L p(·)–L q(·) boundedness of some integral operators obtained by extrapolation techniques

Abstract

Given a matrix A such that AM = I and 0 ≤ α < n, for an exponent p satisfying p(Ax) = p(x) for a.e. x ∈ Rn, using extrapolation techniques,we obtain Lp(·) → Lq(·) boundedness, 1/q(·) = 1/p(·) - α/n, andweak type estimates for integral operators of the form Tαf(x) = ∫ f(y)|x - A1y|α1⋯|x - Amy|αm dy, where A1,⋯, Am are different powers of A such that Ai - Aj is invertible for i ≠ j, α1 + ⋯ + αm = n - α. We give some generalizations of these results.