In this paper, we consider a product of a symmetric stable process in ℝd and a one-dimensional Brownian motion in 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 Hölder continuous. We also argue a result on Littlewood–Paley functions which are obtained by the α-harmonic extension of an L p(ℝd) function.
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Karlı, D. Harnack Inequality and Regularity for a Product of Symmetric Stable Process and Brownian Motion. Potential Anal 38, 95–117 (2013). https://doi.org/10.1007/s11118-011-9265-6
- Harnack inequality
- Symmetric stable process
- Dirichlet problem
- Littlewood–Paley functions
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