f$ in L$^p$. Since $(u_j)_j \subseteq L^p \cap \text{dom}(DB)$ with $itDBu_j = f_j - u_j \in L^p$, we also have $(u_j)_j \subseteq \text{dom}(D_pB)$ with $(1 + it(DB)_p)u_j = f_j \to f$ in L$^p$. Now, $1 + it(DB)_p$ is closed and $R_p(t)f \in \text{dom}(D_pB)$ as well as $(1 + it(DB)_p)R_p(t)f = f$ follow. Hence, $R_p(t)$ is a right-inverse for $1 + it(DB)_p$.

As for injectivity, we pick $u \in \text{dom}(DB)_p$ with $(1 + it(DB)_p)u = 0$ and prove $u = 0$. Given $\varphi \in C_c^\infty$, we write

$$ 
\begin{aligned}
0 &= \langle R_p(t)0, \varphi \rangle \\
  &= \langle (1 + itDB)^{-1}(u + it(DB)_p u), \varphi \rangle.
\end{aligned}
$$

Since $(1 + itDB)^{-1}$ is L$^p$-bounded, the dual family $(1 - itB^*D)^{-1}$ is L$^{p'}$-bounded. We set $v := (1 - itB^*D)^{-1}\varphi$. Hence, $v \in L^{p'} \cap \text{dom}(D)$ with $-itB^*Dv = \varphi - v \in L^{p'}$, so that even $v \in \text{dom}(D_{p'}) \cap \text{dom}(D)$. Through smooth truncation and convolution, we can approximate $v$ by a sequence $(v_j) \subseteq C_c^\infty$ such that $v_j \to v$ and $Dv_j \to Dv$, both in L$^{p'}$. Consequently, we can continue by

$$ 
\begin{aligned}
&= \langle u + it(DB)_p u, v \rangle \\
&= \lim_{j \to \infty} \langle u + it(DB)_p u, v_j \rangle \\
&= \lim_{j \to \infty} \langle u, (1 - itB^*D)v_j \rangle \\
&= \langle u, (1 - itB^*D)v \rangle \\
&= \langle u, \varphi \rangle.
\end{aligned}
$$
Since $\varphi \in C_c^\infty$ was arbitrary, $u = 0$ follows.

(ii) $\implies$ (iii). Let $M \subseteq I(DB)$ be the set of exponents $p$ for which we have compatibility with the $L^2$-resolvent of $DB$ as stated in (iii). $M$ is non-empty because it contains 2. We claim that it is open and closed in $I(DB)$ and hence equal to $I(DB)$.

Openness follows directly from Šnečberg's stability theorem [4, 28]. Next, we take a sequence $(p_j)_j \subseteq M$ with $p_j \to p \in I(DB)$. Again by Šnečberg's theorem and for large enough $j$ we have

$$ (1 + it(DB)_p)^{-1} f = (1 + it(DB)_{p_j})^{-1} f = (1 + itDB)^{-1} f $$

for all $f \in L^p \cap L^{p_j} \cap L^2$. Now, $p \in M$ follows since $L^p \cap L^{p_j} \cap L^2$ is dense in $L^p \cap L^2$. ■

The following algebraic identity links the $DB$-resolvents with the $\tau$-dependent Hodge projectors. Note that the formulation in [9] uses the inhomogeneous gradients $[1, it\nabla_x]^\top = itS_{-t-1}$ in place of $S_t$.

**Lemma 6.8.** Let $\tau \in \mathbb{R}^*$ and $f = [f_\perp, f_\parallel]^\top \in L^2$. Then

$$ (1 - i\tau^{-1}DB)^{-1}f = \overline{AS_\tau L_\tau^{-1}S_\tau^*Mf} + \begin{bmatrix} bf_\parallel \\ f_\parallel \end{bmatrix}, \quad \text{where } M := \begin{bmatrix} 1 & -b \\ 0 & -d \end{bmatrix}. $$

With the algebra in place, we can prove the characterization of $p_+(DB)$ alluded to above.

**Proposition 6.9.** Let $p \in (1, \infty)$. The following are equivalent:

(i) $p \in I(DB)$.

(ii) $(S_\tau L_\tau^{-1}S_\tau^*)_{\tau \in \mathbb{R}^*}$ is $L^p$-bounded.

In particular, the set of exponents $p$, for which (ii) holds, is an open interval with upper endpoint $p_+(DB)$.

**Proof.** In view of Lemma 6.8 and since $\overline{A}$, $\overline{A}^{-1}$ and $b$ are in $L^\infty$, the following two uniform bounds are equivalent:

$$ \|(1 + itDB)^{-1}f\|_p \lesssim \|f\|_p \quad (t \in \mathbb{R}^*, f \in L^p \cap L^2), $$

$$ \|S_\tau L_\tau^{-1} S_\tau^* M f\|_p \lesssim \|f\|_p \quad (\tau \in \mathbb{R}^*, f \in L^p \cap L^2). $$

According to Lemma 6.7, the upper estimate is equivalent to (i). Since $A$ is strongly elliptic, so is $d$ in one dimension lower and therefore $M$ is invertible in $L^\infty$. Thus, the lower estimate is equivalent to the $L^p$-boundedness of the Hodge projectors $(S_\tau L_\tau^{-1} S_\tau^*)_{\tau \in \mathbb{R}^*}$ as stated in (ii). ■
## 7. GLOBAL CHARACTERIZATION OF THE MEYERS EXPONENT

We start with the proof of our main theorem. In this section, we carry out the globalization step in Figure 1.

**Definition 7.1.** Let $\tau \in \mathbb{R}$ and $Q = Q(x, r) \subseteq \mathbb{R}^n$. We say that $u$ is $L_\tau$-harmonic in $Q$ if $u \in W_{\text{loc}}^{1,2}(Q)$ satisfies $L_\tau u = 0$ weakly in $Q$, that is

$$ \int_Q AS_{\tau}u \cdot \overline{S_{\tau}\varphi} dx = 0 \quad (\varphi \in C_c^{\infty}(Q)). $$

**Definition 7.2.** (i) The Meyers exponent $m_+(L_\tau)$ of a fixed operator $L_\tau$ is the supremum of all $p \in [2, \infty)$ for which the following holds true: There is a constant $C > 0$ such that for all open axes-parallel cubes $Q \subseteq \mathbb{R}^n$ and $L_\tau$-harmonic $u$ in $2Q$ it follows that

$$ \left( \int_Q |S_{\tau}u|^p dx \right)^{\frac{1}{p}} \le C \left( \int_{2Q} |S_{\tau}u|^2 dx \right)^{\frac{1}{2}}. $$

(ii) Given $I \subseteq \mathbb{R}$, the Meyers exponent $m_+((L_\tau)_{\tau \in I})$ of the collection of operators $(L_\tau)_{\tau \in I}$ is the supremum of all $p \in [2, \infty)$ for which (i) holds for all $\tau \in I$ and a constant that does not depend on $\tau$.

Refining an argument from [27], we provide the following characterization.

**Proposition 7.3.** Let $n \ge 2$. For any $\tau \in \mathbb{R}$ and $I \subseteq \mathbb{R}$ the following hold true:

(i) We have $m_+(L_\tau) = \sup \{p \ge 2 : S_\tau L_\tau^{-1} S_\tau^* \text{ is } L^p\text{-bounded}\}$.

(ii) We have $m_+((L_\tau)_{\tau \in I}) = \sup \{p \ge 2 : (S_\tau L_\tau^{-1} S_\tau^*)_{\tau \in I} \text{ is } L^p\text{-bounded}\}$.

**Proof.** We provide a proof for (i). Our argument will automatically give the additional uniformity of implicit constants that is required in (ii).

**Step 1: Meyers controls Hodge.** Suppose that $p > 2$ is such that $S_\tau L_\tau^{-1} S_\tau^*$ is $L^p$-bounded. Fix $Q = Q(x, r) \subseteq \mathbb{R}^n$ and an $L_\tau$-harmonic $u$ in $2Q$. We establish a weak reverse Hölder estimate for $S_\tau u$.

