# IMPACT OF YVES MEYER'S WORK ON KATO'S CONJECTURE

Pascal Auscher

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Pascal Auscher. IMPACT OF YVES MEYER'S WORK ON KATO'S CONJECTURE. 2023. hal-04266629

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Distributed under a Creative Commons CC BY 4.0 - Attribution - International License
IMPACT OF YVES MEYER'S WORK ON KATO'S
CONJECTURE

PASCAL AUSCHER

ABSTRACT. We discuss how the works of Yves Meyer, together with Raphy Coifman, on Calderón's program and singular integrals with minimal smoothness in the seventies, paved the way not only to a solution to Kato's conjecture for square roots of elliptic operators, but also to major developments in elliptic and parabolic boundary value problems with rough coefficients on rough domains.

# 1. INTRODUCTION

Tosio Kato's square root conjecture is one example of a question arising from one field, formulated in a second one and finding its solution in a third one. Namely, the question arising in the sixties from the work of T. Kato, motivated by partial differential equations in inhomogeneous media, was set using a functional analysis framework and it was finally methods from real harmonic analysis that put a final end to the problem as posed.

It is likely that R. Coifman and Y. Meyer were not aware of Kato's conjecture before the visit of A. McIntosh to Paris in 1980. McIntosh had remarked the analogies between the multilinear series that he and they were using. McIntosh was indeed trying to attack Kato's conjecture by this tool. R. Coifman and Y. Meyer were trying to prove the boundedness of the Cauchy singular integral operator on Lipschitz curves, as part of the program of A. Calderón on functional analysis and operator algebras. This encounter allowed them to find the way of summing the series for both the Cauchy integral and the square root in one dimension. This appears in the famous article [28] published in 1982. It opened the door to new developments in the above mentioned three fields and other ones, where improvements came from round trips from Cauchy integrals and Kato's conjecture, with R. Coifman, A. McIntosh and Y. Meyer as the first architects.

<footnote>Date: October 31, 2023.</footnote>

<footnote n="2020">2020 Mathematics Subject Classification. 35J25, 42B35, 47A60, 42B30, 42B37. 35J46, 35K40.</footnote>

<footnote>Key words and phrases. Square root problem, Cauchy integral, singular integrals, T(b) theorem, boundary value problems, operator adapted Hardy spaces.</footnote>
From the point of view of Yves, it all comes from Calderón's insight. Let us quote his article on the work of Calderón in complex analysis and operator theory [58]:

"He [Calderón] pointed up a concise question which happened to be the magic key opening all doors."

Motivated by the example of the Cauchy integral on a curve, this concise question was to prove the $L^2$ boundedness of what we now call anti-symmetric singular integrals. It turns out that it is not always true and a necessary and sufficient criterion for a positive answer is the celebrated $T(1)$ theorem of G. David and J.L. Journé [34], followed by the $T(b)$ theorem of the same authors with S. Semmes [35]. These two theorems owe a great deal to the pioneer work of R. Coifman, Y. Meyer and A. McIntosh. Kato's conjecture one dimensional solution was also very much instrumental in the first version of the $T(b)$ theorem [57].

Boundedness of singular integrals could be addressed in several dimensions but Kato's question in higher dimension was more difficult since the direct connection with the Cauchy integral and singular integrals is lost. Although various different proofs for the Cauchy integral's boundedness came soon after [28], it took another twenty years to fully understand Kato's question in all dimension. A new approach was needed. We shall explain how the progress in the understanding of $T(b)$ theorems helped.

Among the applications, the solution to the square root problem could address the original questions of Kato on propagation of waves and stability with respect to the conductivity matrix of the medium. More applications connecting Kato's question and boundary value problems were also brought to light by C. Kenig [49].

But the story was not finished. Almost another twenty years after the solution of Kato's conjecture for elliptic operators, the estimates that come with and the ideas surrounding the argument found unforeseen applications in boundary value problems for elliptic and parabolic operators and Hardy space theory. It is still an active topic.

The objective of this article is to give a non-exhaustive feeling for the advances permitted by the breakthroughs made by Yves Meyer and co-authors towards partial differential equations in the last forty years with Kato's square root problem as a central theme. The kind and friendly shadow of Yves spreads all over these developments.

## 2. THE SQUARE ROOT PROBLEM FOR ELLIPTIC OPERATORS

Let $A = (a_{jk})$ be a matrix of bounded measurable complex functions on $\mathbb{R}^n$ satisfying the accretivity inequality,

$$ (2.1) \quad \operatorname{Re} \sum_{j,k=1}^{n} a_{jk}(x) z_k \bar{z}_j \geq \lambda |z|^2 \quad (z \in \mathbb{C}^n, x \in \mathbb{R}^n), $$
for some $\lambda > 0$. One can then define a maximal accretive operator in $L^2(\mathbb{R}^n)$ by

$$ (2.2) \qquad L(u) := - \operatorname{div}_x (A \nabla_x u) $$

with domain $D(L) = \{u \in W^{1,2}(\mathbb{R}^n); \operatorname{div}_x(A\nabla_x u) \in L^2(\mathbb{R}^n)\}$, where $W^{1,2}(\mathbb{R}^n)$ denotes the usual $L^2$-Sobolev space of order 1. Functional analytic methods furnish fractional powers $L^\alpha$, $\alpha \in \mathbb{R}$, to such operators and the case of interest is $0 < \alpha < 1$. For $\alpha < 1/2$, Kato was able to identify the domain of $L^\alpha$ as the Sobolev space $W^{2\alpha,2}(\mathbb{R}^n)$ by interpolation methods. The situation when $\alpha \ge 1/2$ remained unsolved [46]. For $\alpha = 1/2$,<fnref n="1" /> Kato's conjecture (he claimed to have never explicitly formulated this conjecture but that it was extrapolated from his work by A. McIntosh, see [56]) is that the domain of $L^{1/2}$ identifies to $W^{1,2}(\mathbb{R}^n)$ with the equivalence

$$ (2.3) \qquad \|L^{1/2}u\|_2 \simeq \|\nabla_x u\|_2. $$

An interpretation in physics terms is that the kinetic energy of a wave propagating across a medium with conductivity coefficients given by $A$ is related to its spectral frequency, where spectrum is understood in the sense of the spectral theory for the operator $L$. When it is related to physics, the matrix $A$ is symmetric with real-valued entries and (2.3) is a consequence of spectral theory as $L$ is self-adjoint. But Kato wanted to understand the stability of the upper inequality in (2.3) with respect to $L^\infty$ perturbations of $A$: that is, small perturbation in the medium should imply small perturbation in the spectrum in the sense that

$$ (2.4) \qquad \|L_0^{1/2}u - L_1^{1/2}u\|_2 \lesssim \|A_0 - A_1\|_{\infty} \|\nabla_x u\|_2 $$

if $\|A_0 - A_1\|_{\infty}$ is small. This is related to the program described in his monumental book [47]. This is why the question is formulated for complex matrices: proving (2.3) for complex entries using only $L^\infty$ information on $A$ would automatically give analytic regularity, hence (2.4).

