Harnack Inequality and Regularity for a Product of Symmetric Stable Process and Brownian Motion

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.