Boundedness of oscillatory hyper-Hilbert transform along curves on Sobolev spaces

The oscillatory hyper-hilbert transform along curves is of the following form: h n,β f (x) = ∫ 0 1 f (x - γ (t)) e i t - β t - 1 - d t,where ≥ 0,β ≥ 0,and γ (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. the l p bounds for h n,β have been given by chen et al. (2008 and 2010). in this paper,for some,β,and p,the boundedness of h n,β on sobolev spaces l s p (r n) and the boundedness of this operator from l s 2 (r n) to l 2 (r n) are obtained. © 2014 jun li and guilian gao.