Abstract
We consider the Kato square root problem for nondivergence second order elliptic operators \(L = \sum _{i,j=1}^{n}a_{ij} D_iD_j\), and, especially, the normalized adjoints of such operators. In particular, our results are applicable to the case of real coefficients having sufficiently small BMO norm. We assume that the coefficients of the operator are smooth, but our quantitative estimates do not depend on the assumption of smoothness.
Similar content being viewed by others
Avoid common mistakes on your manuscript.
1 Introduction and Main Result
In [8], the authors resolved in the affirmative the longstanding square root problem of Kato for divergence form complexelliptic operators in \(\mathbb {R}^n\). This was the culmination of a series of previous results: [16] (treating the 1d case); [15] and [26, 27] (treating small perturbations of the constant coefficient case); [37] (the 2d case); [9] (small perturbations of the realsymmetric case, sometimes referred to as the restricted Kato problem); and [33] (the case that the heat kernel satisfies a Gaussian upper bound and Nashtype Hölder continuity estimates). The solution of the Kato problem, and the circle of ideas involved in its proof, led to subsequent breakthroughs in the theory of elliptic boundary value problems, see, e.g., [1,2,3,4,5, 10, 12, 18, 25, 30,31,32, 34,35,36]. See also the significant related groundbreaking work in the parabolic setting: [6, 7, 43].
In this note, we initiate the study of the square root problem in the nondivergence setting in dimensions greater than 1, with the eventual goal of developing applications to the theory of boundary value problems, as has been done in the aforementioned divergence form case. Previously, the nondivergence problem had been treated only in the 1dimensional setting, in [39]; in fact, in that paper the authors treat the more general class of operators of the form \(L=aDbD\), where D denotes the ordinary differentiation operator on the line, and a, b are arbitrary bounded accretive complexvalued functions on the line.
At present, we are able to treat only the case of real coefficients. On the other hand, we point out that in the divergence form case, there is no known proof for real, nonsymmetric coefficients that is fundamentally easier than the proof in the general case, owing to the nonselfadjointness of nonsymmetric divergence form operators. Moreover, it is the real, nonsymmetric case that underlies the breakthrough in the study of the Dirichlet problem obtained in [31]. We observe that in the nondivergence setting, we may assume without loss of generality that the coefficient matrix is symmetric, but in contrast to the divergence form case, operators of nondivergence type are inherently nonselfadjoint, even with symmetric coefficients.
A fundamental difficulty that one encounters in the nondivergence setting, is that the Kato problem for nondivergence elliptic operators seems to be most naturally formulated in a weighted \(L^2\) space, and in general, the weight need not belong to the Muckenhoupt \(A_2\) class. Another difficulty inherent to the nondivergence setting is the lack of uniqueness (see the work of Nadirashvili [42]). On the other hand, working with real coefficients allows us to make use of the pioneering work of Krylov and Safonov [40], as well as the important ideas of Baumann [11] and Escauriaza [24]. We shall return to these matters in the sequel. First, let us set notation and definitions.
We will say that the operator L is a secondorder elliptic operator in nondivergence form on \(\mathbb {R}^n\) if
where \(A = (a_{ij}(\cdot ))\) is a real and measurable coefficient matrix which (without loss of generality) we can take to be symmetric, and for which we also assume, for some \(\lambda > 0\),
For such L, we have also its adjoint operator
Following [24], we say that the function \(u \in L^1_{\textrm{loc}}(\mathbb {R}^n)\) is a solution of the adjoint equation \(L^*u = 0\) if for every \(\varphi \in \mathscr {C}_c^\infty (\mathbb {R}^n)\) we have
Let us recall also the definition of Muckenhoupt weights, that will be used throughout the text because of the properties of some particularly relevant adjoint solution, as we shall see shortly.
Definition 1.2
(Muckenhoupt weights) We say that the function w belongs to the Muckenhoupt class of weights \(A_p\) for some \(1< p < \infty \) if \(w(x) > 0\) a.e. \(x \in \mathbb {R}^n\) and
where the supremum is taken over all the balls \(B \subset \mathbb {R}^n\), and also denote \(A_\infty := \bigcup _{1< p < \infty } A_p\).
We recall that it is well known that the \(A_\infty \) property is equivalent to the Reverse Hölder property, i.e., that there is an exponent \(q>1\), and a uniform constant C such that for every ball B,
For details of the theory of Muckenhoupt weights, the reader may consult, e.g. [20, Chapter 7].
With this definition in mind, we recall a fundamentally important property of equations in nondivergence form.
Lemma 1.3
([24, Theorem 1.1]) Let L be a secondorder elliptic operator in nondivergence form, and \(L^*\) its adjoint. Then there exists a nonnegative solution W of the adjoint equation \(L^*W = 0\) in \(\mathbb {R}^n\), satisfying \(W(B_1(0)) = B_1(0)\), which we call the global nonnegative adjoint solution. Furthermore, W satisfies a Reverse Hölder property with exponent \(\frac{n}{n1}\), so \(W \in A_\infty \) (c.f. Definition 1.2). Moreover, the \(RH_{n/(n1)}\) constants depend only on dimension and ellipticity.
If the coefficients of L are smooth, or even belong to VMO, then W (with the stated normalization) is unique. In general, it need not be unique. On the other hand, for any given L, any choice of such a W will enjoy the same quantitative estimates, with uniform control of all relevant constants. In the case that W is not unique, we may therefore simply fix an arbitrary choice of W.
It is a well known fact that \(A_\infty \) weights are doubling. In our case, this means that there exists a constant \(C_D = C_D([W]_{A_\infty }) \ge 1\) such that \(W(2B) \le C_D W(B)\) for every ball B.
From now on we will work most of the time in the weighted Hilbert space
In the nondivergence setting, this space is more natural in many ways than unweighted \(L^2\). In particular, the following identity holds, as may be seen formally by using \(L^*W = 0\) and integrating by parts:
In fact, one may readily deduce that (1.4) holds when the coefficients are smooth, and more generally, it also holds at least when the coefficients have sufficiently small BMO norm (depending only on dimension and ellipticity), for all \(u\in \mathcal {D}(L)\) (the domain of L), defined by
Indeed, if the coefficients have sufficiently small BMO norm, then \(W\in A_2\) (see [23, Theorem 1.2], and its proof^{Footnote 1}). In turn, using this fact, one may prove^{Footnote 2} the regularity estimate
for solutions of the Poisson problem \(Lu=f\in L^2_W\), and hence, that
The identity (1.4) then follows readily for all \(u\in \mathcal {D}(L)\).
Remark 1.7
We observe that \(H_W^{2}(\mathbb {R}^n)\) is dense in \(L^2_W\) when \(W\in A_2\) (indeed, even \(\mathscr {C}_c^\infty \) is dense in that case), and therefore L is densely defined when the coefficients have sufficiently small BMO norm.
We shall also consider the normalized adjoint of L, which we denote by \(\widetilde{L}\), and which we define to be the adjoint of L with respect to the space \(L^2_W\). Thus, \(\widetilde{L}\) is given, at least for smooth coefficients, by the formula
If the coefficients are merely measurable, then we interpret \(\widetilde{L}\) in the weak sense: we say that \(u\in L^2_W\) belongs to \(\mathcal {D}(\widetilde{L})\), the domain of \(\widetilde{L}\), provided that there is a function \(g\in L^2_W\) such that for every \(v\in \mathcal {D}(L)\),
and in this case we set \(\widetilde{L} u = g\). Just as for (1.4), integrating by parts and using \(L^*W = 0\), we obtain (at least in the case of smooth coefficients, and for \(u\in H^2_W\))