The construction of $L$ follows from the representation

$$ (2.5) \qquad \int_{\mathbb{R}^n} Lu(x) \overline{v(x)} \, dx = \beta(u, v) \quad (u \in D(L), v \in W^{1,2}(\mathbb{R}^n)) $$

with the sesquilinear form

$$ (2.6) \qquad \beta(u, v) := \int_{\mathbb{R}^n} \sum_{j,k=1}^{n} a_{jk}(x) \partial_k u(x) \overline{\partial_j v(x)} \, dx \quad (u, v \in W^{1,2}(\mathbb{R}^n)). $$

This construction of the operator falls within the theory of maximally accretive sesquilinear forms. If one forgets about elliptic operators and deals with an abstract form $B(u, v) = \langle ASu, Sv \rangle$ on a Hilbert space,

<footnote n="1">¹When $\alpha > 1/2$, one does not expect a possible identification with of the domain of $L^\alpha$ with a concrete function space when $A$ is only measurable.</footnote>
where A is a bounded and strictly accretive operator and S a self-adjoint operator, Kato's question reformulates as follows:

*Is the domain of the square root of the maximal accretive operator S*\*\*AS obtained from a maximally accretive sesquilinear form B equal to the domain of the form, that is the domain of S?

A. McIntosh found a counter-example [53], indicating that functional analytic methods would not suffice to prove (2.3) in full generality. But the question for the differential operators in (2.5) coming from the forms in (2.6) remained.

### 3. THE APPROACH OF COIFMAN, MCINTOSH, MEYER

We feel interesting to modestly plagiate the presentation by Yves in [58] as we want to give a stress on the Kato problem while he focused more on singular integrals. He writes

*“Morever, the same key [question posed by A. Calderón] led to striking discoveries in real and complex analysis.”*

As mentioned, this central question included the $L^2$ boundedness of the Cauchy integral operator. Some other striking discoveries came later in geometric measure theory: solution to the Vitushkin conjecture by G. David [33] and to the Painlevé conjecture by X. Tolsa [64]. See the article of J. Verdera in the same volume.

The Cauchy operator $C_Γ$ can be defined as follows: $Γ$ is here a Jordan Lipschitz curve in the plane, $s \mapsto z(s)$ a parametrization and one sets, using the notation p.v. for the principal value,

$$ (3.1) \qquad C_Γf(z(s)) = \text{p.v.} \frac{1}{\pi i} \int_{\mathbb{R}} \frac{f(z(t))z'(t)}{z(s) - z(t)} dt. $$

The approach was to expand the kernel in a multilinear series, to prove $L^2$ boundedness for the general terms and to control the radius of convergence. Multilinear series was a topic that Coifman and Meyer were exploring at this time using their so-called $P_t, Q_t$ spectral approach and which was explained in their Astérisque opus [29], see also [30]. When $Γ$ is the graph of a Lipschitz function A, choosing $z(x) = x + iA(x)$ and forgetting the harmless factor $z'(x)$, for $f$ now defined on $\mathbb{R}$, one studies

$$ C_{A}f(x) = \text{p.v.} \frac{1}{\pi i} \int_{\mathbb{R}} \frac{f(y)}{x-y} \left( 1 + i \frac{A(x) - A(y)}{x-y} \right)^{-1} dy $$

which can be expanded as $\sum_{k=0}^{\infty} (-i)^k T_{k,A}f$ where $T_{k,A}$ is the $k$th iterated Calderón commutator

$$ T_{k,A}f(x) = \text{p.v.} \frac{1}{\pi i} \int_{\mathbb{R}} \frac{f(y)}{x-y} \left( \frac{A(x) - A(y)}{x-y} \right)^k dy. $$

The term $k=0$ is the well-known Hilbert transform. A. Calderón was able to prove $L^2(\mathbb{R})$ operator bounds of the order $C^k \|A'\|_{\infty}^k$ for the $k$th
term with some unspecified, possibly large $C$, thus imposing $\|A'\|_{\infty}$ be small for the convergence of the series.

At the same time, A. McIntosh was exploring the possibility of proving Kato's question for differential operators. One of the many formulas to compute the square root is

$$ (3.2) \qquad L^{1/2}u = \frac{2}{\pi} \int_{0}^{\infty} (1 + t^2 L)^{-1} Lu \, dt, \quad u \in \mathcal{D}(L). $$

In one dimension the operator takes the form $L = DaD$ where $D = -i\frac{d}{dx}$ is self-adjoint (self-adjointess turned out to be important feature of the one dimensional case with no equivalent in higher dimension) and by abuse of notation we identify the function $a$ and the multiplication by $a$, which is here an accretive function, that is a bounded complex-valued function with real part bounded below by $\lambda > 0$. Expanding in power series with $a = 1 - m$, $\|m\|_{\infty} < 1$, yields

$$ L^{1/2}u = \sum_{k=0}^{\infty} \frac{2}{\pi} \int_{0}^{\infty} Q_t(m(1-P_t))^k (aDu) \frac{dt}{t} := \sum_{k=0}^{\infty} S_k(aDu) $$

with $P_t = (1 + t^2 D^2)^{-1}$ and $Q_t = tDP_t$. The strategy is to prove the boundedness of $S_k$ by iteration and to control the growth in $k$. Note that the first term $S_1$ contains $Q_t(m(1-P_t)) = Q_t m - Q_t m P_t$. Handling the term with $Q_t m P_t$ is already difficult and is what Coifman and Meyer could do. Actually, A. McIntosh showed that the $k$th iterated commutator of Calderón can be represented as

$$ T_{k,A} = \text{p.v.} \frac{1}{\pi i} \int_{\mathbb{R}} R_t(A'R_t)^k \frac{dt}{t} $$

where $R_t = P_t - iQ_t = (1 + itD)^{-1}$. Using this and taking into account evenness (in $t$) of $P_t$ and oddness of $Q_t$, this is in fact very close to the formula for $S_k$ so that the same technology can be used. For example,

$$ T_{1,A} = -\text{p.v.} \frac{1}{\pi} \int_{\mathbb{R}} (Q_t A' P_t + P_t A' Q_t) \frac{dt}{t}. $$

This is a term that the Coifman-Meyer technique could handle.

