Regularity properties of commutators and layer potentials associated to the heat equation
Regularity properties of commutators and layer potentials associated to the heat equation
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DOI：
10.1090/s0002-9947-1991-1020043-6
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发表时间：
1991-02
影响因子：
1.3
通讯作者：
John L. Lewis;M. Murray
中科院分区：
文献类型：
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作者：
John L. Lewis;M. Murray
近年来，用层势法求解抛物型边值问题又引起了人们的兴趣。本文考虑图域D = {（x，t）：x > f（t）}，其中边界函数f在I_（112）（BMO）中，这类区域似乎是用层势法求解热方程Dirichlet问题的最小光滑类.我们证明了，当1 < p < oo时，D的边界单层位势算子将LP映射到齐次Sobolev空间I1 I2（Lp）。这一正则性结果是通过研究一类相关算子族的正则性得到的。沿着的方式，我们证明了LP估计的一类奇异积分算子的T 1定理的大卫和Quinne不适用。通过各种实变量方法获得必要的估计。
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 D = {(x, t): x > f(t)} in 9j2, where the boundary function f is in I112(BMO), 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 1 < p < oo, the boundary single-layer potential operator for D maps LP into the homogeneous Sobolev space I1I2(Lp) . This regularity result is obtained by studying the regularity properties of a related family of commutators. Along the way, we prove LP estimates for a class of singular integral operators to which the T 1 Theorem of David and Journe does not apply. The necessary estimates are obtained by a variety of real-variable methods.