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We generalize to the setting of 1-sided chord-arc domains, that is, to domains satisfying the interior Corkscrew and Harnack Chain conditions (these are respectively scale-invariant/quantitative versions of the openness and path-connectedness) and which have an Ahlfors regular boundary, a result of Kenig-Kirchheim-Pipher-Toro, in which Carleson measure estimates for bounded solutions of the equation $ Lu=-\operatorname {div}(A\nabla u) = 0$ with $ A$ being a real (not necessarily symmetric) uniformly elliptic matrix imply that the corresponding elliptic measure belongs to the Muckenhoupt $ A_\infty $ class with respect to surface measure on the boundary. We present two applications of this result. In the first one we extend a perturbation result recently proved by Cavero-Hofmann-Martell presenting a simpler proof and allowing non-symmetric coefficients. Second, we prove that if an operator $ L$ as above has locally Lipschitz coefficients satisfying certain Carleson measure condition, then $ \omega _L\in A_\infty $ if and only if $ \omega _{L^\top }\in A_\infty $. As a consequence, we can remove one of the main assumptions in the non-symmetric case of a result of Hofmann-Martell-Toro and show that if the coefficients satisfy a slightly stronger Carleson measure condition the membership of the elliptic measure associated with $ L$ to the class $ A_\infty $ yields that the domain is indeed a chord-arc domain., The first author was partially supported by “la Caixa”-Severo Ochoa international PhD Programme. The first and third authors acknowledge financial support from the Spanish Ministry of Economy and Competitiveness, through the “Severo Ochoa” Programme for Centres of Excellence in R&D” (SEV-2015-0554). They also acknowledge that the research leading to these results received funding from the European Research Council under the European Union’s Seventh Framework Programme (FP7/2007-2013)/ ERC agreement no. 615112 HAPDEGMT The second author was supported by NSF grant DMS-1664047. The fourth author was partially supported by the Craig McKibben & Sarah Merner Professor in Mathematics, by NSF grant number DMS-1664867, and by the Simons Foundation Fellowship 614610.

We generalize the extrapolation theory of Rubio de Francia to the context of Banach function spaces and modular spaces. Our results are formulated in terms of some natural weighted estimates for the Hardy-Littlewood maximal function and are stated in measure spaces and for general Muckenhoupt bases. Finally, we give several applications in analysis and partial differential equations., The first author is supported by grant FJC2018-038526-I funded by MCIN/AEI/ 10.13039/501100011033. The first and third authors acknowledge financial support from MCIN/AEI/ 10.13039/501100011033 grants CEX2019-000904-S and PID2019-107914GB-I00. The second and third authors also 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.

Let Omega subset of Rn+1, n >= 2, be a 1-sided nontangentially accessible domain, that is, a set which is quantitatively open and path-connected. Assume also that Omega satisfies the capacity density condition. Let L(0)u = - div(A(0)del u), Lu = - div(A del u) be two real (not necessarily symmetric) uniformly elliptic operators in Omega, and write at omega(L0),omega(L) for the respective associated elliptic measures. We establish the equivalence between the following properties: (i) omega(L) is an element of A(infinity)(omega(L0)), (ii) L is L-P (omega(L0))-solvable for some p is an element of (1, infinity), (iii) bounded null solutions of L satisfy Carleson measure estimates with respect to omega(L0), (iv) S < Ar (i.e., the conical square function is controlled by the nontangential maximal function) in L-q (omega(L0)) for some (or for all) q is an element of (0, infinity) for any null solution of L, and (v) L is BMO(omega(L0))-solvable. Moreover, in each of the properties (ii)-(v) it is enough to consider the class of solutions given by characteristic functions of Borel sets (i.e, u (X) = omega(X)(L)(S) (S) for an arbitrary Borel set S subset of partial derivative Omega)., The authors would like to thank the anonymous referees for their comments to improve the presentation of the paper. All the authors acknowledge financial support from MCIN/AEI/ 10.13039/501100011033 grants FJC2018-038526-I (first authors), CEX2019-000904-S (first, third and last authors), PID2019-107914GB-I00 (first and third authors), MTM2017-84058- P (second author) and PID2020-116398GB-I00 (last author). The third author also acknowledges 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.

We prove well-posedness results for the Dirichlet problem in Rn + for homogeneous, second order, constant complex coe cient elliptic systems with boundary data in generalized H older spaces C !(Rn1;CM) and in generalized Morrey-Campanato spaces E !;p(Rn1;CM) under certain assumptions on the growth function !. We also identify a class of growth functions ! for which C !(Rn1;CM) = E !;p(Rn1;CM) and for which the aforementioned well-posedness results are equivalent, in the sense that they have the same unique solution, satisfying natural regularity properties and estimates., Spanish Ministry of Economy and Competitiveness, through the Severo Ochoa Programme for Centres of Excellence in R&D" (SEV-2015-0554), European Research Council under the European Union's Seventh Framework Programme (FP7/2007-2013)/ ERC agreement no. 615112 HAPDEGMT, Simons Foundation grant #281566

We present a unified method to obtain unweighted and weighted estimates of linear and multilinear commutators with BMO functions, that is amenable to a plethora of operators and functional settings. Our approach elaborates on a commonly used Cauchy integral trick, recovering many known results but yielding also numerous new ones. In particular, we solve a problem about the boundedness of the commutators of the bilinear Hilbert transform with functions in BMO., Simons Foundation, grants No. 246024 and No.160427, Spanish Ministry of Economy and Competitiveness, through the “Severo Ochoa” Programme for Centres of Excellence in R&D (SEV-2015-0554), European Research Council under the European Union’s Seventh Framework Programme (FP7/2007-2013)/ ERC agreement no. 615112 HAPDEGMT, NSF grant DMS 1069015

