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The boundedness of singular integral operator along an open curve

Authors:	Wang Xiaolin Min Honglin

Institution:	(1) College of Mathematical Sciences, Wuhan University, 430072 Wuhan, China

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 fromH ^μ(L) intoH ^μ(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. Supported by the National Science Foundation of China Wang Xiaolin: born in Aug. 1950, Professor

Keywords:	singular integral operator " target="_blank"> gif" alt=" $$H_{\lambda _1 \lambda _2 }^\mu $$ " target="_blank">" align="middle" border="0"> space boundedness
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