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

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

Published electronically:
April 2, 2012

MathSciNet review:
2957206

Full-text PDF

Abstract | References | Similar Articles | Additional Information

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.

**[A1]**D. R. Adams,*A note on Riesz potentials,*Duke Math. J.**42**(1975), no. 4, 765-778. MR**0458158 (56:16361)****[AH]**D. R. Adams and L. I. Hedberg,*Function spaces and potential theory,*Grundlehren der Mathematischen Wissenschaften**314**(1996), Springer-Verlag, Berlin. MR**1411441 (97j:46024)****[BK]**R. J. Bagby and D. S. Kurtz,*A rearranged good inequality,*Trans. Amer. Math. Soc.**293**(1986), 71-81 MR**814913 (87g:42033)****[B1]**R. Banuelos,*A sharp good- inequality with an application to Riesz transforms*, Michigan Math. J.**35**(1988), no. 1, 117-125. MR**931943 (89c:42023)****[CWW]**S.-Y. A. Chang, J. M. Wilson and T. H. Wolff,*Some weighted norm inequalities concerning the Schrödinger operators,*Comment. Math. Helv.**60**(1985), no. 2, 217-246. MR**800004 (87d:42027)****[CM]**A. Cianchi and V. G. Maz'ya,*Global Lipschitz regularity for a class of quasilinear elliptic equations*, Comm. PDE**36**(2011), 100-133. MR**2763349****[CV]**W. S. Cohn and I. E. Verbitsky,*Nonlinear potential theory on the ball, with applications to exceptional and boundary interpolation sets*, Michigan Math. J.**42**(1995), no. 1, 79-97. MR**1322190 (96b:32003)****[DMMOP]**G. Dal Maso, F. Murat, L. Orsina, and A. Prignet,*Renormalized solutions of elliptic equations with general measure data*, Ann. Scuola Norm. Sup. Pisa Cl. Sci.**28**(1999), no. 4, 741-808. MR**1760541 (2001d:35190)****[DM1]**F. Duzaar and G. Mingione,*Local Lipschitz regularity for degenerate elliptic systems*, Annales de l'Institut Henri Poincaré, Analyse Non Linéaire**27**(2010), no. 6, 1361-1396. MR**2738325 (2011i:35057)****[DM2]**F. Duzaar and G. Mingione,*Gradient estimates via linear and nonlinear potentials*, J. Funct. Anal.**259**(2010), no. 11, 2961-2998. MR**2719282****[DM3]**F. Duzaar and G. Mingione,*Gradient estimates via nonlinear potentials*, American J. Math. (to appear).**[HW]**L. I. Hedberg and T. H. Wolff,*Thin sets in nonlinear potential theory,*Ann. Inst. Fourier (Grenoble)**33**(1983), 161-187. MR**727526 (85f:31015)****[HKM]**J. Heinonen, T. Kilpeläinen, and O. Martio,*Nonlinear Potential Theory of Degenerate Elliptic Equations*, Dover Publications, 2006 (unabridged republication of 1993 edition, Oxford University Press). MR**1207810 (94e:31003)****[JPW]**B. Jawerth, C. Perez, and G. Welland,*The positive cone in Triebel-Lizorkin spaces and the relation among potential and maximal operators*, Harmonic analysis and partial differential equations (Boca Raton, FL, 1988), 71-91, Contemp. Math.,**107**, Amer. Math. Soc., Providence, RI, 1990. MR**1066471 (91m:42019)****[JV]**B. J. Jaye and I. E. Verbitsky,*Local and global behaviour of nonlinear equations with natural growth terms.*To appear in Arch. Rational Mech. Anal.**[KM]**T. Kilpeläinen and J. Malý,*The Wiener test and potential estimates for quasilinear elliptic equations,*Acta Math.**172**(1994), 137-161 MR**1264000 (95a:35050)****[Min07]**G. Mingione,*The Calderón-Zygmund theory for elliptic problems with measure data*, Ann. Sc. Norm. Super. Pisa**6**(2007), 195-261. MR**2352517 (2008i:35102)****[MW]**B. Muckenhoupt and R. Wheeden,*Weighted norm inequalities for fractional integrals,*Trans. Amer. Math. Soc.**192**(1974), 261-274. MR**0340523 (49:5275)****[PW]**C. Pérez and R. Wheeden,*Potential operators, maximal functions, and generalizations of*, Potential Anal.**19**(2003), no. 1, 1-33. MR**1962949 (2004a:42025)****[S1]**E. T. Sawyer,*Two weight norm inequalities for certain maximal and integral operators*, Harmonic analysis (Minneapolis, Minn., 1981), pp. 102-127, Lecture Notes in Math.,**908**, Springer, Berlin-New York, 1982. MR**654182 (83k:42020b)****[St1]**G. Stampacchia,*The spaces , and interpolation*, Ann. Scuola Norm. Sup. Pisa (3)**19**(1965), 443-462. MR**0199697 (33:7840)****[Ste1]**E. M. Stein,*Singular integrals and differentiability properties of functions*, Princeton Mathematical Series, No. 30, Princeton University Press, Princeton, NJ, 1970. MR**0290095 (44:7280)****[Vyb]**J. Vybíral,*A remark on better -inequality*, Math. Inequal. Appl.**10**(2007), no. 2, 335-341. MR**2312089 (2008d:31009)**

Retrieve articles in *Proceedings of the American Mathematical Society*
with MSC (2010):
42B35,
42B37,
35J92,
35J60

Retrieve articles in all journals with MSC (2010): 42B35, 42B37, 35J92, 35J60

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:
https://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.