The boundedness of singular integral operator along an open curve

Abstract

It is well known that, the singular integral operatorS defined as:\(\left( {S\varphi } \right)\left( {t_0 } \right) = \frac{1}{{\pi i}}\int {_L \frac{{\varphi \left( t \right)}}{{t - t_0 }}} dt \left( {t_0 \in L} \right)\) ifL is a closed smooth contour in the complex plane C, thenS is a bounded linear operator fromHμ(L) intoHμ(L): ifL is an open smooth curve, thenS is just a linear operator fromH* intoH*. In this paper, we define a Banach space\(H_{\lambda _1 , \lambda _2 }^\mu \), and prove that\(S:H_{\lambda _1 , \lambda _2 }^\mu \to H_{\lambda _1 , \lambda _2 }^\mu \) is a bounded linear operator, then verify the boundedness of other kinds of singular integral operators.