Boundedness of Oscillatory Hyper-Hilbert Transform along Curves on Sobolev Spaces

Abstract

The oscillatory hyper-Hilbert transform along curves is of the following form: Hn,α,βf(x)=∫01‍f(x-Γ(t))eit-βt-1-αdt, where α≥0, β≥0, and Γ(t)=(tp1,tp2,…,tpn). The study on this operator is motivated by the hyper-Hilbert transform and the strongly singular integrals. The Lp bounds for Hn,α,β have been given by Chen et al. (2008 and 2010). In this paper, for some α, β, and p, the boundedness of Hn,α,β on Sobolev spaces Lsp(Rn) and the boundedness of this operator from Ls2(Rn) to L2(Rn) are obtained.