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), 815-842

MSC:
Primary 35K05; Secondary 31A20, 42B20, 47F05

DOI:
https://doi.org/10.1090/S0002-9947-1991-1020043-6

MathSciNet review:
1020043

Full-text PDF

Abstract | References | Similar Articles | Additional Information

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 single-layer 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 real-variable methods.

**[BL]**J. Bergh and J. Lofstrom,*Interpolation spaces*:*An introduction*, Springer-Verlag, New York, 1976. MR**0482275 (58:2349)****[Br1]**R. M. Brown,*The method of layer potentials for the heat equation in Lipschitz cylinders*, Amer. J. Math.**111**(1989), 339-379. MR**987761 (90d:35118)****[Br2]**-,*The initial-Neumann problem for the heat equation in Lipschitz cylinders*, Trans. Amer. Math. Soc.**320**(1990), 1-52. MR**1000330 (90k:35112)****[DJ]**G. David and J.-L. Journé,*A boundedness criterion for generalized Calderón-Zygmund operators*, Ann. of Math. (2)**120**(1985), 371-397. MR**763911 (85k:42041)****[FR]**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. 179-196. MR**545307 (81b:35044)****[KW]**R. Kaufman and J. M. G. Wu,*Parabolic measure on domains of class*, Compositio Math.**65**(1988), 201-207. 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), 801-839. MR**982831 (90e:35079)****[Mu1]**M. A. M. Murray,*Commutators with fractional differentiation and**Sobolev spaces*, Indiana Univ. Math. J.**34**(1985), 205-215. MR**773402 (86c:47042)****[Mu2]**-,*Multilinear singular integrals involving a derivative of fractional order*, Studia Math.**87**(1987), 139-165. 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)****[Str]**R. S. Strichartz,*Bounded mean oscillation and Sobolev spaces*, Indiana Univ. Math. J.**29**(1980), 539-558. MR**578205 (82f:46040)****[T]**A. Torchinsky,*Real-variable 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)**

Retrieve articles in *Transactions of the American Mathematical Society*
with MSC:
35K05,
31A20,
42B20,
47F05

Retrieve articles in all journals with MSC: 35K05, 31A20, 42B20, 47F05

Additional Information

DOI:
https://doi.org/10.1090/S0002-9947-1991-1020043-6

Article copyright:
© Copyright 1991
American Mathematical Society