Along The Curve Of Singular Integral Operator Bounded

We mainly study IP bounds of hypersingular integral operatoralong the curve Γ(t) = (t,γ(t)) in R², we also get some results on hypersingular integral operatoralong the variable curve Γ_x(t) — (t,P(x)γ(t)) in R², where P(x) is a real-valued polynomial of degree n.Our paper is composed of three chapters. In the first chapter, we have an introduction to the history of the above two operators, our main results, and some important lemmas and propositions.In the second chapter, we study IP bounds of the operator H_α,β in two methods, this work improves the results of Chandarana [5] and Chen-Fan-Wang [7]. Furthermore, the second method will help us consider the operator T_α,β.In the third chapter, we get some results of hypersingular operator along variable curves.