A. McIntosh told me that the first proof of the $L^2$ boundedness of the Cauchy integral was a reduction from the solution of Kato's question for the square root of $L = DaD$, that is,

$$ (3.3) \qquad \|(DaD)^{1/2}u\|_2 \simeq \|Du\|_2. $$

This is not the way it is presented in [28]. Actually, it has been shown later that the two results are equivalent, and we shall come back to this. R. Coifman, A. McIntosh and Y. Meyer eventually obtained in the bound $C_0(1 + k)^4 \|A'\|_{\infty}^k$ for the operator norm of $T_{k,A}$ using this representation, and a renormalisation showed that the assumption $\|A'\|_{\infty} < 1$ could be removed.
This is no accident that Cauchy integrals and Kato's square root belong to the same family of operators, as evidenced by the following identity. If one considers the operator $-i \frac{d}{dz}$ acting on smooth functions defined on the Lipschitz curve $\Gamma$, and use parametrization $z = z(x)$, $x \in \mathbb{R}$, then it can be shown it is similar to $aD$ with $a(x) := 1/z'(x)$, $D = -i \frac{d}{dx}$, and a calculation shows that

$$ (3.4) \qquad (aDaD)^{1/2}(u)(x) = \text{p.v.} \frac{1}{\pi i} \int_{\mathbb{R}} \frac{Du(y)}{z(y) - z(x)} dy, $$

where the definition of the square root given by (3.2) relies on the theory of sectorial operators that includes maximal accretive operators as a particular case. In different notation, this integral is the same as the one in (3.1).

Pushing the techniques of [28] further, C. Kenig and Y. Meyer wrote in the title of [48] that Cauchy integrals and Kato's square roots are the same (in one dimension). They proved

$$ (3.5) \qquad \|(bDaD)^{1/2}u\|_2 \simeq \|Du\|_2. $$

by showing that the multilinear expansions obtained for $(bDaD)^{1/2}$, with $b$ having the same properties as $a$, can be controlled with polynomial growth.

The Cauchy integral is an example of a Calderón-Zygmund operator. One can therefore push the ressemblance further and ask about the relation between $(bDaD)^{1/2}$ and Calderón-Zygmund operators. Later, P. Tchamitchian and I showed in [18] that even when $b \neq a$ so in particular when $b = 1$, $(bDaD)^{1/2} = RD$ where $R$ is a Calderón-Zygmund operator. The $L^2$ boundedness did not use multilinear expansions; it is a (simple) consequence of the $T(b)$ theorem of G. David, J.L. Journé and S. Semmes. Note that this reproved Kato's conjecture as a particular case, and reversed the chronology of ideas. Indeed, recall that the first version of a $T(b)$ theorem is due to A. McIntosh and Y. Meyer [57] using ideas from the proof of the Kato's conjecture.

The ideas surrounding the $T(b)$ theorem became the core of all further developments as we shall see.

## 4. THE SOLUTION OF THE SQUARE ROOT PROBLEM FOR ELLIPTIC OPERATORS

The solution in higher dimension came in several steps that took twenty years. In his 1990 survey article [56], A. McIntosh wrote "It remains a challenge, however, to prove the multilinear estimates needed to solve the square root problem of Kato for elliptic operators, or to find an alternative approach."

First, one had to give up the idea that the multilinear series that worked in one dimension would do the same in higher dimension. Such techniques were worked out by R. Coifman, D. Deng, Y. Meyer [27]
and independently by E. Fabes, D. Jerison and C. Kenig [36]. They were optimized by J. L. Journé [45]. The best bound he obtained is that, writing $A^{-1} = I - M$ with $||M||_\infty < 1$ (the $L^\infty$ norm of the matrix $M(x)$ as an operator on $\mathbb{C}^n$ with hermitian structure), the $k$th term in the multilinear expansion is controlled by $C^k |||M|||_\infty^k$ with $C$ on the order of $n^{1/2}$. This is not enough.

At the same time, while the Cauchy integral $L^2$ boundedness was given many new proofs essentially because of the tight relation with holomorphic functions, there were no new ones for the one-dimensional Kato problem until the nineties. One was using an adapted wavelet basis [17] but the same idea in higher dimension remains mysterious. The proof in [18] around the mid nineties showed that there was hope for a proof without doing such expansions by incorporating the $T(b)$ technology. But this proof (in one dimension) used the full force of Calderón-Zygmund theory. An example of C. Kenig, see [19], showed that square roots in higher dimension cannot be related to Calderón-Zygmund operators as some $L^p$ estimates usually obtained as a consequence of the Calderón-Zygmund extrapolation method fail.

Nevertheless, in any dimension, S. Semmes [62] proposed a variant of the $T(b)$ theorem requiring less information than the one needed for Calderón-Zygmund theory. Let us explain this and, at the same time, describe the role of $b$. This variant was a criterion to prove square function estimates of the form

$$ (4.1) \qquad \int_0^\infty \|\theta_t f\|_2^2 \frac{dt}{t} \lesssim \|f\|_2^2 $$

when $(\theta_t)_{t>0}$ is a family of operators acting on complex functions, with kernels $\theta_t(x, y)$ having good pointwise bounds and some regularity in the $y$ variable. In that case, it can be shown that (4.1) holds if and only if $|\theta_t 1(x)|^2 \frac{dx dt}{t}$ is a Carleson measure, which means that there is a constant $C < \infty$ such that for all cubes $Q \subset \mathbb{R}^n$ with sides parallel to the axes (Euclidean balls work as well), $\ell(Q)$ being its sidelength and $|Q|$ its volume,

$$ (4.2) \qquad \int_Q \int_0^{\ell(Q)} |\theta_t 1(x)|^2 \frac{dx dt}{t} \le C|Q|. $$

Here, $\theta_t 1$ is the operator $\theta_t$ applied to the constant function 1. But the calculation of $\theta_t 1$ might just be impossible so this could be useless. However, this equivalence contains the fact that the inequality

$$ \int_0^\infty \int_{\mathbb{R}^n} |\theta_t f(x) - \theta_t 1(x) \cdot S_t f(x)|^2 \frac{dx dt}{t} \le C \|f\|_2^2 $$

always holds, where $S_t$ is a dyadic martingale ($S_t f(x)$ is the average of the function $f$ on the dyadic cube containing $x$ with $\ell(Q)/2 \le t < \ell(Q)$). In passing, this reduction to a principal part $\theta_t 1(x) \cdot S_t f(x)$ is done using the $P_t, Q_t$ techniques of Coifman and Meyer. So if one
knows functions b for which $|\theta_t b(x)|^2 \frac{dx dt}{t}$ is a Carleson measure and the
real part of $S_t b(x)$ is bounded below, at least locally on each Carleson
window $Q \times (0, \ell(Q))$, one can conclude that (4.2) holds. In the one
dimensional application towards Kato's problem, where we recall that
$a$ is a bounded complex-valued function with real part bounded below
by a positive number, one has $\theta_t f = (1+t^2L)^{-1}(taf)'$ so that $\theta_t a^{-1} = 0$
and the above algorithm works out perfectly.

