Extrapolation and interpolation in generalized Orlicz spaces

Cruz-Uribe, David; Hästö, Peter

doi:10.1090/tran/7155

Extrapolation and interpolation in generalized Orlicz spaces
HTML articles powered by AMS MathViewer

by David Cruz-Uribe and Peter Hästö PDF

Trans. Amer. Math. Soc. 370 (2018), 4323-4349 Request permission

Abstract:

We prove versions of the Rubio de Francia extrapolation theorem in generalized Orlicz spaces. As a consequence, we obtain boundedness results for several classical operators as well as a Sobolev inequality in this setting. We also study complex interpolation in the same setting and use it to derive a compact embedding theorem. Our results include as special cases classical Lebesgue and Sobolev space estimates and their variable exponent and double phase growth analogs.

References

A. Almeida, P. Harjulehto, P. Hästö, and T. Lukkari, Riesz and Wolff potentials and elliptic equations in variable exponent weak Lebesgue spaces, Ann. Mat. Pura Appl. (4) 194 (2015), no. 2, 405–424. MR 3322428, DOI 10.1007/s10231-013-0382-2
Kenneth F. Andersen and Russel T. John, Weighted inequalities for vector-valued maximal functions and singular integrals, Studia Math. 69 (1980/81), no. 1, 19–31. MR 604351, DOI 10.4064/sm-69-1-19-31
Pascal Auscher and José María Martell, Weighted norm inequalities, off-diagonal estimates and elliptic operators. I. General operator theory and weights, Adv. Math. 212 (2007), no. 1, 225–276. MR 2319768, DOI 10.1016/j.aim.2006.10.002
M. Avci and A. Pankov, Multivalued elliptic operators with nonstandard growth, Advances in Nonlinear Analysis, March 2016, 1–14.
Paolo Baroni, Maria Colombo, and Giuseppe Mingione, Harnack inequalities for double phase functionals, Nonlinear Anal. 121 (2015), 206–222. MR 3348922, DOI 10.1016/j.na.2014.11.001
P. Baroni, M. Colombo, and G. Mingione, Nonautonomous functionals, borderline cases and related function classes, Algebra i Analiz 27 (2015), no. 3, 6–50; English transl., St. Petersburg Math. J. 27 (2016), no. 3, 347–379. MR 3570955, DOI 10.1090/spmj/1392
Colin Bennett and Robert Sharpley, Interpolation of operators, Pure and Applied Mathematics, vol. 129, Academic Press, Inc., Boston, MA, 1988. MR 928802
Jöran Bergh and Jörgen Löfström, Interpolation spaces. An introduction, Grundlehren der Mathematischen Wissenschaften, No. 223, Springer-Verlag, Berlin-New York, 1976. MR 0482275
S. Chanillo, A note on commutators, Indiana Univ. Math. J. 31 (1982), no. 1, 7–16. MR 642611, DOI 10.1512/iumj.1982.31.31002
Yunmei Chen, Stacey Levine, and Murali Rao, Variable exponent, linear growth functionals in image restoration, SIAM J. Appl. Math. 66 (2006), no. 4, 1383–1406. MR 2246061, DOI 10.1137/050624522
R. R. Coifman and C. Fefferman, Weighted norm inequalities for maximal functions and singular integrals, Studia Math. 51 (1974), 241–250. MR 358205, DOI 10.4064/sm-51-3-241-250
R. R. Coifman, R. Rochberg, and Guido Weiss, Factorization theorems for Hardy spaces in several variables, Ann. of Math. (2) 103 (1976), no. 3, 611–635. MR 412721, DOI 10.2307/1970954
Maria Colombo and Giuseppe Mingione, Bounded minimisers of double phase variational integrals, Arch. Ration. Mech. Anal. 218 (2015), no. 1, 219–273. MR 3360738, DOI 10.1007/s00205-015-0859-9
Michael Cowling, José García-Cuerva, and Hendra Gunawan, Weighted estimates for fractional maximal functions related to spherical means, Bull. Austral. Math. Soc. 66 (2002), no. 1, 75–90. MR 1922609, DOI 10.1017/S0004972700020694
Lucas Chaffee and David Cruz-Uribe, Necessary conditions for the boundedness of linear and bilinear commutators on Banach function spaces, Math. Inequal. Appl. 21 (2018), no. 1, 1–16. MR 3716205, DOI 10.7153/mia-2018-21-01
