The Regularity Problem for Uniformly Elliptic Operators in Weighted Spaces

This paper studies the regularity problem for block uniformly elliptic operators in divergence form with complex bounded measurable coefficients. We consider the case where the boundary data belongs to Lebesgue spaces with weights in the Muckenhoupt classes. Our results generalize those of S. Mayboroda (and those of P. Auscher and S. Stahlhut employing the first order method) who considered the unweighted case. To obtain our main results we use the weighted Hardy space theory associated with elliptic operators recently developed by the last two named authors. One of the novel contributions of this paper is the use of an “inhomogeneous” vertical square function which is shown to be controlled by the gradient of the function to which is applied in weighted Lebesgue spaces.

Key words and phrases.Regulatity problem, uniformly elliptic operators in divergence form, Muckenhoupt weights, singular non-integral operators, square functions, heat and Poisson semigroups, a priori estimates, off-diagonal estimates, square roots of elliptic operators, Kato's conjecture.
The research leading to these results has received funding from the European Research Council under the European Union's Seventh Framework Programme (FP7/2007-2013)/ ERC agreement no.615112 HAPDEGMT.The authors also acknowledge financial support from the Spanish Ministry of Economy and Competitiveness, through the "Severo Ochoa Programme for Centres of Excellence in R&D" (SEV-2015-0554).

Introduction
The study of elliptic boundary value problems on upper half spaces and on Lipschitz domains has a long history (see [21] for an introduction of major results for elliptic equations in divergence form with real symmetric coefficients).Recent breakthroughs in the field include [17,18], where the Dirichlet and Regularity problems with real non-symmetric coefficients were considered.The study of elliptic problems with rough complex coefficients has been arousing great interest, particularly after the solution of Kato's conjecture in [4].In this direction, the connections between the Dirichlet, regularity and Neumann problems were studied in [24] for the block case.Some of the tools used in the latter reference include a Calderón-Zygmund theory adapted to singular "non-integral" operators (see [1], and also [5,6,7]), as well as the a Hardy space theory adapted to elliptic operators (see [19,20]).The reader is also referred to the work [8] for the robust first order method to deal with more general cases.
The main purpose of the present article is to continue with this line of research and study the regularity problem for elliptic operators with block structure, and with data in weighted Lebesgue spaces, for Muckenhoupt weights.This is a natural problem to consider, in view of a well-established weighted Calderón-Zygmund theory in the series of papers [5,6,7] and the weighted Hardy space theory adapted to elliptic operators recently developed in [22,23,25].To state our main results, we need to introduce some background.Some notation is taken from [1,22,24].Let A = (a jk ) n j,k=1 be an n × n matrix whose entries are L ∞ -valued complex coefficients defined on R n .We assume that A satisfies the following uniform ellipticity condition: there exist 0 < λ ≤ Λ < ∞ such that j,k a j,k (x) ξ k ζj .Associated with this matrix, consider the second order divergence form uniformly elliptic operator L := −div(A∇).
Given A and L as above we construct the block matrix Clearly, A is uniformly elliptic in R n+1 and, thus, A gives rise to the block divergence form uniformly elliptic operator Here and elsewhere the points in R n+1 are written as (x, t) ∈ R n × R so that ∇ x,t and div x,t denote respectively the full gradient and divergence, while ∇ x and div x stand respectively for the gradient and divergence in the first n variables, and L x = L.
With the previous definition in hand, it is easy to see that whenever f ∈ S, one has that u(x, t) := e −t √ L f (x), where (x, t) ∈ R n+1 + , is a weak solution of Lu = 0 in R n+1 + .By this we mean that u ∈ W 1,2 loc (R n+1 + ) satisfies where 0 < κ < 1 is a fixed small constant, Γ κ (x) := {(y, t) ∈ R n+1 + : |x − y| < κt}, and D((y, t), κt) denotes the R n+1 -ball centered at (y, t) with radius κt, which is by construction contained in R n+1 + .Using the previous definitions, the regularity boundary value problem (R p ), 1 < p < ∞, is said to be solvable for the operator L in R n+1 + if, for every f ∈ S, the weak solution to the equation Lu = 0 (with boundary data f ) given by the Poisson + , satisfies the non-tangential maximal function estimate The solvability of (R p ), 1 < p < ∞, was proved by S. Mayboroda in [24].In particular, the result is the following: Then for any p so that max 1, nq − (L) n+q − (L) < p < q + (L), the regularity problem (R p ) is solvable.
