On the good inequality for nonlinear potentials
Authors:
Petr Honzík and Benjamin J. Jaye
Journal:
Proc. Amer. Math. Soc. 140 (2012), 41674180
MSC (2010):
Primary 42B35, 42B37; Secondary 35J92, 35J60
Published electronically:
April 2, 2012
MathSciNet review:
2957206
Fulltext 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 TriebelLizorkin spaces.
 [A1]
David
R. Adams, A note on Riesz potentials, Duke Math. J.
42 (1975), no. 4, 765–778. MR 0458158
(56 #16361)
 [AH]
David
R. Adams and Lars
Inge Hedberg, Function spaces and potential theory,
Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of
Mathematical Sciences], vol. 314, SpringerVerlag, Berlin, 1996. MR 1411441
(97j:46024)
 [BK]
Richard
J. Bagby and Douglas
S. Kurtz, A rearranged good 𝜆
inequality, Trans. Amer. Math. Soc.
293 (1986), no. 1,
71–81. MR
814913 (87g:42033), 10.1090/S00029947198608149137
 [B1]
Rodrigo
Bañuelos, A sharp good𝜆 inequality with an
application to Riesz transforms, Michigan Math. J. 35
(1988), no. 1, 117–125. MR 931943
(89c:42023), 10.1307/mmj/1029003685
 [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), 10.1007/BF02567411
 [CM]
Andrea
Cianchi and Vladimir
G. Maz’ya, Global Lipschitz regularity for a class of
quasilinear elliptic equations, Comm. Partial Differential Equations
36 (2011), no. 1, 100–133. MR 2763349
(2012f:35147), 10.1080/03605301003657843
 [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), 10.1307/mmj/1029005154
 [DMMOP]
Gianni
Dal Maso, François
Murat, Luigi
Orsina, and Alain
Prignet, Renormalized solutions of elliptic equations with general
measure data, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4)
28 (1999), no. 4, 741–808. MR 1760541
(2001d:35190)
 [DM1]
Frank
Duzaar and Giuseppe
Mingione, Local Lipschitz regularity for degenerate elliptic
systems, Ann. Inst. H. Poincaré Anal. Non Linéaire
27 (2010), no. 6, 1361–1396. MR 2738325
(2011i:35057), 10.1016/j.anihpc.2010.07.002
 [DM2]
Frank
Duzaar and Giuseppe
Mingione, Gradient estimates via linear and nonlinear
potentials, J. Funct. Anal. 259 (2010), no. 11,
2961–2998. MR 2719282
(2012a:31008), 10.1016/j.jfa.2010.08.006
 [DM3]
F. Duzaar and G. Mingione, Gradient estimates via nonlinear potentials, American J. Math. (to appear).
 [HW]
L.
I. Hedberg and Th.
H. Wolff, Thin sets in nonlinear potential theory, Ann. Inst.
Fourier (Grenoble) 33 (1983), no. 4, 161–187.
MR 727526
(85f:31015)
 [HKM]
Juha
Heinonen, Tero
Kilpeläinen, and Olli
Martio, Nonlinear potential theory of degenerate elliptic
equations, Oxford Mathematical Monographs, The Clarendon Press, Oxford
University Press, New York, 1993. Oxford Science Publications. MR 1207810
(94e:31003)
 [JPW]
B.
Jawerth, C.
Perez, and G.
Welland, The positive cone in TriebelLizorkin spaces and the
relation among potential and maximal operators, Harmonic analysis and
partial differential equations (Boca Raton, FL, 1988), Contemp. Math.,
vol. 107, Amer. Math. Soc., Providence, RI, 1990,
pp. 71–91. MR 1066471
(91m:42019), 10.1090/conm/107/1066471
 [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]
Tero
Kilpeläinen and Jan
Malý, The Wiener test and potential estimates for
quasilinear elliptic equations, Acta Math. 172
(1994), no. 1, 137–161. MR 1264000
(95a:35050), 10.1007/BF02392793
 [Min07]
Giuseppe
Mingione, The CalderónZygmund theory for elliptic problems
with measure data, Ann. Sc. Norm. Super. Pisa Cl. Sci. (5)
6 (2007), no. 2, 195–261. MR 2352517
(2008i:35102)
 [MW]
Benjamin
Muckenhoupt and Richard
Wheeden, Weighted norm inequalities for
fractional integrals, Trans. Amer. Math.
