  In this paper, we consider a product of a symmetric stable process in
$\mathbb{R}^d$ and a one-dimensional Brownian motion in $\mathbb{R}^+$. 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 H\"older continuous.
We also argue a result on Littlewood-Paley functions which are obtained by the
$\alpha$-harmonic extension of an $L^p(\mathbb{R}^d)$ function.
