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

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 Kantorovič for integral operators with positive kernel is extended to kernels satisfying two mixed weak ${L^p}$ 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 ${L^p}$ 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 ${L^p}$ spaces can be weakened. Some implications for integral operators in ${R^n}$ 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.