Some remarks on L-p-L-q estimates for some singular fractional integral operators

摘要

Let d >= 2 and let eta: Rd-1 -> R be a smooth function which is supported in [-1, 1.](d-1) Suppose mu is the measure on R-d given by mu (E) = integral(Rd-1) XE (x, phi(x))eta(x)dx with phi(x) = Sigma (d-1)(i=l) +/- vertical bar x(i)vertical bar(ai), 1 not equal a(i) is an element of R. In this paper we study the L-p-L-q estimates for singular fractional integral operators give by Af (x) = integral (Rd) f (x - y) (Pi(d-1)(i=1)vertical bar y(i)vertical bar(gamma i-1)) d mu(y) with 0 < y(i). (c) 2006 Elsevier Inc. All rights reserved.