WIENER-TYPE CRITERION OF BOUNDARY HÖLDER REGULARITY OF FRACTIONAL LAPLACIAN

WIENER-TYPE CRITERION OF BOUNDARY HÖLDER REGULARITY OF FRACTIONAL LAPLACIAN

FENG LI

Abstract. It’s now well-known that with smooth enough boundary assumption the fractional harmonic functions on domains are Hölder continuous up to the boundary. In this tail we characterize a sharp Wiener- type boundary condition of Hölder regularity up to the boundary for fractional harmonic functions through Caffarelli-Silvestre extension. Briefly we correlate the uniformly capacity density of boundary points with fractional harmonic measure decay on regular boundary points. The fractional harmonic measure can be derived in two approaches, one is by Poisson formula, and the other is based on Caffarelli-Silvestre extension method by reducing harmonic measure for weighted laplacian operators to lower dimension fractional laplacian operators. In this talk, we mainly focus on the second way.

This is a joint work with Kaj Nyström in Uppsala University.