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.
Read more about this topic: Singular Integral
Famous quotes containing the words singular and/or type:
“It is singular how soon we lose the impression of what ceases to be constantly before us. A year impairs, a lustre obliterates. There is little distinct left without an effort of memory, then indeed the lights are rekindled for a momentbut who can be sure that the Imagination is not the torch-bearer?”
—George Gordon Noel Byron (17881824)
“The Republican form of government is the highest form of government; but because of this it requires the highest type of human naturea type nowhere at present existing.”
—Herbert Spencer (18201903)