Soc. 192 (1974),
261–274. MR 0340523
(49 #5275), 10.1090/S00029947197403405236
 [PW]
Carlos
Pérez and Richard
L. Wheeden, Potential operators, maximal functions, and
generalizations of 𝐴_{∞}, Potential Anal.
19 (2003), no. 1, 1–33. MR 1962949
(2004a:42025), 10.1023/A:1022449810008
 [S1]
Eric
T. Sawyer, Two weight norm inequalities for certain maximal and
integral operators, Harmonic analysis (Minneapolis, Minn., 1981)
Lecture Notes in Math., vol. 908, Springer, BerlinNew York, 1982,
pp. 102–127. MR 654182
(83k:42020b)
 [St1]
Guido
Stampacchia, The spaces \cal𝐿^{(𝑝,𝜆)},
𝑁^{(𝑝,𝜆)} and interpolation, Ann. Scuola Norm.
Sup. Pisa (3) 19 (1965), 443–462. MR 0199697
(33 #7840)
 [Ste1]
Elias
M. Stein, Singular integrals and differentiability properties of
functions, Princeton Mathematical Series, No. 30, Princeton University
Press, Princeton, N.J., 1970. MR 0290095
(44 #7280)
 [Vyb]
Jan
Vybíral, A remark on better 𝜆inequality, Math.
Inequal. Appl. 10 (2007), no. 2, 335–341. MR 2312089
(2008d:31009), 10.7153/mia1030
 [A1]
 D. R. Adams, A note on Riesz potentials, Duke Math. J. 42 (1975), no. 4, 765778. MR 0458158 (56:16361)
 [AH]
 D. R. Adams and L. I. Hedberg, Function spaces and potential theory, Grundlehren der Mathematischen Wissenschaften 314 (1996), SpringerVerlag, Berlin. MR 1411441 (97j:46024)
 [BK]
 R. J. Bagby and D. S. Kurtz, A rearranged good inequality, Trans. Amer. Math. Soc. 293 (1986), 7181 MR 814913 (87g:42033)
 [B1]
 R. Banuelos, A sharp good inequality with an application to Riesz transforms, Michigan Math. J. 35 (1988), no. 1, 117125. 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, 217246. 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), 100133. 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, 7997. 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, 741808. 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, 13611396. MR 2738325 (2011i:35057)
 [DM2]
 F. Duzaar and G. Mingione, Gradient estimates via linear and nonlinear potentials, J. Funct. Anal. 259 (2010), no. 11, 29612998. 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), 161187. 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 TriebelLizorkin spaces and the relation among potential and maximal operators, Harmonic analysis and partial differential equations (Boca Raton, FL, 1988), 7191, 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), 137161 MR 1264000 (95a:35050)
 [Min07]
 G. Mingione, The CalderónZygmund theory for elliptic problems with measure data, Ann. Sc. Norm. Super. Pisa 6 (2007), 195261. MR 2352517 (2008i:35102)
 [MW]
 B. Muckenhoupt and R. Wheeden, Weighted norm inequalities for fractional integrals, Trans. Amer. Math. Soc. 192 (1974), 261274. MR 0340523 (49:5275)
 [PW]
 C. Pérez and R. Wheeden, Potential operators, maximal functions, and generalizations of , Potential Anal. 19 (2003), no. 1, 133. MR 1962949 (2004a:42025)
 [S1]
 E. T. Sawyer, Two weight norm inequalities for certain maximal and integral operators, Harmonic analysis (Minneapolis, Minn., 1981), pp. 102127, Lecture Notes in Math., 908, Springer, BerlinNew York, 1982. MR 654182 (83k:42020b)
 [St1]
 G. Stampacchia, The spaces , and interpolation, Ann. Scuola Norm. Sup. Pisa (3) 19 (1965), 443462. 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, 335341. MR 2312089 (2008d:31009)
Similar Articles
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:
http://dx.doi.org/10.1090/S000299392012113528
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 DMS0901550.
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.