We shall henceforth make the qualitative assumption that the coefficients \(a_{ij}\) are smooth, with qualitative \(L^\infty \) bounds on \(\nabla a_{ij}\) and \(\nabla ^2 a_{ij}\). Thus, (1.8) will be valid in the sequel, for \(u\in H^2_W\). On the other hand, we emphasize that our quantitative bounds will never depend on smoothness, nor on estimates for the derivatives of \(a_{ij}\).
Remark 1.9
Using (1.4), (1.8), (1.5), and (1.6)^{Footnote 3}, one may then show that L, and \(\widetilde{L}\), viewed as unbounded operators on \(L^2_W\), are each closed, sectorial and maccretive, and hence each has an maccretive square root (see [38, Theorem 3.35, p. 281], or [28, Sections 3 and 7]). Moreover, L generates a heat semigroup \(z\mapsto e^{zL}\), which is welldefined and analytic in a sector containing the positive real axis (hence, the analogous statement is also true for \(\widetilde{L}\)).
Let us now sketch the proofs of the functional analytic facts listed in Remark 1.9.
L and \(\widetilde{L}\) are closed operators. For \(\widetilde{L}\), this follows immediately from the fact that L is densely defined (see [38, p. 168]). Thus we consider L. Suppose that \(\{u_n\}_n \subset \mathcal {D}(L)\), that \(u_n \rightarrow u\) in \(L^2_W\), and that
We need to verify that \(u\in \mathcal {D}(L)\), and that \(L u=f\), i.e., that the graph \(\{(u,Lu):u \in \mathcal {D}(L)\}\) is a closed set in \(L^2_W\times L^2_W\). Applying (1.4) and (1.5) to \(u_nu_m\), we see that \(\{\nabla u_n\}_n\) and \(\{\nabla ^2 u_n\}_n\) are each convergent in \(L^2_W\), thus, \(\{u_n\}_n\) is convergent in \(H^2_W\), and since \(u_n\rightarrow u\) in \(L^2_W\), we see that \(u\in H^2_W =\mathcal {D}(L)\), and that \(u_n \rightarrow u\) in \(H^2_W\). In particular, \(D_iD_j u_n \rightarrow D_iD_j u\) in \(L^2_W\) for each \(i,j = 1,2,\dots ,n\), hence \(Lu_n \rightarrow Lu\), so that \(Lu=f\), as desired.
L and \(\widetilde{L}\) are sectorial. It follows readily from (1.4) (respectively, (1.8)) that the numerical ranges
are each contained in a sector \(S_\omega := \{z\in \mathbb {C}: \textrm{arg}\,z\le \omega \} \cup \{0\}\), with \(0<\omega <\pi /2\), depending only on ellipticity. We omit the standard argument.
L and \(\widetilde{L}\) are maccretive. By [38, Problem 3.31, p. 279], it suffices to show that L is maccretive. To this end, set
and for \(\zeta \in \varDelta \), set \(\delta =\delta (\zeta ):= \textrm{dist}(\zeta ,S_\omega )\). By symmetry, we also have \(\delta = \textrm{dist}(\overline{\zeta },S_\omega )\). We now claim that
Indeed, to verify the claim, we may assume without loss of generality that \(\Vert u\Vert _{L^2_W} = 1\), in which case
since \(\langle Lu,u\rangle \in \varTheta \subset S_\omega \).
Fix \(\zeta \in \varDelta \). Then \(L\zeta \) is 11 on \(\mathcal {D}(L)\), and has closed range (since L is a closed operator). Similarly, \(\widetilde{L}\overline{\zeta }\) is 11 on \(\mathcal {D}(\widetilde{L})\). Since L is densely defined,
i.e., the null space of \(\widetilde{L}\overline{\zeta }\) is the orthogonal complement of the range of \(L\zeta \). Thus, \(L\zeta \) has dense range, since \(\widetilde{L}\overline{\zeta }\) is 11. Hence, \(L\zeta \) is invertible as a mapping from \(\mathcal {D}(L)\) onto \(L^2_W\). Combined with the estimate (1.10), this shows that L is maccretive (see [38, p. 279]).
The Heat SemiGroup
Given the preceeding properties of L, we have existence, uniqueness, and analyticity of a contraction semigroup \(e^{zL}\), for z in the open sector \(S^0_\alpha := \{z\in \mathbb {C}: \textrm{arg}\,z < \alpha \}\), provided \(0<\alpha < \pi /2\omega \). See, e.g., [38, pp. 480–493, especially Theorem 1.24, p. 492].
Our main result is the following:
Theorem 1.11
Let L be a secondorder elliptic operator in nondivergence form with smooth real coefficients satisfying (1.1), and let W be the associated global nonnegative adjoint solution provided by Lemma 1.3. If \(W \in A_2\) (see Definition 1.2), then we have
where the implicit constants depend only on n, \(\lambda \) and \([W]_{A_2}\).
The main goal of this paper is to prove this theorem. Hence, from here on we will always impose the extra assumption that \(W \in A_2\), along with the qualitative assumption that the coefficients are smooth. The result will follow at once from Theorems 3.1 and 4.1.
Some additional remarks are in order.
Remark 1.12
As mentioned above, in general W belongs to the class \(A_\infty \), and thus \(W\in A_p\) for some p depending on dimension and ellipticity, but p may be strictly greater than 2, and in fact in the general case we have no precise upper bound on p. Thus, our result is only a partial one, and does not address the fundamental challenge of treating the non\(A_2\) case. On the other hand, as noted above, if the coefficients have sufficiently small BMO norm, then \(W\in A_2\), and thus our result does apply in that setting.
Remark 1.13
In the case that the coefficient matrix has sufficiently small BMO norm, then as also noted above, we may identify the domain \(\mathcal {D}(L)\) as the weighted Sobolev space \(H^2_W(\mathbb {R}^n)\) (see (1.6)). Hence, by combining several known (or at least implicit) results, we may identify the domain of \(\sqrt{L}\) as the Sobolev space \(H_W^1(\mathbb {R}^n):=\{u\in L^2_W(\mathbb {R}^n): \nabla u \in L^2_W(\mathbb {R}^n)\}\); this corresponds to the estimate \(\Vert \sqrt{L}f\Vert _{L^2_W} \lesssim \Vert \nabla f\Vert _{L^2_W}\). Indeed, in [22] it is shown that the operator L has a bounded holomorphic functional calculus in (unweighted) \(L^2\) (even in \(L^p\)), provided that the BMO norm of the coefficients is sufficiently small. Under the same smallness assumption, the arguments of [22] may be extended to the weighted case considered here, to deduce that L has a bounded holomorphic functional calculus in \(L_W^2\). Combining the results of [45] and [41], we find that \(\mathcal {D}(\sqrt{L})\) is the complex interpolation space midway between \(L^2_W\) and \(\mathcal {D}(L) = H^2_W\), i.e., \(\mathcal {D}(\sqrt{L}) = H^1_W\).
The analogous strategy fails for \(\widetilde{L}\), as we have no idea how to identify \(\mathcal {D}(\widetilde{L})\) (similarly, the square root problem in the divergence form case entailed the same difficulty).
Remark 1.14
The assumption of smoothness of the coefficients is purely qualitative, and our quantitative estimates will not depend on smoothness, but only on the stated parameters n, \(\lambda \) and \([W]_{A_2}\). However, it is not clear at present how to make sense of the identity (1.8) for nonsmooth coefficients, and as a consequence, in the absence of smoothness, we do not know how to prove certain estimates which rely on (1.8), such as Lemma 2.11(viii) (in the case of measurable coefficients, we know how to give only a formal proof of the latter, assuming a priori finiteness of \(\Vert \nabla e^{t^2\widetilde{L}} f\Vert _{L^2_W}\)).
On the other hand, as noted above, identity (1.4) holds without smoothness, in the case that the coefficients have sufficiently small BMO norm. Under the latter scenario, we require identity (1.8) and its consequences (and thus, the qualitative, a priori assumption of smoothness of the coefficients) in two places: 1) in the proof of Theorem 4.1 (the square root problem for \(\widetilde{L}\)), where estimate (1.8) is heavily used, and 2) in the proof of the maccretivity of L given above, where we used (1.8) to establish density of the range of \(L\zeta \). Otherwise, (1.8) is not used in the proof of Theorem 3.1 (the square root problem of L).
Although our operators L and \(\widetilde{L}\) are not of divergence form, there is a nice identity relating these two nondivergence operators with another one which is in divergence form, but degenerate elliptic^{Footnote 4}. Indeed, if we let \(\widetilde{\textrm{div}}\) denote the normalized divergence, defined for an \(\mathbb {R}^n\)valued function \(\textbf{v}\) by \(\widetilde{\textrm{div}}\,\textbf{v}:= \frac{1}{W} \textrm{div}(W\textbf{v})\), then \(\widetilde{\textrm{div}}\) is precisely the adjoint operator to \(\nabla \) inside \(L^2_W\), and we also have, using \(L^*W=0\),
In the case of nonsmooth coefficients, we interpret the latter identity in the weak sense described above: for \(u, \varphi \in H_W^2\),
The identity (1.15) will be of great use to us in the sequel.
The paper is organized as follows:

In Section 2 we give some definitions and estimates for some of the operators, which will appear repeatedly across the paper.

In Section 3 we prove \(\Vert \sqrt{L}f\Vert _{L^2_W} \lesssim \Vert \nabla f\Vert _{L^2_W}\), which turns out to be a relatively easy consequence of Littlewood–Paley theory because of the form of L (since L annihilates not only constants but also first degree monomials).

In Section 4 we prove \(\Vert \sqrt{\widetilde{L}}f\Vert _{L^2_W} \lesssim \Vert \nabla f\Vert _{L^2_W}\), which is in fact the more difficult result in the paper. To treat \(\widetilde{L}\), we follow broadly the scheme provided by [8], first reducing the problem to some square function estimates, which are handled using a T1like argument and then a local Tb argument. Of course, some significant modifications of the arguments in [8] are needed; the identity (1.15) will be useful in this case.
We remark that the square root problem for \(\widetilde{L}\) is significantly more difficult than its analogue for L.
1.1 Notation

We use the notation \(a \lesssim b\) to denote that there exists a positive harmless constant C (which can vary from line to line) such that \(a \le Cb\). We will also denote \(a \approx b\) whenever \(a \lesssim b\) and \(b \lesssim a\).

In the proofs of the results from now on, we will omit dependencies of constants on n, \(\lambda \), \([W]_{A_\infty }\) and \([W]_{A_2}\) – treating them as harmless constants – although we will make these dependencies explicit in the statements.

Euclidean balls are denoted by \(B_t(x) := \{y \in \mathbb {R}^n : yx < t \}\).

If \(B = B_t(x) \subset \mathbb {R}^n\) is a ball and \(\kappa > 0\), by \(\kappa B\) we denote the ball with same radius and scaled by a factor of \(\kappa \), i.e., \(B_{\kappa t}(x)\). The same applies to cubes.

For \(E \subset \mathbb {R}^n\), E denotes the Lebesgue measure of E.

If \(E, F \subset \mathbb {R}^n\) are arbitrary subsets, we write \(\textrm{dist}(E, F) := \inf \{xy : x \in E, y \in F\}\).

For any subset \(E \subset \mathbb {R}^n\), we denote \(\textbf{1}_E\) the characteristic function of E (i.e. \(\textbf{1}_E(x) = 1\) if \(x \in E\) and 0 otherwise). Concretely, we write \(\textbf{1} := \textbf{1}_{\mathbb {R}^n}\), the function constantly 1.

We will denote vectorvalued functions with boldface letters, e.g., \(\textbf{v} := (v_1,\dots ,v_n)\).

\(D_j\) denotes the differentiation operator in the direction of \(x_j\), i.e., \(D_j = \frac{\partial }{\partial x_j}\).

We denote averages with respect to a measure \(\nu \) by ⨏_{E}\(f d\nu := \nu (E)^{1} \int _E f d\nu \). Often the measure with respect to which we take averages will be the weighted measure W(x)dx: it will be clear by the context. For the latter measure, we write \(W(E) := \int _E W(x) dx\).

We will frequently use cubes in our proofs: every time we cover \(\mathbb {R}^n\) (or some portion of it) by cubes, we mean we are using a covering by cubes of the dyadic grid \(\{2^j \textbf{k} + [0, 2^{j})^n : j \in \mathbb {Z}, \textbf{k} \in \mathbb {Z}^n\}\). Anytime we use the letter Q, we will be referring to a dyadic cube. For such a cube Q, we let \(\ell (Q)\) denote its sidelength.

We let \(\mathcal {M}\) and \(\mathcal {M}_W\) denote, respectively, the classical Hardy–Littlewood maximal operator, and the Hardy–Littlewood maximal function with respect to the measure W(x)dx, that is,
and
Since W is doubling (because \(W \in A_\infty \)), \(\mathcal {M}_W\) is bounded on \(L^p_W\) for every \(1< p < \infty \). We will use this fact in the sequel.

As explained before, we set \(L^2_W := L^2(\mathbb {R}^n, W(x)dx)\), and we define the weighted Sobolev space \(H_W^2:=\{u\in L^2_W: \nabla u\in L_W^2, \, \nabla ^2 u \in L_W^2\}\). We will also write \(L_W^2(E) := L^2(E, W(x)dx)\) for any subset \(E \subset \mathbb {R}^n\).

We denote the composition of two operators U and V by \(UV(f) := U(V(f))\)).

For a function \(f\in L^2(\mathbb {R}^n)\), we denote its Fourier transform by \(\hat{f}\).

\(\mathcal {S}\) will denote the usual Schwarz class of smooth, rapidly decaying functions on \(\mathbb {R}^n\).
2 Preliminaries
2.1 Gaussian Bounds for Kernels of Semigroups
From now on, we will use many times the parabolic semigroup (with elliptic homogeneity) \(e^{t^2L}\), whose kernel is the fundamental solution \({\varGamma }_{t^2}(x, y)\); i.e. we have \(e^{t^2L}f(x) = \int _{\mathbb {R}^n} {\varGamma }_{t^2}(x, y)f(y)dy\) for sufficiently regular f. The fundamental solution satisfies the following Gaussian estimate:
Lemma 2.1
([24, Theorem 1.2]) The kernel \({\varGamma }_{t^2}(\cdot , \cdot )\) of \(e^{t^2L}\) satisfies the Gaussian bounds
and
where the implicit constants and c depend on n and \(\lambda \).
Remark 2.4
The results stated above as Lemmas 1.3 and 2.1, are stated in [24] explicitly for smooth coefficients, but as the author points out, “the usual compactness arguments”, and the uniformity of the estimates depending only on n and \(\lambda \), allow one to deduce the existence of (nonunique) W and \({\varGamma }\) verifying the same bounds, in the general case of bounded measurable coefficients.
Remark 2.5
The doubling property of W, combined with the exponential decay factor, allow us to interchange “min” and “max” in (2.2) and (2.3), modulo an adjustment of the constants.
Remark 2.6
The (absolute values of) the kernels of the operators \(t^2Le^{t^2L}\) and \(t^4L^2e^{t^2L}\) also satisfy the upper bound (2.2), by analyticity of the semigroup \(z\mapsto e^{zL}\) in a sector.
2.2 Weighted Littlewood–Paley Theory
The following results are standard. We recall them here for the sake of convenience of exposition.
Lemma 2.7
Let \(W\in A_2\), and let \(K_t f := k_t * f\), with k defined on \(\mathbb {R}^n\) satisfying \(k(x) \le (1+x)^{n1}\), where \(k_t(x) := t^{n}k(x/t)\). Then
where \(\mathcal {M}\) is the classical Hardy–Littlewood maximal operator, and the implicit constant depends on n and \([W]_{A_2}\).
Lemma 2.8
Let \(W\in A_2\), and let \(Q_s f := \psi _s * f\), where \(\psi \in \mathcal {S}\) and satisfies \(\int _{\mathbb {R}^n} \psi = 0\). Then
where the implicit constant depends on n, \(\psi \), and \([W]_{A_2}\). Moreover, if in addition \(\psi \) is radial and nontrivial, then using a slight abuse of notation and then normalizing, we may assume that \( \int _0^\infty \hat{\psi }(s)^2 \,\frac{ds}{s} =1\), in which case we have the Calderón reproducing formula
Remark 2.9
Regarding the last pair of lemmata:

We will often denote by \(Q_t\) the operators satisfying the hypotheses in Lemma 2.8, while we use \(P_t\) for “nice” approximate identities (i.e. \(P_t f := \varphi _t * f\), with \(\varphi \in \mathcal {S}\) radial, and \(\int \varphi = 1\)).

It is easy to check if \(P_t\) is a nice approximate identity, then \(Q_t := t D_i P_t\) satisfies the hypotheses of the first part of Lemma 2.8 (where \(D_i\) denotes the partial derivative in any direction \(x_i\)).

We will frequently further assume that the kernel k in Lemma 2.7 (in particular, \(\varphi \) and \(\psi \) as above) satisfies \(\textrm{supp}\,k \subset B_1(0)\); in this case, we shall refer to \(K_t\) (in particular, \(P_t\) or \(Q_t\)) as having a “compactly supported kernel”.

