A rough hypersingular integral operator with an oscillating factor

We study certain hypersingular integrals TΩ,α,βf defined on all test functions f ∈S(Rn), where the kernel of the operator TΩ,α,β has a strong singularity |y|−n−α (α > 0) at the origin, an oscillating factor ei|y|−β (β > 0) and a distribution Ω ∈ Hr (Sn−1), 0 < r <1. We show that TΩ,α,β extends to a bounded linear operator from the Sobolev space L˙ pγ ∩ Lp to the Lebesgue space Lp for β/(β − α) < p < β/α, if the distribution Ω is in the Hardy space Hr (Sn−1) with 0 < r = (n − 1)/(n − 1 + γ ) (0 < γ α) and β >2α >0. © 2005 Elsevier Inc. All rights reserved