A rough hypersingular integral operator with an oscillating factor

Abstract

We study certain hypersingular integrals T Ω , α , β f defined on all test functions f ∈ S ( R n ) , where the kernel of the operator T Ω , α , β has a strong singularity | y | − n − α ( α > 0 ) at the origin, an oscillating factor e i | y | − β ( β > 0 ) and a distribution Ω ∈ H r ( S n − 1 ) , 0 < r < 1 . We show that T Ω , α , β extends to a bounded linear operator from the Sobolev space L ˙ γ p ∩ L p to the Lebesgue space L p for β / ( β − α ) < p < β / α , if the distribution Ω is in the Hardy space H r ( S n − 1 ) with 0 < r = ( n − 1 ) / ( n − 1 + γ ) ( 0 < γ ⩽ α ) and β > 2 α > 0 .