Singular Integral - Singular Integrals of Convolution Type

Singular Integrals of Convolution Type

A singular integral of convolution type is an operator T defined by convolution with a kernel K that is locally integrable on Rn\{0}, in the sense that

(1)

Suppose that the kernel satisfies:

1. The size condition on the Fourier transform of K

2. The smoothness condition: for some C > 0,

Then it can be shown that T is bounded on Lp(Rn) and satisfies a weak-type (1, 1) estimate.

Property 1. is needed to ensure that convolution (1) with the tempered distribution p.v. K given by the principal value integral

is a well-defined Fourier multiplier on L2. Neither of the properties 1. or 2. is necessarily easy to verify, and a variety of sufficient conditions exist. Typically in applications, one also has a cancellation condition

which is quite easy to check. It is automatic, for instance, if K is an odd function. If, in addition, one assumes 2. and the following size condition

then it can be shown that 1. follows.

The smoothness condition 2. is also often difficult to check in principle, the following sufficient condition of a kernel K can be used:

Observe that these conditions are satisfied for the Hilbert and Riesz transforms, so this result is an extension of those result.