Let $\gamma \in (1, 2)$ be a number that will be fixed later on and $\varphi \in C_c^\infty$ be such that $1_Q \le \varphi \le 1_{\gamma Q}$ and $\|\nabla_x \varphi\|_\infty \lesssim r^{-1}$ with implicit constant depending on $\gamma$ and dimension. Then $v := \varphi(u - \kappa) \in W^{1,2}$ with $\kappa := (u)_{\gamma Q}$ satisfies a global equation

$$ L_{\tau}v = f - \operatorname{div}_{x}(F), $$

on $\mathbb{R}^n$, where

$$ \begin{aligned} f &:= \left( -d\nabla_x u \cdot \nabla_x \varphi - i\tau\kappa c \cdot \nabla_x \varphi \right) - \tau \left( i(u - \kappa)(b + c^T)\nabla_x \varphi + \tau\kappa a\varphi \right) \\ &=: f_1 - \tau f_2, \end{aligned} $$

$$ F := (u - \kappa)d\nabla_x \varphi - i\tau\kappa\varphi c. $$
At this point, we need to start tracking the dependence on $\tau$ carefully. For all $g \in L^p$, we have $\|\tau g\|_{W_{\tau}^{-1,p}} \le \|g\|_p$. The assumptions $n \ge 2$ and $p > 2$ imply $p_* > 1$ and consequently, by the standard Sobolev embedding, we have $\|g\|_{W_{\tau}^{-1,p}} \lesssim \|g\|_{p_*}$ for all $g \in L^{p_*}$. Using the properties of $\varphi$ and then the $L^p$-boundedness of the Hodge projector through its equivalent formulation in Proposition 5.2, we obtain

$$ \begin{aligned} \|S_{\tau}u\|_{L^p(Q)} & \le |\tau\kappa| \|Q\|_{W_{\tau}^{1,p}}^{\frac{1}{p}} + \|v\|_{W_{\tau}^{1,p}} \\ & \lesssim |\tau\kappa| \|Q\|_{W_{\tau}^{1,p}}^{\frac{1}{p}} + \|f - \operatorname{div}_x F\|_{W_{\tau}^{-1,p}} \\ & \lesssim |\tau\kappa| \|Q\|_{W_{\tau}^{1,p}}^{\frac{1}{p}} + \|f_1\|_{p_*} + \|f_2\|_p + \|F\|_p \\ & \lesssim |\tau\kappa| \|Q\|_{W_{\tau}^{1,p}}^{\frac{1}{p}} + r^{-1} \|u - \kappa\|_{L^p(\gamma Q)} + r^{-1} \|\nabla_x u\|_{L^{p_*(\gamma Q)}}. \end{aligned} $$

For the first term, we use Hölder's inequality

$$ |\tau\kappa| \le |\gamma Q|^{-\frac{1}{p}-\frac{1}{n}} \|\tau u\|_{L^{p_*(\gamma Q)}} $$

whereas the second term can be controlled by the third one via the Sobolev-Poincaré inequality. In total, we have shown that

$$ \|S_{\tau}u\|_{L^p(Q)} \lesssim r^{-1} \|S_{\tau}u\|_{L^{p_*(\gamma Q)}} $$

which, after dividing both sides by $|Q|^{1/p}$, becomes

$$ (7.1) \qquad \left( \fint_Q |S_{\tau}u|^p \,dx \right)^{\frac{1}{p}} \lesssim \left( \fint_{\gamma Q} |S_{\tau}u|^{p_*} \,dx \right)^{\frac{1}{p_*}}. $$

If $p_* \le 2$, then the right-hand side is bounded by the $L^2$-average and we are done. Else, we have $2 < p_* < p$ and the previous argument re-applies to the right-hand side of (7.1) with $p_*$ in place of $p$, leading to a new bound by the $(p_*)$-average of $S_{\tau}u$ on $\gamma^2 Q$ as long as $\gamma^2 < 2$. After a finite number of iterations, say $N$, this procedure yields an $L^2$-average on the right. We then choose $\gamma$ a priori such that $\gamma^N < 2$.

**Step 2: Hodge controls Meyers.** Let $2 < p < m_+(L_\tau)$ and let $C$ be an $L^p$ weak reverse Hölder constant for $L_\tau$. It suffices to prove that $S_\tau L_\tau^{-1} S_\tau^*$ is $L^q$-bounded for all $q \in (2, p)$.

From Section 5 we know that $S_\tau L_\tau^{-1} S_\tau^*$ is $L^2$-bounded with norm at most $\lambda^{-1}$. Moreover, if $Q \subseteq \mathbb{R}^n$ is an axes-parallel cube and $f = [f_\perp, f_\parallel]^\top \in C_c^\infty$ is such that $f|_{4Q} = 0$, then $u_\tau := L_\tau^{-1} S_\tau^* f$ satisfies $L_\tau u_\tau = S_\tau^* f = 0$ in $2Q$. Hence, we have

$$ \left( \fint_Q |S_\tau u_\tau|^p \,dx \right)^{\frac{1}{p}} \le C \left( \fint_{2Q} |S_\tau u_\tau|^2 \,dx \right)^{\frac{1}{2}} $$

by assumption. We have verified the assumptions for Shen's extrapolation theorem [27, Thm. 3.1], which in turn yields that $S_\tau L_\tau^{-1} S_\tau^*$ is $L^q$-bounded for all $q \in (2, p)$ with a bound depending on $C$, $\lambda$, $n$ and $q$, see also [29, Thm. 4.1].

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

As usual, the $\tau$-dependent theory with $\tau = 0$ yields an analogous conclusion for the divergence-form operator $\mathcal{L}$ in dimension $n+1$.
**Corollary 7.4.** The Meyers exponent for $\mathcal{L}$ is given by

$$m_+(\mathcal{L}) = \sup \{p \ge 2 : \nabla_{t,x} \mathcal{L}^{-1} \text{ div}_{t,x} \text{ is } \text{L}^p\text{-bounded}\}.$$

The supremum on the right-hand side in Corollary 7.4 was studied extensively in [5]. Multiple characterizations are known and the one that turns out particularly useful for our purpose is as follows.

**Corollary 7.5.** The Meyers exponent for $\mathcal{L}$ is also given by

$$m_+(\mathcal{L}) = \sup \{p \ge 2 : \text{ there is } N \in \mathbb{N} \text{ such that } (s\nabla_{t,x}(1+s^2\mathcal{L})^{-N})_{s>0} \text{ is L}^p\text{-bounded}\}.$$

This is a combination of two results: By [5, Thm. 13.12] one can always take $N = 1$ and by [5, Lem. 6.5] the choice of $N$ does not matter.

## 8. FROM $1+n$ TO $n$ DIMENSIONS

In this section, we carry out the easiest step in Figure 1 by showing that a Meyers estimate for $\mathcal{L}$ implies a Meyers estimate for $L_\tau$ in one dimension lower, uniformly in $\tau$.

**Proposition 8.1.** The inequality $m_+(\mathcal{L}) \le m_+((L_\tau)_{\tau \in \mathbb{R}})$ holds true.

**Proof.** Let $2 < p < m_+(\mathcal{L})$. Given $\tau \in \mathbb{R}$, $Q = Q(x, r) \subseteq \mathbb{R}^n$ and any $L_\tau$-harmonic $u$ in $2Q$, we need to show

$$(8.1) \qquad \left( \int_Q |S_\tau u|^p \,dx \right)^{\frac{1}{p}} \le C \left( \int_{2Q} |S_\tau u|^2 \,dx \right)^{\frac{1}{2}}.$$

Splitting the coefficients of $\mathcal{L}$ as in (3.1), we find

$$\mathcal{L}U = -\operatorname{div}_x(d\nabla_x U) - \operatorname{div}_x(c\partial_t U) - \partial_t(b\nabla_x U) - \partial_t(a\partial_t U).$$

Comparing with (3.2), we see that $U(t, x) := e^{it\tau} u(x)$ defines an $\mathcal{L}$-harmonic function in $(0, 4r) \times 2Q \subseteq \mathbb{R}^{1+n}$. The assumption yields some $C > 0$ not depending on $\tau$ such that

$$\left( \int_r^{3r} \int_Q |\nabla_{t,x} U|^p \,dx\,dt \right)^{\frac{1}{p}} \le C \left( \int_0^{4r} \int_{2Q} |\nabla_{t,x} U|^2 \,dx\,dt \right)^{\frac{1}{2}},$$

but as $|\nabla_{t,x} U| = |S_\tau u|$ is independent of $t$, this estimate collapses to (8.1). ■

<figure><img src="image_4.png" /></figure>

## 9. FROM $n$ TO $n+1$ DIMENSIONS

In this section, we prove the following reverse inequality to Proposition 8.1 in dimension $n \ge 2$.

**Proposition 9.1.** In dimension $n \ge 2$, the inequality $m_+(\mathcal{L}) \ge m_+((L_\tau)_{\tau \in \mathbb{R}^*})$ holds true.

**Remark 9.2.** Since we have $m_+((L_\tau)_{\tau \in \mathbb{R}^*}) \ge m_+((L_\tau)_{\tau \in \mathbb{R}})$ by definition, combining Propositions 9.1 and 8.1 also reveals that $m_+((L_\tau)_{\tau \in \mathbb{R}^*}) = m_+((L_\tau)_{\tau \in \mathbb{R}})$.