D. Cruz-Uribe and A. Fiorenza, Endpoint estimates and weighted norm inequalities for commutators of fractional integrals, Publ. Mat. 47 (2003), no. 1, 103–131. MR 1970896, DOI 10.5565/PUBLMAT_{4}7103_{0}5
D. Cruz-Uribe and A. Fiorenza, $L\log L$ results for the maximal operator in variable $L^p$ spaces, Trans. Amer. Math. Soc. 361 (2009), no. 5, 2631–2647. MR 2471932, DOI 10.1090/S0002-9947-08-04608-4
David V. Cruz-Uribe and Alberto Fiorenza, Variable Lebesgue spaces, Applied and Numerical Harmonic Analysis, Birkhäuser/Springer, Heidelberg, 2013. Foundations and harmonic analysis. MR 3026953, DOI 10.1007/978-3-0348-0548-3
D. Cruz-Uribe, A. Fiorenza, J. M. Martell, and C. Pérez, The boundedness of classical operators on variable $L^p$ spaces, Ann. Acad. Sci. Fenn. Math. 31 (2006), no. 1, 239–264. MR 2210118
David V. Cruz-Uribe, José Maria Martell, and Carlos Pérez, Weights, extrapolation and the theory of Rubio de Francia, Operator Theory: Advances and Applications, vol. 215, Birkhäuser/Springer Basel AG, Basel, 2011. MR 2797562, DOI 10.1007/978-3-0348-0072-3
David Cruz-Uribe and C. J. Neugebauer, The structure of the reverse Hölder classes, Trans. Amer. Math. Soc. 347 (1995), no. 8, 2941–2960. MR 1308005, DOI 10.1090/S0002-9947-1995-1308005-6
David Cruz-Uribe and Li-An Daniel Wang, Extrapolation and weighted norm inequalities in the variable Lebesgue spaces, Trans. Amer. Math. Soc. 369 (2017), no. 2, 1205–1235. MR 3572271, DOI 10.1090/tran/6730
Lars Diening, Maximal function on Musielak-Orlicz spaces and generalized Lebesgue spaces, Bull. Sci. Math. 129 (2005), no. 8, 657–700 (English, with English and French summaries). MR 2166733, DOI 10.1016/j.bulsci.2003.10.003
Lars Diening, Petteri Harjulehto, Peter Hästö, and Michael Růžička, Lebesgue and Sobolev spaces with variable exponents, Lecture Notes in Mathematics, vol. 2017, Springer, Heidelberg, 2011. MR 2790542, DOI 10.1007/978-3-642-18363-8
L. Diening, P. Hästö, and A. Nekvinda, Open problems in variable exponent Lebesgue and Sobolev spaces. In FSDONA04 Proceedings, Drabek and Rakosnik (eds.), Milovy, Czech Republic, Academy of Sciences of the Czech Republic, Prague, 2005, 38–58.
Javier Duoandikoetxea, Fourier analysis, Graduate Studies in Mathematics, vol. 29, American Mathematical Society, Providence, RI, 2001. Translated and revised from the 1995 Spanish original by David Cruz-Uribe. MR 1800316, DOI 10.1090/gsm/029
Flavia Giannetti and Antonia Passarelli di Napoli, Regularity results for a new class of functionals with non-standard growth conditions, J. Differential Equations 254 (2013), no. 3, 1280–1305. MR 2997371, DOI 10.1016/j.jde.2012.10.011
Eleonor Harboure, Roberto A. Macías, and Carlos Segovia, Extrapolation results for classes of weights, Amer. J. Math. 110 (1988), no. 3, 383–397. MR 944321, DOI 10.2307/2374616
P. Harjulehto and P. Hästö, Reflexivity and uniform convexity of generalized Orlicz spaces, preprint.
Petteri Harjulehto and Peter Hästö, The Riesz potential in generalized Orlicz spaces, Forum Math. 29 (2017), no. 1, 229–244. MR 3592600, DOI 10.1515/forum-2015-0239
Petteri Harjulehto, Peter Hästö, and Riku Klén, Generalized Orlicz spaces and related PDE, Nonlinear Anal. 143 (2016), 155–173. MR 3516828, DOI 10.1016/j.na.2016.05.002
P. Harjulehto, P. Hästö, V. Latvala, and O. Toivanen, Critical variable exponent functionals in image restoration, Appl. Math. Lett. 26 (2013), no. 1, 56–60. MR 2971400, DOI 10.1016/j.aml.2012.03.032
Petteri Harjulehto, Peter Hästö, Út V. Lê, and Matti Nuortio, Overview of differential equations with non-standard growth, Nonlinear Anal. 72 (2010), no. 12, 4551–4574. MR 2639204, DOI 10.1016/j.na.2010.02.033
