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 \(\mathbb{R}^n, n\geq 2\) 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.