The proof of this estimate is the centerpiece of our paper and spreads over five subsections, following the strategy outlined in Figure 1.
**9.1. Reduction to a resolvent estimate in 1 + n dimensions.** For the rest of the section, we assume $n \ge 2$ and use arbitrary exponents $r, q$ such that

$$ (9.1) \qquad 2_* < r < 2 < q < \infty \quad \text{and} \quad (S_\tau L_\tau^{-1} S_\tau^*)_{\tau \in \mathbb{R}^*} \text{ is } L^r\text{- and } L^q\text{-bounded.} $$

Such exponents exist thanks to Proposition 6.9 and the assumption is open-ended with respect to $r$ and $q$. (By this we mean that the same assumption holds for a smaller $r$ and a larger $q$.) The role of $r$ will become clear only at the very end of the proof, but it cannot be omitted. With these exponents at hand, we formulate a technical result from which Proposition 9.1 will follow, using the results from the previous sections.

**Proposition 9.3.** Let $q$ be as in (9.1). For any $p \in (2, q)$ there exists $N \in \mathbb{N}$ such that $(s\nabla_{t,x}(1 + s^2\mathcal{L})^{-N})_{s>0}$ is $L^p$-bounded.

**Proof of Proposition 9.1, admitting Proposition 9.3.** Let $p < m_+((L_\tau)_{\tau \in \mathbb{R}^*})$. By the characterization of $m_+((L_\tau)_{\tau \in \mathbb{R}^*})$ in Proposition 7.3, we have (9.1) with an exponent $q > p$ at our disposal. Proposition 9.3 and the characterization of $m_+(\mathcal{L})$ in Corollary 7.5 yield $p \le m_+(\mathcal{L})$. Since $p < m_+((L_\tau)_{\tau \in \mathbb{R}^*})$ was chosen arbitrarily, the claim follows. ■

**9.2. Reduction to an operator-valued multiplier estimate.** In the following, we identify $L^2(\mathbb{R}^{1+n}) \cong L^2(\mathbb{R}; L^2(\mathbb{R}^n)) := L^2(\mathbb{R}; L^2)$ via Fubini's theorem. Since the coefficients of $\mathcal{L}$ are $t$-independent, we obtain the following correspondence to operator-valued Fourier multipliers.

**Lemma 9.4.** Let $s > 0$. Then $(1 + s^2\mathcal{L})^{-1}$ is an $\mathcal{L}(L^2)$-valued Fourier multiplier with symbol

that is

$$ \tau \mapsto (1 + s^2 L_{\tau})^{-1}, $$

$$ (1 + s^2 \mathcal{L})^{-1} = \mathcal{F}_t^{-1} (1 + s^2 L_{\tau})^{-1} \mathcal{F}_t. $$

**Proof.** We let $f \in L^2(\mathbb{R}^{1+n})$ and set $g := (1 + s^2\mathcal{L})^{-1}f \in \text{dom}(\mathcal{L}) \in W^{1,2}(\mathbb{R}^{1+n})$. Then

$$ \begin{aligned} f &= g + s^2 \mathcal{L}g \\ &= g + s^2(-\text{div}_x(d\nabla_x g) - \text{div}_x(c\partial_t g) - \partial_t(b\nabla_x g) - a\partial_t^2 g) \end{aligned} $$

in the weak sense. In particular, for all $h \in \mathcal{S}(\mathbb{R}^{1+n})$, we have

$$ \langle f, h \rangle = \langle g, h \rangle + s^2 \left( \langle d\nabla_x g, \nabla_x h \rangle + \langle c\partial_t g, \nabla_x h \rangle + \langle b\nabla_x g, \partial_t h \rangle + \langle a\partial_t g, \partial_t h \rangle \right). $$

By Plancherel's theorem in the $t$-variable and since $a, b, c, d$ are $t$-independent, we get

$$ \begin{aligned} \langle \mathcal{F}_t f, \mathcal{F}_t h \rangle &= \langle \mathcal{F}_t g, \mathcal{F}_t h \rangle \\ &\quad + s^2 \left( \langle d\nabla_x \mathcal{F}_t g, \nabla_x \mathcal{F}_t h \rangle + \langle i\tau c \mathcal{F}_t g, \nabla_x \mathcal{F}_t h \rangle \right. \\ &\qquad \left. + \langle b\nabla_x \mathcal{F}_t g, i\tau c \mathcal{F}_t h \rangle + \langle a i\tau \mathcal{F}_t g, i\tau \mathcal{F}_t h \rangle \right) \\ &= \langle (1 + s^2(-\text{div}_x d\nabla_x - i\tau \text{div}_x c - i\tau b\nabla_x + a\tau^2)) \mathcal{F}_t g, \mathcal{F}_t h \rangle. \end{aligned} $$

This calculation implies that $(\mathcal{F}_t g)(\tau, \cdot) \in \text{dom}(L_\tau)$ for a.e. $\tau \in \mathbb{R}$ with

$$ (\mathcal{F}_t f)(\tau, \cdot) = (1 + s^2 L_\tau)(\mathcal{F}_t g)(\tau, \cdot). $$
Hence,

$$ (1 + s^2 \mathcal{L})^{-1} f = g = \mathcal{F}_t^{-1} (1 + s^2 L_\tau)^{-1} \mathcal{F}_t f. \quad \blacksquare $$

In the following, we split vectors $f \in \mathbb{C}^{1+n}$ as $f = [f_\perp, f_\|]^\top$ in accordance with our writing for the coefficients $A$ in (3.1).

**Corollary 9.5.** Let $s > 0$ and $N \in \mathbb{N}$. Then $s\nabla_{t,x}(1+s^2\mathcal{L})^{-N}$ is an $\mathcal{L}(L^2; (L^2)^{1+n})$-valued Fourier multiplier with symbol

$$ (9.2) \qquad m_{s,N}(\tau) := \begin{bmatrix} m_{s,N}(\tau)_{\perp} \\ m_{s,N}(\tau)_{\|} \end{bmatrix} := \begin{bmatrix} \mathrm{is}\tau(1 + s^2 L_{\tau})^{-N} \\ s\nabla_x(1 + s^2 L_{\tau})^{-N} \end{bmatrix}. $$

Proposition 9.3 now asks for boundedness of a Fourier multiplier in $L^p(\mathbb{R}; (L^p)^{1+n})$. A sufficient condition is furnished by the celebrated theorem of Weis [30]. We phrase his result in terms of square function estimates rather than $\mathcal{R}$-boundedness, which are equivalent concepts in the Banach space $L^p$, see [22, Thm. 8.1.3 (3)].

**Definition 9.6.** Let $p \in (1, \infty)$ and $I \subseteq \mathbb{R}^*$. We say that $(T(t))_{t \in I} \subseteq \mathcal{L}(L^2)$ satisfies square function estimates on $L^p$, if there is $C \ge 0$ such that

$$ \left\| \left( \sum_{j=1}^{k} |T(t_j)f_j|^2 \right)^{\frac{1}{2}} \right\|_{p} \le C \left\| \left( \sum_{j=1}^{k} |f_j|^2 \right)^{\frac{1}{2}} \right\|_{p} $$

for all choices $k \in \mathbb{N}$, $t_1, \dots, t_k \in I$ and $f_1, \dots, f_k \in L^p \cap L^2$.

**Theorem 9.7 (Weis [30], [22, Cor. 8.3.11]).** Let $p \in (1, \infty)$ and $m \in C^1(\mathbb{R}^*; \mathcal{L}(L^p))$ be such that $(m(\tau))_{\tau \in \mathbb{R}^*}$ and $(\tau m'(\tau))_{\tau \in \mathbb{R}^*}$ satisfy square function estimates on $L^p$ with constant $C$. Then the operator given by

$$ T f = \mathcal{F}_t^{-1}(m \mathcal{F}_t f) \quad (f \in \mathcal{S}(\mathbb{R}^{1+n})) $$

has a unique bounded extension to $L^p(\mathbb{R}; L^p)$ with a bound depending on $p$ and $C$.

We can now further reduce Proposition 9.3 to a square function estimate. Note carefully that the symbol in (9.2) does not even map into $L^p$ for general $p$ and already for this we will use properties of the $\tau$-dependent Hodge family.

**Proposition 9.8.** Let $p \in (2, q)$. There exists $N \in \mathbb{N}$ such that for all $s > 0$ the symbol $m_{s,N}$ is of class $C^1(\mathbb{R}^*; \mathcal{L}(L^p))$, and $(m_{s,N}(\tau))_{\tau \in \mathbb{R}^*}$ and $(\tau m'_{s,N}(\tau))_{\tau \in \mathbb{R}^*}$ satisfy square function estimates on $L^p$ with bounds independent of $s$.