Peter A. Hästö, The maximal operator on generalized Orlicz spaces, J. Funct. Anal. 269 (2015), no. 12, 4038–4048. MR 3418078, DOI 10.1016/j.jfa.2015.10.002
Peter A. Hästö, Corrigendum to “The maximal operator on generalized Orlicz spaces” [J. Funct. Anal. 269 (2015) 4038–4048] [ MR3418078], J. Funct. Anal. 271 (2016), no. 1, 240–243. MR 3494250, DOI 10.1016/j.jfa.2016.04.005
Peter Hästö, Yoshihiro Mizuta, Takao Ohno, and Tetsu Shimomura, Sobolev inequalities for Orlicz spaces of two variable exponents, Glasg. Math. J. 52 (2010), no. 2, 227–240. MR 2610969, DOI 10.1017/S0017089509990292
Mohammed Kbiri Alaoui, Tamer Nabil, and Mohamed Altanji, On some new non-linear diffusion models for the image filtering, Appl. Anal. 93 (2014), no. 2, 269–280. MR 3169204, DOI 10.1080/00036811.2013.769132
M. A. Krasnosel′skiĭ and Ja. B. Rutickiĭ, Convex functions and Orlicz spaces, P. Noordhoff Ltd., Groningen, 1961. Translated from the first Russian edition by Leo F. Boron. MR 0126722
Gary M. Lieberman, On the regularity of the minimizer of a functional with exponential growth, Comment. Math. Univ. Carolin. 33 (1992), no. 1, 45–49. MR 1173745
Fumi-Yuki Maeda, Yoshihiro Mizuta, and Takao Ohno, Approximate identities and Young type inequalities in variable Lebesgue-Orlicz spaces $L^{p(\cdot )}(\log L)^{q(\cdot )}$, Ann. Acad. Sci. Fenn. Math. 35 (2010), no. 2, 405–420. MR 2731699, DOI 10.5186/aasfm.2010.3526
Fumi-Yuki Maeda, Yoshihiro Mizuta, Takao Ohno, and Tetsu Shimomura, Approximate identities and Young type inequalities in Musielak-Orlicz spaces, Czechoslovak Math. J. 63(138) (2013), no. 4, 933–948. MR 3165506, DOI 10.1007/s10587-013-0063-8
Fumi-Yuki Maeda, Yoshihiro Mizuta, Takao Ohno, and Tetsu Shimomura, Boundedness of maximal operators and Sobolev’s inequality on Musielak-Orlicz-Morrey spaces, Bull. Sci. Math. 137 (2013), no. 1, 76–96. MR 3007101, DOI 10.1016/j.bulsci.2012.03.008
Fumi-Yuki Maeda, Yoshihiro Sawano, and Tetsu Shimomura, Some norm inequalities in Musielak-Orlicz spaces, Ann. Acad. Sci. Fenn. Math. 41 (2016), no. 2, 721–744. MR 3525396, DOI 10.5186/aasfm.2016.4148
Julian Musielak, Orlicz spaces and modular spaces, Lecture Notes in Mathematics, vol. 1034, Springer-Verlag, Berlin, 1983. MR 724434, DOI 10.1007/BFb0072210
Hidegorô Nakano, Modulared Semi-Ordered Linear Spaces, Maruzen Co. Ltd., Tokyo, 1950. MR 0038565
Hidegorô Nakano, Topology of linear topological spaces, Maruzen Co. Ltd., Tokyo, 1951. MR 0046560
Takao Ohno and Tetsu Shimomura, Trudinger’s inequality for Riesz potentials of functions in Musielak-Orlicz spaces, Bull. Sci. Math. 138 (2014), no. 2, 225–235. MR 3175020, DOI 10.1016/j.bulsci.2013.05.007
W. Orlicz, Über konjugierte Exponentenfolgen, Stud. Math. 3 (1931), 200–211.
Carlos Pérez, Sharp estimates for commutators of singular integrals via iterations of the Hardy-Littlewood maximal function, J. Fourier Anal. Appl. 3 (1997), no. 6, 743–756. MR 1481632, DOI 10.1007/BF02648265
M. M. Rao and Z. D. Ren, Theory of Orlicz spaces, Monographs and Textbooks in Pure and Applied Mathematics, vol. 146, Marcel Dekker, Inc., New York, 1991. MR 1113700
José Luis Rubio de Francia, Factorization and extrapolation of weights, Bull. Amer. Math. Soc. (N.S.) 7 (1982), no. 2, 393–395. MR 663793, DOI 10.1090/S0273-0979-1982-15047-9
Elias M. Stein, Maximal functions. I. Spherical means, Proc. Nat. Acad. Sci. U.S.A. 73 (1976), no. 7, 2174–2175. MR 420116, DOI 10.1073/pnas.73.7.2174
Agnieszka Świerczewska-Gwiazda, Nonlinear parabolic problems in Musielak-Orlicz spaces, Nonlinear Anal. 98 (2014), 48–65. MR 3158445, DOI 10.1016/j.na.2013.11.026