We will use repeatedly the fact that \(P_t\) and \(Q_s\) commute with derivatives, for they are convolution operators.
The following is an easy consequence of Lemma 2.8, by standard “almostorthogonality” arguments. We omit the wellknown proof.
Lemma 2.10
Let \(\{Q_s\}_{s>0}\) be a family operators satisfying the conditions in Lemma 2.8, and \(R_t\) be a family of operators, bounded on \(L^2_W(\mathbb {R}^n)\) for each fixed \(t>0\), and satisfying, for some \(\alpha > 0\), the almostorthogonality condition
where \(C_1\) is a uniform constant which does not depend on t, s. Then \(R_t\) satisfies the square function estimate
where the implicit constant depends on n, \(C_1\), \(\alpha \) and \([W]_{A_2}\).
2.3 Uniform Bounds for Some Operators
Let us show that some operators related with the semigroups are uniformly bounded, which we will use throughout the text.
Lemma 2.11
The following operators are \(L^2_W \rightarrow L^2_W\) bounded, uniformly on t, with norm depending on n, \(\lambda \) and the doubling constant for W:
(i) \(e^{t^2L}\),  (ii) \(t^2Le^{t^2L}\),  (iii) \(t^4L^2e^{t^2L}\),  (iv) \(t\nabla e^{t^2L}\), 
(v) \(e^{t^2\widetilde{L}}\),  (vi) \(t^2\widetilde{L} e^{t^2\widetilde{L}}\),  (vii) \(t^4\widetilde{L}^2e^{t^2\widetilde{L}}\),  (viii) \(t\nabla e^{t^2\widetilde{L}}\), 
(ix) \(t e^{t^2L} \widetilde{\textrm{div}}\),  (x) \(te^{t^2\widetilde{L}}\widetilde{\textrm{div}}\),  (xi) \(t^3 L e^{t^2L} \widetilde{\textrm{div}}\),  (xii) \(t^3 \widetilde{L} e^{t^2\widetilde{L}} \widetilde{\textrm{div}}\). 
Remark 2.12
Some of the operators are in fact defined for functions \(\textbf{f} \in (L^2_W)^n\). In this case, we obviously mean that each of their components are bounded.
Proof
(i)–(iii). Recall that \(\mathcal {M}_W\) denotes the Hardy–Littlewood maximal operator with respect to the weighted measure W(x)dx. If we let \(T_t\) denote the operator under consideration in (i), (ii) or (iii), it suffices to observe that in each case we have the pointwise bound
using only the Gaussian bounds in Lemma 2.1 (and Remark 2.6 in the case of (ii) and (iii)), and the doubling property of W. We omit the routine details.
To prove (iv), we set \(u := e^{t^2L} f\). Note that by the identity (1.4), we can “interpolate” the estimates in (i) and (ii) using Cauchy–Schwarz as follows:
which shows (iv).
In turn, (v)–(vii) follow by duality from (i)–(iii), while (viii) follows “interpolating” (v) and (vi) in the same way that we did it for (iv), this time using (1.8) instead of (1.4).
Then, (ix) and (x) follow by duality from (viii), and (iv), respectively.
Lastly, (xi) (resp. (xii)) follows by first dualizing, and then “interpolating” (vi) and (vii) using (1.8) (resp. (ii) and (iii) using (1.4)). \(\square \)
2.4 Offdiagonal Estimates
Definition 2.13
We say that the operators \(S_t\) satisfy offdiagonal estimates (aka Gaffney estimates), if there exist \(c, C > 0\) (independent of t) such that it holds for every \(t > 0\), and every pair of measurable sets E and F,
Lemma 2.20
The following operators satisfy offdiagonal estimates, with constants depending only on n, \(\lambda \) and the doubling constant of W:
(i) \(e^{t^2L}\),  (ii) \(t^2Le^{t^2L}\),  (iii) \(t^4L^2e^{t^2L}\),  (iv) \(t\nabla e^{t^2L}\), 
(v) \(e^{t^2\widetilde{L}}\),  (vi) \(t^2\widetilde{L} e^{t^2\widetilde{L}}\),  (vii) \(t^4\widetilde{L}^2e^{t^2\widetilde{L}}\),  (viii) \(t e^{t^2\widetilde{L}} \widetilde{\textrm{div}}\), 
(ix) \(t\nabla e^{t^2\widetilde{L}}\),  (x) \(t e^{t^2L} \widetilde{\textrm{div}}\). 
Proof
Fix sets \(E, F \subset \mathbb {R}^n\). We may assume that \(d := \textrm{dist}(E, F) > 5t\), as otherwise we may invoke Lemma 2.11.
The bound for (i) is a straightforward consequence of the pointwise bounds in Lemma 2.1. We omit the routine details. The offdiagonal estimates for (ii)–(iii) follow in the same way as for (i), using Remark 2.6.
Let us now treat (iv). We argue as in the proof of Caccioppoli’s inequality. Let \(u := e^{t^2L} f\), with f supported in E. Choose \(\psi \in \mathscr {C}^\infty (\mathbb {R}^n)\), where \(0 \le \psi \le 1\) satisfies \(\psi \equiv 1\) on F, \(\textrm{dist}(\textrm{supp}\,\psi , E) \ge d/2\) (denoting \(d := \textrm{dist}(E, F)\)), and \(\Vert \psi \Vert _\infty + d \Vert \nabla \psi \Vert _\infty + d^2 \Vert \nabla ^2 \psi \Vert _\infty \le C\). For future reference, we note that
Clearly, we have
Compute, using (1.4) and the symmetry of A:
Also,
Combining (2.17) and (2.18), we may dominate the right hand side of (2.16) by
For \(\varepsilon > 0\) to be chosen momentarily, we have
where we have used “Cauchy’s inequality with \(\varepsilon \)” in term I, and then (2.15). Choosing \(\varepsilon \) small enough, we may hide the first term, and use the offdiagonal bounds for (i) to obtain the desired bound for the second term, since \(t\le d\). We estimate term II using the Cauchy–Schwarz inequality, along with the offdiagonal bounds for (i) and (iii).
The operators (v)–(viii) are dual to the first four, and (x) is dual to (ix). It therefore remains to treat (ix). To this end, we now set \(u:= e^{t^2\widetilde{L}} f\), with f supported in E, so with \(\psi \) as above, again using (2.16), we have
where summation over i, j is understood. We first note that
We obtain the desired estimate for \(\widetilde{I}_1\) by the offdiagonal bounds for (v) and (vi) like for term II in (2.19). Also, integrating by parts,
We will cancel term \(\widetilde{I}_{2,1}\) momentarily, but it may also be handled directly, exactly like term I in (2.19), using Cauchy’s inequality with \(\varepsilon \), hiding the small term, and bounding the other term using the offdiagonal bounds for (v). We estimate term \(\widetilde{I}_{2,2}\) like terms I and III in (2.19), using (2.15), the fact that \(t\le d\), and the offdiagonal bounds for (v).
Since W is an adjoint solution,
Observe that \(\widetilde{II}_1\equiv \widetilde{I}_{2,1}\), and \(\widetilde{II}_2 \equiv \frac{1}{2} \widetilde{I}_{2,2} \). \(\square \)
Lemma 2.21
Let \(K_t\) be a convolution type operator as in Lemma 2.7, with a compactly supported kernel. If \(h=h(x,t)\) is a function such that \(f\mapsto h(\cdot ,t) K_t f\) is bounded on \(L^2_W\), uniformly in t (i.e. \(\Vert h(\cdot ,t) K_t\Vert _{L^2_W \rightarrow L^2_W} \lesssim 1\) uniformly in t), then \(h(\cdot ,t) K_t\) satisfies offdiagonal estimates.
We omit the trivial proof.
Lemma 2.22
Let \(\{U_t\}_{t>0}\) and \(\{U'_t\}_{t>0}\) be two families of operators, each satisfying offdiagonal estimates, then the composition \(U_t U'_t\) also satisfies offdiagonal estimates for each t.
We omit the routine proof.
2.5 Estimates for Differences and Gradients
Lemma 2.25
For \(f \in \textrm{Lip}\) and \(t \le \ell (Q)\), we have
and
where the implicit constants depend on n, \(\lambda \) and \([W]_{A_2}\).
Proof
Let us show the first estimate. Cover Q by nonoverlapping cubes \(Q_k\) with sidelength \(t/2 < \ell (Q_k) \le t\). Note that \(e^{t^2L} \textbf{1} = \textbf{1}\), since \(L\textbf{1}=0\). Letting \([f]_E\) denote the average of f over the set E,
We then have
Using the boundedness of \(e^{t^2L}\) from Lemma 2.11 and the weighted version of Poincaré’s inequality [29, 15.26], we deduce
For convenience of notation in the rest of this argument, we replace the constant c by 4c in the offdiagonal estimates for \(e^{t^2L}\) from Lemma 2.14. Thus,
Hence, summing on j and interchanging the order of summation we obtain
and therefore we have, by Poincarés inequality,
Thus, going back to \(A_k\) (see (2.23)–(2.24)) we obtain
and therefore, since the cubes \(Q_k\) are nonoverlapping,