The interval (q − (L), q + (L)) represents the maximal open interval on which the gradient of the heat semigroup { √ t∇e −tL } t>0 is uniformly bounded on L p (R n ).More specifically: Similarly, (p − (L), p + (L)) denotes the maximal open interval of p such that the heat semigroup {e −tL } t>0 is uniformly bounded on L p (R n ).As we shall explain in more detail in Section 2, these two intervals are somehow related, and for instance p − (L) = q − (L), see [1,7] where these intervals were deeply studied.
The goal of this paper is to extend Theorem 1.3 to the context of weighted Lebesgue spaces L p (w) with w being a Muckenhoupt weight.To set the stage we first give some definitions and basic properties of Muckenhoupt weights.For further details, see [14,15,16].Given a weight w, that is, a measurable function such that 0 < w < ∞ a.e. and w ∈ L 1 loc (R n ), we say that w and, when p = 1, we say that w Here and below the suprema are taken over all balls B ⊂ R n .We next introduce the reverse Hölder classes.We say that w

Finally, we set
We recall Muckenhoupt's theorem which states that w ∈ A p , 1 < p < ∞, if and only if the Hardy-Littlewood maximal function It is well-known that w ∈ A ∞ implies that w is a doubling measure.Indeed, for any w ∈ A r , 1 ≤ r < ∞, we have Consequently, (R n , dw, | • |) is a space of homogeneous type.
Another important feature of Muckenhoupt classes is their openness.More precisely, the A p and RH s classes have the following self-improving property: if w ∈ A p , there exists ǫ > 0 such that w ∈ A p−ǫ , and similarly if w ∈ RH s , then w ∈ RH s+δ for some δ > 0. These facts motivate the following definitions: Note that, according to our definition, s w is the conjugated exponent of the one defined in [6,Lemma 4.1].Given 0 ≤ p 0 < q 0 ≤ ∞ and w ∈ A ∞ , [6, Lemma 4.1] implies that W w (p 0 , q 0 ) := p ∈ (p 0 , q 0 ) : If p 0 = 0 and q 0 < ∞ it is understood that the only condition that stays is w ∈ RH ( q 0 p ) ′ .Analogously, if 0 < p 0 and q 0 = ∞ the only assumption is w ∈ A p p 0 . Finally After these observations and definitions, we can introduce the notion of solvability in the weighted context.More precisely, given a weight w ∈ A ∞ we say that the weighted regularity boundary value problem (R w p ) is solvable for the operator L in R n+1 + , if, for every f ∈ S, the weak solution to the equation Lu = 0 with the boundary data f given by the Poisson semigroup u(x, t) = e −t √ L f (x), (x, t) ∈ R n+1 + , satisfies the non-tangential maximal function estimate (1.9) and u(•, t) → f in L 2 (R n ) as t → 0 + .Our main result establishes the solvability of (R w p ): Theorem 1.10.Let L be a block elliptic operator in R n+1 + as above and let w ∈ A ∞ be such that W w (q − (L), q + (L)) = Ø.For every p such that (1.11) max r w , nrw q − (L) and for every f ∈ S, if one sets u(x, t) and (R w p ) is solvable.
In the previous result ( q − (L), q + (L)) denotes the maximal open interval where { √ t∇e −tL } t>0 is uniformly bounded on L p (w), or equivalently, where { √ t∇e −tL } t>0 satisfies L p (w)-L q (w) off-diagonal estimates on balls.