**Proof of Propositions 9.1 & 9.3, admitting Proposition 9.8.** Apply Theorem 9.7 to the operators $s\nabla_{t,x}(1 + s^2\mathcal{L})^{-N}$. ■
**9.3. L²-results for auxiliary operators.** We introduce four auxiliary families of operators that will play a key role in the following. Our notation is easy to remember: the superscript on R refers to the power of sτ and the subscript indicates the amount of x-derivatives left and right of the resolvents.

**Definition 9.9.** For $s > 0$ and $\tau \in \mathbb{R}^*$, we set

$$^{\alpha}R_{(0,0)}(s, \tau) := (s\tau)^{\alpha}(1 + s^2 L_{\tau})^{-1} \quad (\alpha \in \{0, 1, 2\}),$$

$$^{\alpha}R_{(1,0)}(s, \tau) := (s\tau)^{\alpha}s\nabla_x(1 + s^2 L_{\tau})^{-1} \quad (\alpha \in \{0, 1\}),$$

$$^{\alpha}R_{(0,1)}(s, \tau) := (s\tau)^{\alpha}(1 + s^2 L_{\tau})^{-1}s \operatorname{div}_x \quad (\alpha \in \{0, 1\}),$$

$$^{\alpha}R_{(1,1)}(s, \tau) := (s\tau)^{\alpha}s\nabla_x(1 + s^2 L_{\tau})^{-1}s \operatorname{div}_x \quad (\alpha = 0).$$

We need L² off-diagonal estimates for these families with respect to the parameter s and implicit constants that do not depend on τ. To this end, we rely on scaling. The rescaled coefficients $A(\tau^{-1}.)$ have the same L∞-norm and satisfy our ellipticity assumption from Section 3 with the same constant λ. From (3.2) we obtain that the corresponding inhomogeneous divergence form operator is given by

$$ (9.3) \qquad \begin{aligned} L^{A(\tau^{-1}.)}u &:= -\operatorname{div}_x(d(\tau^{-1}.)\nabla_x u) - i\operatorname{div}_x(c(\tau^{-1}.)u) - ib(\tau^{-1}.)\nabla_x u + a(\tau^{-1}.)u \\ &= \tau^{-2}\delta_{\tau^{-1}}L_{\tau}\delta_{\tau}. \end{aligned} $$

The following L² off-diagonal estimates for semigroups generated by inhomogeneous operators in divergence form are standard nowadays. For an explicit statement of (i), see [12, Prop. 3.2]. Statement (ii) follows by general duality and composition principles, see [5, Sec. 4.2].

**Lemma 9.10.** There are $C, c > 0$ depending only on $\lambda$ and $\|A\|_{\infty}$ such that for all $s > 0$, $\tau \in \mathbb{R}^*$, measurable sets $E, F \subseteq \mathbb{R}^n$ and $f \in L^2$ with support in E, we have the following off-diagonal bounds:

$$ (i) \quad \| e^{-s^2 L^{A(\tau^.)}} f \|_{L^2(F)} + \|s \nabla_x e^{-s^2 L^{A(\tau^.)}} f \|_{L^2(F)} \le C e^{-c^2 \frac{d(E,F)^2}{s^2} - cs^2} \|f\|_{L^2(E)}, $$

$$ (ii) \quad \begin{aligned} & \| e^{-s^2 L^{A(\tau^.)}} s \operatorname{div}_x f \|_{L^2(F)} + \| s \nabla_x e^{-s^2 L^{A(\tau^.)}} s \operatorname{div}_x f \|_{L^2(F)} \\ & \le C e^{-c^2 \frac{d(E,F)^2}{s^2} - cs^2} \|f\|_{L^2(E)}. \end{aligned} $$

In a next step, we take the Laplace transform to derive estimates for resolvents. For technical reasons, we need an explicit L²-bound for $^0R_{(1,1)}(s, \tau)$ beforehand.

**Lemma 9.11.** For all $s > 0$, $\tau \in \mathbb{R}^*$ and $f \in L^2$, we have

$$ \| {}^0R_{(1,1)}(s, \tau) f \|_2 \le \lambda^{-1} \| f \|_2. $$

**Proof.** We set $u := (1 + s^2 L_{\tau})^{-1} \operatorname{div}_x f$. By ellipticity,

$$ \lambda s^2 \|u\|_{W_{\tau}^{1,2}}^2 \le \operatorname{Re}\langle (1 + s^2 L_{\tau})u, u \rangle \le \|\operatorname{div}_x f\|_{W_{\tau}^{-1,2}} \|u\|_{W_{\tau}^{1,2}}, $$

and therefore

$$ \| {}^0R_{(1,1)}(s, \tau) f \|_2 = s^2 \| \nabla_x u \|_2 \le s^2 \| u \|_{W_{\tau}^{1,2}} \le \lambda^{-1} \| \operatorname{div}_x f \|_{W_{\tau}^{-1,2}} \le \lambda^{-1} \| f \|_2. \blacksquare $$
**Lemma 9.12.** Let $\beta \in \{0, 1\}^2$, $0 \le \alpha \le 2 - |\beta|$ and $\tau \in \mathbb{R}^*$. Then $(^\alpha R_\beta(s, \tau))_{s>0}$ satisfies $L^2$ off-diagonal estimates with implicit constants depending only on $\lambda$ and $\|A\|_\infty$.

**Proof.** We take $s > 0$, $\tau \in \mathbb{R}^*$, measurable sets $E, F \subseteq \mathbb{R}^n$, and $f \in L^2$ supported in $E$ and normalized to $\|f\|_{L^2(E)} = 1$. In view of Lemma 9.11, we can additionally assume that $d(E,F)/s \ge 1$ if $\beta = (1,1)$. Let us write

$$ {}^{\alpha}R_{\beta}^{A(\tau^{-1}.)}(s, 1) $$

for the operator ${}^{\alpha}R_\beta(s, 1)$ with coefficients $A(\tau^{-1}.)$ in place of $A$. By the Laplace transform formula

$$ (1 + s^2 L^{A(\tau^{-1}.)})^{-1} f = \int_0^\infty e^{-t} e^{-ts^2 L^{A(\tau^{-1}.)}} f dt $$

and Lemma 9.10, we get

$$ \begin{aligned} \| {}^{\alpha}R_{\beta}^{A(\tau^{-1}.)}(s, 1) f \|_{L^2(F)} & \le C s^{\alpha} \int_0^{\infty} e^{-t-cts^2-c^2 \frac{d(E,F)^2}{ts^2}} \frac{dt}{t^{|\beta|/2}} \\ & \le C s^{\alpha} \int_0^{\infty} e^{-\frac{t}{2}-cts^2-\left(\frac{t}{2}+c^2 \frac{d(E,F)^2}{2ts^2}\right)} e^{-c^2 \frac{d(E,F)^2}{2ts^2}} \frac{dt}{t^{|\beta|/2}} \\ & \le C e^{-c \frac{d(E,F)}{s}} s^{\alpha} \int_0^{\infty} e^{-\frac{1}{2}(1+2cs^2)t} e^{-c^2 \frac{d(E,F)^2}{ts^2}} t^{-\frac{|\beta|}{2}} dt, \end{aligned} $$

where we used the inequality $2|xy| \le x^2 + y^2$ in the final step. Now, we invoke the assumption $d(E,F)/s \ge 1$ when $\beta = (1,1)$ and substitute $u = (1+2cs^2)t$ in order to arrive at

$$ \begin{aligned} & \le C e^{-c \frac{d(E,F)}{s}} s^{\alpha} \int_0^{\infty} e^{-\frac{1}{2}(1+2cs^2)t} \left( \mathbf{1}_{[|\beta|<2]} t^{-\frac{|\beta|}{2}} + \mathbf{1}_{[|\beta|=2]} \right) dt \\ & = C e^{-c \frac{d(E,F)}{s}} \int_0^{\infty} e^{-\frac{1}{2}u} \left( \mathbf{1}_{[|\beta|<2]} s^{\alpha} (1+2cs^2)^{-\frac{|\beta|}{2}-1} u^{-\frac{|\beta|}{2}} + \mathbf{1}_{[|\beta|=2]} (1+2cs^2)^{-1} \right) du. \end{aligned} $$

The terms in $s$ are uniformly bounded due to the restriction $0 \le \alpha \le 2 - |\beta|$ and the remaining integral in $u$ is finite, leading to an overall estimate

$$ (9.4) \qquad \| {}^{\alpha}R_{\beta}^{A(\tau^{-1}.)}(s, 1) f \|_{L^2(F)} \le C e^{-c \frac{d(E,F)}{s}}. $$

