BMO-regularity in lattices of measurable functions on spaces of homogeneous type
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D. V. Rutsky
Translated by: the author - St. Petersburg Math. J. 23 (2012), 381-412
- DOI: https://doi.org/10.1090/S1061-0022-2012-01201-X
- Published electronically: January 24, 2012
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Abstract:
Let $X$ be a lattice of measurable functions on a space of homogeneous type $(S, \nu )$ (for example, $S = \mathbb R^n$ with Lebesgue measure). Suppose that $X$ has the Fatou property. Let $T$ be either a CalderĂłnâZygmund singular integral operator with a singularity nondegenerate in a certain sense, or the HardyâLittlewood maximal operator. It is proved that $T$ is bounded on the lattice $\bigl (X^\alpha \mathrm {L}_1^{1 - \alpha }\bigr )^\beta$ for some $\beta \in (0, 1)$ and sufficiently small $\alpha \in (0, 1)$ if and only if $X$ has the following simple property: for every $f \in X$ there exists a majorant $g \in X$ such that $\log g \in \mathrm {BMO}$ with proper control on the norms. This property is called $\mathrm {BMO}$-regularity. For the readerâs convenience, a self-contained exposition of the $\mathrm {BMO}$-regularity theory is developed in the new generality, as well as some refinements of the main results.References
- S. V. Kisliakov, Interpolation of $H^p$-spaces: some recent developments, Function spaces, interpolation spaces, and related topics (Haifa, 1995) Israel Math. Conf. Proc., vol. 13, Bar-Ilan Univ., Ramat Gan, 1999, pp. 102â140. MR 1707360
- L. Diening, P. HĂ€stö, and A. Nekvinda, Open problems in variable exponent Lebesgue and Sobolev spaces, FSDONA 2004 Proceedings (Drabek and Rakosnik, eds.), Milovy, Czech Republic, 2004, pp. 38â58.
- Serguei V. Kisliakov and Quan Hua Xu, Interpolation of weighted and vector-valued Hardy spaces, Trans. Amer. Math. Soc. 343 (1994), no. 1, 1â34. MR 1236225, DOI 10.1090/S0002-9947-1994-1236225-7
- Sergei Kisliakov and Quanhua Xu, Partial retractions for weighted Hardy spaces, Studia Math. 138 (2000), no. 3, 251â264. MR 1758858
- Michael Cwikel, John E. McCarthy, and Thomas H. Wolff, Interpolation between weighted Hardy spaces, Proc. Amer. Math. Soc. 116 (1992), no. 2, 381â388. MR 1093595, DOI 10.1090/S0002-9939-1992-1093595-4
- John B. Garnett, Bounded analytic functions, Pure and Applied Mathematics, vol. 96, Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers], New York-London, 1981. MR 628971
- Elias M. Stein, Harmonic analysis: real-variable methods, orthogonality, and oscillatory integrals, Princeton Mathematical Series, vol. 43, Princeton University Press, Princeton, NJ, 1993. With the assistance of Timothy S. Murphy; Monographs in Harmonic Analysis, III. MR 1232192
- Donggao Deng and Yongsheng Han, Harmonic analysis on spaces of homogeneous type, Lecture Notes in Mathematics, vol. 1966, Springer-Verlag, Berlin, 2009. With a preface by Yves Meyer. MR 2467074
- Joram Lindenstrauss and Lior Tzafriri, Classical Banach spaces. I, Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 92, Springer-Verlag, Berlin-New York, 1977. Sequence spaces. MR 0500056
- I. I. Privalov, GraniÄnye svoÄstva analitiÄeskih funkciÄ, Gosudarstv. Izdat. Tehn.-Teor. Lit., Moscow-Leningrad, 1950 (Russian). 2d ed.]. MR 0047765
- S. V. Kislyakov, On BMO-regular lattices of measurable functions. II, Zap. Nauchn. Sem. S.-Peterburg. Otdel. Mat. Inst. Steklov. (POMI) 303 (2003), no. Issled. po LineÄn. Oper. i Teor. Funkts. 31, 161â168, 324 (Russian, with English and Russian summaries); English transl., J. Math. Sci. (N.Y.) 129 (2005), no. 4, 4018â4021. MR 2037537, DOI 10.1007/s10958-005-0338-1
