On $L\sp{p}$ estimates for integral transforms

Abstract

Abstract.In a recent paper R. S. Strichartz has extended and simplified the proofs of a few well-known results about integral operators with positive kernels and singular integral operators. The present paper extends some of his results. An inequality of Kantorovic for integral operators with positive kernel is extended to kernels satisfying two mixed weak V estimates. The "method of rotation" of Calderón and Zygmund is applied to singular integral operators with Banach space valued kernels. Another short proof of the fractional integration theorem in weighted norms is given. It is proved that certain sufficient conditions on the exponents of the V spaces and weight functions involved are necessary. It is shown that the integrability conditions on the kernel required for boundedness of singular integral operators in weighted V spaces can be weakened. Some implications for integral operators in Rn of Young's inequality for convolutions on the multiplicative group of positive real numbers are considered. Throughout special attention is given to restricted weak type estimates at the endpoints of the permissible intervals for the exponents.

On Weighted Norm Inequalities for Fractional and Singular Integrals

“…it follows similarly, by use of the obvious generalization of[19, Lemma 3 and Corollary] from the case of product measures to the more general types of measure denned in (41), that or q = oo , this is still valid provided that the left-hand side is replaced by \\Si(Qi, a>o, coi)||pi t(7co or if, instead, the right-hand side is replaced by ||0|| r .…”

A note on a paper of E.R. Love

“…In case of Lorentz norms there exist quite different proofs for the boundedness of the operator I, (compare [73]) using theorems of where 1/p = l / p 0 + 1/p-, , and B i s a suitable constant, as for example B = {p/p n ) p/(p-l) . For the particular case p = q of (10), that is the Lebesgue case, see for example [12].…”

Weighted norm estimates for the Semyanistyi fractional integrals and Radon transforms

“…Our approach was inspired by a series of works on operators with homogeneous kernels; see, e.g., [8,23,27,29]. A common point for this class of operators and the afore-mentioned operators of integral geometry is a nice behavior with respect to rotations and dilations.…”