Pascal Auscher

JOS É MAR ÍA Li I Chen
email: lichen@lsu.edu

AND Cruz Martell
email: chema.martell@icmat.es

Prisuelos-Arribas

José María Martell

Cruz Prisuelos-Arribas

THE REGULARITY PROBLEM FOR DEGENERATE ELLIPTIC OPERATORS IN WEIGHTED SPACES

Keywords: . 2010 Mathematics Subject Classification. 35J25, 35B65, 35B45, 42B37, 42B25, 47A60, 47D06, 35J15 Regularity problem, degenerate 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

We study the solvability of the regularity problem for degenerate elliptic operators in the block case for data in weighted spaces. More precisely, let L w be a degenerate elliptic operator with degeneracy given by a fixed weight w ∈ A 2 (dx) in R n , and consider the associated block second order degenerate elliptic problem in the upper-half space R n+1

+ . We obtain non-tangential bounds for the full gradient of the solution of the block case operator given by the Poisson semigroup in terms of the gradient of the boundary data. All this is done in the spaces L p (vdw) where v is a Muckenhoupt weight with respect to the underlying natural weighted space (R n , wdx). We recover earlier results in the non-degenerate case (when w ≡ 1, and with or without weight v). Our strategy is also different and more direct thanks in particular to recent observations on change of angles in weighted square function estimates and non-tangential maximal functions. Our method gives as a consequence the (unweighted) L 2 (dx)-solvability of the regularity problem for the block operator

) for any complex-valued uniformly elliptic matrix A and for all -ǫ < α < 2 n n+2 , where ǫ depends just on the dimension and the ellipticity constants of A.

The study of divergence form degenerate elliptic equations was pioneered in the series of papers [START_REF] Fabes | The Wiener test for degenerate elliptic equations[END_REF][START_REF] Fabes | Boundary behavior of solutions to degenerate elliptic equations[END_REF]22], where real symmetric elliptic matrices with some degeneracy expressed in terms of A 2 (dx)-weights were considered (here and elsewhere A 2 (dx) ≡ A 2 (R n , dx)). The goal of this paper is to obtain the solvability of the regularity problem for second order divergence form degenerate elliptic operators with complex coefficients and with boundary data in weighted Lebesgue spaces. To set the stage, let us introduce the class of operators that we consider here. Let A be an n × n matrix of complex L ∞ -valued coefficients defined on R n , n ≥ 2. We assume that this matrix satisfies the following uniform ellipticity (or "accretivity") condition: there exist 0 < λ ≤ Λ < ∞ such that

L w u = -w -1 div(w A ∇u), (1.2)
which is understood as a maximal-accretive operator on L 2 (R n , wdx) ≡ L 2 (w) with domain D(L w ) by means of a sesquilinear form. Note that writing A w = w A, one has that A w is a degenerate elliptic matrix in the sense that

(1.3) λ |ξ| 2 w(x) ≤ Re A w (x) ξ • ξ and |A w (x) ξ • ζ| ≤ Λ |ξ| |ζ| w(x),
for all ξ, ζ ∈ C n and almost every x ∈ R n . Conversely, if A w is degenerate elliptic matrix satisfying the previous conditions one can trivially see that A := w -1 A w is uniformly elliptic. The prominent case w ≡ 1 gives the class of uniformly elliptic operators. The celebrated resolution of the Kato problem in [START_REF] Auscher | The solution of the Kato square root problem for second order elliptic operators on R n[END_REF] established that if L is a uniformly divergence form elliptic operator (that is, L = L w with w ≡ 1) then √ Lf is comparable to ∇f in L 2 (R n , dx) ≡ L 2 (dx). This led to a new Calderón-Zygmund theory developed by the first named author in [START_REF] Auscher | On necessary and sufficient conditions for L p -estimates of Riesz transforms associated to elliptic operators on R n and related estimates[END_REF] to establish the boundedness in Lebesgue spaces of the associated functional calculus, vertical square function, Riesz transforms, reverse inequalities, etc. A key ingredient in that theory is the use of the so-called off-diagonal or Gaffney estimates satisfied by the associated heat semigroup and its gradient. This was later extended in [6, [START_REF] Auscher | Weighted norm inequalities, off-diagonal estimates and elliptic operators. I. General operator theory and weights[END_REF][START_REF] Auscher | Weighted norm inequalities, off-diagonal estimates and elliptic operators. II. Off-diagonal estimates on spaces of homogeneous type[END_REF] where the same operators were shown to satisfy weighted norm inequalities with Muckenhoupt weights. Conical square functions have been also considered in [5,27]. Some of the previous results in conjunction with the theory of Hardy spaces for uniformly elliptic operators from [25,[START_REF] Hofmann | Second order elliptic operators with complex bounded measurable coefficients in L p , Sobolev and Hardy spaces[END_REF] led to [START_REF] Mayboroda | The connections between Dirichlet, regularity and Neumann problems for second order elliptic operators with complex bounded measurable coefficients[END_REF] where the solvability of the regularity problem in the block case for data in Lebesgue spaces was obtained. This amounted to control non-tangentially the full gradient of the solution given by the Poisson semigroup in terms of the gradient of the boundary datum. In turn, using the weighted Hardy space theory developed in [27, [START_REF] Martell | Weighted Hardy spaces associated with elliptic operators. Part II: Characterizations of H 1 L (w)[END_REF][START_REF] Prisuelos-Arribas | Weighted Hardy spaces associated with elliptic operators. Part III: Characterizations of H p L (w) and the weighted Hardy space associated with the Riesz transform[END_REF], the solvability of the regularity problem in the block case for data in Lebesgue spaces with Muckenhoupt weights has been recently studied in [START_REF] Chen | The regularity problem for uniformly elliptic operators in weighted spaces[END_REF].

Concerning the Kato problem in the general case, where L w is a degenerate elliptic operator as above with a generic w ∈ A 2 (dx), [START_REF] Cruz-Uribe | The Kato problem for operators with weighted ellipticity[END_REF] (see also [START_REF] Cruz-Uribe | Gaussian bounds for degenerate parabolic equations[END_REF][START_REF] Cruz-Uribe | The solution of the Kato problem for degenerate elliptic operators with Gaussian bounds[END_REF]) showed that √ L w f is comparable to ∇f in L 2 (w). The boundedness of the associated operators (functional calculus, Riesz transform, reverse inequalities, vertical square functions, etc.), both in the natural Lebesgue spaces L p (w) and also in weighted spaces L p (v dw) with v ∈ A ∞ (w) was considered in [START_REF] Cruz-Uribe | On the Kato problem and extensions for degenerate elliptic operators[END_REF]. A particular case of interest is that on which under further assumptions in w one can show equivalence of √ L w f and ∇f in L 2 (dx) by simply taking v = w -1 , that is, the L 2 (dx)-problem Kato problem was solved for a class of degenerate elliptic operators that goes beyond that of uniformly elliptice.g., one can take

L γ = -| • | γ div(| • | -γ A(•)∇)
where A is a uniformly elliptic matrix, γ ∈ (-ε, 2n/(n + 2)), and ε depends on the dimension and the ellipticity constants of A. Some work has been also done concerning conical square functions with respect to the heat or Poisson semigroup (or theirs gradients) generated by L w . For example, in [START_REF] Chen | Conical square functions for degenerate elliptic operators[END_REF] the last three authors of the present paper established the boundedness and the comparability of some conical square functions extending to the degenerate case the results from [27]. Moreover, in [START_REF] Prisuelos-Arribas | Vertical square functions and other operators associated with an elliptic operator[END_REF], the last named author has made a deeper study of the vertical and conical square functions and some non-tangential maximal functions arising from degenerate elliptic operators. On another direction, in [START_REF] Auscher | Boundary value problems for degenerate elliptic equations and systems[END_REF] the authors considered L 2 -boundary value problems for degenerate elliptic equations and systems. In particular, they initiated the study of Dirichlet and Neumann problems in the degenerate setting using the so-called first order method.

Our goal in this paper to contribute to this theory studying the solvability of the regularity problem for degenerate elliptic operators and also propose other methods. More precisely, consider the degenerate elliptic operator L w = -w -1 div(w A ∇) where w ∈ A 2 (dx) and A is an n × n matrix of complex L ∞ -valued coefficients defined on R n , n ≥ 2, which is uniformly elliptic matrix (see (1.1)). Introduce the (n + 1) × (n + 1) block matrix

A = A 0 0 1 ,
which is (n + 1) × (n + 1) uniformly elliptic. This gives rise to a degenerate elliptic operator in R n+1 , (1.4) L w u = -w -1 div x,t (w A∇ x,t u) = -w -1 div x (w A∇ x u) -∂ 2 t u = (L w ) x u -∂ 2 t u.

Here and elsewhere, ∇ x,t denotes the full gradient, while the symbol ∇ refers just to the spatial derivatives. Note that in the previous equality we have used that w does not depend on the t variable, hence it is trivial to see that with a slight abuse of notation if we write w(x, t) := w(x) for every (x, t) ∈ R n+1 then w ∈ A 2 (R n+1 , dx) since w ∈ A 2 (dx).

The operator -L w generates a C 0 -semigroup {e -tLw } t>0 of contractions on L 2 (w) which is called the heat semigroup. This and the subordination formula (see (3.1) below) yield that {e -t √ Lw } t>0 is a C 0 -semigroup of contractions on L 2 (w). Hence, whenever f ∈ C ∞ c (R n ), one has that u(x, t) := e -t √ L w f (x), with (x, t) ∈ R n+1 + , is a strong solution of L w u = 0 in R n+1 + . It is also a weak solution: by this we mean that u ∈ W 1,2 loc (R n+1 + , dw dt) satisfies

(1.5)

R n+1 + A(x)∇ x,t u(x, t) • ∇ x,t ψ(x, t) dw(x) dt = 0, ∀ ψ ∈ C ∞ 0 (R n+1 + ).
Also, u(•, t) → f in L 2 (w) as t → 0 + by the semigroup representation. As usual, dw(x) ≡ w(x)dx.

Consider the L 2 -non-tangential maximal function N w defined in [START_REF] Auscher | Boundary value problems for degenerate elliptic equations and systems[END_REF]:

(1.6) N w h(x) := sup t>0 -- W (x,t)
|h(y, s)| 2 dw(y) ds

1 2
, h ∈ L 2 loc (R n+1 + , dw dt).

where W (x, t) := (c -1 0 t, c 0 t) × B(x, c 1 t) is a Whitney region and c 0 > 1, c 1 > 0 are fixed parameters throughout the paper.

Given 1 < p < ∞ and v ∈ A ∞ (w) (note that our assumption w ∈ A 2 (dx) implies that w is a doubling measure and hence (R n , w, | • |) is a space of homogeneous type), we say that the weighted regularity problem (R Lw ) L p (vdw) is solvable if for every f ∈ C ∞ c (R n ) the weak solution of L w u = 0 in R n+1

+

given by u(x, t) := e -t √ Lw f (x), (x, t) ∈ R n+1 + , satisfies the following weighted non-tangential maximal function estimate:

(1.7) N w (∇ x,t u) L p (vdw) ≤ C ∇f L p (vdw) .
Once this estimate is under control, one can extend the semigroup to general data. However, the status of convergence to the boundary of the solution needs a specific treatment that is not addressed here.

As in [START_REF] Auscher | On necessary and sufficient conditions for L p -estimates of Riesz transforms associated to elliptic operators on R n and related estimates[END_REF][START_REF] Auscher | Weighted norm inequalities, off-diagonal estimates and elliptic operators. II. Off-diagonal estimates on spaces of homogeneous type[END_REF][START_REF] Cruz-Uribe | On the Kato problem and extensions for degenerate elliptic operators[END_REF], we denote by (p -(L w ), p + (L w )) and by (q -(L w ), q + (L w )) the maximal open intervals on which the heat semigroup {e -tLw } t>0 and the gradient of the heat semigroup { √ t∇e -tLw } t>0 are respectively uniformly bounded on L p (w). That is,

p -(L w ) := inf p ∈ (1, ∞) : sup t>0 e -tLw L p (w)→L p (w) < ∞ , p + (L w ) := sup p ∈ (1, ∞) : sup t>0 e -tLw L p (w)→L p (w) < ∞ , q -(L w ) := inf p ∈ (1, ∞) : sup t>0 √ t∇e -t 2 Lw L p (w)→L p (w) < ∞ , q + (L w ) := sup p ∈ (1, ∞) : sup t>0 √ t∇e -t 2 Lw L p (w)→L p (w) < ∞ .
We need to introduce some extra notation (see Section 2). Set r w := inf p : w ∈ A p (dx) , and note that 1 ≤ r w < 2 since w ∈ A 2 (dx). Given 0

≤ p 0 < q 0 ≤ ∞ and v ∈ A ∞ (w) = A ∞ (R n , w dx) define W w v (p 0 , q 0 ) := p ∈ (p 0 , q 0 ) : v ∈ A p p 0 (w) ∩ RH ( q 0 p ) ′ (w) .
We are now ready to state our main result:

Theorem 1.8. Let w ∈ A 2 (dx) and let L w be a block degenerate elliptic operator in

R n+1 + as above. Let v ∈ A ∞ (w) be such that (1.9) W w v max{r w , q -(L w )}, q + (L w ) = Ø.
Then, for every

p ∈ W w v max r w , nrwq -(Lw) nrw+q -(Lw) , q + (L w ) and every f ∈ C ∞ c (R n ), if one sets u(x, t) = e -t √ L w f (x), (x, t) ∈ R n+1 + , then (1.10) N w (∇ x,t u) L p (vdw) ≤ C ∇f L p (vdw)
and (R Lw ) L p (vdw) is solvable.

Let us compare this result with some previous work. When w ≡ 1 (that is, we are working with the class of uniformly -or non-degenerate-elliptic operators) and v ≡ 1 then, clearly, r w = 1, W w v max{r w , q -(L w )}, q + (L w ) = (q -(L w ), q + (L w )) = Ø, and our result gives (1.10) in the range max 1, nq -(Lw)

n+q - (Lw) , q + (L w ) , hence we fully recover [START_REF] Mayboroda | The connections between Dirichlet, regularity and Neumann problems for second order elliptic operators with complex bounded measurable coefficients[END_REF]Theorem 4.1]. If we still assume that w ≡ 1 and we let v ∈ A ∞ (w) = A ∞ (dx), then our assumption (1.9) agrees with that in [START_REF] Chen | The regularity problem for uniformly elliptic operators in weighted spaces[END_REF]Theorem 1.10] and the range of p's here is slightly worse than the one in that result (the lower end-point in [START_REF] Chen | The regularity problem for uniformly elliptic operators in weighted spaces[END_REF]Theorem 1.10] has been pushed down using an extra technical argument that we have chosen not to follow here).

Our methods to prove Theorem 1.8, in particular the estimate involving ∂ t u, are also novel. The above works used advanced technology of Hardy spaces adapted to operators: developing them in our context is probably a new challenge in itself. Instead, we rely on recent change of angle formulas for weighted conical square function estimates (see Section 2.4) and also the ones we prove for non-tangential weighted maximal functions (see Lemma 3.7) which allow us to implement more directly standard tools in the field.

An important consequence of our method is that we can obtain the solvability of the regularity problem corresponding to data in unweighted Lebesgue spaces. The main idea consists in taking v = w -1 in Theorem 1.8. The following result focuses in the case of the L 2 -solvability (more general results are presented in Section 4, see Corollaries 4.3 and 4.5):

Corollary 1.11. Let w ∈ A 2 (dx) and let L w be a block degenerate elliptic operator in R n+1 + as above. Given Θ ≥ 1 there exists

ǫ 0 = ǫ 0 (Θ, n, Λ/λ) ∈ (0, 1 2n ], such that for every w ∈ A 1+ǫ (dx) ∩ RH max{ 2 1-ǫ ,1+(1+ǫ) n 2 } (dx) with 0 ≤ ǫ < ǫ 0 and [w] A 2 (dx) ≤ Θ, then (1.12) N w (∇ x,t u) L 2 (dx) ≤ C ∇f L 2 (dx) . for every f ∈ C ∞ c (R n ) and where u(x, t) = e -t √ L w f (x), (x, t) ∈ R n+1 + . Hence (R Lw ) L 2 (dx) is solvable.
Furthermore, if we set

L α u(x, t) = -|x| α div x |x| -α A(x)∇ x u(x, t) -∂ 2 t u(x, t) where A is an n × n matrix of complex L ∞ -valued coefficients defined on R n , n ≥ 2,
satisfying the uniform ellipticity condition (1.1), then there exists 0 < ǫ < 1 2 small enough (depending only on the dimension and the ratio Λ/λ) such that if -ǫ < α < 2 n n+2 then (1.12) holds in this scenario and (R Lα ) L 2 (dx) is solvable.

