Nonlinear equations and weighted norm inequalities
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- by N. J. Kalton and I. E. Verbitsky PDF
- Trans. Amer. Math. Soc. 351 (1999), 3441-3497 Request permission
Abstract:
We study connections between the problem of the existence of positive solutions for certain nonlinear equations and weighted norm inequalities. In particular, we obtain explicit criteria for the solvability of the Dirichlet problem \[ \begin {aligned} - \Delta u & = v u^{q} + w, \quad u \ge 0 && \text {on $\Omega $},\\ u &= 0 & \text {on $\partial \Omega $}, \end {aligned} \] on a regular domain $\Omega$ in $\mathbf {R}^{n}$ in the “superlinear case” $q > 1$. The coefficients $v, w$ are arbitrary positive measurable functions (or measures) on $\Omega$. We also consider more general nonlinear differential and integral equations, and study the spaces of coefficients and solutions naturally associated with these problems, as well as the corresponding capacities. Our characterizations of the existence of positive solutions take into account the interplay between $v$, $w$, and the corresponding Green’s kernel. They are not only sufficient, but also necessary, and are established without any a priori regularity assumptions on $v$ and $w$; we also obtain sharp two-sided estimates of solutions up to the boundary. Some of our results are new even if $v \equiv 1$ and $\Omega$ is a ball or half-space. The corresponding weighted norm inequalities are proved for integral operators with kernels satisfying a refined version of the so-called $3 G$-inequality by an elementary “integration by parts” argument. This also gives a new unified proof for some classical inequalities including the Carleson measure theorem for Poisson integrals and trace inequalities for Riesz potentials and Green potentials.References
- David R. Adams, Weighted nonlinear potential theory, Trans. Amer. Math. Soc. 297 (1986), no. 1, 73–94. MR 849468, DOI 10.1090/S0002-9947-1986-0849468-4
- David R. Adams and Lars Inge Hedberg, Function spaces and potential theory, Grundlehren der mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 314, Springer-Verlag, Berlin, 1996. MR 1411441, DOI 10.1007/978-3-662-03282-4
- D. R. Adams and M. Pierre, Capacitary strong type estimates in semilinear problems, Ann. Inst. Fourier (Grenoble) 41 (1991), no. 1, 117–135 (English, with French summary). MR 1112194, DOI 10.5802/aif.1251
- H. Aikawa and M. Essén, Topics in Potential Theory, Lecture Notes, Uppsala Univ., no. 1, 1994.
- A. Ancona, Comparison of Green’s functions for elliptic operators on manifolds or domains, Preprint 96-11, Université de Paris-Sud, 1996, pp. 1-42.
- Pierre Baras and Michel Pierre, Critère d’existence de solutions positives pour des équations semi-linéaires non monotones, Ann. Inst. H. Poincaré Anal. Non Linéaire 2 (1985), no. 3, 185–212 (French, with English summary). MR 797270, DOI 10.1016/S0294-1449(16)30402-4
- Richard F. Bass, Probabilistic techniques in analysis, Probability and its Applications (New York), Springer-Verlag, New York, 1995. MR 1329542
- Haïm Brézis and Louis Nirenberg, Positive solutions of nonlinear elliptic equations involving critical Sobolev exponents, Comm. Pure Appl. Math. 36 (1983), no. 4, 437–477. MR 709644, DOI 10.1002/cpa.3160360405
- H. Brezis and X. Cabre, Some simple nonlinear PDE’s without solutions, preprint (1997), 1-37; Boll. Unione Mat. Ital. Sez. B Artic. Ric. Mat. (8) 1 (1998), 223–262.