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 $w\in A_2(dx)$ in $\mathbb{R}^n$, and consider the associated block second order degenerate elliptic problem in the upper-half space $\mathbb{R}_+^{n+1}$. 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 $(\mathbb{R}^n, wdx)$. We recover earlier results in the non-degenerate case (when $w\equiv 1$, 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 \[ \mathbb{L}_\alpha u(x,t) = -|x|^{\alpha} \mathrm{div}_x \big(|x|^{-\alpha }\,A(x) \nabla_x u(x,t)\big)-\partial_{t}^2 u(x,t) \] for any complex-valued uniformly elliptic matrix $A$ and for all $-\epsilon

Let $\Omega\subset\mathbb{R}^{n+1}$, $n\ge 2$, be a 1-sided non-tangentially accessible domain (aka uniform domain), that is, $\Omega$ satisfies the interior Corkscrew and Harnack chain conditions, which are respectively scale-invariant/quantitative versions of openness and path-connectedness. Let us assume also that $\Omega$ satisfies the so-called capacity density condition, a quantitative version of the fact that all boundary points are Wiener regular. Consider $L_0 u=-\mathrm{div}(A_0\nabla u)$, $Lu=-\mathrm{div}(A\nabla u)$, two real (non-necessarily symmetric) uniformly elliptic operators in $\Omega$, and write $\omega_{L_0}$, $\omega_L$ for the respective associated elliptic measures. The goal of this program is to find sufficient conditions guaranteeing that $\omega_L$ satisfies an $A_\infty$-condition or a $RH_q$-condition with respect to $\omega_{L_0}$. In this paper we are interested in obtaining square function and non-tangential estimates for solutions of operators as before. We establish that bounded weak null-solutions satisfy Carleson measure estimates, with respect to the associated elliptic measure. We also show that for every weak null-solution, the associated square function can be controlled by the non-tangential maximal function in any Lebesgue space with respect to the associated elliptic measure. These results extend previous work of Dahlberg-Jerison-Kenig and are fundamental for the proof of the perturbation results in arXiv:1901.08261., Comment: This paper is part of the earlier submission arXiv:1901.08261(2)

We consider the Riesz transforms ∇L−1/2, where L≡− divA(x)∇, and A is an accretive, n × n matrix with bounded measurable complex entries, defined on Rn. We establish boundedness of these operators on Lp(Rn), for the range pn < p ≤ 2, where pn = 2n/(n + 2), n ≥ 2, and we obtain a weak-type estimate at the endpoint pn. The case p = 2 was already known: it is equivalent to the solution of the square root problem of T. Kato.

In this paper we solve a long standing problem about the multivariable Rubio de Francia extrapolation theorem for the multilinear Muckenhoupt classes A, which were extensively studied by Lerner et al. and which are the natural ones for the class of multilinear Calderón-Zygmund operators as well as for some bilinear rough singular integral operators. Furthermore, we go beyond the classes A and extrapolate within the classes A which appear naturally associated to the weighted norm inequalities for multilinear sparse forms which control fundamental operators such as the bilinear Hilbert transform. We give several applications which can be easily obtained using extrapolation. First, weighted norm inequalities (scalar and vector-valued) for the bilinear rough singular integral operators are established. Second, for the bilinear Hilbert transform one can extrapolate from the recent result of Culiuc et al. who considered the Banach range, extend the estimates to the quasi-Banach range, and furthermore, prove trivially vector-valued inequalities. We also extend recent results of Carando et al. on Marcinkiewicz-Zygmund estimates for multilinear Calderón-Zygmund operators. Finally, our last application gives new weighted estimates (scalar and vector-valued) for the commutators of multilinear Calderón-Zygmund operators, bilinear rough singular integral operators, and for the bilinear Hilbert transform with BMO functions using ideas from Bényi et al., The first author is supported by Juan de la Cierva - Formación 2015 FJCI-2015-24547, by the Basque Government through the BERC 2018-2021 program and by Spanish Ministry of Economy and Competitiveness MINECO through BCAM Severo Ochoa excellence accreditation SEV-2013-0323 and through project MTM2017-82160-C2-1-P funded by (AEI/FEDER, UE) and acronym “HAQMEC”. The second author acknowledges financial support from the Spanish Ministry of Economy and Competitiveness, through the “Severo Ochoa” Programme for Centres of Excellence in R&D” (SEV-2015-0554). He also acknowledges that the research leading to these results has received funding from the European Research Council under the European Union Seventh Framework Programme (FP7/2007-2013)/ERC agreement no. 615112 HAPDEGMT. The third author is supported by CONICET PIP 11220130100329CO, Argentina, by the Basque Government through the BERC 2018-2021 program and by Spanish Ministry of Economy and Competitiveness MINECO through BCAM Severo Ochoa excellence accreditation SEV-2013-0323

Let $\Omega\subset\re^{n+1}$, $n\ge 2$, be a 1-sided chord-arc domain, that is, a domain which satisfies interior Corkscrew and Harnack Chain conditions (these are respectively scale-invariant/quantitative versions of the openness and path-connectedness), and whose boundary $\partial\Omega$ is $n$-dimensional Ahlfors regular. Consider $L_0$ and $L$ two real symmetric divergence form elliptic operators and let $\omega_{L_0}$, $\omega_L$ be the associated elliptic measures. We show that if $\omega_{L_0}\in A_\infty(\sigma)$, where $\sigma=H^n\rest{\partial\Omega}$, and $L$ is a perturbation of $L_0$ (in the sense that the discrepancy between $L_0$ and $L$ satisfies certain Carleson measure condition), then $\omega_L\in A_\infty(\sigma)$. Moreover, if $L$ is a sufficiently small perturbation of $L_0$, then one can preserve the reverse Hölder classes, that is, if for some $1, The first and third authors acknowledge financial support from the Spanish Ministry of Economy and Competitiveness, through the “Severo Ochoa” Programme for Centres of Excellence in R\&D” (SEV-2015-0554). They also 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 second author was supported by NSF grant DMS-1664047.

