On Weighted Norm Inequalities for Fractional and Singular Integrals

Abstract

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.

Weighted Norm Inequalities for Fractional Integrals

“…The result in n dimensions with V(x) = 1 was obtained by Sobolev in [8] and with V(x) = |x|a by Stein and G. Weiss in [10]. T. Walsh in [12] obtained a result for other weight functions and with a more general operator but did not characterize all such V 's.…”

“…In the casej? = 1, (1.1) should be interpreted to mean (12) (a¿IFWr*r(rB»TO)^* this is necessary and sufficient for a weak type inequality. This is also proved in § §2-4.…”