We finish the proof by scaling. By similarity as in (9.3), we have

$$ \operatorname{sgn}(\tau)^{\alpha+|\beta|} {}^{\alpha}R_{\beta}^{A(\tau^{-1}.)}(s|\tau|, 1) \delta_{\tau^{-1}} = \delta_{\tau^{-1}} {}^{\alpha}R_{\beta}(s, \tau). $$

Hence, (9.4), when read with $(\delta_{\tau^{-1}}f, \tau E, \tau F)$ in place of $(f, E, F)$, becomes

$$ \| \delta_{\tau^{-1}} {}^{\alpha}R_{\beta}(s, \tau) f \|_{L^2(\tau F)} \le C e^{-c \frac{d(\tau E, \tau F)}{|s|\tau}} \| \delta_{\tau^{-1}} f \|_{L^2(\tau E)} $$

and by the transformation rule we conclude the required off-diagonal bound

$$ \| {}^{\alpha}R_{\beta}(s, \tau) f \|_{L^2(F)} \le C e^{-c \frac{d(E,F)}{s}} \| f \|_{L^2(E)}. \qquad \blacksquare $$
9.4. **L**<sup>*p*</sup>-theory for the auxiliary operators. We start with the following Sobolev embeddings related to the resolvent. The qualitative result is not new but once again τ-independence of constants matters.

**Lemma 9.13.** If $2_* < p < q^*$, then $(^0R_{(0,0)}(s, \tau))_{s>0}$ is L<sup>*p*</sup>-bounded with implicit constant independent of $\tau \in \mathbb{R}^*$.

**Proof.** We use a bootstrapping argument to increase the exponent *p* step-by-step.

**Base case** $2_* < p < 2^*$. We first consider exponents $p \ge 2$. By the Gagliardo–Nirenberg inequality and the L<sup>2</sup>-estimates from Lemma 9.12, we find for all $s > 0$ and $f \in L^2$ that

$$ \begin{aligned} \|(1 + s^2 L_{\tau})^{-1} f\|_{p} & \lesssim s^{\frac{n}{p} - \frac{n}{2}} \|(s \nabla_x)^{-1} (1 + s^2 L_{\tau})^{-1} f\|_{2}^{\frac{n}{2} - \frac{n}{p}} \|(1 + s^2 L_{\tau})^{-1} f\|_{2}^{1 - \frac{n}{2} + \frac{n}{p}} \\ & \lesssim s^{\frac{n}{p} - \frac{n}{2}} \|f\|_{2}. \end{aligned} $$

This means that $(^0R_{(0,0)}(s, \tau))_{s>0}$ is L<sup>2</sup>–L<sup>*p*</sup>-bounded. Since the restriction on *p* is open-ended, L<sup>*p*</sup>-boundedness follows from Lemma 4.2.

So far, we have only used L<sup>2</sup>-theory for L<sub>τ</sub>. Hence, the same conclusion is valid for the adjoint L<sub>τ</sub><sup>*</sup>, which is an operator in the same class as L<sub>τ</sub> and by duality $(^0R_{(0,0)}(s, \tau))_{s>0}$ is L<sup>*p'</sup> – L<sup>2</sup>-bounded and L<sup>*p'</sup>-bounded. This covers all exponents in $(2_*, 2)$.

**Inductive case.** The next case to consider is $2^* < p < ((2^*)^* \wedge q^*)$. (We do not have to consider $p = 2^*$ explicitly since all assumptions on *p* are open-ended). This scenario can only appear in dimension $n \ge 3$ since $2^* = \infty$ when $n = 2$.

Since $(p_*)_{*} > 2_*$, we know from the base case that $(^0R_{(0,0)}(s, \tau))_{s>0}$ is L<sup>($p_*$)<sup>*</sup></sup>-bounded. Given $f \in L^{($p_*$)<sup>*</sup> \cap L^2$, Sobolev embeddings yield

$$ \begin{aligned} \|(1 + s^2 L_{\tau})^{-1} f\|_{p} & \lesssim s^{-2} \|(s \nabla_x)^{-1} L_{\tau}^{-1} (1 - (1 + s^2 L_{\tau})^{-1}) f\|_{p_*} \\ & \le s^{-2} \|L_{\tau}^{-1} (1 - (1 + s^2 L_{\tau})^{-1}) f\|_{W_{\tau}^{1, p_*}}. \end{aligned} $$

Since $2 < p_* < q$, the L<sup>($p_*$)<sup>*</sup>-boundedness of the Hodge projectors in its equivalent form in Proposition 5.2 and another Sobolev embedding lead to

$$ \begin{aligned} & \lesssim s^{-2} \|(1 - (1 + s^2 L_{\tau})^{-1}) f\|_{W_{\tau}^{-1, p_*}} \\ & \lesssim s^{-2} \|(1 - (1 + s^2 L_{\tau})^{-1}) f\|_{(p_*)_*}. \end{aligned} $$

This means that $(^0R_{(0,0)}(s, \tau))_{s>0}$ is L<sup>($p_*$)<sup>*</sup> – L<sup>*p*</sup>-bounded and Lemma 4.2 yields the desired L<sup>*p*</sup>-boundedness as before. All implicit constants in this argument are independent of $\tau$.

Iterating the procedure covers the full range $2^* < p < q^*$ in a finite number of steps. ■

Our proof in the base case revealed an additional result that we record for later.

**Corollary 9.14.** If $2_* < p < 2$, then $(^0R_{(0,0)}(s, \tau))_{s>0}$ is L<sup>*p*</sup> – L<sup>2</sup>-bounded with implicit constant independent of $\tau \in \mathbb{R}^*$.

We turn to the other operator families.
**Lemma 9.15.** Let $|\beta| \le 1$ and $\alpha = 1$ when $\beta = (0,1)$. Then $(^\alpha R_\beta(s, \tau))_{s>0}$ is $L^r$- and $L^q$-bounded with implicit constant independent of $\tau \in \mathbb{R}^*$.

**Proof.** Since the assumption on $r$ and $q$ in (9.1) is open-ended, it will be enough to prove a respective $L^p$-bound for $p \in (r, q)$. For $^0R_{(0,0)}$ this is Lemma 9.13. The other families can be classified into three groups.

**The family $^\alpha R_{(0,0)}$ with $\alpha \in \{1, 2\}$.** Since $|s\tau| \le 1 + |s\tau|^2$, it suffices to treat $^2R_{(0,0)}$. However,

$$ (s\tau)^2(1 + s^2L_\tau)^{-1} = \tau^2L_\tau^{-1}(1 - (1 + s^2L_\tau)^{-1}), $$

and the claim follows by combining Proposition 5.2 and Lemma 9.13.

**The families $^1R_{(1,0)}$ and $^1R_{(0,1)}$.** We write

$$ \begin{aligned} (s\tau)s\nabla_x(1 + s^2L_\tau)^{-1} &= \tau\nabla_xL_\tau^{-1}(1 - (1 + s^2L_\tau)^{-1}), \\ (s\tau)(1 + s^2L_\tau)^{-1}s\operatorname{div}_x &= (1 - (1 + s^2L_\tau)^{-1})\tau L_\tau^{-1}\operatorname{div}_x \end{aligned} $$

and conclude once again by Proposition 5.2 and Lemma 9.13.

**The family $^0R_{(1,0)}$.** Since we have $q > 2$ and $r > 2_*$, it is suffices to show that for $p \in (2, q)$ both families are $L^p$- and $L^{p_*}$-bounded with $\tau$-independent bound. Note that in this case $p_* > 2_* \ge 1$. As before, we write

$$ s\nabla_x(1 + s^2L_\tau)^{-1} = s^{-1}\nabla_xL_\tau^{-1}(1 - (1 + s^2L_\tau)^{-1}). $$

Proposition 5.2, the Sobolev embedding $L^{p_*} \subseteq W_{\tau}^{-1,p}$ and Lemma 9.13 yield for all $f \in L^{p_*} \cap L^2$ the bound

$$ \begin{aligned} \|s\nabla_x(1 + s^2L_\tau)^{-1}f\|_{p} &\le s^{-1}\|L_\tau^{-1}(1 - (1 + s^2L_\tau)^{-1})f\|_{W_{\tau}^{1,p}} \\ &\lesssim s^{-1}\|(1 - (1 + s^2L_\tau)^{-1})f\|_{W_{\tau}^{-1,p}} \\ &\lesssim s^{-1}\|(1 - (1 + s^2L_\tau)^{-1})f\|_{p_*} \\ &\lesssim s^{-1}\|f\|_{p_*}. \end{aligned} $$

This means that $(^0R_{(1,0)}(s, \tau))_{s>0}$ is $L^{p_*} - L^p$-bounded with $\tau$-independent bound. Open-endedness in $p$ and Lemma 4.2 yield the claim. ■

**9.5. $L^p$-theory for the symbol.** In this section, we assemble estimates for the auxiliary functions in order to derive smoothness and $L^p - L^q$-type bounds for the symbol $m_{s,N}(\tau)$ from (9.2).

