Boundedness of Parametrized Littlewood-Paley Operators with Nondoubling Measures

Abstract

Let be a nonnegative Radon measure on which only satisfies the following growth condition that there exists a positive constant such that for all and some fixed . In this paper, the authors prove that for suitable indexes and , the parametrized function is bounded on for with the assumption that the kernel of the operator satisfies some Hörmander-type condition, and is bounded from into weak with the assumption that the kernel satisfies certain slightly stronger Hörmander-type condition. As a corollary, with the kernel satisfying the above stronger Hörmander-type condition is bounded on for . Moreover, the authors prove that for suitable indexes and is bounded from into (the space of regular bounded lower oscillation functions) if the kernel satisfies the Hörmander-type condition, and from the Hardy space into if the kernel satisfies the above stronger Hörmander-type condition. The corresponding properties for the parametrized area integral are also established in this paper.

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