Remark 1.13.The fact that W w (q − (L), q + (L)) = Ø means that q + (L) q − (L) > r w s w .This is a compatibility condition between L and w.It appears naturally in the theory developed in [5,6,7] and forces the weight to be sufficiently good, depending on how close q − (L) and q + (L) are to each other.As it was shown in [1], q − (L) = p − (L) < 2n n+2 while, in general, we only know that q + (L) > 2, actually, there are examples showing that q + (L) can be found arbitrarily close to two.Thus, q + (L) q − (L) > 1 + 2 n , and if r w s w < 1 + 2 n , we always have that W w (q − (L), q + (L)) = Ø, for every L as above.Of course, this is a very restrictive condition since we would expect to have wider classes of weights as the operator L gets "nicer".A very illustrative example is that of power weights of the form ω α (x) := |x| α .The weight ω α belongs to A ∞ , if and only if, α > −n.It is not hard to see that r w = max{1, 1 + α/n} and s w = max{1, (1 + α/n) −1 } (see the definitions of r w and s w in (1.7)) and hence q + (L) q − (L) > r w s w implies that −n 1 − q − (L) q + (L) < α < n q + (L) q − (L) − 1 .Note that this range of α always contains the interval [− 2n n+2 , 2] and gets bigger as q − (L) decreases to one, and/or q + (L) increases to infinity.
To proof Theorem 1.10 we independently consider the estimate of the weighted norm of the non-tangential maximal function for the time derivative ∂ t e −t √ L f , and the one for the spatial derivatives ∇e −t √ L f .For the former we use ideas from [24,Theorem 4.1] combined with the weighted Hardy space theory associated with elliptic operators developed in [22,23,25]. 1  Regarding, the spatial derivatives we follow a different path, for which we need to introduce the following "inhomogeneous" vertical square function .
Note that compared with the usual vertical square functions (or Littlewood-Paley-Stein functionals), the power of t does not seem to be consistent with the order of the operator in the integral.Indeed, the power of t corresponding to the usual square function would be three instead of two.That is why we use the terminology "inhomogeneous".This has the effect of getting an estimate for G H f in terms of ∇f in L q (w), rather than obtaining an estimate from L q (w) to L q (w) (which is the natural one for the usual square functions).We would like to mention that in the case that L is the Laplace operator, this inhomogeneous square function (as well as its relation to reverse Riesz transform estimates), was studied in [12] in the setting of Riemannian manifolds.
As we shall see below, the estimate regarding the non-tangential maximal function for the spatial derivatives can be inferred from an estimate of the inhomogeneous vertical square function introduced in (1.14).This is one of the novel contributions of this paper even in the unweighted case, and it is stated as follows: Theorem 1.15.Let w ∈ A ∞ and assume that W w (q − (L), q + (L)) = Ø.Given q so that max r w , nrw q − (L) nrw+ q − (L) < q < q + (L) there holds for every f ∈ S.
The plan of the paper is as follows.In Section 2 we present some preliminaries and some auxiliary results needed in Sections 3 and 4. As for these two last sections, the former is devoted to the proof of Theorem 1.15, and the later to proving Theorem 1.10.

Preliminaries and Auxiliary results
Throughout the paper the letters c, C, θ or θ will represent any harmless constant that will not depend on any relevant parameter of the corresponding computation, 1 We note that there is an alternative and independent method developed in the forthcoming paper [2] to deal with the regularity problem for degenerate elliptic operators which is based on some adapted Calderón-Zygmund theory, and does not require to deal with the highly technicality of Hardy spaces.The corresponding result in [2] gives however a more restricted range.and that will may be different in both sides of an inequality.Moreover, we shall write ∇ = (∂ x 1 , . . .∂ xn ) to denote the gradient in R n .We will typically write the elements in R n+1 + as (x, t) with x ∈ R n and t > 0 and hence ∇ x,t = (∇, ∂t) stands for the gradient in R n+1 .Given a ball B, we use the notation C 1 (B) = 4B and hdw.