The plan of the paper is as follows. In Section 2 we introduce notations and definitions, and we recall some known results. We also obtain estimates for some inhomogeneous vertical and conical square functions which are interesting on its own right (see Propositions 2.54 and 2.71). To prove our main result, Theorem 1.8, we split the main estimate into two independent pieces, one regarding N w (∇ x u) and the other one related to N w (∂ t u), see respectively Propositions 3.2 and 3.5 in Section 3. In Section 4 we study the solvability of the regularity problem in unweighted Lebesgue spaces and, in particular, we prove Corollary 1.11.

Preliminaries

We shall use the following notation: dx denotes the usual Lebesgue measure in R n , dw denotes the measure in R n given by the weight w, and vdw or d(vw) denotes the one given by the product weight vw. Besides, throughout the paper n will denote the dimension of the underlying space R n and we shall always assume n ≥ 2.

Given a ball B, let r B denote the radius of B. We write λB for the concentric ball with radius λ r B , λ > 0. Moreover, we set C 1 (B) = 4B and, for j ≥ 2, C j (B) = 2 j+1 B\2 j B.

2.1.

Weights. We need to introduce some classes of Muckenhoupt weights. Namely, A ∞ (dx), on which the underlying measure space is (R n , dx), and then fix w ∈ A ∞ (dx) and consider the class A ∞ (w) where the "weighted" underlying space is (R n , dw).

2.1.1.

A ∞ (dx) weights. By a weight w we mean a non-negative, locally integrable function. For brevity, we will often write dw for w dx. In particular, we write w(E) = E dw and L p (w) = L p (R n , dw). We will use the following notation for averages: given a set E such that 0 < w(E) < ∞,

- E f dw = 1 w(E) E f dw, or, if 0 < |E| < ∞, - E f dx = 1 |E| E f dx.
Abusing slightly the notation, for j ≥ 1, we set

- C j (B) f dw = 1 w(2 j+1 B) C j (B)
f dw.

We state some definitions and basic properties of Muckenhoupt weights. For further details, see [START_REF] Duoandikoetxea | Fourier analysis[END_REF][START_REF] García-Cuerva | Weighted norm inequalities and related topics[END_REF]24]. Consider the Hardy-Littlewood maximal function

Mf (x) := sup B∋x - B |f (y)| dy.
It is well-known that given a weight w, M is bounded on L p (w), if and only if,

w ∈ A p (dx), 1 < p < ∞ where we say that w ∈ A p (dx), 1 < p < ∞, if [w] Ap(dx) := sup B - B w(x) dx - B w(x) 1-p ′ dx p-1 < ∞.
Here and below the sups run over the collection of balls

B ⊂ R n . When p = 1, M is bounded from L 1 (w) to L 1,∞ (w) if and only if w ∈ A 1 (dx), that is, if [w] A 1 (dx) := sup B - B w(x) dx ess sup x∈B w(x) -1 < ∞.
We also introduce the reverse Hölder classes. We say that w

∈ RH s (dx), 1 < s < ∞ if [w] RHs(dx) := sup B - B w(x) dx -1 - B w(x) s dx 1 s < ∞,

It is also well-known that

A ∞ (dx) := 1≤p<∞ A p (dx) = 1<s≤∞ RH s (dx).
Throughout the paper we shall use in several places the following properties.

Namely, if w ∈ RH s (dx), 1 < s ≤ ∞, w(E) w(B) ≤ [w] RHs(dx) |E| |B| 1 s ′ , ∀ E ⊂ B, (2.1) where B is any ball in R n . Analogously, if w ∈ A p (dx), 1 ≤ p < ∞, then |E| |B| p ≤ [w] Ap(dx) w(E) w(B) , ∀ E ⊂ B. (2.2)
This implies in particular that w is a doubling measure, that is,

w(λB) ≤ [w] Ap(dx) λ n p w(B), ∀ B, ∀ λ > 1. (2.3)
We continue by introducing some important notation. Weights in the A p (dx) and RH s (dx) classes have a self-improving property: if w ∈ A p (dx), there exists ǫ > 0 such that w ∈ A p-ǫ (dx), and similarly if w ∈ RH s (dx), then w ∈ RH s+δ (dx) for some δ > 0. Hereafter, given w ∈ A p (dx), let (2.4) r w = inf p : w ∈ A p (dx) , s w = inf q : w ∈ RH q ′ (dx) .

Note that according to our definition s w is the conjugated exponent of the one defined in [START_REF] Auscher | Weighted norm inequalities, off-diagonal estimates and elliptic operators. I. General operator theory and weights[END_REF]Lemma 4.1]. Given 0 ≤ p 0 < q 0 ≤ ∞ and w ∈ A ∞ (dx), [START_REF] Auscher | Weighted norm inequalities, off-diagonal estimates and elliptic operators. I. General operator theory and weights[END_REF]Lemma 4.1] implies that W w (p 0 , q 0 ) := p ∈ (p 0 , q 0 ) : w ∈ A p p 0

(dx) ∩ RH ( q 0 p ) ′ (dx) = p 0 r w , q 0 s w . (2.5) If p 0 = 0 and q 0 < ∞ it is understood that the only condition that stays is w ∈ RH ( q 0 p ) ′ (dx). Analogously, if 0 < p 0 and q 0 = ∞ the only assumption is w ∈ A p p 0 (dx).

Finally W w (0, ∞) = (0, ∞). Furthermore, given p ∈ (0, ∞) and a weight w ∈ A ∞ (dx), we define the following Sobolev exponents with respect to w (p)

v] Ap(w) = sup B - B v(x) dw - B v(x) 1-p ′ dw p-1 < ∞. (2.9)
Analogously, we can define the classes RH s (w) by replacing the Lebesgue measure in the definitions above with dw: v

∈ RH s (w), 1 < s < ∞ if [v] RHs(w) = sup B - B v(x) dw -1 - B v(x) s dw 1 s < ∞. (2.10)
From these definitions, it follows at once that there is a "duality" relationship between the weighted and unweighted A p (dx) and RH s (dx) conditions: w -1 ∈ A p (w) if and only if w ∈ RH p ′ (dx) and w -1 ∈ RH s (w) if and only if w ∈ A s ′ (dx).

For every measurable set E ∈ R n , we write vw(E) = E d(vw) = E vdw = (vdw)(E) and L p (vdw) = L p (R n , v(x) w(x) dx). In this direction, for every w ∈ A p (dx), v ∈ A q (w), 1 ≤ p, q < ∞, it follows that

|E| |B| p q ≤ [w] q Ap(dx) w(E) w(B) q ≤ [w] q Ap(dx) [v] Aq(w) vw(E) vw(B) , ∀ E ⊂ B. (2.11) Analogously, if w ∈ RH p (dx) and v ∈ RH q (w) 1 < p, q ≤ ∞, one has vw(E) vw(B) ≤ [v] RHq(w) w(E) w(B) 1 q ′ ≤ [v] RHq(w) [w] 1 q ′ RHp(dx) |E| |B| 1 p ′ q ′ , ∀ E ⊂ B. (2.12)
As before, for a weight v ∈ A ∞ (w) (recall that w ∈ A ∞ (dx) is fixed) we set

r v (w) := inf r : v ∈ A r (w)
and s v (w) := inf s : v ∈ RH s ′ (w) . (2.13) For 0 ≤ p 0 < q 0 ≤ ∞ and v ∈ A ∞ (w), by a similar argument to that of [START_REF] Auscher | Weighted norm inequalities, off-diagonal estimates and elliptic operators. I. General operator theory and weights[END_REF]Lemma 4.1], we have

W w v (p 0 , q 0 ) := p ∈ (p 0 , q 0 ) : v ∈ A p p 0 (w) ∩ RH ( q 0 p ) ′ (w) = p 0 r v (w), q 0 s v (w)
.

(2.14)

If p 0 = 0 and q 0 < ∞, as before, it is understood that the only condition that stays is v ∈ RH ( q 0 p ) ′ (w). Analogously, if 0 < p 0 and q 0 = ∞ the only assumption is v ∈ A p p 0 (w). Finally W w v (0, ∞) = (0, ∞). Remark 2.15. The proof of our main result will use the Calderón-Zygmund decomposition from Lemma 2.47 with respect to the underlying measure v(x) dw(x) = v(x) w(x) dx where w ∈ A ∞ (dx) and v ∈ A ∞ (w). In that scenario it was shown in [START_REF] Prisuelos-Arribas | Vertical square functions and other operators associated with an elliptic operator[END_REF]Remark 2.15] that wv ∈ A ∞ (dx) and moreover r vw ≤ r w r v (w). The converse inequality is false in general: let w(x) := |x| n and v := w -1 , then one can easily see that r w r v (w) = r w s w = 2 and r vw = 1.

We state some lemma which will be useful in the sequel. Lemma 2.16. Given 0 < p ≤ q < ∞, let B ⊂ R n be a ball and let j ≥ 1, the following holds:

(a) If v ∈ A q p (w) then

- C j (B) |f (x)| p dw(x) 1 p - C j (B) |f (x)| q d(vw)(x) 1 q . (b) If v ∈ RH ( q p ) ′ (w) then - C j (B) |f (x)| p d(vw)(x) 1 p - C j (B) |f (x)| q dw(x) 1 q
.

Proof. We prove the case p < q (when p = q the proof follows similarly and is left to the interested reader). Assume first v ∈ A q p (w). We obtain (a) by applying Hölder's inequality and (2.9)

- C j (B) |f | p dw 1 p = - C j (B) |f | p v p q v -p q dw 1 p - C j (B) |f | q vdw 1 q - 2 j+1 B v 1-( q p ) ′ dw 1 q ( q p -1) - C j (B) |f | q vdw 1 q - 2 j+1 B vdw -1 q = - C j (B) |f | q d(vw) 1 q
.

Next assume v ∈ RH ( q p ) ′ (w), we obtain (b) by applying Hölder's inequality and (2.10)

- C j (B)
|f | p d(vw)

1 p = w(2 j+1 B) vw(2 j+1 B) 1 p - C j (B) |f | p vdw 1 p - 2 j+1 B vdw -1 p - C j (B) |f | q dw 1 q - 2 j+1 B v ( q p ) ′ dw 1 p 1 ( q p ) ′ - C j (B) |f | q dw 1 q .
2.2. Square functions and non-tangential maximal functions.

In this section, we introduce several auxiliary operators (vertical and conical square functions, non-tangential maximal functions) which will be needed at various points along the proofs.

Consider, for κ ≥ 1, the non-tangential maximal function N κ,w defined as

N κ,w F (x) := sup t>0 B(x,κt)
|F (y, t)| 2 dw(y) w(B(x, t))

1 2
. (2.17)

We write N w when κ = 1. We are particularly interested in the non-tangential maximal functions associated with the heat or Poisson semigroup. For f ∈ L 2 (w), define

N κ,w H f (x) := sup t>0 B(x,κt) e -t 2 Lw f (y) 2 dw(y) w(B(x, t)) 1 2 ; (2.18) N κ,w P f (x) := sup t>0 B(x,κt) e -t √ Lw f (y) 2 dw(y) w(B(x, t)) 1 2 . (2.19)
Again, when κ = 1 we write N w P and N w H . We shall obtain weighted boundedness of these operators in Section 4.2.

We also consider several invariants of the vertical square functions associated with the heat semigroup which were studied in [14, Sections 5 and 10]:

g w H f (x) := ∞ 0 t 2 L w e -t 2 Lw f (x) 2 dt t 1 2 ; (2.20) G w 1/2,H f (x) := ∞ 0 t∇(t 2 L w ) 1 2 e -t 2 Lw f (x) 2 dt t 1 2 ; (2.21) G w H f (x) := ∞ 0 t∇t 2 L w e -t 2 Lw f (x)
∈ W w v (q -(L w ), q + (L w )).
Now we recall the following conical square functions studied by the authors in [START_REF] Chen | Conical square functions for degenerate elliptic operators[END_REF]. ∞). Finally, we introduce the following "inhomogeneous" vertical and conical square functions:

S α,w H f (x) := Γ α (x) t 2 L w e -t
G w H f (x) := ∞ 0 ∇t 2 L w e -t 2 Lw f (x) 2 dt t 1 2 ; (2.25) S w H f (x) := Γ(x)
t -1 (t 2 L w )e -t 2 Lw f (y) 2 dw(y) dt tw(B(y, t))

1 2 . (2.26)
By non-homogeneity, we mean that the power of t inside the square functions is not in accordance with the order of the operator L w , we are modifying respectively G w H and S w H by removing one power of t. The analogues of the above two square functions in other settings turn to be very useful in the study of Riesz transform and Hardy space theory, see for instance [START_REF] Coulhon | Riesz transform and related inequalities on noncompact Riemannian manifolds[END_REF]25]. Sections 2.6 and 2.7 below study the boundedness of G w H and S w H on weighted Sobolev spaces, which plays an essential role in the proof of our main results.

We finish this subsection by recalling the results about the reverse inequality of the Riesz transform associated with the operator L w proved in [START_REF] Cruz-Uribe | On the Kato problem and extensions for degenerate elliptic operators[END_REF]. The Riesz transform ∇L -1 2 w associated with the operator L w can be written as

∇L -1 2 w = 2 √ π ∞ 0 t∇e -t 2 Lw dt t ,
consider also the following square root representation (see for instance [START_REF] Auscher | Boundary value problems for degenerate elliptic equations and systems[END_REF][START_REF] Cruz-Uribe | The Kato problem for operators with weighted ellipticity[END_REF]):

L w = 2 √ π ∞ 0 tL w e -t 2 Lw dt t . (2.27) Proposition 2.28 ([14, Proposition 6.1]). Let max{r w , (p -(L w )) w, * } < p < p + (L w ). Then for all f ∈ S, L w f L p (w) ∇f L p (w) . Furthermore, if p ∈ W w v (max{r w , p -(L w )}, p + (L w )). Then for all f ∈ S, L w f L p (vdw)
∇f L p (vdw) .

2.3.

Off-diagonal estimates.

Definition 2.29. Let {T t } t>0 be a family of sublinear operators and let 1 < p < ∞.

Given a doubling measure µ we say that {T t } t>0 satisfies L p (µ)-L p (µ) full off-diagonal estimates, denoted by T t ∈ F (L p (µ) -L p (µ)), if there exist constants C, c > 0 such that for all closed sets E and F , all f ∈ L p (R n ), and all t > 0 we have (2.30)

F |T t (1 E f )| p dµ 1 p ≤ C e -cd(E,F ) 2 t E |f | p dµ 1 p
,

where d(E, F ) = inf{|x -y| : x ∈ E, y ∈ F }. Set Υ(s) = max{s, s -1 } for s > 0.
Recall that, given a ball B, we use the notation C j (B) = 2 j+1 B\2 j B for j ≥ 2, and for any doubling measure µ

- B hdµ = 1 µ(B) B hdµ, - C j (B) hdµ = 1 µ(2 j+1 (B)) C j (B)
hdµ.

Definition 2.31. Given 1 ≤ p ≤ q ≤ ∞ and any doubling measure µ, we say that a family of sublinear operators {T t } t>0 satisfies L p (µ) -L q (µ) off-diagonal estimates on balls, denoted by T t ∈ O(L p (µ) -L q (µ)), if there exist θ 1 , θ 2 > 0 and c > 0 such that for all t > 0 and for all ball B with radius r B ,

(2.32) -

B |T t (f 1 B )| q dµ 1 q Υ r B √ t θ 2 - B |f | p dµ 1 p
, and for j ≥ 2,

(2.33)

- B T t (f 1 C j (B) ) q dµ 1 q 2 jθ 1 Υ 2 j r B √ t θ 2 e -c4 j r 2 B t - C j (B) |f | p dµ 1 p , and (2.34) - C j (B) |T t (f 1 B )| q dµ 1 q 2 jθ 1 Υ 2 j r B √ t θ 2 e -c4 j r 2 B t - B |f | p dµ 1 p .
Let us recall some results about off-diagonal estimates on balls for the heat semigroup associated with L w . Lemma 2.35 ([8, Section 2], [START_REF] Cruz-Uribe | On the Kato problem and extensions for degenerate elliptic operators[END_REF]Sections 3 and 7]). Let L w be a degenerate elliptic operator with w ∈ A 2 (dx).