Given a Muckenhoupt weight $w$ and a second order divergence form elliptic operator $L$, we consider different versions of the weighted Hardy space $H^1_L(w)$ defined by conical square functions and non-tangential maximal functions associated with the heat and Poisson semigroups generated by $L$. We show that all of them are isomorphic and also that $H^1_L(w)$ admits a molecular characterization. One of the advantages of our methods is that our assumptions extend naturally the unweighted theory developed by S. Hofmann and S. Mayboroda and we can immediately recover the unweighted case. Some of our tools consist in establishing weighted norm inequalities for the non-tangential maximal functions, as well as comparing them with some conical square functions in weighted Lebesgue spaces., Comment: The introduction has been updated to include some relevant references describing the development of the topic

It is well-known that quantitative, scale invariant absolute continuity(more precisely, the weak-A∞property) of harmonic measure with respect to sur-face measure, on the boundary of an open setΩ⊂Rn+1with Ahlfors-David regularboundary, is equivalent to the solvability of the Dirichlet problem inΩ, with data inLp(∂Ω) for somep, S.H. was supported by NSF grant DMS-1664047. J.M.M. acknowledges financial support fromthe Spanish Ministry of Economy and Competitiveness, through the “Severo Ochoa” Programme forCentres of Excellence in R&D (SEV-2015- 0554). He also acknowledges that the research leading tothese results has received funding from the European Research Council under the European Union’sSeventh Framework Programme (FP7/2007-2013)/ERC agreement no. 615112 HAPDEGMT. In ad-dition, S.H. and J.M.M. were supported by NSF Grant DMS-1440140 while in residence at the MSRIin Berkeley, California, during Spring semester 2017. M.M. was supported by IKERBASQUE andpartially supported by the grant MTM-2017-82160-C2-2-P of the Ministerio de Econom ́ıa y Com-petitividad (Spain), and by IT-641-13 (Basque Government). X.T. was supported by the ERC grant320501 of the European Research Council (FP7/2007-2013) and partially supported by MTM-2016-77635-P, MDM-2014-044 (MICINN, Spain), 2017-SGR-395 (Catalonia), and by Marie Curie ITNMAnET (FP7-607647)

The present paper establishes the correspondence between the properties of the solutions of a class of PDEs and the geometry of sets in Euclidean space. We settle the question of whether (quantitative) absolute continuity of the elliptic measure with respect to the surface measure and uniform rectifiability of the boundary are equivalent, in an optimal class of divergence form elliptic operators satisfying a suitable Carleson measure condition in uniform domains with Ahlfors regular boundaries. The result can be viewed as a quantitative analogue of the Wiener criterion adapted to the singular Lp data case. The first step is taken in Part I, where we considered the case in which the desired Carleson measure condition on the coefficients holds with sufficiently small constant, using a novel application of techniques developed in geometric measure theory. In Part II we establish the final result, that is, the ¿large constant case¿. The key elements are a powerful extrapolation argument, which provides a general pathway to self-improve scale-invariant small constant estimates, and a new mechanism to transfer quantitative absolute continuity of elliptic measure between a domain and its subdomains., The first author was partially supported by NSF Grants DMS-1664047 and DMS-2000048. The second 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) and Grant MTM PID2019-107914GB-I00. The second author also acknowledges 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 was partially supported by the NSF INSPIRE Award DMS 1344235, the NSF RAISE-TAQ Grant DMS 1839077, and the Simons Foundation Grant 563916, SM. The fourth author was partially supported by the Craig McKibben & Sarah Merner Professor in Mathematics, by NSF Grant Numbers DMS-1664867 and and DMS-1954545, and by the Simons Foundation Fellowship 614610. The fifth author was partially supported by NSF Grants DMS-1361823, DMS-1500098, DMS-1664867, DMS-1902756 and by the Institute for Advanced Study.

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

We prove a structure theorem for any n-rectifiable set E⊂R, n≥1, satisfying a weak version of the lower ADR condition, and having locally finite H (n-dimensional Hausdorff) measure. Namely, that H-almost all of E can be covered by a countable union of boundaries of bounded Lipschitz domains contained in R \E. As a consequence, for harmonic measure in the complement of such a set E, we establish a non-degeneracy condition which amounts to saying that H| is “absolutely continuous” with respect to harmonic measure in the sense that any Borel subset of E with strictly positive H measure has strictly positive harmonic measure in some connected component of R \E. We also provide some counterexamples showing that our result for harmonic measure is optimal. Moreover, we show that if, in addition, a set E as above is the boundary of a connected domain Ω⊂R which satisfies an infinitesimal interior thickness condition, then H| is absolutely continuous (in the usual sense) with respect to harmonic measure for Ω. Local versions of these results are also proved: if just some piece of the boundary is n-rectifiable then we get the corresponding absolute continuity on that piece. As a consequence of this and recent results in [AHMTV], we can decompose the boundary of any open connected set satisfying the previous conditions in two disjoint pieces: one that is n-rectifiable where Hausdorff measure is absolutely continuous with respect to harmonic measure and another purely n-unrectifiable piece having vanishing harmonic measure., The first and last authors acknowledge financial support from the Spanish Ministry of Economy and Competitiveness, through the “Severo Ochoa” Programme for Centres of Excellence in R&D (SEV-2015-0554). They also 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 second and third authors were supported by NSF grant DMS-1361701.