The approach of Semmes was inspiring for square roots in higher
dimension because some easy reductions from functional analysis tell us
that the upper estimate in (2.3) suffices and that this one is equivalent
to (4.1) with $\theta_t F = (1 + t^2L)^{-1}t \operatorname{div}_x (AF)$, $F$ being here $\mathbb{C}^n$-valued,
namely

$$
(4.3) \quad \int_0^\infty \int_{\mathbb{R}^n} |(1+t^2L)^{-1}t \operatorname{div}_x (AF)(x)|^2 \frac{dx dt}{t} \le C \|F\|_2^2.
$$

Despite the fact that formally $\theta_t A^{-1}(x) = 0$, one was missing pointwise
bounds, $y$-regularity in any sense, and the facts that $\theta_t I$ is $\mathbb{C}^n$-valued
($I$ being the identity matrix) and that $A$ is a matrix complicated the
algebra. Many reasons to stop there!

Nevertheless, a possible strategy was developed in the Astérisque
book with P. Tchamitchian [19]. Basically we proved in this rough con-
text (under some further technical hypotheses) that the same reduction
to (4.2) as in Semmes' work holds together with the valid estimate

$$
\int_0^\infty \int_{\mathbb{R}^n} |\theta_t \nabla_x f(x) - \theta_t I(x) \cdot S_t \nabla_x f(x)|^2 \frac{dx \, dt}{t} \le C \|\nabla_x f\|_2^2.
$$

Here, writing $\theta_t I(x) = (\gamma_{t,1}(x), \dots, \gamma_{t,n}(x))$,

$$
\theta_t I(x) \cdot S_t \nabla_x f(x) = \gamma_{t,1}(x) S_t(\partial_{x_1} f)(x) + \dots + \gamma_{t,n}(x) S_t(\partial_{x_n} f)(x).
$$

This led us to formulate a *T*(b) criterion for square roots with the new
idea that the test function to construct could be not just one function
but a family of functions (*F*<sub>Q</sub>) indexed by cubes: one would need

* an $L^2$ control on their gradients on an enlargement of the cube $Q$,

* that $F_Q$ is adapted to $L$, in the sense that $|\theta_t \nabla_x F_Q(x)|^2 \frac{dx dt}{t}$ is controlled on an enlargement of the Carleson window $Q \times (0, \ell(Q))$,

* and that $\theta_t I(x) \cdot S_t \nabla_x F_Q(x)$ controls $|\theta_t I(x)|$ on the Carleson window.

The first two requirements are not so difficult to achieve but the last
one is complicated because $\theta_t I(x)$ is $\mathbb{C}^n$-valued, and also because the
control is not on the Carleson window in general but on a subregion.
Fortunately, the size of missing region where there is no control can be
estimated using stopping-time arguments (this is where dyadic martingales are very useful).
This $T(b)$ criterion for square roots turned out to be the successful approach. S. Hofmann and A. McIntosh [41] were able to apply this criterion to prove Kato's conjecture in dimension $n = 2$. The idea brought by M. Lacey for arbitrary dimension in [40] was to use a sectorial decomposition of $\mathbb{C}^n$ to force $\theta_t I(x)$ to take values in small cones about a finite number of unit vectors $w$ by cut-off and, for each fixed $w$, to choose the $F_Q$ so that $\nabla F_Q$ is close to the direction $\bar{w}$ in average on the cube $Q$. Removal of the assumption of pointwise bounds on the kernels of the heat semigroup $e^{-t^2L}$ in [40] was done in the final assault [12].

Let me mention that M. Christ had already the idea to use varying test functions with $L^\infty$ control to prove a “local” $T(b)$ theorem for singular integrals in his article about analytic capacity [26]. His proof does not allow for a kind of $L^p$ control with $p$ finite though. The possibility of having $L^p$ control with finite $p$ generated some search to refine the exponents and testing conditions in local $T(b)$ theory for singular integrals, motivated by this use in partial differential equations. The best (and nearly optimal) result is due to T. Hytönen and F. Nazarov [42].

The strategy of proof of Kato's square root problem is very flexible and allows generalizations where the proof adapts: elliptic systems of arbitrary high order possibly having lower order terms with bounded measurable coefficients (or even unbounded in some natural spaces in order to preserve ellipticity); Ellipticity can be relaxed to a Gårding inequality (which is weaker than the pointwise ellipticity (2.1)). Lower order terms do bring technical difficulties that are minor compared to the treatment of the highest order terms. Ellipticity can also degenerate in weighted sense, see [31].

## 5. THE FIRST ORDER APPROACH

Even before Kato's question was solved in higher dimension, a first order approach was explored jointly with A. McIntosh and A. Nahmod in the case of the one dimensional setting [16]. The idea is to see the second order operator $L = -b(x)\frac{d}{dx}(a(x)\frac{d}{dx})$ (the same as $bDaD$ above) as (part of) the square $T^2$ of a first order differential system $T$ whose domain is $W^{1,2}(\mathbb{R}; \mathbb{C}^2)$ and whose spectrum belongs to a bisector $|\arg(\pm z)| \le \omega_T < \pi/2$ (such operators belong to the class of bisectorial operators). Then identification of the domain of $L^{1/2}$ follows from knowing the one of $(T^2)^{1/2}$ and one hopes for

$$ (5.1) \qquad \|(T^2)^{1/2}u\|_2 \sim \|Tu\|_2 $$

for all $u$ in the domain of $T$. It amounts to showing that $(T^2)^{1/2}T^{-1}$, restricted to the range of $T$, is bounded on $L^2$. The advantage of this point of view is that all can be seen within the holomorphic functional
calculus of $T$ and for this A. McIntosh had established a criterion using quadratic estimates (see [55]). More precisely, he established that bounded and holomorphic functions of bisectorial operators (this was done for sectorial operators but the proof is the same) $T$ on larger sectors than the one given by $\omega_T$, generate bounded operators if and only if there is equivalence between a certain square function norm and the ambient norm:

$$ (5.2) \qquad \int_0^\infty \|\lambda T(1 + \lambda^2 T^2)^{-1} h\|_2^2 \frac{d\lambda}{\lambda} \approx \|h\|_2^2 $$

for all $h$ in the closure of the range of $T$. The function $(z^2)^{1/2}z^{-1}$ being bounded and holomorphic, this shows that (5.1) follows from (5.2). Here, $T = \mathbf{BD}$ with

$$ \mathbf{B} = \begin{bmatrix} b & 0 \\ 0 & a \end{bmatrix}, \quad \mathbf{D} = \begin{bmatrix} 0 & \frac{d}{dx} \\ -\frac{d}{dx} & 0 \end{bmatrix} $$

so that

$$ T^2 = \begin{bmatrix} -b(x) \frac{d}{dx} (a(x) \frac{d}{dx}) & 0 \\ 0 & -a(x) \frac{d}{dx} (b(x) \frac{d}{dx}) \end{bmatrix} = \begin{bmatrix} b D a D & 0 \\ 0 & a D b D \end{bmatrix}. $$

In one dimension, one can obtain kernel bounds for the operators $\lambda T(1+\lambda^2 T^2)^{-1}$ so that the square function estimate (5.2) can be shown following the Semmes version of $T(b)$ theorem for square functions.