(a) If p -(L w ) < p ≤ q < p + (L w ), then e -tLw and (tL w ) m e -tLw , for every m ∈ N, belong to O(L p (w) -L q (w)).

(b) Let p -(L w ) < p ≤ q < p + (L w ). If v ∈ A p/p -(Lw) (w) ∩ RH (p + (Lw)/q) ′ (w)
, then e -tLw and (tL w ) m e -tLw , for every m ∈ N, belong to O(L p (vdw) -L q (vdw)).

(c) There exists an interval K(L w ) such that if p, q ∈ K(L w ), with p ≤ q, then √ t∇e -tLw ∈ O(L p (w) -L q (w)). Moreover, denoting by q -(L w ) and q + (L w ) the left and right endpoints of

K(L w ), then q -(L w ) = p -(L w ), 2 < q + (L w ) ≤ (q + (L w )) * w ≤ p + (L w ). (d) Let q -(L w ) < p ≤ q < q + (L w ). If v ∈ A p/q -(Lw) (w) ∩ RH (q + (Lw)/q) ′ (w), then √ t∇e -tLw ∈ O(L p (vdw) -L q (vdw)).
(e) If p = q and µ is a doubling measure then

F (L p (µ) -L p (µ)) and O(L p (µ) - L p (µ)) are equivalent.
Remark 2.36. Since off-diagonal estimates on balls are stable under composition (see

[8, Theorem 2.3]), it follows from (b) and (d) that √ t∇tL w e -tLw ∈ O(L p (vdw) - L q (vdw)) for q -(L w ) < p ≤ q < q + (L w ) and v ∈ A p/q -(Lw) (w) ∩ RH (q + (Lw)/q) ′ (w).
Moreover, in the following result, which is a weighted version of [28, (5.12)] (see also [25]), we show 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 , for all M ∈ N. Proposition 2.37. Let p ∈ (p -(L w ), p + (L w )) and let 0 < t, s < ∞. Then, 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 satisfies the following L p (w) -L p (w) off-diagonal estimates: 1 E 2 T t,s f L p (w) s 2 t 2 M e -c d(E 1 ,E 2 ) 2 t 2 +s 2 f 1 E 1 L p (w) . (2.38)
In particular, there holds

(2.39) T t,s f L p (w) s 2 t 2 M f L p (w) .
Proof. Note that we have

1 E 2 T t,s f L p (w) = 1 E 2 e -t 2 Lw -e -(t 2 +s 2 )Lw M f L p (w) = 1 E 2 s 2 0 ∂ r e -(r+t 2 )Lw dr M f L p (w) ≤ s 2 0 • • • s 2 0 1 E 2 M i=1 r i + Mt 2 L w M e -M i=1 r i +M t 2 Lw f L p (w) dr 1 . . . dr M M i=1 r i + Mt 2 M s 2 0 • • • s 2 0 e -c d(E 1 ,E 2 ) 2 M i=1 r i +M t 2 dr 1 . . . dr M M i=1 r i + Mt 2 M 1 E 1 f L p (w) s 2 t 2 M e -c d(E 1 ,E 2 ) 2 t 2 +s 2 1 E 1 f L p (w) ,
where we have applied the fact that (tL w ) M e -tLw ∈ F (L p (w) -L p (w)), for all p ∈ (p -(L w ), p + (L w )), (see Lemma 2.35).

We conclude this section by introducing the following off-diagonal estimates on Sobolev spaces (for non-degenerate elliptic operators see [START_REF] Auscher | On necessary and sufficient conditions for L p -estimates of Riesz transforms associated to elliptic operators on R n and related estimates[END_REF]).

Lemma 2.40. Let q ∈ (q -(L w ), q + (L w )) and α > 0. Assume that p satisfies max {r w , (q -(L w )) w, * } < p ≤ q. Then, for every (x, t) ∈ R n+1 + , there exists θ > 0 such that

(2.41) - B(x,αt) ∇e -t 2 Lw f q dw 1 q Υ(α) θ ∞ j=1 e -c4 j - B(x,2 j+1 αt) |∇f | p dw 1 p
.

Proof. For simplicity, we write B := B(x, αt) and h := ff 4B,w , where, for every λ > 0, f λB,w is the average of f in λB with respect to the measure dw. By the conservation property, that is e -t 2 Lw 1 = 1,

∇e -t 2 Lw f = ∇e -t 2 Lw (f -f 4B,w ) = ∞ j=1 ∇e -t 2 Lw h j , with h j := h1 C j (B)
. By Lemma 2.35, for any q -(L w ) < q 0 < q, we have that

√ t∇e -tLw ∈ O(L q 0 (w) -L q (w)), then - B ∇e -t 2 Lw f q dw 1 q ≤ ∞ j=1 - B ∇e -t 2 Lw h j q dw 1 q Υ(α) θ 2 ∞ j=1 2 j(θ 1 +θ 2 ) e -c4 j t - C j (B) |h| q 0 dw 1 q 0 .
Using the weighted Poincaré-Sobolev inequality (see [START_REF] Cruz-Uribe | On the Kato problem and extensions for degenerate elliptic operators[END_REF]Theorem 2.1] and also [22, Theorem 1.6]), we obtain that for any p > max{r w , (q 0 ) w, * }, -

C j (B) |h| q 0 dw 1 q 0 ≤ - 2 j+1 B f -f 2 j+1 B,w q 0 dw 1 q 0 + j l=2 f 2 l+1 B,w -f 2 l B,w j l=2 - 2 l+1 B f -f 2 l+1 B,w q 0 dw 1 q 0 j l=2 2 l αt - 2 l+1 B |∇f | p dw 1 p . Hence, - B ∇e -t 2 Lw f q dw 1 q Υ(α) θ ∞ j=1 e -c4 j t j l=2 2 l t - 2 l+1 B |∇f | p dw 1 p Υ(α) θ ∞ j=1 e -c4 j - 2 j+1 B |∇f | p dw 1 p
.

This completes the proof.

2.4. Change of angles. We shall use two change of angles results. The first one is a version of [27, Proposition 3.30] in the weighted degenerate case.

Proposition 2.42. [12, Proposition A.2] Let w ∈ A r (dx) and v ∈ RH r ′ (w) with 1 ≤ r, r < ∞.
For every 1 ≤ q ≤ r, 0 < α ≤ 1 and t > 0, there holds

(2.43) R n B(x,αt) |h(y, t)| dw(y) w(B(y, αt)) 1 q v(x)dw(x) α n r( 1 r -1 q ) R n B(x,t)
|h(y, t)| dw(y) w(B(y, t))

1 q v(x)dw(x).
The second result was proved for the unweighted non-degenerate case in [START_REF] Auscher | Change of Angles in tent spaces[END_REF] and for the weighted non-degenerate case in [27, Proposition 3.2]. Consider, for α > 0, the following operator acting over measurable functions F defined in R n+1 + :

A α w F (x) := Γ α (x) |F (y, t)| 2 dw(y) dt tw(B(y, t)) 1 2 , x ∈ R n ,
where Γ α (x) is the cone of aperture α and vertex at x,

Γ α (x) = {(y, t) ∈ R n+1 + : |x -y| < αt}. Proposition 2.44. [12, Proposition 4.9] Let 0 < α ≤ β < ∞. (a) For every w ∈ A r (dx) and v ∈ A r (w), 1 ≤ r, r < ∞, there holds A β w F L p (vdw) ≤ C β α n r r p A α w F L p (vdw) for all 0 < p ≤ 2r, (2.45)
where C ≥ 1 depends on n, p, r, r, [w] A r (dx) , and [v] Ar(w) , but it is independent of α and β.

(b) For every w ∈ RH s ′ (dx) and v ∈ RH s ′ (w), 1 ≤ s, s < ∞, there holds

A α w F L p (vdw) ≤ C α β n s s p A β w F L p (vdw) for all 2 s ≤ p < ∞, (2.46)
where C ≥ 1 depends on n, p, s, s, [w] RH s ′ (dx) , and [v] RH s ′ (w) , but it is independent of α and β.

2.5. Calderón-Zygmund decomposition on Sobolev spaces. Our proofs rely on the following Calderón-Zygmund decomposition on Sobolev spaces.

Lemma 2.47 ([6, Lemma 6.6]). Let n ≥ 1, α > 0, ̟ ∈ A ∞ (dx), and let 1 ≤ p < ∞ be such that ̟ ∈ A p (dx). Assume that f ∈ S is such that ∇f L p (̟) < ∞. Then, there exist a collection of balls {B i } i with radii r B i , smooth functions {b i } i and a function g ∈ L 1 loc (̟) such that (2.48) f = g + i b i
and the following properties hold:

(2.49) |∇g(x)| ≤ Cα, for µ-a.e. x,
(2.50) supp b i ⊂ B i and

B i |∇b i | p d̟ ≤ Cα p ̟(B i ), (2.51) i ̟(B i ) ≤ C α p R n |∇f | p d̟, (2.52) i 1 4B i ≤ N,
where C and N depend only on n, p, and ̟. In addition, for 1 ≤ q < p * ̟ , where p * ̟ is defined in (2.7),

(2.53)

- B i |b i | q d̟ 1 q α r B i .
2.6. Non-homogeneous vertical square function. In this section, we study the weighted boundedness of G w H . Our result is the following. Theorem 2.54. Let w ∈ A 2 (dx) and let L w be a degenerate elliptic operator. Given v ∈ A ∞ (w), assume that W w v (max{r w , q -(L w )}, q + (L w )) = Ø. Then, for every f ∈ S and p ∈ W w v (max{r w , (q

-(L w )) w, * }, q + (L w )), it holds, (2.55) G w H f L p (vdw) ∇f L p (vdw) .
Before starting with the proof, we make some remarks and prove Lemma 2.62 (stated below). These results not only will be useful in this proof but also in the remainder of the paper.

Remark 2.56. Given w ∈ A 2 (dx) and v ∈ A ∞ (w), let 0 < q -< q + < ∞ and p > r v (w) max{r w , (q -) w, * }. Assuming that r v (w) max{r w , q -}, q + /s v (w) = W w v (max{r w , q -}, q + ) = Ø, we claim that

r v (w) max{r w , q -}, min{q + /s v (w), p * vw } = Ø, (2.57)
where we recall that by Remark 2.15 vw ∈ A ∞ (dx), and p * vw is defined in (2.7). Indeed, since by hypothesis r v (w) max{r w , q -} < q + /s v (w), this can be seen from the fact that

r v (w) max{r w , q -} < p * vw . (2.58)
To prove (2.58), we distinguish two cases. If r v (w) max{r w , q -} = r v (w)r w , since we are taking p such that p > r v (w) max{r w , (q -) w, * } and since (q -) w, * ≤ q -(see (2.6)), then r v (w) max{r w , q -} = r v (w) max{r w , (q -) w, * } < p < p * vw . If now r v (w) max{r w , q -} = r v (w)q -, we can assume that nr vw > p (otherwise p * vw = ∞ and the inequality is trivial). Hence, by hypothesis and by (2.7),

1 p * vw = 1 p - 1 nr vw < 1 r v (w)(q -) w, * - 1 nr vw = nr w + q - r v (w)q -nr w - 1 nr vw (2.59) = 1 r v (w)q - - 1 nr vw 1 - r vw r w r v (w) ≤ 1 r v (w)q - = 1 r v (w) max{r w , q -} .
Remark 2.60. Let {B i } i be a collection of balls with bounded overlap, w ∈ A ∞ (dx), and v ∈ A ∞ (w). Besides, consider 1 < p < ∞, u ∈ L p ′ (vdw) such that u L p ′ (vdw) = 1, and M vw the weighted maximal operator defined as

M vw f (x) := sup B∋x - B |f (y)|d(vw)(y),
then, by Kolmogorov's inequality, we have that (2.61)

i B i M vw (|u| p ′ ) 1 p ′ vdw p ∪ i B i M vw (|u| p ′ ) 1 p ′ vdw p vw i B i u p L p ′ (vdw) vw i B i .
We next state a technical lemma which will be used several times. We notice that the statement, which may appear slightly clumsy, is written so that it can be easily invoked in some of our proofs.

Lemma 2.62. Given w ∈ A 2 (dx) and v ∈ A ∞ (w), fix α > 0, 1 < p 1 < ∞, {B i } i a collection of
balls in R n with bounded overlap. Assume that there is a sequence of positive numbers {I ij } i,j (whose significance will become clear when applying the result) so that

I ij ≤ Cαvw(2 j+1 B i ) 1 p 1 2 -j(2M -C) , j ≥ 4, (2.63)
where C, C are fixed (harmless) constants, and 2M > C + nr w r v (w), then

sup u L p ′ 1 (vdw) =1 i j≥4 I ij u1 C j (B i ) L p ′ 1 (vdw) αvw i B i 1 p 1 .
Proof. Fix u so that u L p ′ 1 (vdw) = 1. Note that we can find p > r w , q > r v (w) so that 2M > C + nr with r = pq. In particular, w ∈ A p (dx), v ∈ A q (w) and we have (2.11) at our disposal. This, together with (2.63) and (2.61) with p = p 1 , allows us to show that i j≥4

I ij u1 C j (B i ) L p ′ 1 (vdw) α i j≥4 vw(B i )2 -j(2M -C-nr) - C j (B i ) |u(x)| p ′ 1 d(vw)(x) 1 p ′ 1 α i vw(B i ) inf x∈B i M vw |u| p ′ 1 (x) 1 p ′ 1 α i B i M vw |u| p ′ 1 (x) 1 p ′ 1 v(x)dw(x) αvw i B i 1 p 1 .
This readily leads to the desired estimate.

Proof of Theorem 2.54. Throughout the proof fix w ∈ A 2 (dx) and denote q -:= q -(L w ) and q + := q + (L w ). If p ∈ W w v (max{r w , q -}, q + ), then (2.55) follows easily from Lemma 2.23 and Proposition 2.28. Indeed, we have

(2.64) G w H f L p (vdw) = ∞ 0 t∇(t 2 L w ) 1 2 e -t 2 Lw L w f 2 dt t 1 2 L p (vdw) L w f L p (vdw) ∇f L p (vdw) .
In accordance with (2.14), to go below r v (w) max{r w , q -}, we shall show that if p satisfies r v (w) max {r w , (q -) w, * } < p < r v (w) max{r w , q -}, (2.65) then for any α > 0 and f ∈ S it follows that

vw x ∈ R n : G w H f (x) > α 1 α p R n |∇f | p vdw. (2.66)
Hence using interpolation between Sobolev spaces (see [START_REF] Badr | Real interpolation of Sobolev spaces[END_REF]), we shall conclude the desired estimate.

In order to prove (2.66), we apply to f the Calderón-Zygmund decomposition in Lemma 2.47 at height α > 0 for the product weight vw (recall that r vw ≤ r w r v (w) < p, see Remark 2.15). Thus by (2.48)

vw x ∈ R n : G w H f (x) > α ≤ vw x ∈ R n : G w H g(x) > α 3 + vw x ∈ R n : G w H i b i (x) > 2α 3 =: I + II.
Note that by Remark 2.56 we can pick q such that r v (w) max{r w , q -} < q < min q + s v (w)

, p * vw . (2.67) Keeping this choice of q, by (2.64) we have G w H f L q (vdw) ∇f L q (vdw) . Besides, since p < q (see (2.65)), properties (2.49)-(2.52) yield

I 1 α q R n | G w H g| q vdw 1 α q R n |∇g| q vdw 1 α p R n |∇f | p vdw.
To estimate term II, for every k ∈ Z, let

r i := 2 k if 2 k ≤ r B i < 2 k+1 . Then, II ≤ vw i 16B i + vw x ∈ R n : ∞ 0 t 2 ∇L w e -t 2 Lw i:r i ≤t b i (x) 2 dt t 1 2 > α 3 + vw x ∈ R n \ i 16B i : ∞ 0 t 2 ∇L w e -t 2 Lw i:r i >t b i (x) 2 dt t 1 2 > α 3 1 α p R n |∇f | p vdw + II 1 + II 2 ,
where we have used (2.3) and (2.51).

