This paper is devoted to studying the Rubio de Francia extrapolation for multilinear compact operators. It allows one to extrapolate the compactness of T from just one space to the full range of weighted spaces, whenever an m-linear operator T is bounded on weighted Lebesgue spaces. This result is indeed established in terms of the multilinear Muckenhoupt weights A p, r , and the limited range of the L p scale. To show extrapolation theorems above, by means of a new weighted Fréchet-Kolmogorov theorem, we present the weighted interpolation for multilinear compact operators. As applications, we obtain the weighted compactness of commutators of many multilinear operators, including multilinear ω-Calderón-Zygmund operators, multilinear Fourier multipliers, bilinear rough singular integrals and bilinear Bochner-Riesz means. Beyond that, we establish the weighted compactness of higher order Calderón commutators, and commutators of Riesz transforms related to Schrödinger operators.

This paper is devoted to studying the Rubio de Francia extrapolation for multilinear compact operators. It allows one to extrapolate the compactness of T T from just one space to the full range of weighted spaces, whenever an m m -linear operator T T is bounded on weighted Lebesgue spaces. This result is indeed established in terms of the multilinear Muckenhoupt weights A p → , r → A_{\vec {p}, \vec {r}} , and the limited range of the L p L^p scale. To show extrapolation theorems above, by means of a new weighted Fréchet-Kolmogorov theorem, we present the weighted interpolation for multilinear compact operators. To prove the latter, we also need to build a weighted interpolation theorem in mixed-norm Lebesgue spaces. As applications, we obtain the weighted compactness of commutators of many multilinear operators, including multilinear ω \omega -Calderón-Zygmund operators, multilinear Fourier multipliers, bilinear rough singular integrals and bilinear Bochner-Riesz means. Beyond that, we establish the weighted compactness of higher order Calderón commutators, and commutators of Riesz transforms related to Schrödinger operators.

We study discretized maximal operators associated to averaging over (neighborhoods of) squares in the plane and, more generally, k-skeletons in R n . Although these operators are known not to be bounded on any L p , we obtain nearly sharp L p bounds for every small discretization scale. These results are motivated by, and partially extend, recent results of T. Keleti, D. Nagy and P. Shmerkin, and of R. Thornton, on sets that contain a scaled k-sekeleton of the unit cube with center in every point of R n .

In nice environments, such as Lipschitz or chord-arc domains, it is well-known that the solvability of the Dirichlet problem for an elliptic operator in $$L^p$$ L p , for some finite p, is equivalent to the fact that the associated elliptic measure belongs to the Muckenhoupt class $$A_\infty $$ A ∞ . In turn, any of these conditions occurs if and only if the gradient of every bounded null solution satisfies a Carleson measure estimate. This has been recently extended to much rougher settings such as those of 1-sided chord-arc domains, that is, sets which are quantitatively open and connected with a boundary which is Ahlfors–David regular. In this paper, we work in the same environment and consider a qualitative analog of the latter equivalence showing that one can characterize the absolute continuity of the surface measure with respect to the elliptic measure in terms of the finiteness almost everywhere of the truncated conical square function for any bounded null solution. As a consequence of our main result particularized to the Laplace operator and some previous results, we show that the boundary of the domain is rectifiable if and only if the truncated conical square function is finite almost everywhere for any bounded harmonic function. In addition, we obtain that for two given elliptic operators $$L_1$$ L 1 and $$L_2$$ L 2 , the absolute continuity of the surface measure with respect to the elliptic measure of $$L_1$$ L 1 is equivalent to the same property for $$L_2$$ L 2 provided the disagreement of the coefficients satisfy some quadratic estimate in truncated cones for almost everywhere vertex. Finally, for the case on which $$L_2$$ L 2 is either the transpose of $$L_1$$ L 1 or its symmetric part we show the equivalence of the corresponding absolute continuity upon assuming that the antisymmetric part of the coefficients has some controlled oscillation in truncated cones for almost every vertex.