**Lemma 9.16.** Let $s > 0$ and $N \in \mathbb{N}$. If $r < p < q$, then $\tau \mapsto (1 + s^2L_\tau)^{-N}$ is of class $C^1(\mathbb{R}^*; \mathcal{L}(L^p, W_{1,p}^*))$ with derivative

$$ \frac{d}{d\tau}(1 + s^2L_\tau)^{-N} = s^2 \sum_{k=1}^{N} (1 + s^2L_\tau)^{-k} (i \operatorname{div}_x c + ib\nabla_x - 2\tau a)(1 + s^2L_\tau)^{-(N+1-k)}. $$

In particular, we have $m_{s,N} \in C^1(\mathbb{R}^*; \mathcal{L}(L^p))$ for every $s > 0$.
**Proof.** We begin with the assertion about the resolvent. By the product rule and induction, it suffices to do the case $N = 1$.

For $\sigma \neq \tau$, we have

$$ (1 + s^2 L_{\tau})^{-1} - (1 + s^2 L_{\sigma})^{-1} = s^2(1 + s^2 L_{\tau})^{-1}(L_{\sigma} - L_{\tau})(1 + s^2 L_{\sigma})^{-1} $$

$$ (9.5) \quad \begin{aligned} & = s^2(1 + s^2 L_{\tau})^{-1}(i(\tau - \sigma) \operatorname{div}_x c + i(\tau - \sigma)b \nabla_x - (\tau^2 - \sigma^2)a)(1 + s^2 L_{\sigma})^{-1} \\ & = (\tau - \sigma)s^2(1 + s^2 L_{\tau})^{-1}(i \operatorname{div}_x c + ib \nabla_x - (\tau + \sigma)a)(1 + s^2 L_{\sigma})^{-1} \\ & =: (\tau - \sigma)r(\sigma, \tau). \end{aligned} $$

Thanks to Lemma 9.15, the remainder $r(\sigma, \tau)$ is bounded in $\mathcal{L}(L^p)$-norm, uniformly in $\sigma$ and $\tau$ in compact subsets of $\mathbb{R}^*$. Hence, $\tau \mapsto (1 + s^2 L_{\tau})^{-1}$ is continuous with values in $\mathcal{L}(L^p)$.

The same type of argument can be used to prove continuity with values in $\mathcal{L}(L^p)$ for $\tau \mapsto \nabla_x(1 + s^2 L_{\tau})^{-1}$ and $\tau \mapsto (1 + s^2 L_{\tau})^{-1} \operatorname{div}_x$. In this calculation, the operator $s^2 \nabla_x(1 + s^2 L_{\tau})^{-1} \operatorname{div}_x$ appears, which cannot be handled via Lemma 9.15. However, we can write

$$ s^2 \nabla_x (1 + s^2 L_{\tau})^{-1} \operatorname{div}_x = \nabla_x L_{\tau}^{-1} \operatorname{div}_x - \nabla_x L_{\tau}^{-1} (1 + s^2 L_{\tau})^{-1} \operatorname{div}_x, $$

as a composition of operators that either fall in the scope of Lemma 9.15 or are controlled through the Hodge projector.

Altogether, the remainder function $\tau \mapsto r(\sigma, \tau)$ in (9.5) is continuous with values in $\mathcal{L}(L^p, W_1^{1,p})$. Thus, $\sigma \mapsto (1 + s^2 L_{\sigma})^{-1}$ is of class $C^1(\mathbb{R}^*; \mathcal{L}(L^p, W_1^{1,p}))$ with derivative

$$ \frac{d}{d\sigma} (1 + s^2 L_{\sigma})^{-1} = r(\sigma, \sigma) $$

as claimed.

Continuous differentiability of $m_{s,N}$ follows immediately by the product rule since we have $m_{s,N}(\tau)_{\perp} = \operatorname{is}\tau(1 + s^2 L_{\tau})^{-N}$ and $m_{s,N}(\tau)_{\parallel} = s \nabla_x(1 + s^2 L_{\tau})^{-N}$. ■

We have reached the point in the argument, where we choose $N$ large.

**Lemma 9.17.** There exists $N \in \mathbb{N}$, divisible by 4, such that $((1 + s^2 L_{\tau})^{-N/4})_{s>0}$ is $L^r - L^q$-bounded with implicit constant independent of $\tau \in \mathbb{R}^*$.

**Proof.** We consider the resolvent family $((1 + s^2 L_{\tau})^{-1})_{s>0}$ and the following boundedness properties with implicit constant independent of $\tau \in \mathbb{R}^*$. By Lemma 9.13, we have $L^\rho$-bounds for all $\rho \in (q, q^*)$ and by Corollary 9.14 we have $L^\sigma - L^2$-bounds for all $\sigma \in (2^*, r)$. By interpolation, we get an $L^\sigma - L^q$ bound for some $\sigma$. Again by Lemma 9.13 we also have the $L^\sigma$- and the $L^q$-bound. Now, Lemma 4.3 yields the claim. ■

The proof of the next result clarifies why this choice is appropriate for our purpose.

**Lemma 9.18.** If $N$ is as in Lemma 9.17, then $(m_{s,N}(\tau))_{s>0}$ and $(\tau m'_{s,N}(\tau))_{s>0}$ are $L^r - L^q$-bounded with implicit constants independent of $\tau \in \mathbb{R}^*$. ■
**Proof.** The bound for $m_{s,N}(\tau)$ follows by composition from Lemma 9.17 and the $L^q$- bounds for $^1R_{(0,0)}$ and $^0R_{(1,0)}$ in Lemma 9.15. Let us come to the estimates for $\tau m'_{s,N}(\tau)$.

**The scalar component.** We compute

$$ \tau m'_{s,N}(\tau)_{\perp} = m_{s,N}(\tau)_{\perp} + i s \tau^2 \frac{d}{d\tau} (1 + s^2 L_{\tau})^{-N} $$

and still need to handle the second term on the right. To this end, we write the formula in Lemma 9.16 in the following form:

$$ (9.6) \quad i s \tau^2 \frac{d}{d\tau} (1 + s^2 L_{\tau})^{-N} = \sum_{k=1}^{N} (1 + s^2 L_{\tau})^{-(k-1)} T(s, \tau) (1 + s^2 L_{\tau})^{-(N-k)}, $$

where

$$ \begin{align} T(s, \tau) &:= i s^3 \tau^2 (1 + s^2 L_{\tau})^{-1} (i \operatorname{div}_x c + i b \nabla_x - 2 \tau a) (1 + s^2 L_{\tau})^{-1} \nonumber \\ &= - \left( (s\tau)(1 + s^2 L_{\tau})^{-1} s \operatorname{div}_x \right) \left( c(s\tau)(1 + s^2 L_{\tau})^{-1} \right) \nonumber \\ &\quad - \left( (s\tau)^2 (1 + s^2 L_{\tau})^{-1} \right) \left( b s \nabla_x (1 + s^2 L_{\tau})^{-1} \right) \nonumber \\ &\quad - 2i \left( (s\tau)(1 + s^2 L_{\tau})^{-1} \right) \left( a(s\tau)^2 (1 + s^2 L_{\tau})^{-1} \right) \tag{9.7} \end{align} $$

is a composition of operators $^\alpha R_\beta(s, \tau)$ with $|\beta| \le 1$ and $\alpha = 1$ for $\beta = (0, 1)$. Thus, Lemma 9.15 yields that $(T(s, \tau))_{s>0}$ is $L^r$- and $L^q$-bounded, independently of $\tau$. The upshot is that for each summand in (9.6) the exponents sum up to $(k-1) + (N-k) = N-1 \ge N/2$ and hence one exponent is at least $N/4$. By the choice of $N$, the entire expression, as a family indexed in $s > 0$, is $L^r-L^q$-bounded. This concludes the treatment of the scalar component.

