Abstract: | In this paper, we consider a product of a symmetric stable process in ? d and a one-dimensional Brownian motion in ??+?. Then we define a class of harmonic functions with respect to this product process. We show that bounded non-negative harmonic functions in the upper-half space satisfy Harnack inequality and prove that they are locally Hölder continuous. We also argue a result on Littlewood–Paley functions which are obtained by the α-harmonic extension of an L p (? d ) function. |