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

The aim of this paper is to study singular integrals T generated by holomorphic kernels defined on a natural neighbourhood of the set ${\displaystyle \{z{\zeta }^{-1}:z,\zeta \in ,z\ne \zeta \}}$, where is a star-shaped Lipschitz curve, ${\displaystyle =\{\exp \left(iz\right):z=x+iA\left(x\right),A\prime \in L^{\infty }[-\pi ,\pi ],A(-\pi )=A(\pi )\}}$. Under suitable conditions on F and z, the operators are given by (1) ${\displaystyle TF\left(z\right)=p.v.{\int }_{z{\eta }^{-1}}F(\eta )\left(d\frac{\eta }{\eta }\right).}$ We identify a class of kernels of the stated type that give rise to bounded operators on ${\displaystyle L^2(,\left|d\right|)}$. We establish also transference results relating the boundedness of kernels on closed Lipschitz curves to corresponding results on periodic, unbounded curves.