- Lennart Carleson, Selected problems on exceptional sets, Van Nostrand Mathematical Studies, No. 13, D. Van Nostrand Co., Inc., Princeton, N.J.-Toronto, Ont.-London, 1967. MR 0225986
- Ronald R. Coifman and Guido Weiss, Extensions of Hardy spaces and their use in analysis, Bull. Amer. Math. Soc. 83 (1977), no. 4, 569–645. MR 447954, DOI 10.1090/S0002-9904-1977-14325-5
- M. Cranston, E. Fabes, and Z. Zhao, Conditional gauge and potential theory for the Schrödinger operator, Trans. Amer. Math. Soc. 307 (1988), no. 1, 171–194. MR 936811, DOI 10.1090/S0002-9947-1988-0936811-2
- Kai Lai Chung and Zhong Xin Zhao, From Brownian motion to Schrödinger’s equation, Grundlehren der mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 312, Springer-Verlag, Berlin, 1995. MR 1329992, DOI 10.1007/978-3-642-57856-4
- E. B. Dynkin, Superprocesses and partial differential equations, Ann. Probab. 21 (1993), no. 3, 1185–1262. MR 1235414, DOI 10.1214/aop/1176989116
- E. B. Dynkin and S. E. Kuznetsov, Superdiffusions and removable singularities for quasilinear partial differential equations, Comm. Pure Appl. Math. 49 (1996), no. 2, 125–176. MR 1371926, DOI 10.1002/(SICI)1097-0312(199602)49:2<125::AID-CPA2>3.0.CO;2-G
- C. Fefferman, The uncertainty principle, Bull. Amer. Math. Soc. 9 (1983), 129-206.
- Michael Frazier, Björn Jawerth, and Guido Weiss, Littlewood-Paley theory and the study of function spaces, CBMS Regional Conference Series in Mathematics, vol. 79, Published for the Conference Board of the Mathematical Sciences, Washington, DC; by the American Mathematical Society, Providence, RI, 1991. MR 1107300, DOI 10.1090/cbms/079
- José García-Cuerva and José L. Rubio de Francia, Weighted norm inequalities and related topics, North-Holland Mathematics Studies, vol. 116, North-Holland Publishing Co., Amsterdam, 1985. Notas de Matemática [Mathematical Notes], 104. MR 807149
- J. Garnett, Bounded Analytic Functions, Academic Press, New York-London-Toronto, 1981.
- A. Eduardo Gatto, Carlos Segovia, and Stephen Vági, On fractional differentiation and integration on spaces of homogeneous type, Rev. Mat. Iberoamericana 12 (1996), no. 1, 111–145. MR 1387588, DOI 10.4171/RMI/196
- Kurt Hansson, Imbedding theorems of Sobolev type in potential theory, Math. Scand. 45 (1979), no. 1, 77–102. MR 567435, DOI 10.7146/math.scand.a-11827
- K. Hansson, V. G. Maz$’$ya and I. E. Verbitsky, Criteria of solvability for multidimensional Riccati’s equations, to appear in Arkiv för Matem.
- 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, DOI 10.5802/aif.944
- 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
- H. Hueber and M. Sieveking, Uniform bounds for quotients of Green functions on $C^{1,1}$-domains, Ann. Inst. Fourier (Grenoble) 32 (1982), no. 1, vi, 105–117 (English, with French summary). MR 658944, DOI 10.5802/aif.861
- N. J. Kalton and L. Tzafriri, Legendre and Jacobi polynomials in $L_{p}$ spaces and the $\Lambda _{p}$ problem, in preparation.