**The vectorial component.** Since $\nabla_x : W_1^{1,p} \to (L^p)^n$ is bounded, we obtain from Lemma 9.16 and with $T(s, \tau)$ as in (9.7) that

$$ \begin{align*} \tau m'_{s,N}(\tau)_{\parallel} &= s\tau \nabla_x \frac{d}{d\tau} (1 + s^2 L_{\tau})^{-N} \\ &= \sum_{k=1}^{N} s \nabla_x (1 + s^2 L_{\tau})^{-(k-1)} (i s \tau)^{-1} T(s, \tau) (1 + s^2 L_{\tau})^{-(N-k)} \\ &= \sum_{k=2}^{N} \left( s \nabla_x (1 + s^2 L_{\tau})^{-1} \right) (1 + s^2 L_{\tau})^{-(k-2)} \left( (i s \tau)^{-1} T(s, \tau) \right) (1 + s^2 L_{\tau})^{-(N-k)} \\ &\quad + \left( s \nabla_x (i s \tau)^{-1} T(s, \tau) \right) (1 + s^2 L_{\tau})^{-(N-1)}. \end{align*} $$

Let us first handle the terms in $k = 2, \dots, N$. It follows from (9.7) that $((i s \tau)^{-1} T(s, \tau))_{s>0}$ is a composition of the same type as $(T(s, \tau))_{s>0}$. Hence, this family is $L^r$- and $L^q$- bounded, independently of $\tau$. By Lemma 9.15, the same is true for $(s \nabla_x (1 + s^2 L_{\tau})^{-1})_{s>0}$ appearing on the left. The sum of the exponents still satisfies $(k-2) + (N-k) = N-2 \ge N/2$, so we obtain the required $L^r-L^q$-bound as before.

For the final term, we already know that $((1 + s^2 L_{\tau})^{-(N-1)})_{s>0}$ is $L^r-L^q$-bounded with implicit constants independent of $\tau \in \mathbb{R}^*$. Thus, it suffices to prove $L^q$-boundedness,
uniformly in $\tau$, for

$$ (9.8) \qquad s\nabla_x(is\tau)^{-1}T(s, \tau) = i(s\nabla_x(1 + s^2L_\tau)^{-1}s \operatorname{div}_x)(c(s\tau)(1 + s^2L_\tau)^{-1}) \\ \qquad \qquad + i((s\tau)s\nabla_x(1 + s^2L_\tau)^{-1})(bs\nabla_x(1 + s^2L_\tau)^{-1}) \\ \qquad \qquad - 2(s\nabla_x(1 + s^2L_\tau)^{-1})(a(s\tau)^2(1 + s^2L_\tau)^{-1}). $$

The second and third term on the right are $L^q$-bounded by Lemma 9.15 but this lemma does not cover the family $^0R_{(1,1)}$ that appears in the first term. However, writing

$$ s\nabla_x(1 + s^2L_\tau)^{-1}s \operatorname{div}_x = \nabla_x L_\tau^{-1} s^2 L_\tau (1 + s^2L_\tau)^{-1} \operatorname{div}_x \\ = \nabla_x L_\tau^{-1} \operatorname{div}_x - s\nabla_x(1 + s^2L_\tau)^{-1}\tau L_\tau^{-1} \operatorname{div}_x(s\tau)^{-1}, $$

we see that the first of the three terms on the right in (9.8) can also be decomposed into

$$ i(\nabla_x L_\tau^{-1} \operatorname{div}_x)(c(s\tau)(1 + s^2L_\tau)^{-1}) - i(s\nabla_x(1 + s^2L_\tau)^{-1})(\tau L_\tau^{-1} \operatorname{div}_x)(c(1 + s^2L_\tau)^{-1}). $$

Now, the $L^q$-boundedness follows from Proposition 5.2 and Lemma 9.15. ■

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

The uniform $L^r - L^q$-bounds can be upgraded to off-diagonal estimates upon possibly taking $N$ even larger.

**Proposition 9.19.** There exists a positive integer $N$ such that the families $(m_{s,N}(\tau))_{s>0}$ and $(\tau m'_{s,N}(\tau))_{s>0}$ satisfy $L^r - L^q$ off-diagonal estimates with implicit constants independent of $\tau \in \mathbb{R}^*$.

**Proof.** Since the assumptions on $r$ and $q$ are open-ended, it suffices to prove $L^\sigma - L^p$ off-diagonal estimates whenever $r < \sigma < p < q$.

We already know that $(m_{s,N}(\tau))_{s>0}$ and $(\tau m'_{s,N}(\tau))_{s>0}$ are $L^r - L^q$-bounded. They also satisfy $L^2$ off-diagonal estimates with implicit constants independent of $\tau$. Indeed, we have seen in the proof of Lemma 9.18 that $(m_{s,N}(\tau))_{s>0}$ and $(\tau m'_{s,N}(\tau))_{s>0}$ can be written as a sum and composition of the auxiliary families $(\alpha R_\beta(s, \tau))_{s>0}$, which satisfy $L^2$ off-diagonal estimates by Lemma 9.12. Thus, the claim is a consequence of Lemma 4.2. ■

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

**9.6. Square function bounds for the symbol.** Off-diagonal estimates as in Proposition 9.19 imply a pointwise domination of averages through the Hardy–Littlewood maximal operator $M$ by splitting $\mathbb{R}^n$ into suitable dyadic annuli. For an explicit statement of the following corollary, we refer to [13, Lem. 5.3]. For clarity, we write

$$ (\operatorname{Avg}_{q,s} f)(x) := \left( \int_{B(x,s)} |f|^q dy \right)^{\frac{1}{q}} $$

for $L^q$-averages on balls.

**Lemma 9.20.** There is $C > 0$ such that for all $s > 0$, $\tau \in \mathbb{R}^*$ and $f \in L^r \cap L^2$ we have

$$ \operatorname{Avg}_{q,s} (|m_{s,N}(\tau)f|^q + |\tau m'_{s,N}(\tau)f|^q) \le CM(|f|^r)^{\frac{1}{r}}, $$
everywhere on $\mathbb{R}^n$.
The domination through the maximal function implies square function estimates by a line of reasoning that goes back to Kunstmann-Weis [24] and that we learned from [13, Prop. 5.8]. We use it to provide the missing piece for the proof of Proposition 9.1.

**Proof of Proposition 9.8.** We choose $N \in \mathbb{N}$ as in Proposition 9.19. Given finitely many $f_j \in L^p \cap L^2$ and $\tau_j \in \mathbb{R}^*$, we first use the lower square function estimates for the family $(\text{Avg}_{q,s})_{s>0}$ from [24, Prop. 8.13] in order to write

$$ \left\| \left( \sum_j |(m_{s,N}(\tau_j) + \tau_j m'_{s,N}(\tau_j)) f_j|^2 \right)^{\frac{1}{2}} \right\|_p \lesssim \left\| \left( \sum_j |\text{Avg}_{q,s}((m_{s,N}(\tau_j) + \tau m'_{s,N}(\tau)) f_j)|^2 \right)^{\frac{1}{2}} \right\|_p. $$

Lemma 9.20 controls the right-hand side by

$$ \begin{aligned} & \lesssim \left\| \left( \sum_j |\mathcal{M}(|f_j|^r)|^{\frac{2}{r}} \right)^{\frac{1}{2}} \right\|_p \\ &= \left\| \left( \sum_j |\mathcal{M}(|f_j|^r)|^{\frac{2}{r}} \right)^{\frac{r}{2}} \right\|_{\frac{p}{r}}^{\frac{1}{r}} \end{aligned} $$

with exponents $2/r, p/r > 1$ and the classical Fefferman-Stein inequality [16, Thm. 1] yields

$$ \begin{aligned} & \lesssim \left\| \left( \sum_j |f_j|^2 \right)^{\frac{r}{2}} \right\|_{\frac{p}{r}}^{\frac{1}{r}} \\ &= \left\| \left( \sum_j |f_j|^2 \right)^{\frac{1}{2}} \right\|_p. \end{aligned} $$

■

## 10. PROOF OF THEOREM 1.1 IN DIMENSION $n \ge 2$ AND A GENERALIZATION

According to Proposition 8.1 and 9.1, we have $m_+(\mathcal{L}) = m_+((L_\tau)_{\tau \in \mathbb{R}^*})$ and according to Propositions 7.3 and 6.9 this number coincides with $p_+(DB)$. This concludes the proof of Theorem 1.1.

Our argument also gives a description of the full interval $I(DB)$ from Definition 6.5, not just its upper endpoint. To state the general result, we write $\mathcal{L}^*$ for the adjoint of $\mathcal{L}$ and, with a slight abuse of notation, we call the corresponding operators in one dimension lower $L_\tau^*$ so that $L_{-\tau}^* = (L_\tau)^*$.