We pause to come back to the affirmation that Kato's square root and Cauchy integrals are the same. Using this setup one can indeed show without computing integral formulas as before, but only using functional analytic tools such as interpolation, that (3.3) for all $a$, (3.5) for all $a, b$, and (3.5) for all $a = b$ are equivalent, see [16]. Thanks to (3.4), this is equivalent to the $L^2$ boundedness of the Cauchy integral operator.

As we saw, the solution of Kato's square root problem in all dimension follows from a $T(b)$ strategy even though one does not have pointwise kernel bounds. A. Axelsson, S. Keith and A. McIntosh found a way to extend the first order approach presented above in one dimension to higher dimension using analogy with metric perturbations of Dirac operators and were able to formalise the $T(b)$ argument within this framework [21]. In differential geometry, the Dirac operator is $\Pi = d + d^*$ where $d$ is the exterior derivative on forms and $d^*$ its adjoint with respect to the $L^2$ inner product. What really matters here is that $d$ is a nilpotent first order constant coefficient differential operator and that $\Pi$ is coercive on its range: $\|\nabla u\|_2 \lesssim \|\Pi u\|_2$ for $u$ in the $L^2$ range of $\Pi$. Metric perturbations take the form $\Pi_B = d + B_1 d^* B_2$ where $B_1, B_2$ are bounded operators and $B_1$ is the inverse of $B_2$ (being usually a multiplication operator). In general, it suffices to assume that $B_1, B_2$ enjoy some partial coercivity (a lower bound for the real parts of $\langle B_i h, h \rangle$ for $h$ in appropriate subspaces) and that $d^* B_2 B_1 d^* = 0$
and $dB_1B_2d = 0$. For applications to Kato's problem, it suffices to set this framework on 0 and 1 forms, but further applications to solving partial differential equations for forms may require to use the full exterior algebra. They establish a (sufficient) criterion to prove (5.2) for $T = \Pi_B$ when it is additionally required that $B_1, B_2$ are multiplications by bounded matrix-valued functions. According to A. McIntosh's result, (5.2) for $\Pi_B$ yields the $L^2$ boundedness of $(\Pi_B^2)^{1/2}\Pi_B^{-1}$ defined on the range of $\Pi_B$. This is a stronger statement as it implies as corollaries Kato's square root conjecture for equations and systems in all dimension, the $L^2$ boundedness of the Cauchy integral operator and its Kenig-Meyer extension (3.5) for all $a, b$ and more. In this framework, $d$ can be replaced by a nilpotent constant coefficient first order differential operator provided that the triple $(\Gamma, B_1, B_2)$ satisfies the above requirements.

## 6. THE LIONS PROBLEM

Jacques-Louis Lions asked whether Kato's square root problem can be addressed on a domain for operators with mixed boundary conditions (possibly allowing lower order terms with bounded coefficients) [52]. That is, on a part of the boundary, the operator comes with Dirichlet condition and on the “complement” the operator comes with Neumann condition. The construction of the operator uses the theory of quadratic forms. The first positive results for Lipschitz domains and pure Dirichlet or Neumann conditions are in a work with P. Tchamitchian, roughly using localization and extension techniques to reduce to the case of $\mathbb{R}^n$ [20]. For mixed conditions, such techniques do not apply and it was solved by A. Axelsson, S. Keith and A. McIntosh on Lipschitz domains by implementing their first order approach [22].

The state of the art is a result by S. Betchel, M. Egert and R. Haller-Dintelmann in geometries beyond the Lipschitz domains: Lions question has a positive answer on an open and possibly unbounded set in $\mathbb{R}^n$ under two simple geometric conditions: The Dirichlet boundary part is Ahlfors-David regular and a quantitative connectivity property in the spirit of locally uniform domains holds near the Neumann boundary part. For example, the Neumann boundary condition can be treated on $(\varepsilon, \delta)$-domains, in particular on the interior of the von Koch snowflake. We refer to [24].

## 7. ELLIPTIC BOUNDARY VALUE PROBLEMS

As mentioned earlier, the estimate (2.3) for $L = -\operatorname{div}_x(A(x)\nabla_x)$ has consequences for the boundary value problems of the elliptic equation

$$ (7.1) \qquad \partial_t^2 u(t, x) + \operatorname{div}_x(A(x)\nabla_x u(t, x)) = 0 $$

where $x \in \mathbb{R}^n$ and $t > 0$ and the boundary is identified with $\mathbb{R}^n$. Indeed, if $f$ is a regular Dirichlet data with $\|\nabla_x f\|_2 < \infty$ (this is why this is
tagged as the regularity problem), then the formal solution $u(t, x) = e^{-tL^{1/2}} f(x)$, satisfies $\sup_{t>0} \|\nabla_{t,x}u\|_2 \sim \|\nabla_x f\|_2$; if $g$ is a Neumann data with $\|g\|_2 < \infty$ then the formal solution $u(t, x) = -e^{-tL^{1/2}}(L^{-1/2}g)(x)$, has conormal derivative $g$ at $t = 0$, and satisfies $\sup_{t>0} \|\nabla_{t,x}u\|_2 \sim \|g\|_2$. Actually, much more can be said in terms of non-tangential maximal functions estimates for such solutions and also that they are unique with such requirements. The state of the art, together with extension to other boundary spaces which require the theory of Hardy spaces adapted to $L$, appears in a recent work with M. Egert [9].

Equation (7.1) is a particular occurence, called block form, of the more general elliptic equations

$$ (7.2) \qquad -\operatorname{div}_{t,x}(A(x)\nabla_{t,x}u(t,x)) = 0, $$

where $A$ is a full $(n+1) \times (n+1)$ matrix of bounded functions independent of the transverse direction $t$ to the boundary. The ellipticity is given by (2.1) in $n+1$ dimension.

The motivating example is a simple pullback of the Laplace equation $\Delta u = 0$ above a Lipschitz graph to flatten the boundary: the solvability of the Dirichlet problem for data in $L^2$ in such a situation was proved by B. Dahlberg [32] and the regularity and Neumann problems in $L^2$ by D. Jerison and C. Kenig [44] based on a Rellich identity that became central in the topic. The $L^2$ boundedness of the double layer potential proved by R. Coifman, A. McIntosh and Y. Meyer as a consequence of the method of rotations from the boundedness of the Cauchy integral operator could then be used, as G. Verchota showed in 1984 [65], to reprove such results by the method of layer potentials. For real and symmetric operators of the form (7.2), solvability of the Dirichlet problem in $L^2$ was obtained by D. Jerison and C. Kenig [43], and solvability of the regularity and Neumann problems in $L^2$ by C. Kenig and J. Pipher in the mid nineties [51], while the method of layer potentials was developed a decade ago [1] thanks to the construction of fundamental solutions by S. Hofmann and S. Kim [39]. This allowed additional perturbation results, while the Rellich identities are very rigid and do not allow for perturbations.