In order to estimate term II 1 , write

(2.68) ∞ 0 ∇t 2 L w e -t 2 Lw i:r i ≤t b i (x) 2 dt t 1 2 = ∞ 0 t 3 ∇L w e -t 2 Lw 1 t i:r i ≤t b i (x) 2 dt t 1 2 = ∞ 0 |T t f t (x)| 2 dt t 1 2
, where T t := t 3 ∇L w e -t 2 Lw and f t (x) := 1 t i:r i ≤t b i (x). Moreover, note that 2 ∈ (max{r w , q -}, q + ) , then, by Remark 2.36, for every v 0 ∈ A 2/ max{rw,q -} (w) ∩ RH (q + /2) ′ (w) we have t

3 2 ∇L w e -tLw ∈ O(L 2 (v 0 dw) -L 2 (v 0 dw)). In particular, T t is bounded from L 2 (v 0 dw) to L 2 (v 0 dw). Consequently, ∞ 0 |T t f t | 2 dt t 1 2 2 L 2 (v 0 dw) = ∞ 0 R n |T t f t | 2 v 0 dw dt t ∞ 0 R n |f t | 2 v 0 dw dt t = ∞ 0 |f t | 2 dt t 1 2 2 L 2 (v 0 dw)
. Now, by extrapolation (see [START_REF] Chen | Conical square functions for degenerate elliptic operators[END_REF]Theorem A.1] and also [START_REF] Cruz-Uribe | extrapolation and the theory of Rubio de Francia[END_REF]Theorem 3.31]), we obtain that for v ∈ A ∞ (w) and any q ∈ W w v (max{r w , q -}, q + ),

(2.69)

∞ 0 |T t f t | 2 dt t 1 2 L q ( vdw) ∞ 0 |f t | 2 dt t 1 2
L q ( vdw)

.

In particular the above inequality holds for our choices of q and v.

Next, the proof follows much as in [3, p. 543], but we write the details for the sake of completeness. Consider the following sum:

β k := i:r i =2 k b i r i ,
and note that

f t = 1 t i:r i ≤t b i = k:2 k ≤t 2 k t i:r i =2 k b i r i = k:2 k ≤t 2 k t β k .
By Cauchy-Schwartz inequality, for every t > 0,

|f t | 2 ≤ k:2 k ≤t 2 k t | β k | 2 k:2 k ≤t 2 k t k:2 k ≤t 2 k t | β k | 2 = k∈Z 2 k t | β k | 2 1 [2 k ,∞) (t),
and hence,

∞ 0 |f t | 2 dt t k∈Z ∞ 2 k 2 k t dt t | β k | 2 = k∈Z | β k | 2 .
Using the bounded overlap property (2.52), the fact that r i ≈ r B i , and also (2.53), we have

∞ 0 |f t | 2 dt t 1 2 q L q (vdw) k∈Z | β k | 2 1 2 q L q (vdw) R n i |b i | q r q i vdw α q i vw(B i ) α q-p R n |∇f | p vdw.
This estimate, (2.68), and (2.69) with q and v, yield as desired

II 1 1 α q ∞ 0 |T t f t | 2 dt t 1 2 q L q (vdw) 1 α q k∈Z | β k | 2 1 2 q L q (vdw) 1 α p R n |∇f | p vdw.
In order to estimate term II 2 , notice that

∞ 0 t 2 ∇L w e -t 2 Lw i:r i >t b i 2 dt t 1 2 ≤ i r i 0 t 2 ∇L w e -t 2 Lw b i 2 dt t 1 2 =: i T i b i .
Then, by duality, we have

II 2 1 α q R n \ i 16B i i T i b i (x) q v(x)dw(x) ≤ 1 α q sup u L q ′ (vdw) =1 i R n \ i 16B i |T i b i (x)| |u(x)| v(x)dw(x) q ≤ 1 α q sup u L q ′ (vdw) =1 i j≥4 C j (B i ) |T i b i (x)| |u(x)| v(x)dw(x) q 1 α q sup u L q ′ (vdw) =1 i j≥4 T i b i L q (C j (B i ),vdw) u L q ′ (C j (B i ),vdw) q .
To estimate T i b i L q (C j (B i ),vdw) , we pick p 0 close enough to q -, and q 0 close enough to q + such that (2.70) q -< p 0 < 2 < q 0 < q + , and v ∈ A q p 0 (w) ∩ RH ( q 0 q ) ′ (w).

Note that W w v (q -, q + ) = Ø since by assumption W w v (max{r w , q -}, q + ) = Ø and W w v (max{r w , q -}, q + ) ⊂ W w v (q -, q + ). Notice also that applying Remark 2.36 with v ≡ 1, we have t 3 2 ∇L w e -tLw ∈ O(L p 0 (w) -L q 0 (w)). Then, by Minkowski's integral inequality, Lemma 2.16 (a) and (b) (see (2.70)), (2.53) (see (2.67)), and recalling that r i ≈ r B i , for j ≥ 2,

T i b i L q (C j (B i ),vdw) = vw(2 j+1 B) 1 q - C j (B i ) r i 0 t 3 ∇L w e -t 2 Lw b i 2 dt t 3 q 2 d(vw) 1 q vw(2 j+1 B i ) 1 q - C j (B i ) r i 0 t 3 ∇L w e -t 2 Lw b i 2 dt t 3 q 0 2 dw 1 q 0 ≤ vw(2 j+1 B i ) 1 q   r i 0 - C j (B i ) t 3 ∇L w e -t 2 Lw b i q 0 dw 2 q 0 dt t 3   1 2 2 jθ 1 vw(2 j+1 B i ) 1 q - B i |b i | p 0 dw 1 p 0 r i 0 2 j r B i t 2θ 2 e -c 4 j r 2 B i t 2 dt t 3 1 2 e -c4 j vw(2 j+1 B i ) 1 q - B i b i r B i q d(vw) 1 q e -c4 j α vw(2 j+1 B i ) 1 q .
Now we use Lemma 2.62 with p 1 = q, I ij = T i b i L q (C j (B i ),vdw) , {B i } i the collection of balls given by Lemma 2.47, and with e -c4 j replacing 2 -j(2M -C) (consequently M and C do not play any role here). Therefore, Lemma 2.62 and (2.51) imply

II 2 vw i B i 1 α p R n |∇f | p vdw.
Collecting the above estimates, we get the desired result.

2.7. Non-homogeneous conical square function. In this section, we shall prove weighted boundedness in Sobolev spaces for the inhomogeneous conical square function S w H defined in (2.26). The analogous result for elliptic operators was studied in [25] for the Riesz transform characterization of Hardy spaces. See also [START_REF] Prisuelos-Arribas | Weighted Hardy spaces associated with elliptic operators. Part III: Characterizations of H p L (w) and the weighted Hardy space associated with the Riesz transform[END_REF] for the the Riesz transform characterization of weighted Hardy spaces. Our result is stated as follows.

Theorem 2.71. Given w ∈ A 2 (dx), v ∈ A ∞ (w), assume that

W w v (max {r w , q -(L w )} , q + (L w )) = Ø. (2.72)
Then, for every h ∈ S and p ∈ W w v (max {r w , (p

-(L w )) w, * } , p + (L w )), it holds S w H h L p (w) ∇h L p (w) . (2.73)
In order to prove this theorem, we shall use Lemma 2.74 and Proposition 2.75. Lemma 2.74 will be also useful in the proof of Proposition 3.5 (all these results are stated below).

Lemma 2.74. Let w ∈ A 2 (dx) and v ∈ A ∞ (w) be such that W w v (q -(L w ), q + (L w )) = Ø, and let p ∈ (r v (w) max{r w , (q -(L w )) w, * }, r v (w) max{r w , q -(L w )}).
Given α > 0 and f ∈ S such that ∇f L p (vdw) < ∞, let {b i } i be the collection of smooth functions from Lemma 2.47 (applied to f , p, α, and

̟ = vw). Write b := ∞ i=1 A r B i b i , where A r B i := I -(I -e -r 2 B i
Lw ) M and M ∈ N is arbitrarily large. Then, for p 1 ∈ W w v (q -(L w ), q + (L w )) such that 1 ≤ p 1 < p * vw (note that following (2.59) we get that r v (w)q -(L w ) < p * vw ), there holds

∇ b p 1 L p 1 (vdw) α p 1 -p ∇f p L p (vdw) .
Proof. First of all denote q -:= q -(L w ) and q + := q + (L w ). By duality and expanding A r B i , we have

∇ b p 1 L p 1 (vdw) = R n ∇ i M k=1 C k,M e -kr 2 B i Lw b i p 1 vdw sup u L p ′ 1 (vdw) =1 M k=1 i R n √ kr B i ∇e -kr 2 B i Lw b i r B i |u|vdw p 1
.

Besides, recall that by hypothesis

v ∈ A p 1 q - (w) ∩ RH q + p 1 ′ (w) (see (2.14)) and hence √ τ ∇e -τ Lw ∈ O(L p 1 (vdw) -L p 1 (vdw)).
Using this, (2.53), and also (2.3),

R n √ kr B i ∇e -kr 2 B i Lw b i r B i |u|vdw j≥1 vw(2 j+1 B i ) - C j (B i ) √ kr B i ∇e -kr 2 B i Lw b i r B i p 1 d(vw) 1 p 1 - C j (B i ) |u| p ′ 1 d(vw) 1 p ′ 1 j≥1 e -c4 j vw(B i ) - B i b i r B i p 1 d(vw) 1 p 1 inf x∈B i M vw (|u| p ′ 1 )(x) 1 p ′ 1 α B i M vw (|u| p ′ 1 ) 1 p ′ 1 vdw.
Consequently, (2.61) with p = p 1 and (2.51) imply

∇ b p 1 L p 1 (vdw) α p 1 sup u L p ′ 1 (vdw) =1 i B i M vw (|u| p ′ 1 ) 1 p ′ 1 vdw p 1 α p 1 vw i B i α p 1 -p R n |∇f | p vdw.
To formulate our next result (proceeding similarly as in [25,[START_REF] Prisuelos-Arribas | Weighted Hardy spaces associated with elliptic operators. Part III: Characterizations of H p L (w) and the weighted Hardy space associated with the Riesz transform[END_REF]), we introduce the following conical square function

S w 1/2,H f (x) := B(x,t) ∞ 0 t L w e -t 2 Lw f (y) 2 dw(y) dt tw(B(y, t)) 1 2
.

Observe that S w H f = S w 1/2,H √ L w f . Our goal is to see that S w 1/2,H f compares with S w H f (defined in (2.24)) in some weighted spaces (see [START_REF] Prisuelos-Arribas | Vertical square functions and other operators associated with an elliptic operator[END_REF]Proposition 4.5] for a general version of this result). For the following statement we recall that p + (L w ) k, * w was defined in (2.7).

Proposition 2.75. Given w ∈ A 2 (dx), v ∈ A ∞ (w), and f ∈ L 2 (w), there hold

(a) S w H f L p (vdw) S w 1/2,H f L p (vdw) , for all p ∈ W w v (0, p + (L w ) 2, * w ); (b) S w 1/2,H f L p (vdw) S w H f L p (vdw) , for all p ∈ W w v (0, p + (L w ) * w ).
In particular, if p ∈ W w v (0, p + (L w ) * w ), we have

S w 1/2,H f L p (vdw) ≈ S w H f L p (vdw) .
Proof. We shall use extrapolation to prove both inequalities. Indeed, [

S w H f 2 L 2 (v 0 dw) S w 1/2,H f 2 L 2 (v 0 dw) , ∀v 0 ∈ RH p + (Lw ) 2, * w 2 ′ (w) (2.76) and (b) from S w 1/2,H f 2 L 2 (v 0 dw) S w H f 2 L 2 (v 0 dw) , ∀v 0 ∈ RH p + (Lw ) * w 2 ′ (w). (2.77) To set the stage, fix w ∈ A 2 (dx) and v 0 ∈ RH p + (Lw ) k, * w 2 ′ (w).
Here and below k is either 1 or 2, depending whether we are proving (2.77) or (2.76) respectively.

Note that we can find r, q 0 , r, and M ∈ N such that r w < r < 2, 2 < q 0 < p + (L w ), q 0 /2 ≤ r < ∞, v 0 ∈ RH r ′ (w), and

k + n r 2r - n r q 0 - 1 M > 0. (2.78)
Indeed, if n r w > kp + (L w ), note that we can take r w < r < 2 close enough to r w , ε 0 > 0 small enough, and 2 < q 0 < p + (L w ), close enough to p + (L w ) so that for r := q 0 n r 2(1+ε 0 )(n r-kq 0 ) we have that q 0 /2 ≤ r < ∞, v 0 ∈ RH r ′ (w), and

k + n r 2r - n r q 0 = ε 0 q 0 (n r -kq 0 ) > ε 0 q 0 (nr w -kp + (L w )) > 0.
Then, taking M ∈ N large enough we obtain (2.78).

If now n r w ≤ kp + (L w ), our condition on the weight v 0 becomes v 0 ∈ A ∞ (w). Then, we take r > s v 0 (w) and q 0 satisfying max 2, 2rp + (Lw) p + (Lw)+2r < q 0 < min {p + (L w ), 2r} if p + (L w ) < ∞, or q 0 = 2r if p + (L w ) = ∞. Therefore, we have that 2 < q 0 < p + (L w ), q 0 /2 ≤ r < ∞, v 0 ∈ RH r ′ (w), and

k + n r w 2r - n r w q 0 > k - n r w p + (L w ) ≥ 0.
Taking further r w < r < 2 close enough to r w and M ∈ N large enough, we obtain that

k + n r 2r - n r q 0 - 1 M > 0.
After this observation we show the desired estimate.

We first prove (2.76). Use (2.27), Minkowski's integral inequality, and also (2.3) after noticing that B(x, t) ⊂ B(y, 2t), for all y ∈ B(x, t). Thus,

S w H f (x)   ∞ 0 t 0 B(x,t)
|sL w e -s 2 Lw t 2 L w e -t 2 Lw f (y)| 2 dw(y)

1 2 ds s 2 dt tw(B(x, t))   1 2 +   ∞ 0 ∞ t B(x,t)
sL w e -s 2 Lw t 2 L w e -t 2 Lw f (y) For I, consider F (y, t) := t √ L w e -t 2 2 Lw f (y). Using the fact that τ L w e -τ Lw ∈ F (L 2 (w) -L 2 (w)), we get

I ≤ ∞ 0 t 0 st t 2 2 + s 2 B(x,t) s 2 + t 2 2 L w e -s 2 + t 2 2 Lw F (y, t) 2 dw(y) 1 2 ds s 2 dt tw(B(x, t)) 1 2 j≥1 e -c4 j ∞ 0 t 0 s t ds s 2 B(x,2 j+1 t) |F (y, t)| 2 dw(y) dt tw(B(x, t)) 1 2 j≥1 e -c4 j ∞ 0 B(x,2 j+2 t) |F (y, √ 2t)| 2 dw(y) dt tw(B(y, t)) 1 2
, where in the last inequality we have changed the variable t into √ 2t and used (2.3). Then, applying change of angles (Proposition 2.44), we conclude that

I L 2 (v 0 dw) j≥1 e -c4 j 2 jθv 0 ,w S w 1/2,H f L 2 (v 0 dw) S w 1/2,H f L 2 (v 0 dw) .
For the estimate of II, consider F (y, s) := (s √ L w ) 3 e -s 2 Lw f (y). We apply Cauchy-Schwartz's inequality in the integral in s, the fact that e -τ Lw ∈ F (L 2 (w) -L 2 (w)), Jensen's inequality in the integral in y, Fubini's theorem, and (2.3). Hence,

II   ∞ 0 ∞ t t s 1 M t s 2-1 M - B(x,t) e -t 2 Lw F (y, s) 2 dw(y) 1 2 ds s 2 dt t   1 2 ∞ 0 ∞ t t s 2 M ds s ∞ t t s 4-2 M - B(x,t) e -t 2 Lw F (y, s) 2 dw(y) ds s dt t 1 2 j≥1 e -c4 j ∞ 0 ∞ t t s 4-2 M - B(x,2 j+1 t) F (y, s) 2 dw(y) ds s dt t 1 2 j≥1 e -c4 j ∞ 0 ∞ t t s 4-2 M - B(x,2 j+1 t)
F (y, s) q 0 dw(y)

2 q 0 ds s dt t 1 2 j≥1 e -c4 j ∞ 0 s 0 t s 4-2 M B(x,2 j+1 st/s) F (y, s) q 0 dw(y) w(B(y, 2 j+1 st/s)) 2 q 0 dt t ds s 1 2
.

