The Kato Square Root Problem for Mixed Boundary Conditions

We consider the negative Laplacian subject to mixed boundary conditions on a bounded domain. We prove under very general geometric assumptions that slightly above the critical exponent $\frac{1}{2}$ its fractional power domains still coincide with suitable Sobolev spaces of optimal regularity. In combination with a reduction theorem recently obtained by the authors, this solves the Kato Square Root Problem for elliptic second order operators and systems in divergence form under the same geometric assumptions.Comment: Inconsistencies in Section 6 remove

Non-local Gehring lemmas in spaces of homogeneous type and applications

We prove a self-improving property for reverse H{\"o}lder inequalities with non-local right hand side. We attempt to cover all the most important situations that one encounters when studying elliptic and parabolic partial differential equations as well as certain fractional equations. We also consider non-local extensions of A$\infty$ weights. We write our results in spaces of homogeneous type.Comment: Revised version. Changed title. Application to a more relevant fractional elliptic equation given in the final section. 40 page

Explicit improvements for Lp\mathrm{L}^p-estimates related to elliptic systems

We give a simple argument to obtain $\mathrm{L}^p$-boundedness for heat semigroups associated to uniformly strongly elliptic systems on $\mathbb{R}^d$ by using Stein interpolation between Gaussian estimates and hypercontractivity. Our results give $p$ explicitly in terms of ellipticity. It is optimal at the endpoint $p=\infty$. We also obtain $\mathrm{L}^p$-estimates for the gradient of the semigroup, where $p>2$ depends on ellipticity but not on dimension.Comment: 16 pages, improved readability, accepted for publication in Bulletin of the LM

On regularity of weak solutions to linear parabolic systems with measurable coefficients

We establish a new regularity property for weak solutions of parabolic systems with coefficients depending measurably on time as well as on all spatial variables. Namely, weak solutions are locally H{\"o}lder continuous Lp valued functions for some p > 2.Comment: 23 pages. Proof of Lemma 3.3 corrected. Final version to appear in J. Math. Pures App