In the generic situation of (7.2), one could ask what happens when the coefficients also depend on $t$ but counter-examples by A. Caffarelli, E. Fabes and C. Kenig [23] show that there is no hope for positive results without perturbative assumptions with respect to $t$. We refer to the book by C. Kenig [49] for the state of the art until the mid nineties but the topic has grown since.

It was not clear at all whether the connexion between square roots and boundary value problems extends to non-block form equations. In fact, the first order approach appears to be more adapted, in the spirit of the Cauchy-Riemann system in two dimension and the div-curl
systems of E. Stein and G. Weiss in higher dimension for the Laplace equation [63].

The work with A. Axelsson and S. Hofmann brought the first stone [5]: the strategy in that paper begins by writing (7.2) in the equivalent form

$$ (7.3) \qquad \begin{cases} \operatorname{div}_{t,x} A(x)F(t,x) = 0, \\ \operatorname{curl}_{t,x} F(t,x) = 0, \end{cases} $$

with $F(t,x) = \nabla_{t,x}u(t,x)$ as unknown. Note that the second equation is a dummy one if one knows $u$; however, this allows one to recover a gradient among all $F$ solving the first equation. Working out the system reveals the evolution equation $\partial_t F + T_A F = 0$, where $T_A$ is a first order differential operator related to the $\Pi_B$ operator as in [21]. Solving the boundary value problems can be done in two steps. First solve the initial value problem for the evolution equation when $t > 0$. This can be done only for initial data $F_0$ in a spectral subspace $\mathcal{H}_{T_A}^+$, called Hardy space adapted to $T_A$: $F$ is then obtained by applying an abstract Cauchy operator for $T_A$ given by the semigroup evolution for $-[T_A] = -(T_A^2)^{1/2}$ and the projection $\chi_+(T_A)$ onto that subspace:

$$ F = e^{-t[T_A]} \chi_+(T_A) F_0. $$

The second step is to construct a Hardy data $F_0$ from the knowledge of the (Dirichlet or Neumann) boundary data: this amounts to inverting operators from the Hardy space into the boundary spaces. Roughly, this is an abstract way of thinking the Cauchy extension from a Lipschitz graph $\Gamma$: the Cauchy integral of an $L^2(\Gamma)$ function furnishes a holomorphic function on the complement of the graph. Its traces on the graph from above and below furnish two functions in subspaces of $L^2$, called the (holomophic) Hardy space $\mathcal{H}^+(\Gamma)$ by analogy with what happens in the flat case of the real line. The $L^2$ boundedness of the Cauchy integral operator is precisely what makes the splitting $L^2(\Gamma) = \mathcal{H}^+(\Gamma) + \mathcal{H}^-(\Gamma)$ topological.

Each step applies only for $A$'s being a $L^\infty$ perturbation of well understood cases such as constant coefficients or real symmetric (or even complex hermitian). This is because $T_A$ contains extra multiplication terms compared to $\Pi_B$ and this complicates the analysis.

An important simplification made with A. Axelsson and A. McIntosh around 2010 in [6] is that there is a better choice of unknown function than the gradient of $u$ to do the first order approach: it is the conormal gradient $\tilde{F} := \nabla_A u$. The latter is a vector function obtained by replacing in $F$ the time derivative $\partial_t u$ by the conormal derivative $e_0 \cdot A \nabla_{t,x} u$ where $e_0$ is the unit vector normal and upward to the boundary. The tangential part of $\tilde{F}$ is still of gradient form (or is tangential curl free). Then (7.2) is equivalent to an evolution equation $\partial_t \tilde{F} + DB\tilde{F} = 0$
where

$$ (7.4) \qquad D = \begin{bmatrix} 0 & \operatorname{div}_x \\ -\nabla_x & 0 \end{bmatrix}, \quad B = \begin{bmatrix} I & 0 \\ A_{\parallel\perp} & A_{\parallel\parallel} \end{bmatrix} \begin{bmatrix} A_{\perp\perp} & A_{\perp\parallel} \\ 0 & I \end{bmatrix}^{-1} $$

when writing $A = \begin{bmatrix} A_{\perp\perp} & A_{\perp\parallel} \\ A_{\parallel\perp} & A_{\parallel\parallel} \end{bmatrix}$. The representation as $2 \times 2$ matrices corresponds to the splitting $\mathbb{C}^{1+n} = \mathbb{C} \times \mathbb{C}^n$ in (scalar) normal and (vector) tangential coordinates and where $\tilde{F}(t, \cdot)$ is required to belong to the closure of the range of $D$ for all $t > 0$. The operator $D$ is self-adjoint on $L^2(\mathbb{R}^n; \mathbb{C}^{1+n})$ and $A \mapsto B$ is a self-inverse map on bounded and accretive matrices, so that $DB$ is again defined as bisectorial operator. The square function estimate for $DB$ can be proved directly once again by a $T(b)$ argument or deduced from the $\Pi_B$ formalism of [21] (actually, it is proved in [7] by functional analytic arguments that they are equivalent up to isomorphisms,) and the gain of this $DB$ formalism over the $T_A$ formalism is that the first step of the strategy described above can be shown for all complex matrices $A$ (in retrospect, it does too for $T_A$ but it could not be seen directly); it is only for the second step that things can be more subtle and, so far, there is no full understanding of it.

In the last ten years, the $DB$ formalism has been explored with great success in its relation to elliptic boundary value problems for equations or systems with new results on extrapolation to other boundary spaces [15], solvability results and uniqueness [8, 14], solvability and uniqueness for data with fractional regularity [2], perturbation results with $t$-dependent coefficients [4], etc. Compared to the technology developed for second order equations with real coefficients, an advantage is that it completely avoids use of maximum principle to establish the needed square functions. It also imposed to streamline the overall intrication of various estimates in the field.

## 8. LINK BETWEEN FIRST ORDER APPROACH AND LAYER POTENTIALS

We wish to isolate a spectacular success of the first order $DB$ approach, due to A. Rosén [61].

As said earlier, there is a construction of fundamental solutions for the equations (7.2) in dimension $n + 1 \ge 3$ assuming that solutions of such equations enjoy local boundedness and Hölder regularity in the spirit of de Giorgi's results for real equations. Basically, the fundamental solution should be a function $\Gamma(t, x, s, y)$ with

$$ -\operatorname{div}_{t,x}(A(x)\nabla_{t,x}\Gamma(t, x, s, y)) = \delta(s, y), $$

$$ -\operatorname{div}_{s,y}(A^*(y)\nabla_{s,y}\Gamma^*(s, y, t, x)) = \delta(t, x), $$

where $\Gamma^*(s, y, t, x) = \overline{\Gamma(t, x, s, y)}$ and $\delta(X)$ denotes the Dirac mass at the point $X$. It is constructed in such a way to obtain bounds at $\infty$.
The t-independence of the coefficients, which we assume from now on, further implies that $\Gamma(t, x, s, y) = \Gamma(t - s, x, 0, y)$.