Then, for t < s, applying Propositions 2.42 and (2.3), R n B(x,2 j+1 st/s) F (y, s) q 0 dw(y) w(B(y, 2 j+1 st/s))

2 q 0 v 0 (x)dw(x) t s n r 1 r -2 q 0 R n - B(x,2 j+1 s)
F (y, s) q 0 dw(y)

2 q 0 v 0 (x)dw(x).
Denote F (y, s) := s √ L w e -s 2 2 Lw . Since τ L w e -τ Lw ∈ O(L 2 (w) -L q 0 (w)), then we have

- B(x,2 j+1 t)
F (y, s) q 0 dw(y)

2 q 0 - B(x,2 j+1 s) s 2 2 L w e -s 2 2 Lw F (y, s) q 0 dw(y) 2 q 0 l≥1 e -c4
:= 4 + n r 1 r -2 q 0 -2 M , II L 2 (v 0 dw) j≥1 e -c4 j l≥1 e -c4 l 2 jθ 2 R n ∞ 0 s 0 t s C dt t - B(x,2 l+j+2 s) F (y, s) 2 dw(y) ds s v 0 (x)dw(x) 1 2 j≥1 e -c4 j l≥1 e -c4 l R n ∞ 0 B(x,2 l+j+3 s)
F (y, √ 2s) 2 dw(y)ds sw(B(y, s)) v 0 (x)dw(x)

1 2 j≥1 e -c4 j l≥1 e -c4 l 2 (l+j)θv 0 ,w S w 1/2,H f L 2 (v 0 dw) S w 1/2,H f L 2 (v 0 dw) .
As for proving (2.77), using again (2.3), (2.27), and Minkowski's integral inequality, we obtain

S w 1/2,H f (x)   ∞ 0 t 0 B(x,t)
|tsL w e -s 2 Lw e -t 2 Lw f (y)| 2 dw(y)

1 2 ds s 2 dt tw(B(x, t))   1 2 +   ∞ 0 ∞ t B(x,t)
|tsL w e -s 2 Lw e -t 2 Lw f (y)| 2 dw(y) We first estimate I. Using that s < t and applying the fact that e -τ Lw ∈ F (L 2 (w)-L 2 (w)), and (2.3), we have

I =   ∞ 0 t 0 s t B(x,t) |e -s 2 Lw t 2 L w e -t 2 Lw f (y)| 2 dw(y) 1 2 ds s 2 dt tw(B(x, t))   1 2 j≥1 e -c4 j ∞ 0 t 0 s t ds s 2 B(x,2 j+1 t) |t 2 L w e -t 2 Lw f (y)| 2 dw(y) dt tw(B(y, t)) 1 2 j≥1 e -c4 j ∞ 0 B(x,2 j+1 t) |t 2 L w e -t 2 Lw f (y)| 2 dw(y) dt tw(B(y, t)) 1 2
.

Therefore, applying change of angles (Proposition 2.44), we get

I L 2 (v 0 dw) j≥1 e -c4 j 2 jθv 0 ,w S w H f L 2 (v 0 dw) S w H f L 2 (v 0 dw) .
The estimate of II is very similar to that of II (in the proof of (2.76)), so we skip some details. We apply again the fact that e -τ Lw ∈ F (L 2 (w) -L 2 (w)), Cauchy-Schwartz's inequality in the integral in s, Jensen's inequality in the integral in y, Fubini's theorem, and (2.3). Hence, we have

II j≥1 e -c4 j ∞ 0 ∞ t t s 1 M t s 1-1 M - B(x,2 j+1 t) |s 2 L w e -s 2 Lw f (y)| 2 dw(y) 1 2 ds s 2 dt t 1 2 j≥1 e -c4 j ∞ 0 ∞ t t s 2-2 M - B(x,2 j+1 t) |s 2 L w e -s 2 Lw f (y)| q 0 dw(y) 2 q 0 ds s dt t 1 2 j≥1 e -c4 j ∞ 0 s 0 t s 2-2 M B(x,2 j+1 st/s) |s 2 L w e -s 2 Lw f (y)| q 0 dw(y) w(B(y, 2 j+1 st/s)) 2 q 0 dt t ds s 1 2 .
Note that, for t < s, Proposition 2.42, and (2.3) imply

R n B(x,2 j+1 st/s) |s 2 L w e -s 2 Lw f (y)| q 0 dw(y) B(y, 2 j+1 st/s) 2 q 0 v 0 (x)dw(x) t s n r 1 r -2 q 0 R n - B(x,2 j+1 s)
|s 2 L w e -s 2 Lw f (y)| q 0 dw(y)

2 q 0 v 0 (x)dw(x).
Besides, since e -τ Lw ∈ O(L 2 (w) → L q 0 (w))

- B(x,2 j+1 s)
|s 2 L w e -s 2 Lw f (y)| q 0 dw(y)

2 q 0 l≥1 e -c4 l 2 jθ 2 - B(x,2 j+l+2 s) |s 2 L w e -s 2 2 Lw f (y)| 2 dw(y).
Hence, applying Fubini's theorem, (2.78) with k = 1, changing the variable s into √ 2s, by (2.3) and Proposition 2.44 we have

II L 2 (v 0 dw) j≥1 e -c4 j l≥1 e -c4 l R n ∞ 0 B(x,2 j+l+3 s)
|s 2 L w e -s 2 Lw f (y)| 2 dw(y) ds sw(B(y, s)) v 0 (x)dw(x)

1 2 j≥1 e -c4 j l≥1 e -c4 l 2 (l+j)θw,v 0 S w H f L 2 (v 0 dw) S w H f L 2 (v 0 dw) .
This and the estimate obtained for I L 2 (v 0 dw) give us (2.77).

Proof of Theorem 2.71. First of all fix w ∈ A 2 (dx), and denote q -:= p -(L w ) = q -(L w ) (see Lemma 2.35), q + := q + (L w ), and p + := p + (L w ). We claim that for all p ∈ W w v (max{r w , q -}, p + ) and h ∈ S, S w H h L p (vdw) ∇h L p (vdw) . (2.79) Indeed, applying Proposition 2.75, [12, Theorem 3.1], and Proposition 2.28, we have that

S w H h L p (vdw) = S w 1/2,H L w h L p (vdw) ≈ S w H L w h L p (vdw) L w h L p (vdw) ∇h L p (vdw) .
Note that W w v (max{r w , q -}, p + ) = r v (w) max{r w , q -}, p + /s v (w) . Therefore, for every p satisfying r v (w) max {r w , (q -) w, * } < p < r v (w) max{r w , q -}, (2.80) if we show that S w H h L p,∞ (vdw) ∇h L p (vdw) , ∀h ∈ S, (2.81) then, by interpolation (see [START_REF] Badr | Real interpolation of Sobolev spaces[END_REF] and Remark 2.15) we would conclude (2.73). Now fix p as in (2.80), and note that vw ∈ A p (dx), since r vw ≤ r w r v (w) < p (see Remark 2.15). Given α > 0, we apply Lemma 2.47 to h ∈ S, α, the product weight ̟ = vw and p. Let {B i } i be the collection of balls given by Lemma 2.47. Consider for M ∈ N arbitrarily large,

B r B i := (I -e -r 2 B i Lw ) M , A r B i := I -B r B i = M k=1 C k,M e -kr 2 B i
Lw .

Then h = g + i A r B i b i + i B r B i b i =: g + b + b. It follows that (2.82) vw x ∈ R n : S w H h(x) > α ≤ vw x ∈ R n : S w H g(x) > α 3 + vw x ∈ R n : S w H b(x) > α 3 + vw x ∈ R n : S w H b(x) > α 3 =: I + II + III.
Now, since W w v (max{r w , q -}, q + ) = Ø by assumption and p > r v (w) max{r w , (q -) w, * } (see (2.80)), by Remark 2.56 we can pick p 1 such that r v (w) max{r w , q -} < p 1 < min q + s v (w)

, p * vw . (2.83) Observe that p 1 > p, and p 1 ∈ W w v (max{r w , q -}, q + ) . Note also that in particular r v (w) q -< p 1 < q + sv(w) , that is,

v ∈ A p 1 q - (w) ∩ RH q + p 1 ′ (w). ( 2
I 1 α p 1 R n | S w H g| p 1 vdw 1 α p 1 R n |∇g| p 1 vdw 1 α p R n |∇h| p vdw.
In order to estimate II, apply Chebyshev's inequality, (2.79), and Lemma 2.74 (with f = h). Then

II 1 α p 1 R n S w H b p 1 vdw 1 α p 1 R n ∇ b p 1 vdw 1 α p R n |∇h| p vdw. (2.86)
Next, we estimate III. Note that, by (2.51)

III vw i 16B i + vw x ∈ R n \ i 16B i : S w H b(x) > α 3 (2.87) 1 α p R n |∇h| p vdw + III 1 ,
where

III 1 := vw x ∈ R n \ i 16B i : S w H b(x) > α 3 .
By Chebyshev's inequality, duality, splitting the integral in x, and applying Hölder's inequality:

III 1 1 α p 1 R n \∪ i 16B i S w H b p 1 vdw (2.88) 1 α p 1 sup u L p ′ 1 (vdw) =1 i j≥4 C j (B i ) S w H B r B i b i p 1 vdw 1 p 1 u1 C j (B i ) L p ′ 1 (vdw) p 1 =: 1 α p 1 sup u L p ′ 1 (vdw) =1 i j≥4 III ij u1 C j (B i ) L p ′ 1 (vdw) p 1 .
Splitting the integral in t (recall that j ≥ 4), we have

III ij C j (B i ) 2 j-2 r B i 0 B(x,t)
tL w e -t 2 Lw B r B i b i (y) 2 dw(y) dt tw(B(y, t))

p 1 2 v(x)dw(x) 1 p 1 (2.89) + C j (B i ) ∞ 2 j-2 r B i B(x,t) t 2 L w e -t 2 Lw B r B i b i r B i (y) 2 dw(y) dt tw(B(y, t)) p 1 2 v(x)dw(x) 1 p 1 =: III 1 ij + III 2 ij .
Before estimating III 1 ij and III 2 ij , we take p 0 close enough to q -, and q 0 close enough to q + so that

(2.90) q -< p 0 < min{2, p 1 }, max{2, p 1 } < q 0 < q + , v ∈ A p 1 p 0 (w) ∩ RH q 0 p 1 ′ (w).
Hence, by Lemma 2.16 (b),

(2.91) III 1 ij vw(2 j+1 B i ) 1 p 1 - C j (B i ) 2 j-2 r B i 0 B(x,t) tL w e -t 2 Lw B r B i b i (y) 2 dw(y) dt tw(B(y, t)) q 0 2 dw(x) 1 q 0 .
Besides, note that for x ∈ C j (B i ) and 0 < t ≤ 2 j-2 r B i we have that B(x, t) ⊂ 2 j+2 B i \ 2 j-1 B i . Then, by (2.3), recalling that q 0 > 2, applying Jensen's inequality with respect to dw(y) dt, and Fubini's theorem, we get

- C j (B i ) 2 j-2 r B i 0 B(x,t)
tL w e -t 2 Lw B r B i b i (y) 2 dw(y) dt tw(B(y, t))

q 0 2 dw(x)

1 q 0 (2.92) (2 j r B i ) 1 2 - C j (B i ) - 2 j-2 r B i 0 - B(x,t) 1 t tL w e -t 2 Lw B r B i b i (y) 2 dw(y) dt q 0 2 dw(x) 1 q 0 - C j (B i ) 2 j-2 r B i 0 2 j r B i t q 0 2 -1 - B(x,t)
tL w e -t 2 Lw B r B i b i (y) q 0 dw(y) dt t dw(x)

1 q 0 - C j (B i ) 2 j-2 r B i 0 2 j r B i t q 0 2 -1 B(x,t)
tL w e -t 2 Lw B r B i b i (y) q 0 dw(y) dt tw(B(y, t)) dw(x)

1 q 0 2 j-2 r B i 0 2 j r B i t q 0 2 -1 t -q 0 - 2 j+2 B i \2 j-1 B i t 2 L w e -t 2 Lw B r B i b i (y) q 0 dw(y) dt t 1 q 0 .
We estimate the integral in y by using functional calculus. The notation is taken from [START_REF] Auscher | On necessary and sufficient conditions for L p -estimates of Riesz transforms associated to elliptic operators on R n and related estimates[END_REF] and [6, Section 7]. We write ϑ ∈ [0, π/2) for the supremum of |arg( L w f, f L 2 (w) )| over all f in the domain of L w . Let 0 < ϑ < θ < ν < µ < π/2 and note that, for a fixed t > 0, φ(z, t) := e -t 2 z (1e -r Here Γ = ∂Σ π 2 -θ with positive orientation (although orientation is irrelevant for our computations) and γ = R + e i sign(Im(z)) ν . It is not difficult to see that for every z ∈ Γ,

|η(z, t)| r 2M B i (|z| + t 2 ) M +1 . Moreover, observe that 2 j+2 B i \ 2 j-1 B i = 3 l=1 C l+j-2 (B i ), ∀j ≥ 4.
Also, our choices of p 0 and q 0 in (2.90) yield that zL w e -zLw ∈ O(L p 0 (w) -L q 0 (w)). Thus, by these facts and Minkowski's integral inequality, we obtain -

2 j+2 B i \2 j-1 B i t 2 L w e -t 2 Lw B r B i b i q 0 dw 1 q 0 Γ - 2 j+2 B i \2 j-1 B i zL w e -zLw b i q 0 dw 1 q 0 t 2 |z| r 2M B i (|z| + t 2 ) M +1 |dz| 2 jθ 1 - B i |b i | p 0 dw 1 p 0 Γ Υ 2 j r B i |z| 1 2 θ 2 e -c 4 j r 2 B i |z| t 2 |z| r 2M B i (|z| + t 2 ) M +1 |dz| 2 jθ 1 t 2 - B i |b i | p 1 d(vw) 1 p 1 ∞ 0 Υ 2 j r B i s 1 2 θ 2 e -c 4 j r 2 B i s r 2M B i s M +1 ds s α r -1 B i 2 -j(2M +2-θ 1 ) t 2 ∞ 0 Υ (s) θ 2 e -cs 2 s 2M +2 ds s α r -1 B i 2 -j(2M +2-θ 1 ) t 2 ,
where, to obtain the last inequality, we need to take M ∈ N large enough so that 2M + 2 > θ 2 . Besides, we use Lemma 2.16 (a) in the third inequality; and the forth inequality follows from (3.7) (see (3.36)) and the change of variable s into 4 j r 2 B i /s 2 . Plugging the above estimate into (2.92) and changing the variable t into 2 j r B i t, allows us to obtain

  - C j (B i ) 2 j-2 r B i 0 B(x,t) tL w e -t 2 Lw B r B i b i (y) 2 dw(y) dt tw(B(y, t)) q 0 2 dw(x)   1 q 0 α r -1 B i 2 -j(2M +2-θ 1 ) 2 j-2 r B i 0 2 j r B i t q 0 2 -1 t q 0 dt t 1 q 0 α 2 -j(2M +1-θ 1 ) .
This and (2.91) yield, for M ∈ N such that 2M + 2 > θ 2 ,

III 1 ij α vw(2 j+1 B i ) 1 p 1 2 -j(2M
:= t √ M + 1. Then ∞ 2 j-2 r B i B(x,t) t 2 L w e -t 2 Lw B r B i b i r B i (y) 2 dw(y) dt tw(B(y, t)) 1 2
(2.95)

∞ 2 j-2 r B i - B(x,t) t 2 L w e -t 2 Lw B r B i b i r B i (y) 2 dw(y) dt t 1 2 ∞ 2 j-2 r B i θ M - B(x,θ M t) T t,r B i t 2 L w e -t 2 Lw b i r B i (y) 2 dw(y) dt t 1 2
, where we recall that T t,r B i := (e -t 2 Lwe -(t 2 +r 2 B i

)Lw ) M .