The corresponding estimate for \(e^{t^2\widetilde{L}}\) holds with exactly the same proof, for the operator is also bounded and satisfies offdiagonal estimates (see Lemmas 2.11 and 2.14), and \(e^{t^2\widetilde{L}} \textbf{1} = \textbf{1}\) since \(\widetilde{L}\textbf{1} = \frac{1}{W} L^*W = 0\). \(\square \)
Lemma 3.2
For \(f \in \textrm{Lip}\) and \(t \le \ell (Q)\), we have
and
where the implicit constant depends on n, \(\lambda \) and \([W]_{A_2}\).
Proof
The reader can check that the proof is the same as for Lemma 2.22, this time using the operators \(t\nabla e^{t^2L}\) and \(t\nabla e^{t^2\widetilde{L}}\). Indeed, the proof of the preceeding Lemma used only the boundedness of the operator (Lemma 2.11), the offdiagonal estimates (Lemma 2.14), and the conservation property which allows us to subtract constants. In this case,
and similarly for \(\widetilde{L}\), because \(e^{t^2L} \textbf{1} = \textbf{1} =e^{t^2\widetilde{L}}\textbf{1}\), as before. After this, the proof is the same as that of Lemma 2.22. \(\square \)
3 The Kato Problem for L
In this section our goal is to prove the following result:
Theorem 3.1
It holds
where the hidden constant depends on n, \(\lambda \) and \([W]_{A_2}\).
As noted above (see Remark 1.13), in the case that the coefficient matrix has small enough BMO norm, we could deduce Theorem 3.1 as an easy consequence of certain known results. Instead, following the easier part of our proof of Theorem 4.1 below, we shall give a selfcontained, direct argument, which does not rely on an explicit assumption of smallness in BMO, but only on the validity of (1.4), and the assumption that \(W\in A_2\).
To prove the theorem, let us use the representation of the square root operator via the Functional Calculus formula
where \(a = (\int _0^\infty t^3e^{2t^2} \frac{dt}{t})^{1} = \sqrt{\frac{128}{\pi }}\) is just a normalizing constant, to estimate, using duality and later Cauchy–Schwarz,
With this decomposition, we will finish the proof of Theorem 3.1 by duality once we prove the following Lemmas 3.2 and 3.5.
The desired bound for the second factor is the following.
Lemma 3.5
It holds
where the implicit constant depends only on n, \(\lambda \) and the doubling constant for W.
We could obtain the conclusion of the lemma by invoking the abstract McIntosh and Yagi theorem [41, 45], but instead, we will give a selfcontained and more elementary proof, using quasiorthogonality arguments.
Proof
We abbreviate \(V_t := t^2 L e^{t^2L}\). Its adjoint within \(L^2_W\) is \(\widetilde{V}_t := t^2 \widetilde{L} e^{t^2\widetilde{L}}\). To make the argument rigorous, given a small positive \(\varepsilon \), we set \(V_t \equiv 0 \equiv \widetilde{V}_t\) whenever \(t\le \varepsilon \) or \(t\ge 1/\varepsilon \), and we obtain quantitative bounds that are uniform in \(\varepsilon \). We compute, using duality, Fubini and Cauchy–Schwarz,
To deal with the second term, let us first establish a useful fact:
Claim 3.4
It holds, for any \(t, s > 0\),
Proof of the claim We may assume that \(\varepsilon< s,t <1/\varepsilon \). If \(s \le t\), we can compute using (1.15) and the uniform bounds from Lemma 2.11,
In a similar fashion we can compute, when \(s > t\), using again (1.15) and Lemma 2.11,
\(\square \)
With this almostorthogonality result, we can estimate the last term in (3.3) as follows:
Plugging this estimate into (3.3), we have
from which the result readily follows (recall that we have effectively truncated so \(\varepsilon< t <1/\varepsilon \), hence the integrals are finite).\(\square \)
Let us turn our attention to the other square function estimate.
Lemma 3.7
It holds
where the implicit constant depends on n, \(\lambda \) and \([W]_{A_2}\).
We will devote the rest of the section to the proof of Lemma 3.5. As above, set \(V_t := t^2Le^{t^2L}\) and decompose, with the help of an approximate identity \(P_t\),
The proof of Lemma 3.5, and hence of Theorem 3.1, will come immediately from the next two lemmas.
Lemma 3.8
With the notations of (3.6), we have
where the implicit constant depends on n, \(\lambda \) and \([W]_{A_2}\).
Proof
Choose the approximate identity \(P_t := e^{t^2(\varDelta )}\). With it, we can compute, using the Fundamental Theorem of Calculus,
Now, using the boundedness on \(L^2_W\) of \(V_t = t^2Le^{t^2L}\) (see Lemma 2.11), Hardy’s inequality and the fact that \(\textbf{Q}_s\) satisfies the square function estimate of Lemma 2.8 (see Remark 2.9), we obtain the desired estimate:
\(\square \)
Lemma 4.2
With the notations of (3.6), we have
where the hidden constant depends on n, \(\lambda \) and \([W]_{A_2}\).
Proof
Simply compute, using the boundedness of \(e^{t^2L}\) from Lemma 2.11, and the square function bounds of Lemma 2.8 (see Remark 2.9),
\(\square \)
4 The Kato Problem for \(\widetilde{L}\)
In this section our goal is to prove the following result, which is really the main result in this paper:
Theorem 4.1
It holds
where the implicit constant depends on n, \(\lambda \) and \([W]_{A_2}\).
4.1 Reduction to a Quadratic Estimate
To prove Theorem 4.1, let us again use the representation of the square root operator via the formula
so that
Theorem 4.1 then follows immediately from Lemma 4.2 and Theorem 4.3 below.
We estimate the second square function via the following lemma.
Lemma 4.5
It holds
where the implicit constant depends on n, \(\lambda \) and \([W]_{A_\infty }\).
Proof
The lemma can be proved either by invoking the McIntosh and Yagi theorem [41, 45], or via a selfcontained elementary proof using quasiorthogonality. For the latter path, the proof follows that of Lemma 3.2mutatis mutandis, simply reversing the roles of \(V_t\) and \(\widetilde{V}_t\). We omit the details. \(\square \)
Let us now turn the attention to the other square function estimate, which is in fact the core of this paper.
Theorem 4.3
It holds
where the implicit constant depends on n, \(\lambda \) and \([W]_{A_2}\).
The rest of this section is devoted to the proof of Theorem 4.3. We start by splitting
where \(P_t\) is a nice approximate identity with a smooth compactly supported convolution kernel \(\varphi _t(x)= t^{n} \varphi (x/t)\), which we take to be even. For future reference, let us record the following wellknown observation:
where c is a harmless constant, and \(\psi ^{(1)} (x):= x\varphi (x)\) and \(\psi ^{(2)}:= \nabla \varphi \) are both \(\mathscr {C}_c^\infty \) functions with mean value zero (here we are using that \(\varphi \) is even, in the case of \(\psi ^{(1)}\)). Hence,
where \(Q_t^{(k)}\) is the convolution kernel with kernel \(\psi _t^{(k)}(x):=t^{n}\psi ^{(k)}(x/t)\), \(k=1,2\), and therefore each of \(Q_t^{(1)}\), \(Q_t^{(2)}\) satisfies the square function bound of Lemma 2.8 (and each is bounded on \(L^2_W\), uniformly in t).
Lemma 4.8
With the notations of (4.4), we have
where the implicit constant depends on n, \(\lambda \) and \([W]_{A_2}\).
Proof
Using the preceeding observations, we may follow the proof of Lemma 3.7, invoking Lemma 2.11 to obtain that \(t^2 \widetilde{L} e^{t^2\widetilde{L}}\) is \(L^2_W\) bounded, to obtain (4.6). \(\square \)
Applying now (1.15) to \(u = P_t^2 f\) we obtain
Lemma 4.14
We have
where the implicit constant depends on n, \(\lambda \) and \([W]_{A_2}\).
Proof
The proof is the same as that in Lemma 3.8 once we use that \(e^{t^2\widetilde{L}} : L^2_W \rightarrow L^2_W\) is uniformly bounded by Lemma 2.11, and that \(P_t\) are uniformly bounded on \(L^2_W\) by Lemma 2.7. \(\square \)
Therefore, to finish the proof of Theorem 4.3 (and hence of Theorem 4.1), it remains to show
4.2 Reduction to a Carleson Measure Estimate
For \(\textbf{g} = (g_1,g_2,\dots ,g_n)\), write
With this notation, the remaining estimate (4.9) becomes