This can be used to write out the single and double layer potentials by formally setting

$$ 
\begin{aligned}
S_t f(x) &= \int_{\mathbb{R}^n} \Gamma(t, x, 0, y) f(y) \, dy \quad (t \neq 0) \\
D_t f(x) &= \int_{\mathbb{R}^n} f(y) e_0 \cdot \overline{A^*(y) \nabla_{s,y} \Gamma^*(s, y, t, x)|_{s=0}} \, dy \quad (t > 0).
\end{aligned}
$$

These expressions naturally arise for smooth coefficients when invoking the Green's formula. In his 2006 address to the ICM [37], S. Hofmann formulated the conjecture that $\nabla_{t,x}S_t$ and $D_t$ should be uniformly bounded as operators on $L^2(\mathbb{R}^n)$ when $t > 0$ and have strong limits at $t \to 0$ with bounds only involving dimension and the ellipticity constants. It was known that such bounds hold provided the Dirichlet problem for equations (7.2) is solvable for $L^2$ data (thus, the bounds also depend on this information). The question was to eliminate this a priori information since it is not known in full generality.

This conjecture was first proved by A. Rosén around 2010. He showed that $S_t$ and $D_t$ can be respectively identified to operators $S_t$ and $D_t$ arising from the holomorphic functional calculus of DB and BD respectively, with $D$ and $B$ defined in (7.4). Consequently, he showed that $\nabla_{t,x}S_t$ and $D_t$ are bounded for all complex $A(x)$ as in (7.2). He is neither assuming local boundedness nor Hölder regularity of solutions. It is only when these assumptions are made that the single and double layer operators have an integral representation as above.

Let us explain Rosén's idea. Writing $\mathcal{L}u = -\operatorname{div}_{t,x}(A(x)\nabla_{t,x}u)$ as an operator from the homogeneous Sobolev space $\dot{W}^{1,2}(\mathbb{R}^{n+1})$ into its dual, it is invertible and the fundamental solution can somehow be interpreted as a distributional kernel of $\mathcal{L}^{-1}$. But $\mathcal{L}^{-1}$ and the single layer operator $S_t$ of Rosén are related as follows. Assume $f \in C_0^\infty(\mathbb{R}^{n+1})$. If $n = 1, 2$ assume furthermore that $f = \operatorname{div}_x F$ for some $F \in C_0^\infty(\mathbb{R}^{n+1}; \mathbb{C}^n)$. Then, he proves

$$ (8.1) \qquad (\mathcal{L}^{-1}f)(t, x) = \operatorname{p.v.} \int_{\mathbb{R}} S_{t-s} f_s(x) \, ds, \quad \text{in } \dot{W}^{1,2}(\mathbb{R}^{n+1}), $$

where $f_s(x) = f(s, x)$. Thus, one can construct $\mathcal{L}^{-1}$ (and prove estimates for the fundamental solution) from knowledge of $S_t$. Conversely, $\mathcal{L}^{-1}$ completely determines $S_t$ (recall we assume t-independence here) exactly as the fundamental solution determines the single layer potential by fixing $s = 0$. More precisely, let $f \in C_0^\infty(\mathbb{R}^n)$ with $\int_{\mathbb{R}^n} f(x) \, dx = 0$ if $n = 1, 2$ and $\chi_\varepsilon(s) = \frac{1}{\varepsilon}\chi(\frac{s}{\varepsilon})$ with $\varepsilon > 0$ and $\chi \in C_0^\infty(\mathbb{R})$ satisfying $\int_{\mathbb{R}} \chi(s) \, ds = 1$. Set $f_\varepsilon(s, x) := \chi_\varepsilon(s)f(x)$. Then for all $t \in \mathbb{R}$, $(\mathcal{L}^{-1}f_\varepsilon)_t$ converges in $\dot{W}^{1,2}(\mathbb{R}^n)$ to $S_t f$ as $\varepsilon \to 0$ [8]. Hence, knowledge of $\mathcal{L}^{-1}$ is dictated by the DB-functional calculus, and the
approach of Rosén is natural once one knows that this holomorphic
functional calculus is bounded!

9. THE DIRICHLET PROBLEM FOR REAL BUT NON-SYMMETRIC ELLIPTIC OPERATORS

As said above, the $L^2$ solvability of the Dirichlet problem for real and symmetric elliptic operators as in (7.2) had been known since the early eighties in [43]. The symmetry assumption was used to prove by a clever argument and integration by parts an integral identity which implies control of the conormal derivative by the tangential derivative in $L^2(d\sigma)$ norm on the boundary. By known manipulations, this identity yields the desired estimates.

C. Kenig [49] asked whether one can drop the symmetry assumption. But then, the integral identity is no longer available. At least, can one prove that $L$-harmonic measure is $A_\infty$ with respect to Lebesgue measure on the boundary? This fact implies $L^p$ solvability of the Dirichlet problem for some large unspecified exponent $p$. The $L$-harmonic measure is not just one measure. It is in fact the family of Borel measures $\omega_X$, indexed by points $X$ of the upper-half space, obtained by applying the Riesz representation theorem to the maps sending any bounded non-negative compactly supported continuous function $f$ on the boundary to the value at $X$ of the (unique) non-negative solution with data $f$. These measures are all mutually absolutely continuous thanks to the Harnack principle, but this is not (yet) enough to conclude for the $A_\infty$ property with respect to surface (Lebesgue) measure.

With H. Koch, J. Pipher and T. Toro [50], he was able to give a positive answer in dimension $n + 1 = 2$ in 2000, transposing ideas on $\varepsilon$-approximability of harmonic functions to solutions of the equation and using transformations of the operator that can only be done in two dimension.

In 2014, C. Kenig, S. Hofmann, S. Mayboroda and J. Pipher settled this question in all dimension [38]. Let us quote their article: "The non-symmetric case, however, was not achievable via previously devised methods (...), while being important for several independent reasons. First, the well-posed results for equations with real non-symmetric coefficients and associated estimates on solutions provide the first step towards understanding the operators with complex coefficients, in the non-Hermitian case. The latter is absolutely necessary to establish analyticity of the solution as a function of the coefficients even when the coefficients are real, one of Kato's long-time goals, which would then allow one to study also hyperbolic problems (see [13] for the "block matrix" case) [see also [54]]. From the analytic point of view, this can be viewed as a far-reaching extension of the Kato square root problem and Kato's program. Furthermore, the equations with complex coefficients offer the simplest model of elliptic systems retaining their major
difficulties, with multiple entry points to the theory of elasticity and other applications.”