We recall the notion of full off-diagonal estimates: Definition 2.1.Let {T t } t>0 be a family of sublinear operators and let 1 ≤ p ≤ q ≤ ∞.We say that {T t } t>0 satisfies L p -L q full off-diagonal estimates, denoted by , if there exists a constant c > 0 such that for all closed sets E and F , all f and all t > 0 we have Definition 2.3.Given 1 ≤ p ≤ q ≤ ∞ and any weight w ∈ A ∞ .We say that a family of sublinear operators {T t } t>0 satisfies L p -L q off-diagonal estimates on balls, denoted by T t ∈ O(L p (w) − L q (w)), if there exist θ 1 , θ 2 > 0 and c > 0 such that for all t > 0 and for any ball B with radius r B , (2.4) , and, for all j ≥ 2, (2.5) For the following results we recall that (p for every m ∈ N. In addition, we have p − (L) = q − (L) and (q − (L) ) and is a bounded set in L(L p (w)).(ii) Assume W w (q − (L), q + (L)) = Ø.There exists a maximal open interval denoted by In the above proposition (q + (L)) * and ( q + (L)) * w are defined as follows: for all 0 < q < ∞, (q) * w := qnrw nrw−q , nr w > q, ∞, otherwise, and (q) * := The following result, which appears in a more general way in [25, Proposition 4.1], and it is a weighted version of [23, (5.12)] (see also [19]), contains some off-diagonal estimates for the family {T t,s } s,t>0 := {(e −t 2 Lw − e −(t 2 +s 2 )Lw ) M } s,t>0 , where M ∈ N.
, and let 0 < t, s < ∞.Given M ∈ N, for all sets E 1 , E 2 ⊂ R n and f ∈ L p (w) such that supp(f ) ⊂ E 1 , we have that {T t,s } s,t>0 := {(e −t 2 Lw − e −(t 2 +s 2 )Lw ) M } s,t>0 satisfies the following L p (w)-L p (w) off-diagonal estimates: In particular, We introduced before the Hardy-Littlewood maximal function and here we present some weighted maximal version which will be used throughout the paper.Given w ∈ A ∞ and 0 < q < ∞ we set Since w ∈ A ∞ implies that w is a doubling measure (see (1.6)) then M w q is bounded on L p (w) for every q < p ≤ ∞ and is bounded from L q (w) to L q,∞ (w).
The following result contains a Calderón-Zygmund decomposition where the function is split according to the level sets of its gradient and to its norm in weighted Lebesgue spaces.
where C and N depend only on the dimension, the doubling constant of µ, and p 0 .In addition, for 1 ≤ q < (p 0 ) * w , where (p 0 ) * w is defined in (2.9), we have The estimates contained in the following auxiliary result follows easily from Hölder's inequality along with the definitions of the A p and RH s classes: Lemma 2.20.For every 0 < p ≤ q < ∞, every ball B ⊂ R n and every j ≥ 1, there hold where the implicit constants are independent of j.
We also need the following off-diagonal estimate on Sobolev spaces, which can be proved as [1, (4.6)].Nevertheless, we include the proof for the sake of completeness.
Proof.It is straightforward to see that it suffices to prove the first estimate.Indeed assuming that, we can write t = s/ √ 2 and .
In order to prove the first estimate, fix (x, s) ∈ R n+1 + and define B := B(x, λs).
Hence, by Poincaré inequality, which holds since p > r w and thus w ∈ A p , we conclude that .
This finishes the proof.

Proof of Theorem 1.15
In this section, we shall prove Theorem 1.15, which establishes weighted norm estimates for G H (defined in (1.14)).To this end, we introduce two results that will be used in that proof.
for every f ∈ S.
Note that recalling (1.8), and using Propositions 2.7 and 2.8, it is not difficult to see that the range where (3.2) holds contains W w (p − (L), p + (L)) and hence also W w (q − (L), q + (L)).
The following result deals with some vertical square functions.It can be proved with an argument similar to that of [5,Theorem 7.2], or also combining [13, Proof of Proposition 10.1] and [22,Theorem 1.12]).Further details are left to the interested reader.
