1.1 INTRODUCTION This paper gives a survey on additive Schwarz methods applied to the h and p versions of the boundary element method. We consider weakly singular and hypersingular integral equations, namely the single layer potential operator and the normal derivative of the double layer potential operator. That means we have to consider weak formulations of the form

We show that the L(w) operator norm of the composition M◦TΩ, where M is the maximal operator and TΩ is a rough homogeneous singular integral with angular part Ω ∈ L∞(Sn−1), depends quadratically on [w]A2 , and this dependence is sharp.

Suppose that Ω is an integrable function on S. We study the singular integral operator TΩf = p.v. f ∗ Ω(x/|x|) |x|2 . We show that for α > 0 the condition

We establish a sharp maximal function estimate for some vector-valued multilinear singular integral operators. As an application, we obtain the $(L^p, L^q)$-norm inequality for vector-valued multilinear operators.

* Correspondence: yanpingch@126. com Department of Applied Mathematics, School of Mathematics and Physics, University of Science and Technology Beijing, Beijing 100083, China Full list of author information is available at the end of the article Abstract Let 1 < p <∞ and n ≥ 2. The authors establish the L(R) boundedness for a class of parabolic singular integral operators with rough kernels. MR...

In this paper, an oscillatory singular integral operator T deened by T f (x) = Z IR e ixP(y) f (x ? y) y dy is showed to be bounded on a weighted Sobolev space H

Optimal sufficient conditions are found in weighted Lorentz spaces for weight functions which provide the boundedness of the Calderón– Zygmund singular integral operator defined on spaces of homogeneous and nonhomogeneous type. 2000 Mathematics Subject Classification: 42B20, 42B25.