On the Good-$\lambda$ inequality for nonlinear potentials

TL;DR

This paper extends the good-$\lambda$ inequality to nonlinear potentials, showing exponential decay of constants and deriving new integrability and embedding results for solutions to quasilinear elliptic equations.

Contribution

It generalizes the good-$\lambda$ inequality to nonlinear potentials and proves exponential decay of the constant, leading to new integrability and embedding results.

Findings

Exponential decay of the inequality constant.

Exponential integrability of gradients in quasilinear elliptic equations.

Extension of Morrey space embeddings for nonlinear potentials.

Abstract

This note concerns an extension of the good-$\lambda$ inequality for fractional integrals, due to B. Muckenhoupt and R. Wheeden. The classical result is refined in two aspects. Firstly, general nonlinear potentials are considered; and secondly, the constant in the inequality is proven to decay exponentially. As a consequence, the exponential integrability of the gradient of solutions to certain quasilinear elliptic equations is deduced. This in turn is a consequence of certain Morrey space embeddings which extend classical results for the Riesz potential. In addition, the good-$\lambda$ inequality proved here provides an elementary proof of the result of Jawerth, Perez and Welland regarding the positive cone in certain weighted Triebel-Lizorkin spaces.