- R. Kerman and E. Sawyer, The trace inequality and eigenvalue estimates for Schrödinger operators, Ann. Inst. Fourier (Grenoble) 36 (1986), no. 4, 207–228 (English, with French summary). MR 867921, DOI 10.5802/aif.1074
- J. Komlós, A generalization of a problem of Steinhaus, Acta Math. Acad. Sci. Hungar. 18 (1967), 217–229. MR 210177, DOI 10.1007/BF02020976
- M. A. Krasnosel′skiĭ and P. P. Zabreĭko, Geometrical methods of nonlinear analysis, Grundlehren der mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 263, Springer-Verlag, Berlin, 1984. Translated from the Russian by Christian C. Fenske. MR 736839, DOI 10.1007/978-3-642-69409-7
- J.-F. Le Gall, A class of path-valued Markov processes and its applications to superprocesses, Probab. Theory Related Fields 95 (1993), no. 1, 25–46. MR 1207305, DOI 10.1007/BF01197336
- Joram Lindenstrauss and Lior Tzafriri, Classical Banach spaces. II, Ergebnisse der Mathematik und ihrer Grenzgebiete [Results in Mathematics and Related Areas], vol. 97, Springer-Verlag, Berlin-New York, 1979. Function spaces. MR 540367, DOI 10.1007/978-3-662-35347-9
- P.-L. Lions, On the existence of positive solutions of semilinear elliptic equations, SIAM Rev. 24 (1982), no. 4, 441–467. MR 678562, DOI 10.1137/1024101
- Bernard Maurey, Théorèmes de factorisation pour les opérateurs linéaires à valeurs dans les espaces $L^{p}$, Astérisque, No. 11, Société Mathématique de France, Paris, 1974 (French). With an English summary. MR 0344931
- V. G. Maz$’$ya, On the theory of the $n$-dimensional Schrödinger operator, Izv. Akad. Nauk SSSR, Ser. Matem. 28 (1964), 1145-1172.
- Vladimir G. Maz’ja, Sobolev spaces, Springer Series in Soviet Mathematics, Springer-Verlag, Berlin, 1985. Translated from the Russian by T. O. Shaposhnikova. MR 817985, DOI 10.1007/978-3-662-09922-3
- V. G. Maz$’$ya, Beurling’s theorem on a minimum principle for positive harmonic functions, Zapiski Nauchn. Sem. LOMI 30 (1972), 76–90; English transl. in. J. Soviet Math. 4 (1972), 367–379.
- Vladimir G. Maz′ya and Igor E. Verbitsky, Capacitary inequalities for fractional integrals, with applications to partial differential equations and Sobolev multipliers, Ark. Mat. 33 (1995), no. 1, 81–115. MR 1340271, DOI 10.1007/BF02559606
- Linda Naïm, Sur le rôle de la frontière de R. S. Martin dans la théorie du potentiel, Ann. Inst. Fourier (Grenoble) 7 (1957), 183–281 (French). MR 100174, DOI 10.5802/aif.70
- E. M. Nikishin, Resonance theorems and superlinear operators, Russian Math. Surveys 25 (1970), 124-187.
- N. K. Nikol′skiĭ, Treatise on the shift operator, Grundlehren der mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 273, Springer-Verlag, Berlin, 1986. Spectral function theory; With an appendix by S. V. Hruščev [S. V. Khrushchëv] and V. V. Peller; Translated from the Russian by Jaak Peetre. MR 827223, DOI 10.1007/978-3-642-70151-1
- Carlos Pérez, Two weighted norm inequalities for Riesz potentials and uniform $L^p$-weighted Sobolev inequalities, Indiana Univ. Math. J. 39 (1990), no. 1, 31–44. MR 1052009, DOI 10.1512/iumj.1990.39.39004
- José L. Rubio de Francia, Weighted norm inequalities and vector valued inequalities, Harmonic analysis (Minneapolis, Minn., 1981) Lecture Notes in Math., vol. 908, Springer, Berlin-New York, 1982, pp. 86–101. MR 654181
- Eric T. Sawyer, Weighted norm inequalities for fractional maximal operators, 1980 Seminar on Harmonic Analysis (Montreal, Que., 1980) CMS Conf. Proc., vol. 1, Amer. Math. Soc., Providence, R.I., 1981, pp. 283–309. MR 670111
