ABSTRACT

S O M E M O D I F I C A T I O N S O F H Y P E R S I N G U L A R

I N T E G R A L S A N D T H E I R A P P L I C A T I O N S

In all of the previous chapters we interpreted the hypersingular integrals as limits of the truncated integrals of the form (5.6), that is, as the limits

where Xe{y) — XRn\B(o,e){y) is the characteristic function of the exterior of the ball J3(0, e) = {y £ Rn : \y\ < c}. The first question arising is whether it is possible to use any other truncation than the spherical one. That is, may one take

Xe{y) - XR»\Ge{y),

where Ge is an arbitrary small neighbourhood of the origin, which tends to it when e —»• 0? The question of equivalence of this approach to the case of the spherical truncation is not trivial since we deal with a non-absolute convergence of the integrals. In Emgusheva and Nogin [52]-[53], [56] it was shown that in the case £l(y) EE 1 the convergence in (11.1) does not depend on the choice of the sets Ge 3 0 under the only assumption that

lim\G€C\K\ =0

for any compact set K C Rn (within the framework of the spaces Lpr(R n) studied

in Section 3 of Chapter 7).