Vol. 8, No. 1, 2015

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Symbol calculus for operators of layer potential type on Lipschitz surfaces with VMO normals, and related pseudodifferential operator calculus

Steve Hofmann, Marius Mitrea and Michael E. Taylor

Vol. 8 (2015), No. 1, 115–181
Abstract

We show that operators of layer potential type on surfaces that are locally graphs of Lipschitz functions with gradients in vmo are equal, modulo compacts, to pseudodifferential operators (with rough symbols), for which a symbol calculus is available. We build further on the calculus of operators whose symbols have coefficients in ${L}^{\infty }\cap$vmo, and apply these results to elliptic boundary problems on domains with such boundaries, which in turn we identify with the class of Lipschitz domains with normals in vmo. This work simultaneously extends and refines classical work of Fabes, Jodeit and Rivière, and also work of Lewis, Salvaggi and Sisto, in the context of ${\mathsc{C}}^{1}$ surfaces.

Keywords
singular integral operator, compact operator, pseudodifferential operator, rough symbol, symbol calculus, single layer potential operator, strongly elliptic system, boundary value problem, nontangential maximal function, nontangential boundary trace, Dirichlet problem, regularity problem, Poisson problem, oblique derivative problem, regular elliptic boundary problem, elliptic first-order system, Hodge–Dirac operator, Cauchy integral, Hardy spaces, Sobolev spaces, Besov spaces, Lipschitz domain, SKT domain
Mathematical Subject Classification 2010
Primary: 31B10, 35S05, 35S15, 42B20, 35J57
Secondary: 42B37, 45B05, 58J05, 58J32