Certain convolution type operators on L_1 are nicely invertible

We will consider convolution operators acting on the L_1 space of functions defined on the unit circle equipped with the Lebesgue measure. The kernels will be related to a wide class of random variables having just a one "decent", e.g. absolutely continuous, bit. We will prove that the identity minus such an operator is nicely invertible on the subspace of functions with mean zero.