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.
In a recent paper [12] Muckenhoupt and Wheeden have established necessary and sufficient conditions for the validity of norm inequalities of the form ‖ |x|αTƒ ‖q ≦ C‖ |x|αƒ ‖p, where Tƒ denotes a Calderón and Zygmund singular integral of ƒ or a fractional integral with variable kernel. The purpose of the present paper is to prove, by somewhat different methods, similar inequalities for more general weight functions.In what follows, for p ≧ 1, p′ is the exponent conjugate to p, given by l/p + l/p′ = 1. Ω will always denote a locally integrable function on Rn which is homogeneous of degree 0, Ω∼ will denote a measurable function on Rn × Rn such that for each x ∈ Rn, Ω∼(x, .) is locally integrable and homogeneous of degree 0.
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