  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.