- S. V. Kislyakov, On BMO-regular lattices of measurable functions, Algebra i Analiz 14 (2002), no. 2, 117â135 (Russian); English transl., St. Petersburg Math. J. 14 (2003), no. 2, 273â286. MR 1925883
- S. V. Kislyakov, On BMO-regular couples of lattices of measurable functions, Studia Math. 159 (2003), no. 2, 277â290. Dedicated to Professor Aleksander PeĆczyĆski on the occasion of his 70th birthday (Polish). MR 2052223, DOI 10.4064/sm159-2-8
- S. V. Kislyakov, Bourgainâs analytic projection revisited, Proc. Amer. Math. Soc. 126 (1998), no. 11, 3307â3314. MR 1458882, DOI 10.1090/S0002-9939-98-04502-X
- B. S. Mitjagin, An interpolation theorem for modular spaces, Mat. Sb. (N.S.) 66 (108) (1965), 473â482 (Russian). MR 0177299
- Jöran Bergh and Jörgen Löfström, Interpolation spaces. An introduction, Grundlehren der Mathematischen Wissenschaften, No. 223, Springer-Verlag, Berlin-New York, 1976. MR 0482275
- Kenneth Hoffman, Banach spaces of analytic functions, Prentice-Hall Series in Modern Analysis, Prentice-Hall, Inc., Englewood Cliffs, N.J., 1962. MR 0133008
- L. V. Kantorovich and G. P. Akilov, Functional analysis, 2nd ed., Pergamon Press, Oxford-Elmsford, N.Y., 1982. Translated from the Russian by Howard L. Silcock. MR 664597
- Walter Rudin, Function theory in polydiscs, W. A. Benjamin, Inc., New York-Amsterdam, 1969. MR 0255841
- N. K. NikolâČskiÄ, Lektsii ob operatore sdviga, âNaukaâ, Moscow, 1980 (Russian). MR 575166
- L. Diening, P. Harjulehto, P. HĂ€stö, and M. Ruzicka, Lebesgue and Sobolev spaces with variable exponents (a book manuscript in preparation, see http://www.helsinki.fi/Ëpharjule/varsob/ publications.shtml).
- N. J. Kalton, Complex interpolation of Hardy-type subspaces, Math. Nachr. 171 (1995), 227â258. MR 1316360, DOI 10.1002/mana.19951710114
- Ky. Fan, Fixed-point and minimax theorems in locally convex topological linear spaces, Proc. Nat. Acad. Sci. U.S.A. 38 (1952), 121â126. MR 47317, DOI 10.1073/pnas.38.2.121
- G. Ja. LozanovskiÄ, Certain Banach lattices, Sibirsk. Mat. Ćœ. 10 (1969), 584â599 (Russian). MR 0241949
- D. V. RutskiÄ, Two remarks on the relationship between BMO regularity and the analytic stability of interpolation for lattices of measurable functions, Zap. Nauchn. Sem. S.-Peterburg. Otdel. Mat. Inst. Steklov. (POMI) 366 (2009), no. Issledovaniya po LineÄnym Operatoram i Teorii FunktsiÄ. 37, 102â115, 130 (Russian, with English and Russian summaries); English transl., J. Math. Sci. (N.Y.) 165 (2010), no. 4, 483â490. MR 2749153, DOI 10.1007/s10958-010-9816-1
- D. V. RutskiÄ, Remarks on BMO-regularity and AK-stability, Zap. Nauchn. Sem. S.-Peterburg. Otdel. Mat. Inst. Steklov. (POMI) 376 (2010), no. Issledovaniya po LineÄnym Operatoram i Teoriya FunktsiÄ. 38, 116â166, 182 (Russian, with English and Russian summaries); English transl., J. Math. Sci. (N.Y.) 172 (2011), no. 2, 243â269. MR 2749288, DOI 10.1007/s10958-010-0196-3
- â, Thesis (C. Sc.), S.-Petersburg. Otdel. Mat. Inst. Steklov (POMI), St. Petersburg, 2011. (Russian)
- Y. Benyamini and Y. Sternfeld, Spheres in infinite-dimensional normed spaces are Lipschitz contractible, Proc. Amer. Math. Soc. 88 (1983), no. 3, 439â445. MR 699410, DOI 10.1090/S0002-9939-1983-0699410-7
- D. Cruz-Uribe, A. Fiorenza, J. M. Martell, and C. PĂ©rez, The boundedness of classical operators on variable $L^p$ spaces, Ann. Acad. Sci. Fenn. Math. 31 (2006), no. 1, 239â264. MR 2210118
- JosĂ© L. Rubio de Francia, Factorization theory and $A_{p}$ weights, Amer. J. Math. 106 (1984), no. 3, 533â547. MR 745140, DOI 10.2307/2374284
- P. HĂ€stö and L. Diening, Muckenhoupt weights in variable exponent spaces, Preprint, see http://www.helsinki.fi/Ëpharjule/varsob/publications.shtml, 2010.