Let us also define the operator
so that, taking \(\textbf{g} = \nabla u\), and using (1.15),
It will be convenient to use both operators at different stages of the proof. Note that trivially, \(\widetilde{\theta }_t \textbf{e} = \theta _t \textbf{e}\), for any constant vector \(\textbf{e}\). In particular, if \(\mathbbm {1}\) denotes the \(n\times n\) identity matrix, then
where we naturally define \(\widetilde{\theta }_t \mathbbm {1}= \theta _t \mathbbm {1}\) as a vectorvalued function whose \(k^{th}\) entry is \(\widetilde{\theta }_t \textbf{e}^k = \theta _t \textbf{e}^k\), with \(\textbf{e}^k \) equal to the standard unit basis vector in the \(x_k\) direction.
To prove (4.9), as in the divergence form case treated in [8], we begin with a “T1” reduction.
Lemma 4.27
We have
where the implicit constant depends on n, \(\lambda \) and \([W]_{A_2}\).
Proof
Write \(U_t \textbf{g} := \theta _t P_t^2 \textbf{g}  (\theta _t \mathbbm {1}) \cdot (P_t^2 \textbf{g})\). By Lemma 2.10, it suffices to show that
uniformly on t, and for some \(\alpha > 0\), and for any nice operator \(Q_s\) as in Lemma 2.8, with a compactly supported kernel,
These two estimates, and hence the conclusion of Lemma 4.14, will follow at once from the next claims and Lemma 2.7.
Claim 4.15
We have, uniformly on t,
where the implicit constant depends on n, \(\lambda \) and \([W]_{A_2}\).
Proof of the claim Just compute, with the aid of Lemmas 2.7 and 2.11,
\(\square \)
Claim 4.16
We have, uniformly on t,
where the implicit constant depends on n, \(\lambda \) and \([W]_{A_2}\).
Proof of the claim This proof will follow that of [18, (4.10)]. Let us cover \(\mathbb {R}^n\) by cubes \(Q_k\) satisfying \(t/2 < \ell (Q_k) \le t\). In this way, we obtain
We first establish an \(L^\infty \) bound for \(P_t \textbf{g}(x)\), in the cube \(Q_k\). Note that for \(x \in Q_k\) we have that \(P_t\textbf{g}(x) = P_t(\textbf{g} \textbf{1}_{3Q_k})(x)\) because \(t \le 2 \ell (Q_k)\) (and \(\textrm{supp}\,\varphi \subset B(0, 1)\)).
where we used the definition of an \(A_2\) weight in the last step (see Definition 1.2).
We next claim that
Taking this claim for granted momentarily, we obtain
where in the last two steps we used first (4.18), and then the bounded overlap property of the cubes \(3Q_k\).
It remains to verify (4.18). We dualize: choose \(\textbf{h}=(h_1,h_2,\dots ,h_n) \in L^2_W\), with \(\textrm{supp}\,\textbf{h} \subset Q_k\), and write
For the first term we may simply compute, using Jensen’s inequality and the boundedness of \(t \nabla e^{t^2L}\) from Lemma 2.11,
For the second term we use Jensen again, and later the offdiagonal estimates from Lemma 2.14 (taking advantage of \(\ell (Q_k) \approx t\)) to obtain
With the estimates for \(I^{(k)}\) and \(II^{(k)}\), we can substitute back in (4.19) and obtain
which after squaring gives (4.18) by duality, as desired. This completes the proof of Claim 4.16. \(\square \)
Claim 4.20
Suppose \(s\le t\). Then
Proof of the claim Note that we have the pointwise estimate
where \(\mathcal {M}^2 := \mathcal {M}\circ \mathcal M\) is the iterated Hardy–Littlewood maximal operator (with respect to Lebesgue measure). One may verify (4.21) by a standard argument using the size estimates and compact support of the kernels of \(P_t\) and \(Q_s\), along with the smoothness of the former, and the cancellation property of the latter. We omit the wellknown details. Since \(\mathcal {M}\) is bounded on \(L^2_W\) (recall that \(W\in A_2\)), we find, with the aid of Claims 4.15 and 4.16, and (4.21):
\(\square \)
Claim 4.22
We have, uniformly on t,
Proof
The proof is inspired by [1, Lemma 3.5], and in fact is similar in spirit to that of Lemma 2.22, relying strongly in a decomposition in subcubes of the right size to use Poincaré’s inequality, and some boundedness and offdiagonal estimates. Nevertheless, let us show it in detail, because some parts of it will be reused later. Cover \(\mathbb {R}^n\) by a grid of nonoverlapping dyadic cubes \(Q_k\) with sidelength \(t/2 < \ell (Q_k) \le t\). Using the easy fact that \(U_t \mathbbm {1} = 0\) we compute
Let us first deal with A, denoting \(S_t := \theta _t P_t^2\) because we intend to reuse some computations later on. For each term in the series, simply using linearity and the triangle inequality
Using the boundedness of \(S_t\) (in this case, this follows from Lemmas 2.7 and 2.11) and Poincaré’s inequality, we deduce
And for the other terms, we can use the offdiagonal estimates for \(S_t\) (in this case, this follows from Lemmas 2.14 and 2.21), and taking advantage of \(\ell (Q_k)\approx t\) and Poincaré, we obtain, similarly to the situation in Lemma 2.22,
Thus, going back to (4.24) we obtain
and hence
By bounded overlap of the cubes \(2Q_k\) we easily get
For the other term, we note that \(xy\lesssim 2^j \ell (Q_k) \approx 2^j t\), whenever \(x\in Q_k\), and \(y\in 2^{j+1}Q_k\). We further note that \(W(Q_k) \approx W(B_t(x))\), for \(x\in Q_k\), and that for all \(x\in {\mathbb {R}^n}\),
by the doubling property of W. We now use these observations, along with Cauchy–Schwarz, Fubini’s theorem, and the fact that the cubes \(Q_k\) are nonoverlapping, to obtain
Consequently, we have shown that
We can apply a similar, but simpler argument to handle term B in (4.23). We now set \(S_t := (\theta _t \mathbbm {1}) \cdot P_t^2\), and note that \(S_t\) is uniformly bounded on \(L^2_W\), by Claim 4.16 and Lemma 2.7. Moreover, the kernel of \(P^2_t\) is compactly supported in the ball of radius 2t, so the same is true for \(S_t\). Hence, for the current version of \(S_t\), we obtain a simplified variant of (4.24), in which only the term \(I^{(k)}\) appears, enjoying the same bound as in (4.25). Thus,
The proof of Claim 4.22 is now complete. \(\square \)
Claim 4.26
For \(t\le s\), we have
Proof of the claim Using Claim 4.22 and Lemma 2.7, we have
as desired. \(\square \)
As noted above, the preceding claims conclude the proof of Lemma 4.14. \(\square \)
We are now ready to reduce matters to a Carleson measure estimate. Recall that to prove Theorem 4.3 (and hence Theorem 4.1), it suffices to verify estimate (4.11) (equivalently, (4.9)).
Lemma 4.29
Theorem 4.3 (and hence Theorem 4.1) follows from the Carleson measure estimate
Proof
Recalling that \(\widetilde{\theta }_t \mathbbm {1} = \theta _t \mathbbm {1}\), we see that by Lemma 4.14 and a weighted version of Carleson’s embedding inequality (see [18, Lemma 2.2]), the estimate (4.28) implies (4.11). \(\square \)
Our goal then, is to prove (4.28). To this end, let us first establish a few more estimates to be used in the sequel. We define the dyadic averaging operator by
where \(Q_{x, t}\) is the halfopen dyadic cube containing x for which \(t/2 < \ell (Q_{x, t}) \le t\).
Lemma 4.39
We have
where the implicit constant depends on n, \(\lambda \) and \([W]_{A_2}\).
Proof
The proof of this estimate will be very similar to that of Lemma 4.14. We set
and note that it is enough to show that \(\widetilde{U}_t\) satisfies the hypotheses of the weighted Littlewood–Paley almostorthogonality result Lemma 2.10. The uniform boundedness of \(\widetilde{U}_t\) arises immediately from that of \((\theta _t \mathbbm {1}) \cdot P_t^2\) (see Claim 4.16 and Lemma 2.7), along with the following result:
Claim 4.30
We have, uniformly on t,
Proof of the claim The proof is the same as that of Claim 4.16, which treated \((\theta _t \mathbbm {1}) \cdot P_t\). Indeed, the only properties of \(P_t\) that were used in that argument were the size and support condition of its kernel. The kernel of \(A_t\) enjoys similar properties, in fact
