Boundedness properties of the maximal operator in a nonsymmetric inverse Gaussian setting

We introduce a generalized inverse Gaussian setting and consider the maximal operator associated with the natural analogue of a nonsymmetric Ornstein–Uhlenbeck semigroup. We prove that it is bounded on Lp when p∈(1,∞] and that it is of weak type (1, 1), with respect to the relevant measure. For small values of the time parameter t, the proof hinges on the “forbidden zones” method previously introduced in the Gaussian context. But for large times the proof requires new tools.