L p (w) L p (w) We next define some auxiliary square functions that will be also useful in the proof of Theorem 1.15.In particular, consider and, for m = 1, 2, 15.First of all fix f ∈ S and consider T t := ∇e −tL and F (y, t) := t 2 Lf (y), for (y, t) ∈ R n+1 + .Then, fix (x, t) ∈ R n+1 + , and note that for B := B(x, t) and by the second estimate in Proposition 2.21 with w = 1 and λ = 1, we have that, for all 2 < q 0 < q + (L), .
To finish the proof, note that in view of Lemma 3.3 and Theorem 3.1, we can improve the upper bound of the interval where the above inequality holds up to q + (L) (assuming that q + (L)/s w < q + (L), otherwise there is nothing to prove).Indeed, we just need to observe that for q such that q + (L)/s w ≤ q < q + (L), we have that q falls within the scope of those results.This follows since we assume that W w (q − (L), q + (L)) = Ø, and by definition, it holds that W w (q − (L), q + (L)) ⊂ W w (p − (L), p + (L)).Then, we have that W w (p − (L), p + (L)) = Ø and max r w , nr w p − (L) Remark 3.8.The result given by Theorem 1.15 is, as far as we know, new even in the unweighted case, that is, when w ≡ 1.In that scenario it says that for every q such that max 1, nq − (L) n+q − (L) < q < q + (L) and f ∈ S, there holds The condition W w (q − (L), q + (L)) = Ø always holds when w ≡ 1, since q − (L) < 2 < q + (L), and, by definition, (q − (L), q + (L)) = W w (q − (L), q + (L)).

Proof of Theorem 1.10
In this section we prove Theorem 1.10.Given f ∈ S we set u(x, t) := e −t √ L f (x), for each (x, t) ∈ R n+1 + .It is well-known that u(•, t) ∈ L 2 (R n ) uniformly in t > 0 since the Poisson semigroup is uniformly bounded on L 2 (R n ) (this latter fact can be seen directly from the subordination formula (4.24) below along with the uniform L 2boundedness of the heat semigroup, see Section 2).This and Caccioppoli's inequality readily imply that u ∈ W 1,2 loc (R n+1 + ) and also Lu = 0 in the weak sense.Using standard holomorphic functional calculus techniques one can also see that u(•, t) → f in L 2 (R n ) as t → 0 + .Thus we are left with showing (1.12) and to this end, it suffices to individually bound the operators N (∂ t e −t √ L f ) and N (∇e We first deal with N (∂ t e −t √ L f ).In the unweighted case, in [24], the estimate of this operator relies on the characterization of Hardy spaces associated with L via the non-tangential maximal function associated with the Poisson semigroup and the Riesz transform (characterization established in [19,20]).Recently, the weighted Hardy spaces have been carefully studied in [10,11,23,25], including various characterization of weighted Hardy spaces via molecules, square functions, non-tangential maximal functions, Riesz transform etc.This weighted Hardy space theory enables us to treat the weighted estimate of N (∂ t e −t √ L f ) by following the path laid down in [24].More precisely we obtain the following result whose proof is given in Section 4.1: and let p be chosen so that max r w , nrw p − (L) sw .Then, for any f ∈ S, Regarding the spatial derivatives, the following result establish the desired bound for N (∇e −t √ L f ): Proposition 4.3.Let w ∈ A ∞ be such that W w (q − (L), q − (L)) = Ø.Then, for max r w , nrw q − (L) nrw+ q − (L) < p < q + (L) sw and f ∈ S, there holds The proof of this result is in Section 4.2.Our method differs from the one in [24] and it is of independent interest.More precisely, when w ≡ 1, our proof provides an alternative approach to [24, Proof of Theorem 4.1, Steps II-VIII], in which matters can be essentially reduced to estimate the inhomogeneous vertical square function G H (see Theorem 1.15) along with some similar "homogeneous" conical square function estimates proved in [26] (see (3.5) and (3.7)).We are now ready to prove Proposition 4.1.First, it was shown in [24, (4.25)] that Then by Lemma 4.6, we have, for p ∈ W w (p − (L), (p + (L)) * ), (4.8) Besides, [25,Theorem 1.1] shows that for any p, q ∈ W w (p − (L), p + (L)), the weighted Lebesgue space L p (w) and the Hardy space H p S H ,q (w) (see the definition in [25]) are isomorphic with equivalent norms.This and Lemma 4.6 readily give that for any fixed q ∈ W w (p − (L), p + (L)), (4.9) N P : H p S H ,q (w) → L p (w), ∀p ∈ W w (p − (L), p + (L)).Furthermore, by [23,Theorems 3.9 and 3.11] we also have that (4.10) N P : H 1 S H ,q (w) → L 1 (w).Hence, in view of the interpolation result [25,Theorem 5.1], by (4.9) and (4.10), we get and thus (4.11) Next, from the weighted Hardy space Riesz transform characterization (see [25,Propositions 9.1]), it follows that, for all f ∈ S, Finally, combining (4.7), (4.11), and (4.12), we obtain that, for max r w , nrw p − (L) for all f ∈ S.This completes the proof.

