A functional analytic approach to obtaining self-improving properties of solutions to linear non-local elliptic equations is presented. It yields conceptually simple and very short proofs of some previous results due to Kuusi-Mingione-Sire and Bass-Ren. Its flexibility is demonstrated by new applications to non-autonomous parabolic equations with non-local elliptic part and questions related to maximal regularity.the left-hand side is associated with a sesquilinear form on the Hilbert space W α,2 (R n ) and thanks to ellipticity (1.1) the Lax-Milgram lemma applies and yields invertibility of 1 + L α,A onto the dual 1 2 PASCAL AUSCHER, SIMON BORTZ, MORITZ EGERT, AND OLLI SAARI space. Now, the main difference compared with second order elliptic equations is that we can transfer regularity requirements between u and φ without interfering with the coefficients A: Without making any further assumption we may writeThen the ubiquitous analytic perturbation lemma ofŠneȋberg [21] allows one to extrapolate invertibility to ε > 0 small enough. Compared to [3,14] we can also work in an L p -setting without hardly any additional difficulties. In this way, we shall recover some of their results on global weak solutions in Section 4 and discuss some new and sharpened local self-improvement properties in Section 5.Finally, in Section 6 we demonstrate the simplicity and flexibility of our approach by proving that for each f ∈ L 2 (0, T ; L 2 (R n )) the unique solution u ∈ H 1 (0, T ; W α,2 (R n ) * ) ∩ L 2 (0, T ; W α,2 (R n )) of the non-autonomous Cauchy problem

We prove that coronizations with respect to arbitrary d-regular sets (not necessarily graphs) imply big pieces squared of these (approximating) sets. This is known (and due to David and Semmes in the case of sufficiently large co-dimension, and to Azzam and Schul in general) in the (classical) setting of Euclidean spaces with Hausdorff measure of integer dimension, where the approximating sets are Lipschitz graphs. Our result is a far reaching generalization of these results and we prove that coronizations imply big pieces squared is a generic property. In particular, our result applies, when suitably interpreted, in metric spaces having a fixed positive (perhaps non-integer) dimension, equipped with a Borel regular measure and with arbitrary approximating sets. As a novel application we highlight how to utilize this general setting in the context of parabolic uniform rectifiability.

We establish a new regularity property for weak solutions of linear parabolic systems with coefficients depending measurably on time as well as on all spatial variables. Namely, weak solutions are locally Hölder continuous L p valued functions for some p > 2.Résumé. On démontre une nouvelle propriété de régularité des solutions faibles des systèmes paraboliques dont les coefficients dépendent de façon mesurable du temps et des variables spatiales. Précisément, on montre que ces solutions sont localement Hölder continues comme fonctionsà valeurs dans un espace L p pour un p > 2.

We present an alternative proof of a result of Kenig and Toro [KT4], which states that if Ω ⊂ R n+1 is a two sided NTA domain, with Ahlfors-David regular boundary, and the log of the Poisson kernel associated to Ω as well as the log of the Poisson kernel associated to Ω ext are in VMO, then the outer unit normal ν is in VMO. Our proof exploits the usual jump relation formula for the non-tangential limit of the gradient of the single layer potential. We are also able to relax the assumptions of Kenig and Toro in the case that the pole for the Poisson kernel is finite: in this case, we assume only that ∂Ω is uniformly rectifiable, and that ∂Ω coincides with the measure theoretic boundary of Ω a.e. with respect to Hausdorff H n measure.

Let E ⊂ R n+1 , n ≥ 1, be a uniformly rectifiable set of dimension n.We show E that has big pieces of boundaries of a class of domains which satisfy a 2-sided corkscrew condition, and whose connected components are all chordarc domains (with uniform control of the various constants). As a consequence, we deduce that E has big pieces of sets for which harmonic measure belongs to weak-A ∞ .