Singular integrals and positive kernels

L Bounds for Singular Integrals and Maximal Singular Integrals with Rough Kernels

Convolution type Calderón-Zygmund singular integral operators with rough kernels p.v. Ω(x)/|x| are studied. A condition on Ω implying that the corresponding singular integrals and maximal singular integrals map L → L for 1 < p < ∞ is obtained. This condition is shown to be different from the condition Ω ∈ H1(Sn−1).

Singular Integrals

In this paper I will a t tempt to describe the subject as it has developed in the last fifteen years, outlining the methods by which its problems have been approached and discussing its connections with other branches of Analysis. I t will perhaps be best to start by considering certain classical situations which lead naturally to singular integrals and which contain the seeds of some of the me...

Evaluating Singular and Nearly Singular Integrals Numerically

Multi-parameter singular integrals

An efficient algorithm for the evaluation of certain convolution integrals with singular kernels

In this short note, we are concerned with the evaluation of certain convolution integrals with singular kernels. The problem is as follows. Given a kernel function K(t) which is very often singular at the origin and a density function σ(t), evaluate C(t) = ∫ t 0 K(t − τ)σ(τ)dτ for t = ∆t, 2∆t, · · · , T = N∆t. We will oftern write C(tk) = C(k∆t) = Ck in short, and similarly for σ. Direct comput...