4.2.
Proof of Proposition 4.3.We split the proof into two steps.In Step 1 we obtain (4.4) for all max{r w , q − (L)} < p < q + (L) sw ; and, in Step 2 we show that the same estimate holds in a bigger range, namely for all max r w , nrw q − (L) sw .The following claims are common to both steps, so we start by proving them.Claim 4.13.Under the hypothesis of Proposition 4.3, for any max{r w , q − (L)} < q 0 < q + (L)/s w , there holds , where S t/ √ 2 could be equal to e − t 2 4 L or the identity.
Claim 4.14.Under the hypothesis of Proposition 4.3, for any max{r w , q − (L)} < q 0 < q + (L)/s w , there holds Under the hypothesis of Proposition 4.3, there holds where .
In order to prove these claims, fix p 0 and q such that q − (L) < p 0 < 2, r w p 0 < q + (L)/s w , and max{q 0 , r w p 0 } < q < q + (L)/s w .
Once we have proved the claims, we can start discussing the two cases into which we split the proof of Proposition 4.3.
Step 1: max{r w , q − (L)} < p < q + (L) sw .First of all, note that proceeding similarly as in [24, (4.25)] one can show that for every fixed Since max{r w , q − (L)} < p < q + (L) sw , we can choose q 0 so that max{r w , q − (L)} < q 0 < p < q + (L) sw .Hence by Claim 4.13 with S t equal to the identity, we obtain On the other hand, note that applying the following subordination formula:  e −u u In order to estimate II 1 , we apply Claim 4.14: where G H is the inhomogeneous vertical square function defined in (1.14).On the other hand, for 1/4 ≤ u < ∞, one can see that Next, applying Claim 4.15 we get This, (4.22), (4.23), (4.25), and (4.26) imply that, for max {r w , q − (L)} < q 0 < p, Consequently, for all max {r w , q − (L)} < p < q + sw , by the boundedness of M w q 0 on L p (w), recall that p/q 0 > 1, and change of angles (see [22,Proposition 3.2]), we conclude that where the last inequality follows from Theorem 1.15, (3.5), and (3.7).This completes the proof of Step 1.