Our main result is an abstract good-λ inequality that allows us to consider three self-improving properties related to oscillation estimates in a very general context. The novelty of our approach is that there is one principle behind these self-improving phenomena. First, we obtain higher integrability properties for functions belonging to the so-called John-Nirenberg spaces. Second, and as a consequence of the previous fact, we present very easy proofs of some of the self-improving properties of the generalized Poincaré inequalities studied by B. Franchi, C. Pérez and R.L. Wheeden in [9], and by P. MacManus and C. Pérez in [21]. Finally, we show that a weak Gurov-Reshetnyak condition implies higher integrability with asymptotically sharp estimates. We discuss these questions both in Euclidean spaces with dyadic cubes and in spaces of homogeneous type with metric balls. We develop new techniques that apply to more general oscillations than the standard mean oscillation and to overlapping balls instead of dyadic cubes., The rst author is grateful for the hospitality of ICMAT in Madrid, Spain, where parts of this research was conducted. The visit was supported by the Finnish Academy of Science and Letters, Vilho, Yrj o and Kalle V ais al a foundation. The second author was supported by the Academy of Finland. The third author was supported by ICMAT Severo Ochoa project SEV-2011-0087. He also acknowledges 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.

In this paper we present the results announced in the recent work by the first, second, and fourth authors of the current paper concerning Rubio de Francia extrapolation for the so-called multilinear Muckenhoupt classes. Here we consider the situations where some of the exponents of the Lebesgue spaces appearing in the hypotheses and/or in the conclusion can be possibly infinity. The scheme we follow is similar, but, in doing so, we need to develop a one-variable end-point off-diagonal extrapolation result. This complements the corresponding ``finite'' case obtained by Duoandikoetxea, which was one of the main tools in the aforementioned paper. The second goal of this paper is to present some applications. For example, we obtain the full range of mixed-norm estimates for tensor products of bilinear Calder\'on-Zygmund operators with a proof based on extrapolation and on some estimates with weights in some mixed-norm classes. The same occurs with the multilinear Calder\'on-Zygmund operators, the bilinear Hilbert transform, and the corresponding commutators with BMO functions. Extrapolation along with the already established weighted norm inequalities easily give scalar and vector-valued inequalities with multilinear weights and these include the end-point cases., Comment: v2: final version, incorporated referee comments, to appear in Trans. Amer. Math. Soc. 42 pages

Let L be a second-order, homogeneous, constant (complex) coefficient elliptic system in \({\mathbb {R}}^n\). The goal of this article is to prove a Fatou-type result, regarding the a.e. existence of the nontangential boundary limits of any null-solution u of L in the upper half-space, whose nontangential maximal function satisfies an integrability condition with respect to the weighted Lebesgue measure (1 + |x′|n−1)−1dx′ in \({\mathbb {R}}^{n-1}\equiv \partial {\mathbb {R}}^n_{+}\). This is the best result of its kind in the literature. In addition, we establish a naturally accompanying integral representation formula involving the Agmon-Douglis-Nirenberg Poisson kernel for the system L. Finally, we use this machinery to derive well-posedness results for the Dirichlet boundary value problem for L in \({\mathbb {R}}^n_{+}\) formulated in a manner which allows for the simultaneous treatment of a variety of function spaces.

We prove that for any homogeneous, second-order, constant complex coefficient elliptic system L in ℝ, the Dirichlet problem in ℝ with boundary data in BMO.ℝ/ is well-posed in the class of functions u for which the Littlewood-Paley measure associated with u, namely dμ(x', t):=| ∇u(x',t)| t dx' dt; is a Carleson measure in ℝ. In the process we establish a Fatou-type theorem guaranteeing the existence of the pointwise nontangential boundary trace for smooth null-solutions u of such systems satisfying the said Carleson measure condition. In concert, these results imply that the space BMO(ℝ) can be characterized as the collection of nontangential pointwise traces of smooth null-solutions u to the elliptic system L with the property that μ is a Carleson measure in ℝ. We also establish a regularity result for the BMO-Dirichlet problem in the upper half-space, to the effect that the nontangential pointwise trace on the boundary of ℝ of any given smooth nullsolutions u of L in ℝ satisfying the above Carleson measure condition actually belongs to Sarason's space VMO.ℝ/ if and only if μ(T(Q))/|Q|→0 as |Q|→0, uniformly with respect to the location of the cubeQ⊂ℝ (where T(Q) is the Carleson box associated withQ, and |Q| denotes the Euclidean volume of Q). Moreover, we are able to establish the well-posedness of the Dirichlet problem in ℝ C for a system L as above in the case when the boundary data are prescribed in Morrey-Campanato spaces in ℝ. In such a scenario, the solution u is required to satisfy a vanishing Carleson measure condition of fractional order. By relying on these well-posedness and regularity results we succeed in producing characterizations of the space VMO as the closure in BMO of classes of smooth functions contained in BMO within which uniform continuity may be suitably quantified (such as the class of smooth functions satisfying a Hölder or Lipschitz condition). This improves on Sarason's classical result describing VMO as the closure in BMO of the space of uniformly continuous functions with bounded mean oscillations. In turn, this allows us to show that any Calderón-Zygmund operator T satisfying T (1) = 0 extends as a linear and bounded mapping from VMO (modulo constants) into itself. In turn, this is used to describe algebras of singular integral operators on VMO, and to characterize the membership to VMO via the action of various classes of singular integral operators., The first author would like to express his gratitude to the University of Missouri-Columbia (USA), for its support and hospitality while he was visiting this institution. The first author acknowledges financial support from the Spanish Ministry of Economy and Competitiveness, through the \Severo Ochoa Programme for Centres of Excellence in R&D" (SEV-2015-0554). He also acknowledges 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 second author has been supported in part by the Simons Foundation grant #426669, the third author has been supported in part by the Simons Foundation grant #318658, while the fourth author has been supported in part by the Simons Foundation grant #281566, and by a University of Missouri Research Leave grant.

