Singular Integrals and Potential Theory

Abstract

In this paper I consider a class of non-standard singular integrals motivated by potential theoretic and probabilistic considerations. The probabilistic applications, which are by far the most interesting part of this circle of ideas, are only outlined in Section 1.5: They give the best approximation of the solution of the classical Dirichlet problem in a Lipschitz domain $$\Omega\subset {\mathbb{R}}^{n+1}$$ by the corresponding solution by finite differences. The potential theoretic estimate needed for this gives rise to a natural duality between the L p functions on the boundary ∂Ω and a class $$D_q\, ( \frac{1}{p} + \frac{1}{q}=1)$$ of functions A on Ω that was first considered by Dahlberg. The actual duality is given by ∫Ω $$\nabla^2$$ S f(x)A(x)dx = (f, A) where S f(x) = ∫∂Ω |x − y|1−n f(y)dy is the Newtonian potential. We can identify the upper half Lipschitz space $$\Omega$$ with $${\mathbb{R}}^{n+1}_+$$ in the obvious way and express $$(f,A) = \int K(x, y)f(x)A(y)dxdy\, (x\, \in {\mathbb{R}}^n, y \in {\mathbb{R}}^{n+1}_+)$$ for an appropriate kernel K. It is the boundedness properties of the above (for $$f \in L_p$$ , $$A \in D_q$$ ) that is the essential part of this work. This relates with more classical (but still “rough”) singular integrals that have been considered by Christ and Journe.