Super Singularities Along The Curve Hilbert Transform Boundedness

This paper is a summary report which is focused on introducing the boundedness of hyper singular Hilbert transform along curves in function space.As we know that the boundedness of singular integral operators is a very hot topic in Harmonic analysis:hyper singular integral operators is one of the important operators which many scientists study.Firstly, we come from the Hilbert transform, then discuss the Hilbert transform along curves We display the new work about hyper singular integral operators along curves in low and high Euclidean spaces At last, we put an extension to modulation spaces.