Let $\Omega\subset\mathbb{R}^{n+1}$, $n\ge 2$, be a 1-sided non-tangentially accessible domain (aka uniform domain), i.e., a set which satisfies the interior Corkscrew and Harnack chain conditions, respectively scale-invariant/quantitative versions of openness and path-connectedness. Assume that $\Omega$ satisfies the so-called capacity density condition. Let $L_0u=-\mathrm{div}(A_0\nabla u)$, $Lu=-\mathrm{div}(A\nabla u)$ be two real (non-necessarily symmetric) uniformly elliptic operators, and write $\omega_{L_0}$, $\omega_L$ for the associated elliptic measures. The goal of this program is to find sufficient conditions guaranteeing that $\omega_L$ satisfies an $A_\infty$-condition or a $RH_q$-condition with respect to $\omega_{L_0}$. We show that if the discrepancy of the two matrices satisfies a natural Carleson measure condition with respect to $\omega_{L_0}$, then $\omega_L\in A_\infty(\omega_{L_0})$. Moreover, $\omega_L\in RH_q(\omega_{L_0})$ for any given $1, Comment: This paper is part of the earlier submission arXiv:1901.08261v2

We survey recent progress in a program which to date has produced [18]- [25], aimed at proving general Fatou-type results and establishing the well-posedness of a variety of boundary value problems in the upper half-space Rn + for second-order, homogeneous, constant complex coe cient, elliptic systems L, formulated in a manner that emphasizes pointwise nontangential boundary traces of the null-solutions of L in Rn +. 1, The rst author acknowledges 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. He also acknowledges nancial support from the Spanish Ministry of Economy and Competitiveness, through the \Severo Ochoa Programme for Centres of Excellence in R&D" (SEV-2015-0554).

We show that the boundedness of the Hardy-Littlewood maximal operator on a Kothe function space X and on its Kothe dual X is equivalent to the well-posedness of the X-Dirichlet and X-Dirichlet problems in Rn + in the class of all second-order, homogeneous, elliptic systems, with constant complex coefficients. As a consequence, we obtain that the Dirichlet problem for such systems is well-posed for boundary data in Lebesgue spaces, variable exponent Lebesgue spaces, Lorentz spaces, Zygmund spaces, as well as their weighted versions. We also discuss a version of the aforementioned result which contains, as a particular case, the Dirichlet problem for elliptic systems with data in the classical Hardy space H1, and the Beurling-Hardy space HAp for p € (1,∞). Based on the well-posedness of the Lp-Dirichlet problem we then prove the uniqueness of the Poisson kernel associated with such systems, as well as the fact that they generate a strongly continuous semigroup in natural settings. Finally, we establish a general Fatou type theorem guaranteeing the existence of the pointwise nontangential boundary trace for null-solutions of such systems., The First author has been supported in part by MINECO Grant MTM2010-16518, ICMAT Severo Ochoa project SEV-2011-0087. He also acknowledges 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 second author has been supported in part by a Simons Foundation grant #200750, the third author has been supported in part by US NSF grant #0547944, while the fourth author has been supported in part by the Simons Foundation grant #281566, and by a University of Missouri Research Leave grant

We prove a limited range, off-diagonal extrapolation theorem that generalizes a number of results in the theory of Rubio de Francia extrapolation, and use this to prove a limited range, multilinear extrapolation theorem. We give two applications of this result to the bilinear Hilbert transform. First, we give sufficient conditions on a pair of weights w1,w2 for the bilinear Hilbert transform to satisfy weighted norm inequalities of the form (Formula presented.),where w= ww and 1p=1p1+1p2, The rst author is supported by NSF Grant DMS-1362425 and research funds from the Dean of the College of Arts & Sciences, University of Alabama. The second author acknowledges nancial support from the Spanish Ministry of Economy and Competitiveness, through the \Severo Ochoa" Programme for Centres of Excellence in R&D (SEV-2015-0554). He also acknowledges 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

We characterize the Borel measures μ on R for which the associated dyadic Hilbert transform, or its adjoint, is of weak-type (1, 1) and/or strong-type (p, p) with respect to μ. Surprisingly, the class of such measures is strictly bigger than the traditional class of dyadically doubling measures and strictly smaller than the whole Borel class. In higher dimensions, we provide a complete characterization of the weak-type (1, 1) for arbitrary Haar shift operators, cancellative or not, written in terms of two generalized Haar systems and these include the dyadic paraproducts. Our main tool is a new Calderón-Zygmund decomposition valid for arbitrary Borel measures which is of independent interest., Supported in part by ERC Grant StG- 256997-CZOSQP, by MINECO Spanish Grant MTM-2010-16518 and by ICMAT Severo Ochoa project SEV-2011-0087. 1

We study self-improving properties in the scale of Lebesgue spaces of generalized Poincar e inequalities in the Euclidean space. We present an abstract setting where oscillations are given by certain operators (e.g., approximations of the identity, semigroups or mean value operators) that have o -diagonal decay in some range. Our results provide a uni ed theory that is applicable to the classical Poincar e inequalities and furthermore it includes oscillations de ned in terms of semigroups associated with second order elliptic operators as those in the Kato conjecture. In this latter situation we obtain a direct proof of the John-Nirenberg inequality for the associated BMO and Lipschitz spaces of [HMay, HMM], The second author was supported by MINECO Grant MTM2010-16518 and ICMAT Severo Ochoa project SEV-2011-0087

We present a higher dimensional, scale-invariant version of a classical theorem of F. and M. Riesz [37]. More precisely, we establish scale invariant absolute continuity of harmonic measure with respect to surface measure, along with higher integrability of the Poisson kernel, for a domain Ω ⊂ Rn+1; n 2, with a uniformly rectiable boundary, which satises the Harnack chain condition plus an interior (but not exterior) Corkscrew condition. In a companion paper to this one [28], we also establish a converse, in which we deduce uniform rectifiability of the boundary, assuming scale invariant Lq bounds, with q > 1, on the Poisson kernel., The first author was supported by NSF grant DMS-0801079. The second author was supported by MINECO Grant MTM2010-16518 and ICMAT Severo Ochoa project SEV-2011-0087.