- Eric T. Sawyer, A characterization of two weight norm inequalities for fractional and Poisson integrals, Trans. Amer. Math. Soc. 308 (1988), no. 2, 533–545. MR 930072, DOI 10.1090/S0002-9947-1988-0930072-6
- E. Sawyer and R. L. Wheeden, Weighted inequalities for fractional integrals on Euclidean and homogeneous spaces, Amer. J. Math. 114 (1992), no. 4, 813–874. MR 1175693, DOI 10.2307/2374799
- Eric T. Sawyer, Richard L. Wheeden, and Shiying Zhao, Weighted norm inequalities for operators of potential type and fractional maximal functions, Potential Anal. 5 (1996), no. 6, 523–580. MR 1437584, DOI 10.1007/BF00275794
- Martin Schechter, Superlinear elliptic boundary value problems, Manuscripta Math. 86 (1995), no. 3, 253–265. MR 1323791, DOI 10.1007/BF02567993
- Cahit Arf, Untersuchungen über reinverzweigte Erweiterungen diskret bewerteter perfekter Körper, J. Reine Angew. Math. 181 (1939), 1–44 (German). MR 18, DOI 10.1515/crll.1940.181.1
- Guido Sweers, Positivity for a strongly coupled elliptic system by Green function estimates, J. Geom. Anal. 4 (1994), no. 1, 121–142. MR 1274141, DOI 10.1007/BF02921596
- S. R. Treil and A. L. Volberg, Weighted embeddings and weighted norm inequalities for the Hilbert transform and the maximal operator, Algebra i Analiz 7 (1995), no. 6, 205–226; English transl., St. Petersburg Math. J. 7 (1996), no. 6, 1017–1032. MR 1381983
- B. O. Turesson, Nonlinear Potential Theory and Weighted Sobolev Spaces, Linköping Studies in Sci. and Techn., Dissert. 387, Linköping University, 1995.
- I. E. Verbitsky and R. L. Wheeden, Weighted trace inequalities for fractional integrals and applications to semilinear equations, J. Funct. Anal. 129 (1995), no. 1, 221–241. MR 1322649, DOI 10.1006/jfan.1995.1049
- Igor E. Verbitsky and Richard L. Wheeden, Weighted norm inequalities for integral operators, Trans. Amer. Math. Soc. 350 (1998), no. 8, 3371–3391. MR 1443202, DOI 10.1090/S0002-9947-98-02017-0
- Richard L. Wheeden and Shiying Zhao, Weak type estimates for operators of potential type, Studia Math. 119 (1996), no. 2, 149–160. MR 1391473
- Kjell-Ove Widman, Inequalities for the Green function and boundary continuity of the gradient of solutions of elliptic differential equations, Math. Scand. 21 (1967), 17–37 (1968). MR 239264, DOI 10.7146/math.scand.a-10841
- Zhong Xin Zhao, Green function for Schrödinger operator and conditioned Feynman-Kac gauge, J. Math. Anal. Appl. 116 (1986), no. 2, 309–334. MR 842803, DOI 10.1016/S0022-247X(86)80001-4
- William P. Ziemer, Weakly differentiable functions, Graduate Texts in Mathematics, vol. 120, Springer-Verlag, New York, 1989. Sobolev spaces and functions of bounded variation. MR 1014685, DOI 10.1007/978-1-4612-1015-3
Additional Information
- N. J. Kalton
- Affiliation: Department of Mathematics, University of Missouri, Columbia, Missouri 65211
- Email: nigel@math.missouri.edu
- I. E. Verbitsky
- Affiliation: Department of Mathematics, University of Missouri, Columbia, Missouri 65211
- Email: igor@math.missouri.edu
- Received by editor(s): March 1, 1997
- Received by editor(s) in revised form: August 15, 1997
- Published electronically: April 27, 1999
- Additional Notes: The first author was partially supported by NSF grant DMS-9500125, and the second by NSF grant DMS-9401493 and the University of Missouri Research Board grant RB-96029.
- © Copyright 1999 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 351 (1999), 3441-3497
- MSC (1991): Primary 35J60, 42B25, 47H15; Secondary 31B15
- DOI: https://doi.org/10.1090/S0002-9947-99-02215-1
- MathSciNet review: 1475688