Abstract: | Here we consider the following strongly singular integral TΩ,γ,α,βf(x,t)=∫Rnei|y|-βΩ(y/|y|)f(x-y,t-y(|y|))dy,where Ω∈ LP (Sn-1),p > 1,n > 1,α > 0 and γ is convex on (0,co).We prove that there exists A(p,n) > 0 such that if β > A(p,n) (1 +α),then TΩ,γ,α,β is bounded from L2 (Rn+1) to itself and the constant is independent of γ.Furthermore,when Ω ∈ C∞(sn-1),we will show that TΩ,γ,α,β is bounded from L2 (Rn+1) to itself only if β > 2α and the constant is independent of γ. |