The aim of this paper is to study the boundedness of di erent conical square functions that arise naturally from second order divergence form degenerate elliptic operators. More precisely, let Lw = w1 div(w A r) where w 2 A2 and A is an n n bounded, complex-valued, uniformly elliptic matrix. D. Cruz-Uribe and C. Rios solved the L2(w)-Kato square root problem obtaining that p Lw is equivalent to the gradient on L2(w). The same authors in collaboration with the second named author of this paper studied the Lp(w)-boundedness of operators that are naturally associated with Lw, such as the functional calculus, Riesz transforms, or vertical square functions. The theory developed admitted also weighted estimates (i.e., estimates in Lp(vdw) for v 2 A1(w)), and in particular a class of “degeneracy” weights w was found in such a way that the classical L2-Kato problem can be solved. In this paper, continuing this line of research, and also that originated in some recent results by the second and third named authors of the current paper, we study the boundedness on Lp(w) and on Lp(vdw), with v 2 A1(w), of the conical square functions that one can construct using the heat or Poisson semigroup associated with Lw. As a consequence of our methods, we find a class of degeneracy weights w for which L2-estimates for these conical square functions hold. This opens the door to the study of weighted and unweighted Hardy spaces and of boundary value problems associated with Lw., European Research Council under the European Union’s Seventh Framework Programme (FP7/2007-2013)/ ERC agreement no. 615112 HAPDEGMT, Spanish Ministry of Economy and Competitiveness, through the “Severo Ochoa Programme for Centres of Excellence in R&D” (SEV-2015-0554)

We study the infinitesimal generator of the Poisson semigroup in L associated with homogeneous, second-order, strongly elliptic systems with constant complex coefficients in the upper-half space, which is proved to be the Dirichlet-to-Normal mapping in this setting. Also, its domain is identified as the linear subspace of the L-based Sobolev space of order one on the boundary of the upper-half space consisting of functions for which the Regularity problem is solvable. Moreover, for a class of systems containing the Lamé system, as well as all second-order, scalar elliptic operators, with constant complex coefficients, the action of the infinitesimal generator is explicitly described in terms of singular integral operators whose kernels involve first-order derivatives of the canonical fundamental solution of the given system. Furthermore, arbitrary powers of the infinitesimal generator of the said Poisson semigroup are also described in terms of higher order Sobolev spaces and a higher order Regularity problem for the system in question. Finally, we indicate how our techniques may be adapted to treat the case of higher order systems in graph Lipschitz domains., The first author acknowledges financial support from the Spanish Ministry of Economy and Competitiveness, through the “Severo Ochoa Programme for Centres of Excellence in R&D” (SEV-2015-0554). He also acknowledges 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 second author has been supported in part by a Simons Foundation grant # 426669, the third author has been supported in part by the Simons Foundation grant #318658, while the fourth author has been supported in part by the Simons Foundation grant # 281566, and by a University of Missouri Research Leave grant.

Let $ \Omega \subset \mathbb{R}^{n+1}$, $ n\geq 2$, be a 1-sided NTA domain (also known as a uniform domain), i.e., a domain which satisfies interior corkscrew and Harnack chain conditions, and assume that $ \partial \Omega $ is $ n$-dimensional Ahlfors-David regular. We characterize the rectifiability of $ \partial \Omega $ in terms of the absolute continuity of surface measure with respect to harmonic measure. We also show that these are equivalent to the fact that $ \partial \Omega $ can be covered $ \mathcal {H}^n$-a.e. by a countable union of portions of boundaries of bounded chord-arc subdomains of $ \Omega $ and to the fact that $ \partial \Omega $ possesses exterior corkscrew points in a qualitative way $ \mathcal {H}^n$-a.e. Our methods apply to harmonic measure and also to elliptic measures associated with real symmetric second order divergence form elliptic operators with locally Lipschitz coefficients whose derivatives satisfy a natural qualitative Carleson condition., The first and last authors have been supported in part by the Spanish Ministry of Economy and Competitiveness, through the “Severo Ochoa Programme for Centres of Excellence in R&D” (SEV-2015-0554), and they 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 second author was partially supported by an NSF postdoctoral fellowship, DMS 1203497, and by NSF grant DMS 1500382. The third author was partially supported by NSF grant DMS 1361701.

We study the difference between vertical and conical square functions in the abstract and also in the specific case where the square functions come from an elliptic operator., Comment: 21 pages

The purpose of this paper is to present a general method that allows us to study self-improving properties of generalized Poincare inequalities. When measuring the oscillation in a given cube, we replace the average by an approximation of the identity or a semigroup scaled to that cube and whose kernel decays fast enough. We apply the method to obtain self-improvement in the scale of Lebesgue spaces of Poincare type inequalities. In particular, we propose some expanded Poincare estimates that take into account the lack of localization of the approximation of the identity or the semigroup. As a consequence of this method we are able to obtain global pseudo-Poincare inequalities.

We revisit the “extrapolation method” for Carleson measures, introduced by Lewis and Murray (1995), to prove A ∞ A_\infty estimates for certain caloric measures, and we present a purely real variable version of the method suitable for establishing A ∞ A_\infty estimates. To illustrate the use of this technique, we then reprove a well-known result of Fefferman, Kenig, and Pipher (1991).