hence, the same proof may be repeated. \(\square \)
To prove the quasiorthogonality with the \(Q_s\) operators, the next result will be useful.
Claim 4.31
We have, uniformly on t,
Proof
The proof is similar to that of Claim 4.22, but simpler: now one has to deal only with terms like “B” associated to \(S_t = (\theta _t \mathbbm {1}) \cdot A_t\) in (4.23), so that there is no “tail” as in (4.24), but rather only a local term analogous to \(I^{(k)}\). We omit the routine details. \(\square \)
The following two claims finish the proof of Lemma 4.29, and are analogous to those in the proof of Lemma 4.14.
Claim 4.32
We have, uniformly for \(t\le s\),
Proof
In view of Claim 4.31, repeating the proof of Claim 4.26, we simply write
\(\square \)
Claim 4.33
We have, uniformly in \(s\le t\), and for some fixed \(\alpha > 0\),
Proof
On the one hand, as in Claim 4.20, and using the boundedness of Claim 4.16,
On the other hand, by [8, Lemma 4.7 and its proof], we have the unweighted quasiorthogonality estimate
for some exponent \(\alpha >0\), uniformly for \(s\le t\). Consequently, we may use the technique of Duoandikoetxea and Rubio de Francia [21], in which one first selfimproves the weight W, and then uses Stein–Weiss interpolation with change of measure [44], to deduce the weighted quasiorthogonality estimate
for some positive \(\beta < \alpha \) (see Lemma 2.5 in [18] for more details). Hence, by Claim 4.30,
\(\square \)
Collecting all the above claims, the proof of Lemma 4.29 is completed. \(\square \)
Corollary 4.34
We have the square function bound
where the implicit constant depends on n, \(\lambda \) and \([W]_{A_2}\).
Proof
With Lemma 4.29 in hand, since \(\widetilde{\theta }_t \mathbbm {1} = \theta _t \mathbbm {1}\), it is enough to prove the following:
To this end, we write
By (4.12),
where \(\widetilde{R}_t\) is precisely the same operator defined in (4.4), enjoying the square function bound established in Lemma 4.5. In addition, again using (4.12),
where \(\widetilde{T}_t\) is precisely the same operator defined in (4.4). We now repeat the splitting of \(\widetilde{T}_t\), exactly as in (4.7):
Note that the the second term equals \(2 \theta _t \nabla P^2_t f\) (see (4.10)). Combining these observations, we see that
where the term \(E_tf\) (which is actually the error \((\widetilde{\theta }_t \theta _t)\nabla P_t^2 f\)), satisfies the desired square function bound, by Lemma 4.8. The last term also enjoys the desired square function bound, by Lemma 4.14. This concludes the proof of the corollary. \(\square \)
4.3 The T(b) Argument
Recall that our goal is to prove the Carleson measure estimate (4.28). We now turn to this task, which will finish the proof of Theorem 4.1 (and therefore also the proof of Theorem 1.11). Our arguments here will be an adaptation of the proof of the Kato conjecture in the divergence form setting, see [8], and in particular, the extension of that proof to the degenerate elliptic case in [18].
We note that by the doubling property of W, we may assume that the supremum in (4.28) is taken over dyadic cubes Q. Given any such cube Q, a sufficiently small number \(\varepsilon \in (0, 1)\) to be chosen, and \(v \in {\mathbb {R}^n}\) with \(v = 1\), we define
where \({\varvec{\varPhi }}_Q(x) = x  x_Q\), \(x_Q\) denotes the center of Q, and \(\chi _Q \in \mathscr {C}_0^\infty \) is a cutoff function such that \(\chi _Q \equiv 1\) in 2Q, \(\textrm{supp}\,\chi _Q \subset 4Q\) and \(\Vert \chi _Q\Vert _\infty + \ell (Q) \Vert \nabla \chi _Q\Vert _\infty + \ell (Q)^2 \Vert \nabla ^2 \chi _Q\Vert _\infty \lesssim 1\). Clearly,
and also
The following estimates hold for \(f_{Q, v}^\varepsilon \), with constants that are uniform on Q, v and \(\varepsilon \):
These estimates follow at once from (4.35), Lemmas 2.22 and 2.25 (with \(t=\varepsilon \ell (Q)\)), and the doubling property of W.
The proof of (4.28) (and hence of Theorem 4.1 by Corollary 4.27) follows from the next two lemmas.
Lemma 4.40
There exists \(0 < \varepsilon = \varepsilon (\lambda , n, [W]_{A_2}) \ll 1\) and a finite set V of unit vectors in \({\mathbb {R}^n}\), whose cardinality depends only on \(\varepsilon \) and n, such that
where the implicit constant depends on n, \(\lambda \) and \([W]_{A_2}\).
Proof
The reader may check that the proof of [18, Lemma 5.1] (which in turn is an adaptation to the weighted case of [8, Lemma 5.4]) works perfectly well in our situation: as long as \(W \in A_2\) and \(f_{Q, v}^\varepsilon \) satisfies the estimates (4.37) and (4.38), the proof in [18] goes through^{Footnote 5}. \(\square \)
With Lemma 4.39 in hand, estimate (4.28) will follow immediately from the next lemma.
Lemma 1.3
For every cube Q and unit vector v, we have
where the implicit constant depends on n, \(\lambda \), \([W]_{A_2}\) and \(\varepsilon \), but is uniform on Q and v.
Proof
Fix Q and v, and abbreviate \(f := f_{Q, v}^\varepsilon \). By Corollary 4.34, we have
where in the last step, we have used (4.38) to obtain the desired bound for term I.
Term II can be treated as follows, using (4.12), Lemma 2.11, and the definition of \(f =f_{Q,v}^{\varepsilon }\),
where in the last two steps we have first used Lemma 2.11 (vi) with \(t =\varepsilon \ell (Q)\), and then (4.36). Since \(\varepsilon \) has been fixed depending only on allowable parameters, the dependence on \(\varepsilon \) is harmless.
Collecting all the preceeding estimates, we have finished the proof. \(\square \)
Notes
We caution the reader that the proof of (1.5) is a somewhat nontrivial matter, and is based on the boundedness of the commutator [T, b], where T is a singular integral and \(b\in \) BMO [17], along with a suitable expansion in terms of spherical harmonics; see [13, 14] for related results in the unweighted case.
Thus, our results here are somewhat related to those of CruzUribe and Rios [18].
To clarify a possible point of confusion, we mention that in [8] and [18], the unit vectors were taken in \(\mathbb {C}^n\), because in the divergence form setting of those papers, one treats the case of complex coefficients; at present, our results in the nondivergence form case treat only the case of real coefficients, so we need only consider real unit vectors.
References
Alfonseca, M.A., Auscher, P., Axelsson, A., Hofmann, S., Kim, S.: Analyticity of layer potentials and \(L^2\) solvability of boundary value problems for divergence form elliptic equations with complex \(L^\infty \) coefficients. Adv. Math. 226, 4533–4606 (2011)
Auscher, P., Axelsson, A.: Weighted maximal regularity estimates and solvability of nonsmooth elliptic systems I. Invent. Math. 184, 47–115 (2011)
Auscher, P., Axelsson, A., Hofmann, S.: Functional calculus of Dirac operators and complex perturbations of Neumann and Dirichlet problems. J. Funct. Anal. 255, 374–448 (2008)
Auscher, P., Axelsson, A., McIntosh, A.: Solvability of elliptic systems with square integrable boundary data. Ark. Mat. 48, 253–287 (2010)
Auscher, P., Axelsson, A., McIntosh, A.: On a quadratic estimate related to the Kato conjecture and boundary value problems. Contemp. Math. 505, 105–129 (2010)
Auscher, P., Egert, M., Nyström, K.: The Dirichlet problem for second order parabolic operators in divergence form. J. Éc. Polytech. Math. 5, 407–441 (2018)
Auscher, P., Egert, M., Nyström, K.: \(L^2\) wellposedness of boundary value problems for parabolic systems with measurable coefficients. J. Eur. Math. Soc. 22, 2943–3058 (2020)
Auscher, P., Hofmann, S., Lacey, M., McIntosh, A., Tchamitchian, P.: The solution of the Kato square root problem for second order elliptic operators on \({\mathbb{R} }^n\). Ann. Math. (2) 156, 633–654 (2002)
