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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 |
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Vol. 8 (2015), No. 1, 115–181
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Abstract |
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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 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 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
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Mathematical Subject Classification 2010
Primary: 31B10, 35S05, 35S15, 42B20, 35J57
Secondary: 42B37, 45B05, 58J05, 58J32
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Milestones
Received: 12 March 2014
Accepted: 5 January 2015
Published: 15 April 2015
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Authors |
| Steve Hofmann | |
| Department of Mathematics University of Missouri at Columbia Math. Building University of Missouri Columbia, MO 65211 United States |
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| Marius Mitrea | |
| Department of Mathematics University of Missouri at Columbia Math. Building University of Missouri Columbia, MO 65211 United States |
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| Michael E. Taylor | |
| Department of Mathematics University of North Carolina Phillips Hall UNC Chapel Hill, NC 27599-3250 United States |
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