Vol. 8, No. 1, 2015

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ISSN: 1948-206X (e-only)
ISSN: 2157-5045 (print)
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 Lvmo, 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 C1 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
Milestones
Received: 12 March 2014
Accepted: 5 January 2015
Published: 15 April 2015
Authors
Steve Hofmann
Department of Mathematics
University of Missouri at Columbia
Math. Building
University of Missouri
Columbia, MO 65211
United States
Marius Mitrea
Department of Mathematics
University of Missouri at Columbia
Math. Building
University of Missouri
Columbia, MO 65211
United States
Michael E. Taylor
Department of Mathematics
University of North Carolina
Phillips Hall
UNC
Chapel Hill, NC 27599-3250
United States