The boundary Harnack inequality for variable exponent $p$-Laplacian, Carleson estimates, barrier functions and $p(\cdot )$-harmonic measures

Abstract

We investigate various boundary decay estimates for $p(\cdot )$-harmonic functions. For domains in ${\mathbb{R}}^n,n\ge 2$ satisfying the ball condition ($C^{1,1}$-domains) we show the boundary Harnack inequality for $p(\cdot )$-harmonic functions under the assumption that the variable exponent $p$ is a bounded Lipschitz function. The proof involves barrier functions and chaining arguments. Moreover, we prove a Carleson type estimate for $p(\cdot )$-harmonic functions in NTA domains in ${\mathbb{R}}^n$ and provide lower- and upper- growth estimates and a doubling property for a $p(\cdot )$-harmonic measure.

31 pages, 1 figure