In this paper we study self-improving properties in the scale of Lebesgue spaces of generalized Poincare inequalities in spaces of homogeneous type. In contrast with the classical situation, the oscillations involve approximation of the identities or semigroups whose kernels decay fast enough and the resulting estimates take into account their lack of localization. The techniques used do not involve any classical Poincare or Sobolev–Poincare inequalities and therefore they can be used in general settings where these estimates do not hold or are unknown. We apply our results to the case of Riemannian manifolds with doubling volume form and assuming Gaussian upper bounds for the heat kernel of the semigroup e − t Δ with Δ being the Laplace–Beltrami operator. We obtain generalized Poincare inequalities with oscillations that involve the semigroup e − t Δ and with right hand sides containing either ∇ or Δ 1 / 2 .

International audience; Let $(\Gamma,\mu)$ be an infinite graph endowed with a reversible Markov kernel $p$ and let $P$ be the corresponding operator. We also consider the associated discrete gradient $\nabla$. We assume that $\mu$ is doubling, a uniform lower bound for $p(x,y)$ when $p(x,y)>0$, and gaussian upper estimates for the iterates of $p$. Under these conditions (and in some cases assuming further some Poincaré inequality) we study the comparability of $(I-P)^{1/2} f$ and $\nabla f$ in Lebesgue spaces with Muckenhoupt weights. Also, we establish weighted norm inequalities for a Littlewood-Paley-Stein square function, its formal adjoint, and commutators of the Riesz transform with bounded mean oscillation functions.

We study the efficiency of the greedy algorithm for wavelet bases in Lorentz spaces in order to give the near best approximation. The result is used to give sharp inclusions for the approximation spaces in terms of discrete Lorentz sequence spaces.

We present two extrapolation methods for multi-sublinear operators that allow us to derive estimates for general functions from the corresponding estimates on characteristic functions. Of these methods, the first is applicable to general multi-sublinear operators while the second requires working with the so-called (ε,δ)-atomic operators. Among the applications, we discuss some new endpoint estimates for the bilinear Hilbert transform.

We study the efficiency of greedy algorithms for N-term wavelet approximation in Orlicz spaces LΦ(Rd). We compute the left and right democracy functions in terms of the fundamental function of LΦ, recovering a recent result of Wojtaszczyk [P. Wojtaszczyk, Greediness of the Haar system in rearrangement invariant spaces, in: T. Figiel, A. Kamont (Eds.), Approximation and Probability, vol. 72, Banach Center Publications, Warszawa, 2006, pp. 385–395], which establishes that wavelet bases can only be greedy when LΦ=Lp for some 1

Let E ⊂ ℝ, n ≥ 2, be a uniformly rectifiable set of dimension n. Then bounded harmonic functions in Ω := ℝ \ E satisfy Carleson measure estimates and are ε-approximable. Our results may be viewed as generalized versions of the classical F. and M. Riesz theorem, since the estimates that we prove are equivalent, in more topologically friendly settings, to quantitative mutual absolute continuity of harmonic measure and surface measure., The first author was supported by NSF grant DMS-1361701. The second author was supported in part by MINECO Grant MTM2010-16518, ICMAT Severo Ochoa project SEV-2011-0087. He also acknowledges 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 was supported by the Alfred P. Sloan Fellowship, the NSF CAREER Award DMS 1056004, the NSF INSPIRE Award DMS 1344235, and the NSF Materials Research Science and Engineering Center Seed Grant DMR 0212302.

We consider a certain class of second order, variable coefficient divergence form elliptic operators, in a uniform domain Ω with Ahlfors regular boundary, and we show that the A property of the elliptic measure associated to any such operator and its transpose imply that the domain is in fact NTA (and hence chord-arc). The converse was already known, and follows from work of Kenig and Pipher [30]., The first author was supported by NSF grant DMS-1664047. The second author acknowledges 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. He also acknowledges financial support from the Spanish Ministry of Economy and Competitiveness, through the “Severo Ochoa Programme for Centres of Excellence in R&D” (SEV-2015-0554). The last author was partially supported by the Robert R. & Elaine F. Phelps Professorship in Mathematics and NSF grant DMS-1361823.

In the present paper we prove that for any open connected set Ω ⊂ R, n≥ 1 , and any E⊂ ∂Ω with H(E) < ∞, absolute continuity of the harmonic measure ω with respect to the Hausdorff measure on E implies that ω| is rectifiable. This solves an open problem on harmonic measure which turns out to be an old conjecture even in the planar case n= 1., The second author was supported in part by NSF grant DMS 1361701. The third author has been partially supported by ICMAT Severo Ochoa project SEV-2011- 0087 and he acknowledges 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 fourth author is supported in part by the Alfred P. Sloan Fellowship, the NSF INSPIRE Award DMS 1344235, NSF CAREER Award DMS 1220089 and NSF UMN MRSEC Seed grant DMR 0212302. The sixth author was supported by the ERC grant 320501 of the European Research Council (FP7/2007-2013) (which also funded the first and fifth authors), by 2014- SGR-75 (Catalonia), MTM2013-44304-P (Spain), and by the Marie Curie ITN MAnET (FP7-607647). The last author was partially supported by the NSF grant DMS-1265549

In the present paper we sketch the proof of the fact that for any open connected set Rn+1, n 1, and any E @ with 0 < Hn(E) < 1, absolute continuity of the harmonic measure ! with respect to the Hausdorff measure on E implies that !jE is rectifiable., We acknowledge the support of the following grants: the second author – NSF DMS 1361701, the third author – SEV- 2011-0087 and (FP7/2007-2013)/ ERC agreement no. 615112 HAPDEGMT, the fourth author – Sloan Fellowship, NSF DMS 1344235, NSF DMS 1220089, NSF DMR 0212302, the sixth author – ERC 320501 FP7/2007-2013 (which also funded the first and fifth authors), 2014-SGR-75, MTM2013-44304-P, ITN MAnET (FP7-607647), the last author – NSF DMS-1265549. The results of this paper were obtained at the Institut Henri Poincar´e, and at the 2015 ICMAT program Analysis and Geometry in Metric Spaces. All authors would like to express their gratitude to these institutions for their support and nice working environments.

