Some remarks on Lp–Lq estimates for some singular fractional integral operators

Let d 2 and let η :Rd−1 →R be a smooth function which is supported in [−1, 1]d−1. Suppose μ is the measure on Rd given by μ(E) = Rd−1 χE x,ϕ(x) η(x) dx with ϕ(x) = d−1 i=1 ±|xi |ai, 1 = ai ∈ R. In this paper we study the Lp–Lq estimates for singular fractional integral operators given by Af (x) = Rd f (x − y) d−1 i=1 |yi |γi−1 dμ(y) with 0<γi . © 2006 Elsevier Inc. All rights reserved