On the good- inequality for nonlinear potentials

Authors:
Petr Honzík and Benjamin J. Jaye

Journal:
Proc. Amer. Math. Soc. **140** (2012), 4167-4180

MSC (2010):
Primary 42B35, 42B37; Secondary 35J92, 35J60

Published electronically:
April 2, 2012

MathSciNet review:
2957206

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Abstract: This paper concerns an extension of the good- 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- 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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Additional Information

**Petr Honzík**

Affiliation:
Institute of Mathematics, AS CR, Žitná 25, CZ - 115 67 Praha 1, Czech Republic

Email:
honzik@gmail.com

**Benjamin J. Jaye**

Affiliation:
Department of Mathematics, Kent State University, Kent, Ohio 44240

Email:
bjaye@kent.edu

DOI:
http://dx.doi.org/10.1090/S0002-9939-2012-11352-8

Keywords:
Weighted norm inequalities,
good-$𝜆$ inequality,
elliptic equations

Received by editor(s):
May 17, 2011

Published electronically:
April 2, 2012

Additional Notes:
The first author was supported by the Institutional Research Plan No. AV0Z10190503 and by grant KJB100190901 GAAV

The second author was partially supported by NSF grant DMS-0901550.

Communicated by:
Michael T. Lacey

Article copyright:
© Copyright 2012
American Mathematical Society

The copyright for this article reverts to public domain 28 years after publication.