**Definition 10.1.** We introduce the following quantities and sets:

$$ q_+(\mathcal{L}) := \sup \left\{ p \ge 2 : (s \nabla_{t,x} (1 + s^2 \mathcal{L})^{-1})_{s>0} \text{ is } L^p\text{-bounded} \right\}, $$

$$ \mathcal{P}(\mathcal{L}) := \left\{ p \in (1, \infty) : \nabla_{t,x} \mathcal{L}^{-1} \text{ div}_{t,x} \text{ is } L^p\text{-bounded} \right\}, $$

$$ \mathcal{P}((L_\tau)_{\tau \in \mathbb{R}^*}) := \left\{ p \in (1, \infty) : (S_\tau L_\tau^{-1} S_\tau^*)_{\tau \in \mathbb{R}^*} \text{ is } L^p\text{-bounded} \right\}. $$
**Theorem 10.2** (General version of Theorem 1.1). In dimension $n \ge 2$, we have

$$ 
\begin{aligned}
I(DB) &= \mathcal{P}((L_{\tau})_{\tau \in \mathbb{R}^*}) \\
&= \mathcal{P}(\mathcal{L}) \\
&= (m_+(\mathcal{L}^*)', m_+(\mathcal{L})) \\
&= (m_+((L_{\tau}^*)_{\tau \in \mathbb{R}^*})', m_+((L_{\tau})_{\tau \in \mathbb{R}^*})) \\
&= (q_+(\mathcal{L}^*)', q_+(\mathcal{L})).
\end{aligned}
$$

**Proof.** The first equality is due to Proposition 6.9 and both sets are open intervals. The statements of Proposition 7.3, Corollary 7.4, Proposition 8.1 and Proposition 9.1 is that the sets in the first four lines have the same upper endpoint. The equality of line two with line five is due to [5, Thm. 13.12]. Since these results are also true for $\mathcal{L}^*$ in place of $\mathcal{L}$, the sets in lines two to five coincide. Finally, the duality relation $(S_{\tau}L_{\tau}^{-1}S_{\tau}^*)^* = S_{\tau}L_{-\tau}^*S_{\tau}^*$ implies

$$ \mathcal{P}((L_{\tau}^*)_{\tau \in \mathbb{R}^*}) = \{p' \in (1, \infty) : p \in \mathcal{P}((L_{\tau})_{\tau \in \mathbb{R}^*})\}. $$

Consequently, the lower endpoint of $\mathcal{P}((L_{\tau})_{\tau \in \mathbb{R}^*})$ is the Hölder conjugate of the upper endpoint of $\mathcal{P}((L_{\tau}^*)_{\tau \in \mathbb{R}^*})$ as claimed. ■

**Remark 10.3.** The identity $I(DB) = (q_+(\mathcal{L}^*)', q_+(\mathcal{L}))$ has a non-trivial consequence due to [14, Thm. 1.2]. Namely, not only is this set open but there is some $\varepsilon > 0$ depending only on $\lambda$ and $\|A\|_{\infty}$ (but especially not on the dimension $n$) such that $(2 - \varepsilon, 2 + \varepsilon) \subseteq I(DB)$.

**Remark 10.4.** In the block case $b = 0, c = 0$, it was previously shown in [5, Prop. 15.1] that $p_+(DB) = q_+(L_0)$. Hence, Theorem 10.2 yields $q_+(L_0) = q_+(\mathcal{L})$. It is possible to prove this equality directly.

## 11. PROOF OF THEOREM 1.1 IN DIMENSION $n = 1$

In this section, we include the proof of Theorem 1.1 in dimension $n = 1$. In this case, it is known that $p_+(DB) = \infty$, see [9, Prop. 3.11]. Hence, we need to show $m_+(\mathcal{L}) = \infty$. In fact, we obtain the slightly stronger result that the weak reverse Hölder bound with $p = \infty$ holds.

**Proposition 11.1.** Let $n = 1$. There is $C > 0$ that only depends on $\lambda$ and $\|A\|_{\infty}$, such that for all axes-parallel cubes $Q \subseteq \mathbb{R}^{1+n}$ and every $\mathcal{L}$-harmonic $U$ in $2Q$ we have

$$ \|\nabla_{t,x} U\|_{L^{\infty}(Q)} \le C \left( \fint_{2Q} |\nabla_{t,x} U|^2 d(t,x) \right)^{\frac{1}{2}}. $$

**Proof.** For any fixed $x_0 \in \mathbb{R}$ and $r > 0$, the transformed coefficients $A(x_0 + rx)$ are of the same class as $A$ with the same ellipticity bounds. Thus, it suffices to treat the case $Q = Q(0, 1)$. We split vectors $f \in \mathbb{C}^{1+1}$ as $f = [f_{\perp}, f_{\parallel}]^{\top}$ and, as in Section 6, we write

$$ \underline{A} = \begin{bmatrix} 1 & 0 \\ c & d \end{bmatrix}, \quad \text{so that} \quad \underline{A}^{-1} = \begin{bmatrix} 1 & 0 \\ -d^{-1}c & d^{-1} \end{bmatrix}. $$
Note that $d$ is invertible with $\|d^{-1}\|_{\infty} \le \lambda^{-1}$ since $A$ is elliptic. We introduce $V := A\nabla_{t,x}U$ and aim for the bound

$$ (11.1) \qquad \|V\|_{L^{\infty}(Q)} \le C\|\nabla_{t,x}U\|_{L^2(2Q)}. $$

This yields the claim because $|Q| = 1$ and we have the pointwise comparability $|V| \simeq |\nabla_{t,x}U|$ with implicit constants depending only on $\lambda$ and $\|A\|_{\infty}$.

The function $U$ is qualitatively smooth in $t$ since $A$ is $t$-independent (see [10, App. B, Lem. 1]). Within $2Q$, we compute

$$ \nabla_{t,x} V_{\|} = \nabla_{t,x}(A\nabla_{t,x}U)_{\|} = \left[ \frac{\partial_t(A\nabla_{t,x}U)_{\|}}{\partial_x(A\nabla_{t,x}U)_{\|}} \right] = \left[ \frac{(A\nabla_{t,x}\partial_t U)_{\|}}{-(A\nabla_{t,x}\partial_t U)_{\perp}} \right], $$

where we have used the equation $\mathcal{L}U = 0$ in the final step. We also have $\nabla_{t,x}V_{\perp} = \nabla_{t,x}\partial_t U$, which altogether leads to the pointwise control $|\nabla_{t,x}V| \lesssim |\nabla_{t,x}\partial_t U|$. Since we are working in dimension $1+n=2$, Sobolev embeddings yield for any $p > 2$ a bound

$$ \begin{aligned} \|V\|_{L^{\infty}(Q)} &\lesssim \|V\|_{L^p(Q)} + \|\nabla_{t,x}V\|_{L^p(Q)} \\ &\lesssim \|\nabla_{t,x}U\|_{L^p(Q)} + \|\nabla_{t,x}\partial_t U\|_{L^p(Q)}. \end{aligned} $$

We pick $p > 2$ such that we have the classical Meyers estimate [25] for $\mathcal{L}$-harmonic functions at our disposal. Since $\partial_t U$ is $\mathcal{L}$-harmonic by $t$-independence of the coefficients, Meyers estimate applies to $U$ and $\partial_t U$, allowing us to continue by

$$ \lesssim \|\nabla_{t,x}U\|_{L^2(\frac{3}{2}Q)} + \|\nabla_{t,x}\partial_t U\|_{L^2(\frac{3}{2}Q)}. $$

Now, (11.1) follows from the Caccioppoli inequality for $\partial_t U$.

■

## 12. SYSTEMS AND OPEN PROBLEMS

Interested readers can check that all results in this paper remain valid for elliptic systems as long as one requires a pointwise (also known as uniformly strong) ellipticity condition.

Several references including [1, 2, 7, 9] introduce the first-order approach under a weaker ellipticity assumption that does not imply that $B$ is invertible in $L^{\infty}$. In this case, $p_+(DB)$ has a slightly more complicated definition, compare with Remark 6.6 and the proof of Proposition 6.9 fails. It remains as an open question whether Theorem 1.1 is still true for elliptic systems under this weaker algebraic requirement.

Another open question is whether one can prove a characterization similar to Theorem 1.1 if one replaces $DB$ in $n$ dimensions by more general Dirac-type operators as in [11, 17]. In this case, it is even unclear, what the corresponding equation in $(1+n)$ dimensions should be.

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*Email address: pascal.auscher@universite-paris-saclay.fr*

UNIVERSITÉ PARIS-SACLAY, CNRS, LABORATOIRE DE MATHÉMATIQUES D'ORSAY, 91405 ORSAY

*Email address: boehnlein@mathematik.tu-darmstadt.de, egert@mathematik.tu-darmstadt.de*

FACHBEREICH MATHEMATIK, TECHNISCHE UNIVERSITÄT DARMSTADT, SCHLOSSGARTENSTR. 7, 64289 DARMSTADT, GERMANY