This is the third part of a series of four articles on weighted norm inequalities, off-diagonal estimates and elliptic operators. For $L$ in some class of elliptic operators, we study weighted norm $L^p$ inequalities for singular 'non-integral' operators arising from $L$ ; those are the operators $\phi(L)$ for bounded holomorphic functions $\phi$, the Riesz transforms $\nabla L^{-1/2}$ (or $(-\Delta)^{1/2}L^{-1/2}$) and its inverse $L^{1/2}(-\Delta)^{-1/2}$, some quadratic functionals $g\_{L}$ and $G\_{L}$ of Littlewood-Paley-Stein type and also some vector-valued inequalities such as the ones involved for maximal $L^p$-regularity. For each, we obtain sharp or nearly sharp ranges of $p$ using the general theory for boundedness of Part I and the off-diagonal estimates of Part II. We also obtain commutator results with BMO functions., Comment: 38 pages. Third of 4 papers

We generalize the Ap extrapolation theorem of Rubio de Francia to A∞ weights in the context of Muckenhoupt bases. Our result has several important features. First, it can be used to prove weak endpoint inequalities starting from strong-type inequalities, something which is impossible using the classical result. Second, it provides an alternative to the technique of good-λ inequalities for proving Lp norm inequalities relating operators. Third, it yields vector-valued inequalities without having to use the theory of Banach space valued operators. We give a number of applications to maximal functions, singular integrals, potential operators, commutators, multilinear Calder´on-Zygmund operators, and multiparameter fractional integrals. In particular, we give new proofs, which completely avoid the good-λ inequalities, of Coifman’s inequality relating singular integrals and the maximal operator, of the Fefferman-Stein inequality relating the maximal operator and the sharp maximal operator, and the Muckenhoupt-Wheeden inequality relating the fractional integral operator and the fractional maximal operator. Ministerio de Ciencia y Tecnología

Two versions of Rubio de Francia’s extrapolation theorem for multivariable operators of functions are obtained. One version assumes an initial estimate with different weights in each space and implies boundedness on all products of Lebesgue spaces. Another version assumes an initial estimate with the same weight but yields boundedness on a product of Lebesgue spaces whose indices lie on a line. Applications are given in the context of multilinear Calderon-Zygmund operators. Vector-valued inequalities are automatically obtained for them without developing a multilinear Banach-valued theory. A multilinear extension of the Marcinkiewicz and Zygmund theorem on l2-valued extensions of bounded linear operators is also obtained.

In the context of the spaces of homogeneous type, given a family of op- erators that look like approximations of the identity, new sharp maximal functions are considered. We prove a good- inequality for Muckenhoupt weights, which leads to an analog of the Feerman{Stein estimate for the classical sharp maximal function. As a consequence, we establish weighted norm estimates for certain singular integrals, dened on irregular domains, with Hormander conditions replaced by some estimates which do not involve the regularity of the kernel. We apply these results to prove the bounded- ness of holomorphic functional calculi on Lebesgue spaces with Muckenhoupt weights. In particular, some applications are given to second order elliptic operators with dieren t boundary conditions.

Let $E\subset \mathbb{R}^{n+1}$, $n\ge 2$, be an Ahlfors-David regular set of dimension $n$. We show that the weak-$A_\infty$ property of harmonic measure, for the open set $\Omega:= \mathbb{R}^{n+1}\setminus E$, implies uniform rectifiability of $E$. More generally, we establish a similar result for the Riesz measure, $p$-harmonic measure, associated to the $p$-Laplace operator, $1, Comment: arXiv admin note: substantial text overlap with arXiv:1505.06499

We study the Kato problem for divergence form operators whose ellipticity may be degenerate. The study of the Kato conjecture for degenerate elliptic equations was begun by Cruz-Uribe and Rios (2008, 2012, 2015). In these papers the authors proved that given an operator L = -wdiv(A∇) where w is in the Muckenhoupt class A and A is a w-degenerate elliptic measure (that is, A = wB with B(x) an n×n bounded, complex-valued, uniformly elliptic matrix), then L satisfies the weighted estimate ‖√Lwf‖L(w)≈‖∇f‖L(w)˙ In the present paper we solve the L-Kato problem for a family of degenerate elliptic operators. We prove that under some additional conditions on the weight w, the following unweighted L-Kato estimates hold:‖L f‖L(ℝ)˙This extends the celebrated solution to the Kato conjecture by Auscher, Hofmann, Lacey, McIntosh,and Tchamitchian, allowing the differential operator to have some degree of degeneracy in its ellipticity.For example, we consider the family of operators L=-|x| div(|x|B(x)∇), where B is any bounded,complex-valued, uniformly elliptic matrix. We prove that there exists ε > 0, depending only on dimension and the ellipticity constants, such that‖L f ‖L(ℝ),-ε, The first author is supported by NSF grant 1362425 and research funds provided by the Dean of Arts & Sciences at the University of Alabama. While substantial portions of this work was done he was supported by the Stewart-Dorwart faculty development fund at Trinity College. The second authors acknowledges fi nancial support from the Spanish Ministry of Economy and Competitiveness, through the \Severo Ochoa Programme for Centres of Excellence in R&D" (SEV-2015-0554). He also acknowledges 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 is supported by the Natural Sciences and Engineering Research Council of Canada Discovery Grant RT733901.