- V. Kokilashvili and V. Paatashvili, On variable Hardy and Smirnov classes of analytic functions, Georgian Internat. J. 1 (2008), no. 2, 181â195.
- V. I. Gavrilov and A. V. Subbotin, A maximal version of the multidimensional Khinchin-Ostrowski theorem and applications, Dokl. Akad. Nauk 405 (2005), no. 2, 158â160 (Russian). MR 2262530
- Tengiz Kopaliani, Interpolation theorems for variable exponent Lebesgue spaces, J. Funct. Anal. 257 (2009), no. 11, 3541â3551. MR 2572260, DOI 10.1016/j.jfa.2009.06.009
- Quan Hua Xu, Some properties of the quotient space $L^1(\textbf {T}^d)/H^1(D^d)$, Illinois J. Math. 37 (1993), no. 3, 437â454. MR 1219649
- Quan Hua Xu, Notes on interpolation of Hardy spaces, Ann. Inst. Fourier (Grenoble) 42 (1992), no. 4, 875â889 (English, with English and French summaries). MR 1196097
- D. S. Anisimov and S. V. Kislyakov, Double singular integrals: interpolation and correction, Algebra i Analiz 16 (2004), no. 5, 1â33 (Russian); English transl., St. Petersburg Math. J. 16 (2005), no. 5, 749â772. MR 2106665, DOI 10.1090/S1061-0022-05-00877-0
- S. V. Kislyakov and Kuankhua Shu, Real interpolation and singular integrals, Algebra i Analiz 8 (1996), no. 4, 75â109 (Russian, with Russian summary); English transl., St. Petersburg Math. J. 8 (1997), no. 4, 593â615. MR 1418256
- A. K. Lerner, Weighted norm inequalities for the local sharp maximal function, J. Fourier Anal. Appl. 10 (2004), no. 5, 465â474. MR 2093912, DOI 10.1007/s00041-004-0987-3
- B. Jawerth and A. Torchinsky, Local sharp maximal functions, J. Approx. Theory 43 (1985), no. 3, 231â270. MR 779906, DOI 10.1016/0021-9045(85)90102-9
- J. L. Krivine, ThĂ©orĂšmes de factorisation dans les espaces rĂ©ticulĂ©s, SĂ©minaire Maurey-Schwartz 1973â1974: Espaces $L^p$, applications radonifiantes et gĂ©omĂ©trie des espaces de Banach, Centre de Math., Ăcole Polytech., Paris, 1974, pp. Exp. Nos. 22 et 23, 22 (French). MR 0440334
- E. M. DynâČkin and B. P. Osilenker, Weighted estimates for singular integrals and their applications, Mathematical analysis, Vol. 21, Itogi Nauki i Tekhniki, Akad. Nauk SSSR, Vsesoyuz. Inst. Nauchn. i Tekhn. Inform., Moscow, 1983, pp. 42â129 (Russian). MR 736522
- T. P. Hytönen, The sharp weighted bound for general CalderĂłnâZygmund operators, Preprint, http://arxiv.org/abs/1007.4330, 2010, jul.
- T. P. Hytönen, C. Pérez, S. Treil, and A. Volberg, Sharp weighted estimates for dyadic shifts and the $A_2$ conjecture, Preprint, http://arxiv.org/abs/1010.0755v2, 2010, dec.
- Joan Orobitg and Carlos PĂ©rez, $A_p$ weights for nondoubling measures in $\textbf {R}^n$ and applications, Trans. Amer. Math. Soc. 354 (2002), no. 5, 2013â2033. MR 1881028, DOI 10.1090/S0002-9947-02-02922-7
- JosĂš L. Rubio de Francia, Linear operators in Banach lattices and weighted $L^2$ inequalities, Math. Nachr. 133 (1987), 197â209. MR 912429, DOI 10.1002/mana.19871330114
Bibliographic Information
- D. V. Rutsky
- Affiliation: St. Petersburg Branch, Steklov Mathematical Institute, Fontanka 27, St. Petersburg 191023, Russia
- Email: rutsky@pdmi.ras.ru
- Received by editor(s): October 21, 2010
- Published electronically: January 24, 2012
- © Copyright 2012 American Mathematical Society
- Journal: St. Petersburg Math. J. 23 (2012), 381-412
- MSC (2010): Primary 42B35; Secondary 42B20
- DOI: https://doi.org/10.1090/S1061-0022-2012-01201-X
- MathSciNet review: 2841677