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Harmonic functions on balls for x-dependent rectilinear stable processes

Tadeusz Kulczycki, Michał Ryznar

Abstract

We obtain sharp estimates for functions harmonic with respect to $x$-dependent rectilinear stable processes in balls, under the assumption that the Dirichlet exterior data are radial about the center. The main idea of the proof is based on the construction of global barrier functions for the $x$-dependent rectilinear fractional Laplacian in balls.

Harmonic functions on balls for x-dependent rectilinear stable processes

Abstract

We obtain sharp estimates for functions harmonic with respect to -dependent rectilinear stable processes in balls, under the assumption that the Dirichlet exterior data are radial about the center. The main idea of the proof is based on the construction of global barrier functions for the -dependent rectilinear fractional Laplacian in balls.
Paper Structure (5 sections, 23 theorems, 259 equations)

This paper contains 5 sections, 23 theorems, 259 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Proof of the main result
  4. Comparison inequalities for $\mathcal{L}_{e_d}$
  5. Appendix

Key Result

Theorem 1.1

Let $z \in \mathds{R}^d$, $r > 0$, $D = B(z,r)$ and $g: D^c \to \mathds{R}$ be a bounded Borel function, which is radial with respect to $z$. Denote by $\tilde{g}$ the radial profile function of $g$. Let $u$ be the function harmonic with respect to $X$ on $D$, which is given by (Dirichlet). For each Moreover, there exist constants $C_1, C_2 >0$ which depend only on $\alpha$, $d$, $\eta_1$, $\eta_2

Theorems & Definitions (47)

  • Theorem 1.1
  • Remark 1.2
  • Lemma 2.1
  • proof
  • Theorem 3.1
  • proof
  • Corollary 3.2
  • proof
  • Corollary 3.3
  • Definition 3.4
  • ...and 37 more