On the good-𝜆 inequality for nonlinear potentials

Honzík, Petr; Jaye, Benjamin

doi:10.1090/S0002-9939-2012-11352-8

On the good-$\lambda$ inequality for nonlinear potentials
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by Petr Honzík and Benjamin J. Jaye PDF

Proc. Amer. Math. Soc. 140 (2012), 4167-4180 Request permission

Abstract:

This paper 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.

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