Next, fix x ∈ C j (B i ) and t > 2 j-2 r B i θ M . In this case, B i ⊂ B(x, 12θ M t). Thus, by (2.39) and the fact that τ L w e -τ Lw ∈ O(L p 0 (w) -L 2 (w)), for τ > 0, we get

- B(x,θ M t) T t,r B i t 2 L w e -t 2 Lw b i r B i (y) 2 dw(y), r 2
B i t 2 2M 1 w(B(x, θ M t)) R n t 2 L w e -t 2 Lw 1 B(x,12θ M t) b i r B i (y) 2 dw(y) r 2 B i t 2 2M l≥1 2 2nl - C l (B(x,12θ M t)) t 2 L w e -t 2 Lw 1 B(x,12θ M t) b i r B i (y) 2 dw(y) r 2 B i t 2 2M l≥1 e -c4 l - B(x,12θ M t) b i (y) r B i p 0 dw(y) 2 p 0 r 2 B i t 2 2M w(B i ) w(B(x, 12θ M t)) 2 p 0 - B i b i r B i p 0 dw 2 p 0 r 2 B i t 2 2M - B i b i r B i p 1 d(vw) 2 p 1 r 2 B i t 2 2M α 2 .
Here the next-to-last inequality is due to Lemma 2.16 (a) and the fact that B i ⊂ B(x, 12θ M t), and the last inequality follows from (2.53).

Plugging the above estimate into (2.95) and recalling the definition of III 2 ij in (2.89) allows us to obtain

III 2 ij α C j (B i ) ∞ 2 j-2 r B i θ M r 2 B i t 2 2M dt t p 1 2 vdw 1 p 1 αvw(2 j+1 B i ) 1 p 1 2 -j2M .
By this and (2.94), for M ∈ N such that 2M > θ 2 -2, we have

III ij αvw(2 j+1 B i ) 1 p 1 2 -j(2M -θ 1 ) .
Then, by Lemma 2.62 with I ij = III ij , C = θ 1 , and {B i } i the collection of balls given by Lemma 2.47, and by (2.51) and (2.88), for M ∈ N so that 2M > max{θ 2 -2, θ 1 + r w r v (w)n}, we conclude that

III 1 vw i B i 1 α p R n |∇h| p vdw.
This, together with (2.85)-(2.87) and (2.82), yields (2.81).

3. Proof of Theorem 1.8

In this section, we prove Theorem 1.8 for functions f ∈ S, and conclude the result by a density argument. Fix w ∈ A 2 (dx), v ∈ A ∞ (w) and f ∈ S, and note that for every (x, t) ∈ R n+1

+ and u(x, t) := ∇ x,t e -t √ Lw f (x), |u(x, t)| 2 = |∇e -t √ Lw f (x)| 2 + |∂ t e -t √ Lw f (x)| 2 ,
where we define the Poisson semigroup {e -t √ Lw } t>0 using the classical subordination formula, or the functional calculus for L w (see [START_REF] Auscher | On necessary and sufficient conditions for L p -estimates of Riesz transforms associated to elliptic operators on R n and related estimates[END_REF][START_REF] Cruz-Uribe | On the Kato problem and extensions for degenerate elliptic operators[END_REF]):

e -t √ Lw = C ∞ 0 e -λ λ 1 2 e -t 2 4λ Lw dλ λ . (3.1) Therefore, it suffices to see that if W w v (max {r w , q -(L w )} , q + (L w )) = Ø, then N w (∇e -t √ Lw f ) L p (vdw) ∇f L p (vdw) and N w (∂ t e -t √ Lw f ) L p (vdw) ∇f L p (vdw) ,
for all p ∈ W w v (max {r w , (q -(L w )) w, * } , q + (L w )). We shall see this in Propositions 3.2 and 3.5 below.

3.1.

Non-tangential maximal function estimate for the spatial derivatives. Proposition 3.2. Given w ∈ A 2 (dx) and v ∈ A ∞ (w) such that W w v (max {r w , q -(L w )} , q + (L w )) = Ø. Then, for all f ∈ S and p ∈ W w v (max {r w , (q -(L w )) w, * } , q + (L w )), we have

(3.3) N w (∇e -t √ Lw f ) L p (vdw) ∇f L p (vdw) .
Proof. First of all, fix w ∈ A 2 (dx) and define q -:= q -(L w ) and q + := q + (L w ).

In the context of (1.6) we set α := c 0 c 1 . We claim that

N w (∇e -t √ Lw f )(x) sup t>0 - B(x,αt) ∇e -t √ Lw f (z) 2 dw(z) 1 2 . (3.4) Indeed, by (2.3), N w (∇e -t √ Lw f )(x) = sup t>0 - c -1 0 t<s<c 0 t - B(x,c 1 t) ∇e -s √ Lw f (z) 2 dw(z)ds 1 2 sup t>0 sup c -1 0 t<s<c 0 t - B(x,αs) ∇e -s √ Lw f (z) 2 dw(z) 1 2 sup t>0 - B(x,αt) ∇e -t √ Lw f (z) 2 dw(z) 1 2
.

Besides, by the subordination formula (3.1) and Minkowski's integral inequality,

- B(x,αt) ∇e -t √ Lw f (z) 2 dw(z) 1 2 1 4 0 e -λ λ 1 2 - B(x,αt) ∇e -t 2 4λ Lw f (z) 2 dw(z) 1 2 dλ λ + ∞ 1 4 e -λ λ 1 2 - B(x,αt) ∇e -t 2 4λ Lw f (z) 2 dw(z) 1 2 dλ λ =: I + II.
Dealing first with term I, note that

I ≤ 1 4 0 λ 1 2 - B(x,αt) ∇e -t 2 Lw f (z) 2 dw(z) 1 2 dλ λ + 1 4 0 λ 1 2 - B(x,αt) ∇e -t 2 4λ Lw -∇e -t 2 Lw f (z) 2 dw(z) 1 2 dλ λ =: I 1 + I 2 .
In order to estimate term I 1 , for any p ∈ W w v (max {r w , (q -) w, * } , q + ), we pick p 0 in the interval (max {r w , (q -) w, * } , min{2, p}), close enough to max {r w , (q -) w, * } so that

v ∈ A p p 0 (w) (see (2.13) and (2.14)). Therefore M w p 0 (f ) := M w p 0 (|f | p 0 ) 1 p 0 is bounded on L p (vdw).
∇e -t 2 Lw f (z) 2 dw(z) 1 2 L p (vdw) sup t>0 j≥1 e -c4 j - B(•,2 j+1 αt) |∇f (z)| p 0 dw(z) 1 p 0 L p (vdw) j≥1 e -c4 j M w p 0 (∇f ) L p (vdw) ∇f L p (vdw) .
Consequently, by Minkowski's integral inequality,

sup t>0 I 1 L p (vdw) ≤ 1 4 0 λ 1 2 sup t>0 - B(•,αt) ∇e -t 2 Lw f (z) 2 dw(z) 1 2 L p (vdw) dλ λ ∇f L p (vdw) .
Now we turn to the estimate of term I 2 . Write

∇e -t 2 4λ Lw -∇e -t 2 Lw = ∇e -t 2 2 Lw e -( 1 4λ -1 2 )t 2 Lw -e -t 2 2 Lw
and use again Lemma 2.40 and (2.3). Then,

I 2 = 1 4 0 λ 1 2 - B(x,αt) ∇e -t 2 2 Lw e -( 1 4λ -1 2 )t 2 Lw -e -t 2 2 Lw f (z) 2 dw(z) 1 2 dλ λ ∞ j=1 e -c4 j 1 4 0 λ 1 2 - B(x,2 j+2 αt) ∇ e -( 1 4λ -1 2 )t 2 Lw -e -t 2 2 Lw f (z) p 0 dw(z) 1 p 0 dλ λ .
Since 0 < λ ≤ 1/4, there holds

∇e -( 1 4λ -1 2 )t 2 Lw -∇e -t 2 2 Lw f (z) ≤ t √ 1 4λ -1 2 t √ 2 ∂ s ∇e -s 2 Lw f (z) ds t √ 1 4λ -1 2 t √ 2 s 2 ∇L w e -s 2 Lw f (z) ds s ∞ 0 s 2 ∇L w e -s 2 Lw f (z) 2 ds s 1 2 log(2λ) -1 2 1 2 log λ -1 1 2 G w H f (z),
where G w H is the vertical square function defined in (2.25). Then, we get

sup t>0 I 2 j≥1 e -c4 j 1 4 0 λ 1 2 log λ -1 1 2 sup t>0 - B(x,2 j+2 αt) G w H f (z) p 0 dw(z) 1 p 0 dλ λ M w p 0 G w H f (x).
Then, since M w p 0 is bounded on L p (vdw), the above computation and Theorem 2.54 imply

sup t>0 I 2 L p (vdw) ≤ M w p 0 G w H f L p (vdw) G w H f L p (vdw) ∇f L p (vdw) .
We finally estimate term II. Applying Lemma 2.40, we have that, for every t > 0,

- B(x,α2 √ λt) |∇e -t 2 Lw f (z)| 2 dw(z) 1 2 Υ( √ λ) θ ∞ j=1 e -c4 j - B(x,2 j+2 α √ λt)
|∇f (z)| p 0 dw(z)

1 p 0 Υ( √ λ) θ M w p 0 (∇f )(x).
Hence,

sup t>0 II L p (vdw) ∞ 1 4 e -λ Υ( √ λ) θ dλ λ M w p 0 (∇f ) L p (vdw) ∇f L p (vdw) .
Collecting the above estimates, we conclude (3.3).

3.2.

Non-tangential maximal function estimate for the time derivative.

Proposition 3.5. Given w ∈ A 2 (dx) and v ∈ A ∞ (w), assume that

W w v (max {r w , q -(L w )} , q + (L w )) = Ø. Then, for all p ∈ W w v (max {r w , (p -(L w )) w, * } , p + (L w )) and f ∈ S, we have (3.6) N w (∂ t e -t √ Lw f ) L p (vdw)
∇f L p (vdw) .

To prove this result, we need Theorem 2.71, a change of angles result in L p (vdw) for the operator defined in (2.17), and the boundedness of the non-tangential maximal square functions defined in (2.18) and (2.19). We obtain these results in Lemma 3.7, and Proposition 3.12 below.

Our next result is an extension of [25, Lemma 6.2] (see also [START_REF] Martell | Weighted Hardy spaces associated with elliptic operators. Part II: Characterizations of H 1 L (w)[END_REF]Lemma 7.3]).

Lemma 3.7. Given w ∈ A r (dx) and v ∈ A r (w), 1 ≤ r, r < ∞, let 0 < p < ∞ and κ ≥ 1. There hold

(3.8) N κ,w F L p,∞ (vdw) κ n( r+1 2 + r r p ) N w F L p,∞ (vdw) , and
(3.9) N κ,w F L p (vdw) κ n( r+1 2 + r r p ) N w F L p (vdw) .
Proof. We will just prove (3.8), the proof of (3.9) follows analogously by writing the L p (vdw)-norm as an integral of the level sets. Details are left to the interested reader. Consider, for any λ > 0,

O λ := {x ∈ R n : N w F (x) > λ}, E λ := R n \O λ ,
and, for γ = 1 -

1 [w] Ar(dx) (11κ) rn , the set of γ-density E * λ := x ∈ R n : ∀ r > 0, w(E λ ∩ B(x, r)) w(B(x, r)) ≥ γ . Note that O * λ := R n \ E * λ = x ∈ R n : M w (1 O λ )(x) > 1 [w] Ar(dx) (11κ) rn . We claim that for every λ > 0, N κ,w F (x) ≤ [w] Ar(dx) 2 nr 2 (9κ) n(r+1) 2 λ, ∀x ∈ E * λ . (3.10)
Assuming this momentarily, let 0 < p < ∞. Since M w : L r (vdw) → L r,∞ (vdw), as we are assuming that v ∈ A r (w), we get

N κ,w F p L p,∞ (vdw) = sup λ>0 λ p vw({x ∈ R n : N κ,w F (x) > λ}) = sup λ>0 [w] Ar(dx) 2 nr 2 (9κ) n(r+1) 2 λ p vw x ∈ R n : N κ,w F (x) > [w] Ar(dx) 2 nr 2 (9κ) n(r+1) 2 λ ≤ [w] p Ar(dx) 2 pnr 2 (9κ) n(r+1)p 2 sup λ>0 λ p vw(O * λ ) κ n( (r+1)p 2 +r r) sup λ>0 λ p vw(O λ ) = κ n( (r+1)p 2 +r r) N w F p L p,∞ (vdw)
, which would finish the proof.

It remains to show (3.10). First, note that if x ∈ E * λ and t > 0, for every y ∈ B(x, 2κt) we have B(y, t/2)∩E λ = Ø. To prove this, suppose by way of contradiction that B(y, t/2) ⊂ O λ . Then, by (2.2), since B(y, t/2) ⊂ B(x, 3κt) and B(x, 3κt) ⊂ B(y, 5κt),

M w (1 O λ )(x) ≥ w(B(y, t/2)) w(B(x, 3κt)) ≥ w(B(y, t/2)) w(B(y, 5κt)) ≥ 1 [w] Ar(dx) (10κ) rn > 1 [w] Ar(dx) (11κ) rn .
This implies that x ∈ O * λ , which contradicts our assumption. Fix now x ∈ E * λ and t > 0, and note that if y ∈ B(x, 2κt) there exists x 0 ∈ B(y, t/2) ∩ E λ , hence N w F (x 0 ) ≤ λ. Besides, since B(y, t/2) ⊂ B(x 0 , t) and by (2.2), for every y ∈ B(x, 2κt), (3.11)

B(y,t/2) |F (z, t)| 2 dw(z) w(B(y, t/2)) 1 2 ≤ [w] 1 2 Ar(dx) 2 nr 2 sup s>0 B(x 0 ,s)
|F (z, s)| 2 dw(z) w(B(x 0 , s))

1 2 = [w] 1 2
Ar(dx) 2

nr 2 N w F (x 0 ) ≤ [w] 1 2
Ar(dx) 2 nr 2 λ.

On the other hand, for every x ∈ R n and t > 0, we have that B(x, κt) ⊂ i B(x i , t/2), where {B(x i , t/2)} i is a collection of at most (9κ) n balls such that, for every i, we have that x i ∈ B(x, 2κt). In particular, B(x i , t/2), B(x, t) ⊂ B(x i , 3κt).

Therefore, by the above observations and (2.2), we conclude that

B(x,κt) |F (y, t)| 2 dw(y) w(B(x, t)) ≤ [w] Ar(dx) (3κ) nr i B(x i ,t/2) |F (y, t)| 2 dw(y) w(B(x i , t/2)) ≤ (9κ) n(r+1) [w] 2
Ar(dx) 2 nr λ 2 , where we have used (3.11), since x i ∈ B(x, 2κt). Finally, taking the supremum over all t > 0, we obtain

N κ,w F (x) 2 ≤ [w] 2
Ar(dx) 2 nr (9κ) n(r+1) λ 2 , ∀ x ∈ E * λ . This readily gives (3.10) and the proof is complete. Proposition 3.12. Let L w be a degenerate elliptic operator with w ∈ A 2 (dx) and let v ∈ A ∞ (w). Then Proof. We first prove part (a). Fix p ∈ W w v (p -(L w ), ∞) and choose p 0 close enough to p -(L w ) so that (3.13) p -(L w ) < p 0 < min{2, p} and v ∈ A p p 0 (w).

Then e -τ Lw ∈ O(L p 0 (w) -L 2 (w)). This fact and (2.3) yield

N w H f (x) sup t>0 j≥1 - B(x,t) e -t 2 Lw 1 C j (B(x,t)) f (z) 2 dw(z) 1 2 sup t>0 j≥1 2 jθ 1 Υ 2 j+1 θ 2 e -c4 j - C j (B(x,t)) |f (z)| p 0 dw(z) 1 p 0 M w p 0 f (x). Consequently, N w H f L p (vdw) M w p 0 f L p (vdw) f L p (vdw) ,
since M w p 0 is bounded on L p (vdw) by our choice of p 0 .

