Boundedness of Oscillatory Hyper-Hilbert Transform along Curves on Sobolev Spaces. (19th May 2014) - Li, Jun; Gao, Guilian
Abstract : The oscillatory hyper-Hilbert transform along curves is of the following form:H n, α, β f ( x ) = ∫ 0 1 ‍ f ( x - Γ ( t ) ) e i t - β t - 1 - α d t, whereα ≥ 0, β ≥ 0, andΓ ( t ) = ( t p 1, t p 2, …, t p n ) . The study on this operator is motivated by the hyper-Hilbert transform and the strongly singular integrals. TheL p bounds forH n, α, β have been given by Chen et al. (2008 and 2010). In this paper, for someα, β, andp, the boundedness ofH n, α, β on Sobolev spacesL s p ( R n ) and the boundedness of this operator fromL s 2 ( R n ) toL 2 ( R n ) are obtained.