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Boundedness of singular integral operators with holomorphic kernels on star-shaped closed Lipschitz curves

Garth GaudryTao QianSilei Wang — 1996

Colloquium Mathematicae

The aim of this paper is to study singular integrals T generated by holomorphic kernels defined on a natural neighbourhood of the set z ζ - 1 : z , ζ , z ζ , where is a star-shaped Lipschitz curve, = e x p ( i z ) : z = x + i A ( x ) , A ' L [ - π , π ] , A ( - π ) = A ( π ) . Under suitable conditions on F and z, the operators are given by (1) T F ( z ) = p . v . ( z η - 1 ) F ( η ) ( d η / η ) . We identify a class of kernels of the stated type that give rise to bounded operators on L 2 ( , | d | ) . We establish also transference results relating the boundedness of kernels on closed Lipschitz curves to corresponding results on periodic, unbounded curves.

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