A Multiplier Related to Symmetric Stable Processes

In two recent papers [5] and [6], we generalized some classical results of Harmonic Analysis using probabilistic approach by means of a d- dimensional rotationally symmetric stable process. These results allow one to discuss some boundedness conditions with weaker hypotheses. In this paper, we study a multiplier theorem using these more general results. We consider a product process consisting of a d-dimensional symmetric stable process and a 1-dimensional Brownian motion, and use properties of jump processes to obtain bounds on jump terms and the L^p(R^d)-norm of a new operator.