The boundary Harnack inequality for variable exponent $$p$$ p -Laplacian, Carleson estimates, barrier functions and $${p(\cdot )}$$ p ( · ) -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.