We now prove part (b). Note that

N w P f (x) ≤ N w H f (x)+sup t>0 - B(x,t) e -t √ Lw -e -t 2 Lw f (z) 2 dw(z) 1 2 =: N w H f (x)+sup t>0 I.
By the subordination formula (3.1) and Minkowski's integral inequality,

I = - B(x,t) ∞ 0 e -λ λ 1 2 e -t 2 4λ Lw -e -t 2 Lw f (z) dλ λ 2 dw(z) 1 2 ≤ ∞ 0 e -λ λ 1 2 - B(x,t) e -t 2 4λ Lw -e -t 2 Lw f (z) 2 dw(z) 1 2 dλ λ = 1 4 0 e -λ λ 1 2 - B(x,t) e -t 2 4λ Lw -e -t 2 Lw f (z) 2 dw(z) 1 2 dλ λ + ∞ 1 4 e -λ λ 1 2 - B(x,t) e -t 2 4λ Lw -e -t 2 Lw f (z) 2 dw(z) 1 2 dλ λ =: I 1 + I 2 .
In order to estimate I 1 and I 2 , for each t > 0 consider

h t := e -( 1 4λ -1 2 )t 2 Lw -e -t 2 2 Lw f . Next fix p ∈ W w v (p -(L w ), p + (L w ))
, and choose p 0 as in (3.13). Then, applying the fact that e -τ Lw ∈ O(L p 0 (w) -L 2 (w)), we have

I 1 ≤ 1 4 0 λ 1 2 - B(x,t) e -t 2 2 Lw h t (z) 2 dw(z) 1 2 dλ λ ≤ 1 4 0 λ 1 2 j≥1 - B(x,t) e -t 2 2 Lw 1 C j (B) h t (z) 2 dw(z) 1 2 dλ λ ≤ 1 4 0 λ 1 2 j≥1 2 jθ 1 Υ 2 j+1 θ 2 e -c4 j - B(x,2 j+1 t)
|h t (z)| p 0 dw(z) (log(4λ))

1 p 0 dλ λ j≥1 e -c4 j 1 4 0 λ 1 2 - B(x,2 j+1 t) |h t (z)| p 0 dw(z) 1 p 0 dλ λ . When 0 < λ ≤ 1/4, Cauchy-Schwartz inequality implies e -( 1 4λ -1 2 )t 2 Lw -e -t 2 2 Lw f (z) ≤ t √ 1 4λ -1 2 t √ 2 ∂ s e -s 2 Lw f (z) ds t √ 1 4λ -1 2 t √ 2 s 2 L w e -s 2 Lw f (z) ds s g w H,t f (z) (log λ -1 ) 1
1 2 .
Then, by (2.3),

I 2 ∞ 1 4 e -cλ t t 2 √ λ B(x,t) s 2 L w e -s 2 Lw f (z) 2 dw(z)ds sw(B(z, t)) 1 2 dλ λ ∞ 1 4 e -cλ t t 2 √ λ B(x,2 √ λs)
s 2 L w e -s 2 Lw f (z) 2 dw(z) ds s w(B(z, s))

1 2 dλ λ ∞ 1 4 e -cλ S 2 √ λ,w H f (x) dλ λ ,
where S 2 √ λ,w H is defined in (2.24). From the above estimates for I 1 and I 2 , we obtain that, for all x ∈ R n ,

N w P f (x) N w H f (x) + sup t>0 j≥1
e -c4 j -B(x,2 j+1 t)

|g w H,t f (z)| p 0 dw(z)

1 p 0 (3.15) + ∞ 1 4 e -cλ S 2 √ λ,w H f (x) dλ λ N w H f (x) + M w p 0 (g w H f )(x) + ∞ 1 4 e -cλ S 2 √ λ,w H f (x) dλ λ ,
where g w H is defined in (2.20). To proceed, we first note that M w p 0 is bounded on L p (vdw), since v ∈ A p p 0 (w), and so is g w H (see [START_REF] Cruz-Uribe | On the Kato problem and extensions for degenerate elliptic operators[END_REF]) since p ∈ W w v (p -(L w ), p + (L w )). Using this and invoking Proposition 2.44 and [12, Theorem 3.1], for some θ > 0 depending on v, w, and n, we conclude that

N w P f L p (vdw) N w H f L p (vdw) + M w p 0 (g w H f ) L p (vdw) + ∞ 1 4 e -cλ S 2 √ λ,w H f L p (vdw) dλ λ f L p (vdw) + g w H f L p (vdw) + f L p (vdw) ∞ 1 4
λ θ e -cλ dλ λ f L p (vdw) .

This completes the proof.

Proof of Proposition 3.5. First of all, fix w ∈ A 2 (dx) and denote q -:= p -(L w ) = q -(L w ), q + := q + (L w ), p + := p + (L w ), and u(x, t)

:= ∂ t e -t √ Lw f (x) = -L w e -t √ Lw f (x).
From the definitions of N w and N w P (see (1.6) and (2.19)), proceeding as in the proof of (3.4) we have that (3.16) N w u(x) N α,w P ( L w f )(x), ∀x ∈ R n , with α = c 0 c 1 . Consequently, Lemma 3.7, and Propositions 3.12 (b) and 2.28 imply

N w u L p (vdw) N w P ( L w f ) L p (vdw) L w f L p (vdw) ∇f L p (vdw) ,
for all p ∈ W w v (max{r w , q -}, p + ) = r v (w) max{r w , q -}, p + /s v (w) and f ∈ S. Our goal is to obtain (3.6) for all p ∈ W w v (max{r w , (q -) w, * }, p + ) = r v (w) max{r w , (q -) w, * }, p + /s v (w) . Recall that (q -) w, * < q -(see (2.6)). Hence, fix p such that r v (w) max {r w , (q -) w, * } < p < r v (w) max {r w , q -} . (3.17) Then, in view of inequality (3.16) and Lemma 3.7, if we show that, for all f ∈ S,

N w P L w f L p,∞ (vdw)
∇f L p (vdw) , (3.18) by interpolation, see [START_REF] Badr | Real interpolation of Sobolev spaces[END_REF] and Remark 2.15, we would conclude the desired estimate.

Given α > 0, take a function f ∈ S. We apply Lemma 2.47 to f , α, and the product weight ̟ = vw (note that vw ∈ A p (dx) since r vw ≤ r w r v (w) < p, see Remark 2.15). Let {B i } i be the collection of balls given by Lemma 2.47. Consider for M ∈ N arbitrarily large,

B r B i := (I -e -r 2 B i Lw ) M , A r B i := I -B r B i = M k=1 C k,M e -kr 2 B i
Lw .

Hence,

f = g + i A r B i b i + i B r B i b i =: g + b + b. (3.19)
To prove the weak-type estimates for g, b, and b, we need some preparations. On the one hand, since we assume that W w v (max{r w , q -}, q + ) = Ø, by (3.17) and (2.57) we can take p 1 satisfying r v (w) max {r w , q -} < p 1 < min q + s v (w)

, p * vw . (3.20) In particular, r v (w)q -< p 1 < q + sv(w) , that is, p 1 ∈ W w v (q -, q + ). This can be written as

v ∈ A p 1 q - (w) ∩ RH q + p 1 ′ (w). (3.21)
On the other hand, take p 0 satisfying q -< p 0 < min{2, p 1 } close enough to q -, and q 0 satisfying max{2, p 1 } < q 0 < q + close enough to q + , so that

v ∈ A p 1 p 0 (w) ∩ RH q 0 p 1 ′ (w). (3.22)
Next, by (3.15) and (2.3), we have that for any function h ∈ L 2 (w)

N w P h(x) N w H h(x) + l≥1 e -c4 l sup t>0 - B(x,2 l+1 t)
g w H,t h(y) p 0 dw(y)

1 p 0 + ∞ 1 4 e -cλ S 2 √ λ,w H h(x) dλ λ =: N w H h(x) + l≥1 e -c4 l O 2,l h(x) + O 3 h(x),
where we recall that g w H,t is defined in (3.14). Besides, note that, the fact that e -τ Lw ∈ O(L p 0 (w) -L 2 (w)), and (2.3) yield

N w H h(x) = sup t>0 - B(x,t) e -t 2 Lw h(y) 2 dw(y) 1 2 l≥1 e -c4 l sup t>0 - B(x,2 l+1 t) e -t 2 2 Lw h(y) p 0 dw(y) 1 p 0 l≥1 e -c4 l sup t>0 - B(x,2 l+2 t) e -t 2 Lw h(y) p 0 dw(y) 1 p 0 =: l≥1 e -c4 l O 1,l h(x).
Therefore, for any function h ∈ L 2 (w), we have that

N w P h(x) ≤ C l≥1 e -c4 l O 1,l h(x) + l≥1 e -c4 l O 2,l h(x) + O 3 h(x) , ∀x ∈ R n .
Using this and (3.19), we get

vw x ∈ R n : N w P L w f (x) > α (3.23) ≤ vw x ∈ R n : N w P L w g (x) > α 5 + vw x ∈ R n : N w P L w b (x) > α 5 + 2 m=1 vw x ∈ R n : C l≥1 e -c4 l O m,l L w b (x) > α 5 + vw x ∈ R n : CO 3 L w b (x) > α 5 =: I + II + 2 m=1 III m + IV.
In order to estimate I, first note that p < p 1 (see (3.17) and (3.20)). Then, apply Chebyshev's inequality, Propositions 3.12 (b) and 2.28, and properties (2.48)-(2.52) to get

I 1 α p 1 R n N w P L w g p 1 vdw 1 α p 1 R n |∇g| p 1 vdw 1 α p R n |∇f | p vdw. (3.24)
Now we estimate II. To this end, apply Chebyshev's inequality, Propositions 3.12 (b) and 2.28, and Lemma 2.74. Then,

(3.25) II 1 α p 1 R n N w P L w b p 1 vdw 1 α p 1 R n ∇ b p 1 vdw 1 α p R n |∇f | p vdw.
We next estimate IV . With this aim, we write b = i b i so that b = bb, and note that

IV ≤ vw x ∈ R n : CO 3 L w b (x) > α 10 + vw x ∈ R n : CO 3 L w b (x) > α 10 =: IV 1 + IV 2 .
In order to estimate IV 1 apply Chebyshev's inequality, Minkowski's integral inequality, and Proposition 2.44, then

IV 1 1 α p ∞ 1 4 e -cu S 2 √ u,w H L w b L p (vdw) du u p 1 α p S w H L w b p L p (vdw) 1 α p S w 1/2,H L w b p L p (vdw) = 1 α p S w H b p L p (vdw)
IV 2 1 α p 1 ∞ 1 4 e -cu S 2 √ u,w H L w b L p 1 (vdw) du u p 1 1 α p 1 S w H L w b p 1 L p 1 (vdw) 1 α p 1 ∇ b p 1 L p 1 (vdw) 1 α p R n |∇f | p vdw.
Therefore, we conclude that

IV 1 α p R n |∇f | p vdw. (3.26)
Now, it remains to estimate III m , for m = 1, 2. Note that by (2.51),

III m ≤ vw i 16B i +vw x ∈ R n \ ∪ i 16B i : C l≥1 e -c4 l O m,l L w b (x) > α 5 (3.27) 1 α p R n |∇f | p vdw+ l≥1 vw x ∈ R n \ ∪ i 16B i : O m,l L w b (x) > e c4 l α C2 l =: 1 α p R n |∇f | p vdw + l≥1 III m,l .
Applying Chebyshev's inequality, duality, and Hölder's inequality, it follows that

III m,l e -c4 l α p 1 R n \∪ i 16B i O m,l L w b p 1 vdw (3.28) e -c4 l α p 1 sup u L p ′ 1 (vdw) =1 i j≥4 C j (B i ) O m,l L w B r B i b i p 1 vdw 1 p 1 u1 C j (B i ) L p ′ 1 (vdw) p 1 =: e -c4 l α p 1 sup u L p ′ 1 (vdw) =1 i j≥4 I ij m,l u1 C j (B i ) L p ′ 1 (vdw) p 1 .
Then, for m = 1, we have that

I ij 1,l C j (B i ) sup 0<t<2 j-l-3 r B i - B(x,2 l+2 t) e -t 2 Lw L w B r B i b i (y) p 0 dw(y) 1 p 0 p 1 v(x)dw(x) 1 p 1 + C j (B i ) sup t≥2 j-l-3 r B i - B(x,2 l+2 t) e -t 2 Lw L w B r B i b i (y) p 0 dw(y) 1 p 0 p 1 v(x)dw(x) 1 p 1 =: C 1 + C 2 .
In order to estimate C 1 , we use functional calculus as in the proof of Theorem 2.71. Recall (2.93) and take φ(z, t) := tz

1 2 e -t 2 z (1 -e -r 2 B i z ) M . Then φ(z, t) is holomorphic in the open sector Σ µ = {z ∈ C \ {0} : |arg(z)| < µ} and satisfies |φ(z, t)| |z| M (1 + |z|) -2M
(with implicit constant depending on µ, t > 0, r B i , and M) for every z ∈ Σ µ . We can check that for every z

∈ Γ = ∂Σ π 2 -θ , |η(z, t)| tr 2M B i (|z| + t 2 ) M + 3 2 . Now fix x ∈ C j (B i ), j ≥ 4, and 0 < t < 2 j-l-3 r B i , then B(x, 2 l+2 t) ⊂ 2 j+2 B i \ 2 j-1 B i .
This and Minkowski's integral inequality imply

- B(x,2 l+2 t) e -t 2 Lw L w B r B i b i (y) p 0 dw(y) 1 p 0 = - B(x,2 l+2 t) φ(L w , t) b i t (y) p 0 dw(y) 1 p 0 Γ - B(x,2 l+2 t) e -zLw b i t (y) p 0 dw(y) 1 p 0 tr 2M B i (|z| + t 2 ) M + 3 2 |dz| 1 p 1 2 -j(2M +1-θ 1 ) , provided 2M + 1 > θ 2 .
We continue by estimating C 2 . To this end, first change the variable t into t √ M + 1 =: tθ M . Next, for any x ∈ C j (B i ) and t ≥

2 j-1 r B i 2 l+2 θ M , note that B i ⊂ B(x B i , θ M 2 l+2 t) =: B l i ⊂ B(x, θ M 2 l+2 5t) (x B i denotes the center of B i ). Then, C 2 C j (B i ) sup t≥ 2 j-l-3 r B i θ M - B(x,θ M 2 l+2 t) T t,r B i L w e -t 2 Lw 1 B l i b i (y) p 0 dw(y) p 1 p 0 d(vw)(x) 1 p 1 C j (B i ) sup t≥ 2 j-l-3 r B i θ M w(B(x, θ M 2 l+2 t)) -1 R n T t,r B i L w e -t 2 Lw 1 B l i b i (y) p 0 dw(y) p 1 p 0 d(vw)(x) 1 p 1 , where T t,r B i := e -t 2 Lw -e -(t 2 +r 2 B i )Lw M .
In the above setting, (2.39), Proposition 2.28, the fact that √ τ ∇e -τ Lw ∈ O(L p 0 (w)-L p 0 (w)), (2.3), and Lemma 2.16 (a) (see (3.22)), imply

R n T t,r B i L w e -t 2 Lw 1 B l i b i p 0 dw 1 p 0 (3.29) r 2 B i t 2 M R n ∇e -t 2 Lw 1 B l i b i p 0 dw 1 p 0
where in the first inequality, we have used that w

(B(x, θ M 2 l+2 t)) -1 w(B l i ) ≤ C, since B l i ⊂ B(x, θ M 2 l+2 5t).
Collecting the estimates obtained for C 1 and C 2 , we conclude that, for M ∈ N such that 2M + 1 > θ 2 ,

I ij 1,l α vw(2 j+1 B i ) 1 p 1 2 -j(2M -θ 1 ) 2 l(2M +θ) . (3.30)
Next, let us estimate term I ij 2,l . Splitting the supremum in t, we have

I ij 2,l C j (B i ) sup 0<t<2 j-l-2 r B i - B(x,2 l+1 t) g w H L w B r B i b i (y) p 0 dw(y) p 1 p 0 v(x)dw(x) 1 p 1 + C j (B i ) sup t≥2 j-l-2 r B i - B(x,2 l+1 t) g w H,t L w B r B i b i (y) p 0 dw(y) p 1 p 0 v(x)dw(x) 1 p 1 =: D ij 1 + D ij 2 . Regarding D ij 1 , we claim that D ij 1 αvw(2 j+1 B i ) 1 p 1 2 -j(2M +1-θ 1 ) . (3.31)
To this end, first note that for 0 < t < 2 j-l-2 r B i and

x ∈ C j (B i ), then B(x, 2 l+1 t) ⊂ 2 j+2 B i \ 2 j-1 B i . Next recall that M w p 0 is L p 1 (vdw) bounded since v ∈ A p 1 p 0 (w) (see (3.22)). Hence D ij 1 C j (B i ) M w p 0 1 2 j+2 B i \2 j-1 B i g w H L w B r B i b i p 1 vdw 1 p 1 vw(2 j+1 B i ) 1 p 1 - 2 j+2 B i \2 j-1 B i g w H L w B r B i b i p 1 d(vw) 1 p 1 .
For any t ≥ 2 j-l-2 r B i and f ∈ L 2 (w), we have that

g w H,t f (x) = ∞ t 2 |r 2 L w e -r 2 Lw f (x)| 2 dr r 1 2 ≤ ∞ 2 j-l-3 r B i |r 2 L w e -r 2 Lw f (x)| 2 dr r 1 2
.