We start by observing that Step 1 leads to N (∇e −t √ L ) : Ẇ 1,q (w) → L q (w), max{r w , q − (L)} < q < q + (L) s w , (4.28) where Ẇ 1,1 (w) = f ∈ S : ∇f ∈ L q (w) .With this in hand, by interpolation (see [9]), we can conclude (4.4) for all max r w , rwn q − (L) rwn+ q − (L) < p < q + (L) sw provided that for every p and r 0 > r w such that q − (L) < p < q + (L)/s w , r w q − (L) < r 0 q − (L) < q + (L)/s w , and q := max r 0 , nr 0 p nr 0 + p , we show that In order to prove (4.29) fix f ∈ S, and fix p and r 0 satisfying the above restrictions.Furthermore, fix q 2 and p 1 so that q − (L) < q 2 < 2, r 0 q 2 < q + (L) sw , and max{ p, r 0 q 2 } < p 1 < q + (L)/s w .Note that in particular ′ , and p 1 > q. (4.30)Moreover, note by Claim 4.13 with S t = e − t 2 4 L , and Claims 4.14 and 4.15, we obtain that, for any q 0 such that max{r w , q − (L)} < q 0 < max{r 0 , p} < q + (L)/s w and any Next we observe that to obtain (4.29) it suffices to prove for every α > 0 that In order to obtain this inequality, fix α > 0 and consider the Calderón-Zygmund decomposition for the function f at height α given by Lemma 2.14 with p 0 = q.Let {B i } i be the corresponding collection of balls, and define, for M ∈ N arbitrarily large, Hence, Before starting with the estimate of the above terms, we make a couple of observations for later use.First, take 1 < q < ∞ and h ∈ L q ′ (w) such that h L q ′ (w) = 1 and recall the definition of M w in (2.13).Then, using a Kolmogorov type inequality (see [16,Exercise 2.1.5]),the fact that M w is bounded from L 1 (w) to L 1,∞ (w) since w ∈ A ∞ and hence it gives rise to a doubling measure, and (2.17), we have that (4.35) Second, note that for p 2 := max{r 0 , p}, we have that q 0 < p 2 .Assuming momentarily that p 2 < ( q) * w , by (2.19), we get that In order to see that p 2 < ( q) * w , we first consider the case p 2 = r 0 .Then, q = max r 0 , nr 0 p nr 0 + p = r 0 , and thus p 2 = r 0 < (r 0 ) * w = ( q) * w .On the other had, if p 2 = p, we may assume that nr w > q -otherwise we trivially have p = p 2 < ( q) * w = ∞.Besides, Now, we are ready to estimate the terms in (4.34).In order to estimate I, first recall that q < p 1 and max{r w , q − (L)} < p 1 < q + (L)/s w (see (4.30)).Then, by Chebyshev's inequality, (4.28) and Lemma 2.14, we get Next we estimate II.Consider p 2 defined as in (4.36) and apply Chebyshev's inequality and (4.28) (recall that max{r w , q − (L)} < p 2 < p 1 < q + (L)/s w ).Thus Besides, using that { √ t∇e −tL } t>0 satisfies L p 2 (w)-L p 2 (w) off-diagonal estimates on balls; and by (1.6) and (4.36), we have Hence, by (4.35) with q = p 2 , we have Consequently, (4.39) holds.In view of (4.38), that implies In order to estimate IV 1 apply Chebyshev's inequality, Minkowski's integral inequality, change of angles ([22, Proposition 3.2]), (3.5), and (3.7), to get 1 α q ∇b q L q (w) .
Besides, note that by (2.16) and (2.17) Hence, As for the estimate of IV 2 , apply again Chebyshev's inequality, Minkowski's integral inequality, change of angles, (3.5), and (3.7), to get Thus, by (4.39), Collecting the estimates for IV 1 and IV 2 we conclude that  .
For m = 1, we have that )L M .
In the above setting, first recall that max{r w , q − (L)} < q 0 < p 2 < q + (L)/s w , consequently t∇e −t 2 L is bounded on L q 0 (w).Besides, applying the L q 0 (w)-L q 0 (w) off-diagonal estimates that T t,r B i satisfies (see (2.11)), and (1.6) to obtain w(C N (B l i )) 2 lθ w(B l i ) where in the last inequality we have used (4.36) since q 0 < p 2 .Consequently, where in the first inequality, we have used that w(B(x, θ M 2 l+2 t)) −1 w(B i ) ≤ C, since B i ⊂ B(x, θ M 2 l+5 t) and by (1.6).
Collecting the estimates obtained for C 1 and C 2 , we conclude that, for M ∈ N such that 2M + 1 > θ 2 , (see (4.30)), by Lemma 2.20, and the L p 2 (w)-L p 1 (w) off-diagonal estimates on balls satisfied by e −τ L , we have that