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Abstract  We study the interplay between the geometry of Hardy spaces and functional analytic properties of singular integral operators (SIO’s), such as the Riesz transforms as well as Cauchy–Clifford and harmonic double-layer operator, on the one hand and, on the other hand, the regularity and geometric properties of domains of locally finite perimeter. Among other things, we give several characterizations of Euclidean balls, their complements, and half-spaces, in terms of the aforementioned SIO’s. [ABSTRACT FROM AUTHOR]

We study the analyticity of the semigroup generated by the Stokes operator equipped with Neumann-type boundary conditions on $L^p$ spaces in Lipschitz domains. Our strategy is to regularize this operator by considering the Hodge Laplacian, which has the additional property that it commutes with the Leray projection. [ABSTRACT FROM AUTHOR]

As a limiting case of the classical Calderón-Zygmund theory, in this note we study the Besov regularity of differential forms $u$ for which $du$ and $delta u$ have absolutely integrable coefficients in ${mathbb R}^n$. [ABSTRACT FROM AUTHOR]

Abstract  We study the method of layer potentials for manifolds with boundary and cylindrical ends. The fact that the boundary is non-compact prevents us from using the standard characterization of Fredholm or compact pseudo-differential operators between Sobolev spaces, as, for example, in the works of Fabes-Jodeit-Lewis [10] and Kral-Wedland [18]. We first study the layer potentials depending on a parameter on compact manifolds. This then yields the invertibility of the relevant boundary integral operators in the global, non-compact setting. As an application, we prove a well-posedness result for the non-homogeneous Dirichlet problem on manifolds with boundary and cylindrical ends. We also prove the existence of the Dirichlet-to-Neumann map, which we show to be a pseudodifferential operator in the calculus of pseudodifferential operators that are “almost translation invariant at infinity.” [ABSTRACT FROM AUTHOR]

We study the applicability of the method of layer potentials in the treatment of boundary value problems for the Laplace-Beltrami operator on Lipschitz sub-domains of Riemannian manifolds, in the case when the metric tensor $g_{jk} dx_j\otimes dx_k$ has low regularity. Under the assumption that \[|g_{jk}(x)-g_{jk}(y)|\leq C\omega(|x-y|),] where the modulus of continuity $\omega$ satisfies a Dini-type condition, we prove the well-posedness of the classical Dirichlet and Neumann problems with $L^p$ boundary data, for sharp ranges of $p$'s and with optimal nontangential maximal function estimates. [ABSTRACT FROM AUTHOR]

Let $L$ be a second order, (variable coefficient) elliptic differential operator and let $u\in B^{p,p}_\alpha(\Omega)$, $10$, satisfy $Lu=0$ in the Lipschitz domain $\Omega$. We show that $u$ can exhibit more regularity on Besov scales for which smoothness is measured in $L^\tau$ with $\tau