Moreover, recall that p 0 < q 0 (see (3.22)), this implies the boundedness of the maximal operator M w p 0 on L q 0 (w). This, together with Lemma 2.16 (b) and Minkowski's integral inequality, allows us to obtain

D ij 2 vw(2 j+1 B i ) 1 p 1 - C j (B i ) M w p 0 g w H,2 j-l-2 r B i L w B r B i b i q 0 dw 1 q 0 (3.34) vw(2 j+1 B i ) 1 p 1 w(2 j+1 B i ) -1 q 0   R n ∞ 2 j-l-3 r B i r 2 L w e -r 2 Lw L w B r B i b i 2 dr r q 0 2 dw   1 q 0 vw(2 j+1 B i ) 1 p 1 w(2 j+1 B i ) -1 q 0 ∞ 2 j-l-3 r B i θ M R n T r,r B i L w r 2 L w e -r 2 Lw 1 B l i b i q 0 dw 2 q 0 dr r 1 2
, where in the last inequality we have changed the variable r into rθ )Lw ) M .

M := r √ M + 1, used that B i ⊂ B(x B i , θ M 2 l+1
Proceeding as in the estimate of (3.29), but using now the fact that √ τ ∇τL w e -τ Lw ∈ O(L p 0 (w) -L q 0 (w)) instead of √ τ ∇e -τ Lw ∈ O(L p 0 (w) -L p 0 (w)), we get R n T r,r B i L w r 2 L w e -r 2 Lw 1 B l i b i q 0 dw 1 q 0 2 l θ αw(B l i )

1 q 0 r 2 B i r 2 M 2 l θ+ 2n
q 0 αw(2 j+1 B i )

1 q 0 2 -2jn q 0 r 2 B i r 2 M -n q 0
, where in the last inequality we have used that for r > q 0 α2 -2jn q 0 vw(2 j+1 B i )

1 p 1 ∞ 2 j-l-3 r B i θ M r 2 B i r 2
2M -2n q 0 dr r 1 2

2 l(2M + θ) αvw(2 j+1 B i )

1 p 1 2 -2jM ,
provided 2M > 2n q 0 . Gather (3.31) and (3.33), then for M ∈ N such that 2M > max{ θ 2 , 2n/q 0 }, I ij 2,l 2 l(2M + θ) αvw(2 j+1 B i ) 1 p 1 2 -j(2M -θ 1 ) . This and (3.30) yield, for 2M > max{ θ 2 , 2n/q 0 , θ 2 -1},

I ij m,l ≤ C 1 αvw(2 j+1 B i ) 1 p 1 2 -j(2M -C 2 ) , m = 1,

The regularity problem in unweighted Lebesgue spaces

Our main result, Theorem 1.8, establishes the solvability of the regularity problem in L p (v dw) of the block operator L w . Recall that w ∈ A 2 (dx) is fixed and controls the degeneracy of the operator and that v ∈ A ∞ (w). This means that we can establish the solvability of the regularity problem in unweighted Lebesgue spaces by taking v = w -1 . In this section our goal is to explore this idea and study ranges for which we can solve the regularity problem in terms of the weight w. A particular case of interest, where we can be more explicit, is that of power weights.

To start fix w ∈ A 2 (dx) and recall the definitions of r w and s w in (2.4). As just mentioned, we let v = w -1 and observe that from the definitions it is clear that for every 1 ≤ r < ∞ one has w -1 ∈ A r (w) if and only if w ∈ RH r ′ (dx), and w -1 ∈ RH r ′ (w) if and only if w ∈ A r (dx). Hence, according to (2.13) we have r w -1 (w) = s w and s w -1 (w) = r w . Then looking at Theorem 1.8 and using (2.14), we see that (1.9) is equivalent to (4.1) max{r w , q -(L w )} s w < q + (L w ) r w , and if that holds we have (R Lw ) L p (dx) solvability for p so that (4.2) max r w , n r w q -(L w ) n r w + q -(L w ) s w < p < q + (L w ) r w .

It is important to note that q -(L w ) and q + (L w ) are defined in an abstract way and depend intrinsically on w. From [14, Propositions 3.1 and 7.1] and recalling that n ≥ 2, we know that q -(L w ) = p -(L w ) ≤ 2 n rw n rw+2 , hence we have an estimate for q -(L w ) in terms of n and r w . On the other hand q + (L w ) > 2 and can be arbitrarily close to 2 (even in the case w ≡ 1), and we do not have an explicit bound in terms of w (see [START_REF] Cruz-Uribe | On the Kato problem and extensions for degenerate elliptic operators[END_REF]Proof of Theorem 11.8] in this regard). Taking this into account and in order to check that (4.1) holds we will replace its right-hand side with 2 rw . Our first result for general weights is as follows: (c) If w ∈ A r (dx) ∩ RH s(r) ′ (dx) with 1 < r < r 0 and s(r) = min{ 2 r 2 , n r+2 n r 2 }, and max r w , n r w q -(L w ) n r w + q -(L w ) s w < p < q + (L w ) r w , in particular, in the range max r w , 2 n r w n r w + 4 s w < p ≤ 2 r w .

(d) Given Θ ≥ 1 there exists ǫ 0 = ǫ 0 (Θ, n, Λ/λ) ∈ (0, 1 2n ], such that for every w ∈ A 1+ǫ (dx) ∩ RH max{ 2 1-ǫ ,1+(1+ǫ) n 2 } (dx) with 0 ≤ ǫ < ǫ 0 and [w] A 2 (dx) ≤ Θ, then (4.4) holds with p = 2, or equivalently (R Lw ) L 2 (dx) is solvable.

Proof. We first consider (a). Let w ∈ A 1 (dx) ∩ RH 1+ n 2 (dx) then r w = 1 and s w < (1 + n 2 ) ′ = 1 + 2 n . Using that q -(L w ) ≤ 2 n n+2 (since n ≥ 2) we have max{r w , q -(L w )} s w < max 1, 2 n n + 2 1 + 2 n = 2 < q + (L w ) = q + (L w ) r w .

That is, (4.1) holds and according to (4.2) we have (R Lw ) L p (dx) -solvability for p so that max 1, n q -(L w ) n + q -(L w ) s w < p < q + (L w ) and, in particular, in the range and s(r) = min{ 2 r 2 , n r+2 n r 2 } and note that the restriction on r gives s(r) ∈ [1, ∞). In particular, r w < r, s w ≤ s(r), and r w max{r w , q -(L w )} s w ≤ r w max r w , 2 n r w n r w + 2

s w provided n n + 2 < β < min √ 2, 1+ √ 1+ 8 n 2 .
Moreover, there exists ǫ 1 = ǫ 1 (n, Λ/λ) ∈ (0, 1 2 n ) such that if n n + 2 < β < 1 + ǫ 1 then (4.7) holds with p = 2, or equivalently (R Lw β ) L 2 (dx) is solvable.

Proof. Write w β (x) = |x| n (β-1) with 0 < β < 2 so that w β ∈ A 2 (dx). It is not difficult to see that r w β = max{1, β} and s w β = max{1, β -1 }.

Consider first the case 0 < β ≤ 1 so that w β ∈ A 1 (dx), r w β = 1, and

s w β = β -1 . If β ≥ n n+2 then max{r w β , q -(L w β )} s w ≤ 2 n n + 2 1 β ≤ 2 < q + (L w β ) = q + (L w β ) r w β .
Thus (4.1) holds and if n n+2 ≤ β ≤ 1 we have (R L wβ ) L p (dx) -solvability for p such that max 1, n q -(L w β ) n + q -(L w β ) 1 β < p < q + (L w β ).

In particular, if n n+2 < β ≤ 1, the solvability holds in the range max{1, 2 n n+4 } β -1 < p ≤ 2.

Let us treat the case 1 < β < 2, so that we have r w β = β and s w β = 1. If Hence what we have proved so far gives the (R Lw β ) L 2 (dx) -solvability. To consider the case β > 1 we first assume that β < 2 n+1 2 n so that w β ∈ A 1+ 1 2 n (dx). Note that one can easily see that there exists Θ ≥ 1 depending just on n (and independent of β) such

(1. 1 ) λ |ξ| 2 ≤

 12 Re A(x) ξ • ξ and |A(x) ξ • ζ| ≤ Λ |ξ| |ζ|, for all ξ, ζ ∈ C n and almost every x ∈ R n . We have used the notation ξ • ζ = ξ 1 ζ 1 + • • • + ξ n ζ n and therefore ξ • ζ is the usual inner product in C n . Associated with this matrix and a given weight w ∈ A 2 (dx) (which is fixed from now on, unless stated otherwise) we define the second order divergence form degenerate elliptic operator

  12, Theorem A.1, (b)] (or [12, Theorem A.1, (c)], if p + (L w ) * w = ∞) allows us to obtain (a) from

=

  : I + II.

  (a) N w H is bounded on L p (vdw) for all p ∈ W w v (p -(L w ), ∞). (b) N w P is bounded on L p (vdw) for all p ∈ W w v (p -(L w ), p + (L w )).

  r) =: B l i , for r > and j ≥ 4 (x B i denotes the center of B i ), and we recall that T r,r B i := (e -r 2 Lwe -(r 2 +r 2 B i

2

  j-l-3 r B i θ M and j ≥ 4, 2 j+1 B i ⊂ 2 3 B l i ,and (2.2). Plugging this into (3.34) leads to

Corollary 4 . 3 . 2 2 and

 4322 Let w ∈ A 2 (dx) and let L w be a block degenerate elliptic operator in R n+1 + as in(1.4). Associated with L w consider the regularity problem (RLw ) L p (dx) as in Section 1. Given f ∈ C ∞ c (R n ) if one sets u(x, t) = e -t √ L w f (x), (x, t) ∈ R n+1 + , then (4.4) N w (∇ x,t u) L p (dx) ≤ C ∇f L p (dx)in any of the following scenarios:(a) If w ∈ A 1 (dx) ∩ RH 1+ n 2 (dx) and max 1, n q -(L w ) n + q -(L w ) s w < p < q + (L w ), If w ∈ A r 0 (dx) ∩ RH ∞ (dx) with r 0 := min √ max r w , n r w q -(L w ) n r w + q -(L w ) < p < q + (L w ) r w ,in particular, in the range max r w , 2 n r w n r w + 4 < p ≤ 2 r w .

s w < p ≤ 2 .

 2 To prove (b) and (c) assume that w ∈ A r (dx) ∩ RH s(r) ′ (dx) with 1 < r ≤

  l 2 jθ 2 -

	F (y, s)	2 dw(y).
	B(x,2 l+j+2 s)
	Consequently, applying Fubini's theorem, (2.78) with k = 2, changing the variable s into √ 2s, and by (2.3) and change of angles (Proposition 2.44), we get, for C

  +1-θ 1 ) . (2.94) In order to estimate III 2 ij , we first change the variable t into tθ M

  2, with C 2 := max{θ 1 , θ 1 } and C 1 := C2 lC M . Then, in view of (3.28), applying Lemma 2.62 with I ij = I ij m,l and {B i } i the collection of balls given by Lemma 2.47, and (2.51), for 2M> max C 2 +nr w r v (w), θ 2 , 2n q 0 , θ 2 -1 , we get III m,l e -c4 l vw Collecting this estimate and (3.23)-(3.26), the proof is complete.

			i	B i		e -c4 l 1 α p	R n	|∇f | p vdw, m = 1, 2.
	Therefore, by (3.27)
	III m	l≥1	e -c4 l 1 α p	R n	|∇f | p vdw		1 α p	R n	|∇f | p vdw.

  max{r w β , q -(L w β )} ≤ max β 2 , 2 n β 2 n β + 2 ≤ 2 < q + (L w β ).This implies that (4.1) holds and, as a consequence, (4.2) yields that if 1 < β ≤ L wβ ) L p (dx) is solvable in the range Let us finally focus on the (R Lw β ) L 2 -solvability. Consider first the case when n

	1 < β ≤ min{	√	2,	1+	√ 2 1+ 8 n	} then
	r w β min √ 2, 1+ √ 1+ 8 n 2 In particular, if 1 < β < min then (R max β, √ n β + q -(L w ) n β q -(L w ) 2, 1+ √ 1+ 8 n 2 one can solve (R L wβ ) L p (dx) for p satisfying < p < q + (L w ) . β
										max β,	2 n β n β + 4	< p ≤	2 β	.
	β ≤ 1 then max 1, β,	2 n n + 4	,	2 n β n β + 4	max{1, β -1 } = max 1,	2 n n + 4	1 β	< 2 =	n+2 < max{1, β} 2 .

p 0

j-l-3 r B i θ M

The secondauthors acknowledge that 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 third author acknowledges financial support from the Spanish Ministry of Science and Innovation, through the "Severo Ochoa Programme for Centres of Excellence in R&D" (CEX2019-000904-S). The third and last authors were partially supported by the Spanish Ministry of Science and Innovation, MTM PID2019-107914GB-I00. 1 2 PASCAL AUSCHER, LI CHEN, JOS É MAR ÍA MARTELL, AND CRUZ PRISUELOS-ARRIBAS 2.

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In view of (3.22), we apply Lemma 2.16 (b) and Minkowski's integral inequality to get

.

(3.32)

In order to estimate the integral in x, we use functional calculus as in the estimate of C 1 . Apply the fact that zL w e -zLw ∈ O(L p 0 (w) -L q 0 (w)), Lemma 2.16 (a), and (2.53). Thus, -

Plugging this into (3.32) and changing the variable r into 2 j r B i r, we obtain, for

. Now turning to the estimate of D ij 2 , we claim

This implies (4.1) and we have (R Lw ) L p (dx) -solvability for p in the range given by (4.2), and in particular for those p's satisfying max r w , 2 n r w n r w + 4 s w < p ≤ 2 r w .

All these show (b) by taking r = r 0 so that s(r) = 1 and hence s w = 1. Also (c) follows from the case 1 < r < r 0 .

To deal with (d) we proceed as in [14, pp. 654-655]. There, it is shown that given Θ ≥ 1 there exists

rw . On the other hand, if we additionally assume that w ∈ RH max{ 2 1-ǫ ,1+(1+ǫ) n 2 } (dx) then

Altogether we have obtained that max r w , 2 n rw n rw+2 s w < 2 < q + (Lw) rw . This implies (4.1) and also that p = 2 satisfies (4.2). Consequently, (R Lw ) L 2 (dx) is solvable as desired.

Concerning power weights we have the following result: Corollary 4.5. Consider the power weight w β (x) = |x| n (β-1) with 0 < β < 2, and let L w β be the associated block operator (4.6) L w β u(x, t) = -|x| -n (β-1) div x |x| n (β-1) A(x)∇ x u(x, t) -∂ 2 t u(x, t) where A is an n × n matrix of complex L ∞ -valued coefficients defined on R n , n ≥ 2 satisfying the uniform ellipticity condition (1.1).

Assume that

for every p satisfying max 1, β, n q -(L w ) n + q -(L w ) , n β q -(L w ) n β + q -(L w ) max{1, β -1 } < p < q + (L w ) max{1, β} .

In particular, in the non-empty range max 1, β, 2 n n + 4 , 2 n β n β + 4 max{1, β -1 } < p ≤ 2 max{1, β}

Conflict of interest

On behalf of all authors, the corresponding author states that there is no conflict of interest.