Regularity properties of commutators and layer potentials associated to the heat equation
Authors:
John L. Lewis and Margaret A. M. Murray
Journal:
Trans. Amer. Math. Soc. 328 (1991), 815842
MSC:
Primary 35K05; Secondary 31A20, 42B20, 47F05
MathSciNet review:
1020043
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Abstract: In recent years there has been renewed interest in the solution of parabolic boundary value problems by the method of layer potentials. In this paper we consider graph domains in , where the boundary function is in . This class of domains would appear to be the minimal smoothness class for the solvability of the Dirichlet problem for the heat equation by the method of layer potentials. We show that, for , the boundary singlelayer potential operator for maps into the homogeneous Sobolev space . This regularity result is obtained by studying the regularity properties of a related family of commutators. Along the way, we prove estimates for a class of singular integral operators to which the Theorem of David and Journé does not apply. The necessary estimates are obtained by a variety of realvariable methods.
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 , The initialNeumann problem for the heat equation in Lipschitz cylinders, Trans. Amer. Math. Soc. 320 (1990), 152. MR 1000330 (90k:35112)
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 E. B. Fabes and N. M. Riviere, Dirichlet and Neumann problems for the heat equation in cylinders, Proc. Sympos. Pure Math., vol. 35, part 2, Amer. Math. Soc., Providence, R.I., 1979, pp. 179196. MR 545307 (81b:35044)
 [KW]
 R. Kaufman and J. M. G. Wu, Parabolic measure on domains of class , Compositio Math. 65 (1988), 201207. MR 932644 (89g:31001)
 [LeS]
 J. L. Lewis and J. Silver, Parabolic measure and the Dirichlet problem for the heat equation in two dimensions, Indiana Univ. Math. J. 37 (1988), 801839. MR 982831 (90e:35079)
 [Mu1]
 M. A. M. Murray, Commutators with fractional differentiation and Sobolev spaces, Indiana Univ. Math. J. 34 (1985), 205215. MR 773402 (86c:47042)
 [Mu2]
 , Multilinear singular integrals involving a derivative of fractional order, Studia Math. 87 (1987), 139165. MR 928573 (89g:42020)
 [St]
 E. M. Stein, Singular integrals and differentiability properties of functions, Princeton Univ. Press, Princeton, N. J., 1970. MR 0290095 (44:7280)
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 R. S. Strichartz, Bounded mean oscillation and Sobolev spaces, Indiana Univ. Math. J. 29 (1980), 539558. MR 578205 (82f:46040)
 [T]
 A. Torchinsky, Realvariable methods in harmonic analysis, Academic Press, Orlando, 1986. MR 869816 (88e:42001)
 [Z]
 A. Zygmund, Trigonometric series, vol. I, Cambridge Univ. Press, Cambridge, 1959. MR 0107776 (21:6498)
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DOI:
http://dx.doi.org/10.1090/S00029947199110200436
PII:
S 00029947(1991)10200436
Article copyright:
© Copyright 1991
American Mathematical Society