Here, the main new ingredients are the square functions related to Kato's square root problem for $-\operatorname{div}_x A_{\parallel\parallel}(x)\nabla_x$, where $A_{\parallel\parallel}(x)$ appears in the $2 \times 2$ matrix form as in Section 8. The heuristic and vague idea behind the use of such estimates is roughly to treat (7.2) in perturbation form

$$-\operatorname{div}_x(A_{\parallel\parallel}(x)\nabla_x u(t, x)) = \operatorname{div}_x(A_{\parallel\perp}(x)\partial_t u(t, x)) \\ + \partial_t(A_{\perp\parallel}(x)\nabla_x u(t, x)) + \partial_t(A_{\perp\perp}(x)\partial_t u(t, x)).$$

However, there is no smallness attached to such a decomposition. The term with $A_{\parallel\perp}$ is handled thanks to the square function bounds (4.1) for $\theta_t = (1 - t^2 \operatorname{div}_x(A_{\parallel\parallel}(x)\nabla_x))^{-1}t \operatorname{div}_x$, while the term with $A_{\perp\parallel}$ requires these bounds with $\theta_t = (1 - t^2 \operatorname{div}_x(A_{\parallel\parallel}^*(x)\nabla_x))^{-1}t \operatorname{div}_x$. The last term offers no special difficulty.

## 10. THE SQUARE ROOT PROBLEM FOR PARABOLIC OPERATORS

Parabolic equations and elliptic equations usually are closely related. For example, if one considers

$$(10.1) \qquad H(u) := \partial_t u - \operatorname{div}_x(A(x, t)\nabla_x u)$$

for $t \in \mathbb{R}$ and $x \in \mathbb{R}^n$, with $A$ bounded and satisfying (2.1) also uniformly with respect to $t$, one may ask whether the technology seen above for elliptic operators $L$ transfers to parabolic operators $H$. Until recently, no results were available when the $t$-regularity of the coefficients is merely measurable and one often has to assume at least a control on half-order derivatives with respect to $t$. In a series of papers, K. Nyström and collaborators explored such situations in order to construct tools for solving boundary value problems for parabolic equations [25, 59, 60]. It is assumed time-independence and they show that the square function estimates extend in parabolic setting, i.e., replacing $L$ by $H$ in (4.3) (and the $t$ variable integration should be renamed to avoid confusion with time), using again a $T(b)$ technology. These works opened new doors.

The challenge was to understand whether these square function estimates hold without the time-independent assumption. In fact, these estimates are akin to show a Kato's square root estimate for $H$.<fnref n="2" /> What does that mean?

In order to set the context, define the energy space

$$V := H^{1/2}(\mathbb{R}; L^2(\mathbb{R}^n)) \cap L^2(\mathbb{R}; W^{1,2}(\mathbb{R}^n)),$$

<footnote n="2">²E. Ouhabaz (private communication) told us afterwards that the time-independence allows one to use rather well-known techniques of functional analysis involving interpolation and maximal regularity to obtain (10.2) below. But this no longer applies in the time-independent case, and it was again fortunate that one did not know this abstract argument before.</footnote>
contained in $L^2(\mathbb{R}^{n+1})$ with norm

$$ \|u\|_V = \left( \|u\|_2^2 + \|\nabla_x u\|_2^2 + \|D_t^{1/2} u\|_2^2 \right)^{1/2} $$

where $D_t^{1/2}$ is the half-order derivative in the $t$ variable defined through the Fourier transform with the multiplier $|\tau|^{1/2}$. Then define the parabolic operator $H$ as an operator $V \to V^*$ via a sesquilinear form,

$$ \langle Hu, v \rangle := \iint_{\mathbb{R}^{n+1}} A \nabla_x u \cdot \overline{\nabla_x v} + H_t D_t^{1/2} u \cdot \overline{D_t^{1/2} v} \, dx \, dt \quad (u, v \in V), $$

where $H_t$ is the Hilbert transform in the $t$ variable defined in such a way that $\partial_t = D_t^{1/2} H_t D_t^{1/2}$. Carving out the surprising analogy with elliptic operators even further, one can show that $H$ with maximal domain $\{u \in V : Hu \in L^2(\mathbb{R}^{n+1})\}$ in $L^2(\mathbb{R}^{n+1})$ is a maximal-accretive operator, hence it possesses a square root. With M. Egert and K. Nyström [10], we showed that the domain of $H^{1/2}$ is that of the accretive form, that is $V$, with the two-sided estimate

$$ (10.2) \qquad \|H^{1/2} u\|_2 \sim \|\nabla_x u\|_2 + \|D_t^{1/2} u\|_2 \quad (u \in V) $$

and implicit constants depending only upon $n$ and ellipticity constants of $A$. The reader should remark that not even the case $A^* = A$ can be treated by abstract functional analysis (while this is the case for Kato's problem for elliptic operators) because $H$ is never self-adjoint.

The original proof uses $T(b)$ technology applied with a “parabolic” first order formalism of the type DB: $D$ is replaced by a non self-adjoint operator that involves the non-local $D_t^{1/2}$ and it is therefore very delicate to handle for these two reasons. It yields the boundedness of a certain square function and, hence, the boundedness of the holomorphic functional calculus for this first order operator. Having this at hand opens again more doors to handle boundary value problems for parabolic operators on $\mathbb{R}_+^{n+2}$. Afterwards, a different proof of the square root estimate not using the first order approach has been recently obtained in [3], even under degenerate ellipticity conditions on $A$.

As a side product, these square functions bounds allowed us to solve the Dirichlet problem for $\partial_t u - \operatorname{div}_{\lambda,x}(A(x,t)\nabla_{\lambda,x}u) = 0$ when $t \in \mathbb{R}, x \in \mathbb{R}^n, \lambda > 0$ with $A$ being an $(n+1) \times (n+1)$ bounded and real matrix satisfying (2.1) in dimension $n+1$, for data in $L^p(\mathbb{R}^{n+1})$ for large $p$ [11]. As is in the elliptic case, this is akin to proving that the caloric measure for this equation is $A_\infty$ with respect to Lebesgue measure $dx \, dt$ on $\mathbb{R}^{n+1}$, and this is obtained by showing a (parabolic) Carleson measure estimate for $\lambda|\nabla_{\lambda,x}u|^2 dx \, dt \, d\lambda$ whenever $u$ is a bounded solution. For this, the argument from [38] in the elliptic case was streamlined to avoid some unnecessary steps and transposed to parabolic setting.
## 11. CONCLUSION

In retrospect, although it has been abandoned later on, the multi-linear approach taken by Yves Meyer to prove the boundedness of the Cauchy integral can be considered as a miracle. Indeed, later proofs of that result have been devised within complex function theory, without using multilinear analysis and real harmonic analysis. The multilinear approach was not only useful to solve a fundamental question, but also allowed him to formulate the $T(b)$ theorem that would become a central tool. This had tremendous consequences in singular integral operator theory and geometrical measure theory, and also unsuspected and fruitful developments towards functional calculus and boundary value problems. This tells us how profound is the impact of Yves Meyer's original contributions on the topics evoked in this note.

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

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