Auscher, P., Hofmann, S., Lewis, J.L., Tchamitchian, P.: Extrapolation of Carleson measures and the analyticity of Kato’s squareroot operators. Acta Math. 187, 161–190 (2001)
Axelsson, A., Keith, S., McIntosh, A.: Quadratic estimates and functional calculi of perturbed Dirac operators. Invent. Math. 163, 455–497 (2006)
Bauman, P.: Positive solutions of elliptic equations in nondivergence form and their adjoints. Ark. Mat. 22, 153–173 (1984)
Bortz, S., Hofmann, S., Luna García, J.L., Mayboroda, S., Poggi, B.: Critical perturbations for secondorder elliptic operators, I: Square function bounds for layer potentials. Anal. PDE 15, 1215–1286 (2022)
Chiarenza, F., Frasca, M., Longo, P.: Interior \(W^{2, p}\) estimates for nondivergence elliptic equations with discontinuous coefficients. Ric. Mat. 40, 149–168 (1991)
Chiarenza, F., Frasca, M., Longo, P.: \(W^{2, p}\)solvability of the Dirichlet problem for nondivergence elliptic equations with VMO coefficients. Trans. Amer. Math. Soc. 336, 841–853 (1993)
Coifman, R., Deng, D., Meyer, Y.: Domaine de la racine carrée de certains operateurs différentiels accrétifs. Ann. l’Inst. Fourier 33, 123–134 (1983)
Coifman, R., McIntosh, A., Meyer, Y.: L’intégrale de Cauchy définit un opérateur borné sur \(L^2\) pour les courbes lipschitziennes. Ann. Math. 116, 361–387 (1982)
Coifman, R., Rochberg, R., Weiss, G.: Factorization theorems for Hardy spaces in several variables. Ann. Math. (2) 103, 611–635 (1976)
CruzUribe, D., Ríos, C.: The Kato problem for operators with weighted ellipticity. Trans. Amer. Math. Soc. 367, 4727–4756 (2015)
CruzUribe, D., Ríos, C.: The solution of the Kato problem for degenerate elliptic operators with Gaussian bounds. Trans. Amer. Math. Soc. 364, 3449–3478 (2012)
Duoandikoetxea, J.: Fourier Analysis. Graduate Studies in Mathematics, vol. 29. American Mathematical Society, Providence, RI (2001)
Duoandikoetxea, J., Rubio de Francia, J.L.: Maximal and singular integral operators via Fourier transform estimates. Invent. Math. 84, 541–561 (1986)
Duong, X.T., Yan, L.X.: Bounded holomorphic functional calculus for nondivergence form differential operators. Differ. Integral Equ. 15, 709–730 (2002)
Escauriaza, L.: Weak type\((1,1)\) inequalities and regularity properties of adjoint and normalized adjoint solutions to linear nondivergence form operators with VMO coefficients. Duke Math J. 74, 177–201 (1994)
Escauriaza, L.: Bounds for the fundamental solution of elliptic and parabolic equations in nondivergence form. Commun. Partial Differ. Equ. 25, 821–845 (2000)
Escauriaza, L., Hofmann, S.: Kato square root problem with unbounded leading coefficients. Proc. Amer. Math. Soc. 146, 5295–5310 (2018)
Fabes, E., Jerison, D., Kenig, C.: Multilinear square functions and partial differential equations. Amer. J. Math. 107, 1325–1367 (1985)
Fabes, E., Jerison, D., Kenig, C.: Necessary and sufficient conditions for absolute continuity of ellipticharmonic measure. Ann. Math. (2) 119, 121–141 (1984)
Haase, M.: The Functional Calculus for Sectorial Operators. Operator Theory: Advances and Applications, vol. 169. Birkhäuser Verlag, Basel (2006)
Heinonen, J., Kilpeläinen, T., Martio, O.: Nonlinear Potential Theory of Degenerate Elliptic Equations. Courier Dover Publ. (2018)
Hofmann, S.: Local \(Tb\) theorems and applications in PDE. Proceedings of the ICM Madrid, Vol. II, pp. 1375–1392. European Math. Soc. (2006)
Hofmann, S., Kenig, C., Mayboroda, S., Pipher, J.: Square function/nontangential maximal function estimates and the Dirichlet problem for nonsymmetric elliptic operators. J. Amer. Math. Soc. 28, 483–529 (2015)
Hofmann, S., Kenig, C., Mayboroda, S., Pipher, J.: The regularity problem for second order elliptic operators with complexvalued bounded measurable coefficients. Math. Ann. 361, 863–907 (2015)
Hofmann, S., Lacey, M., McIntosh, A.: The solution of the Kato problem for divergence form elliptic operators with Gaussian heat kernel bounds. Ann. Math. (2) 156, 623–631 (2002)
Hofmann, S., Le, P., Morris, A.J.: Carleson measure estimates and the Dirichlet problem for degenerate elliptic equations. Anal. PDE 12, 2095–2146 (2019)
Hofmann, S., Li, L., Mayboroda, S., Pipher, J.: \(L^p\) theory for the square roots and square functions of elliptic operators having a BMO antisymmetric part. Math. Z. 301, 935–976 (2022)
Hofmann, S., Li, L., Mayboroda, S., Pipher, J.: The Dirichlet problem for elliptic operators having a BMO antisymmetric part. Math. Ann. 382, 103–168 (2022)
Hofmann, S., McIntosh, A.: The solution of the Kato problem in two dimensions. Proceedings of the Conference on Harmonic Analysis and PDE held in El Escorial, Spain in July 2000. Publ. Mat. Vol. extra, pp. 143–160 (2002)
Kato, T.: Perturbation Theory for Linear Operators. SpringerVerlag, New York (1966)
Kenig, C., Meyer, Y.: The Cauchy integral on Lipschitz curves and the square root of second order accretive operators are the same. In: Peral, I., de Francia, J.L.R. (eds.) Recent Progress in Fourier Analysis. North Holland Math Studies, vol. 111, pp. 123–145. North Holland (1985)
Krylov, N., Safonov, M.: A certain property of solutions of parabolic equations with measurable coefficients. Math. USSR Izv. 16, 151–164 (1981)
McIntosh, A.: Operators which have an \(H_\infty \) functional calculus. In: Miniconference on Operator Theory and Partial Differential Equations, 1986. Proceedings of the Centre for Mathematical Analysis, ANU, Canberra, pp. 210–231 (1986)
Nadirashvili, N.S.: Nonuniqueness in the martingale problem and the Dirichlet problem for uniformly elliptic operators. Ann. Sc. Norm. Super. Pisa Cl. Sci. (4) 24, 537–550 (1997)
Nyström, K.: \(L^2\) Solvability of boundary value problems for divergence form parabolic equations with complex coefficients. J. Differ. Equ. 262, 2808–2939 (2017)
Stein, E.M., Weiss, G.: Interpolation of operators with change of measures. Trans. Amer. Math. Soc. 87, 159–172 (1958)
Yagi, A.: Coincidence entre des espaces d’interpolation et des domaines de puissances fractionnaires d’opérateurs. C. R. Acad. Sc. Paris, Sér. I 299, 173–176 (1984)
Acknowledgements
The second author is supported by the grant CEX2019000904S203, funded by MCIN/AEI/ 10.13039/501100011033, and acknowledges financial support from MCIN/AEI/ 10.13039/501100011033 grants CEX2019000904S and PID2019107914GBI00. The third author was supported by NSF grant DMS2000048. Part of this work was carried out while the first and third authors were visiting ICMAT in Madrid, and part of this work was carried out while the second author was visiting the third author at the University of Missouri  Columbia. The authors express their gratitude to these institutions. The third author thanks Prof. X. T. Duong for an interesting conversation concerning the latter’s joint work with L. Yan [22], and in particular for pointing out to us the argument sketched in Remark 1.13.
Author information
Authors and Affiliations
Corresponding author
Additional information
Dedicated to Prof. Carlos Kenig on the occasion of his 70th birthday, and to the memory of Luis Escauriaza.
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author selfarchiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.
About this article
Cite this article
Escauriaza, L., HidalgoPalencia, P. & Hofmann, S. On the Kato Problem for Elliptic Operators in NonDivergence Form. Vietnam J. Math. 52, 1067–1096 (2024). https://doi.org/10.1007/s10013024006831
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10013024006831
Keywords
 Elliptic operators in nondivergence form
 Kato square root problem
 Muckenhoupt weights
 LittlewoodPaley theory
 Functional Calculus