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SHARP ESTIMATES FOR FUNCTIONS OF BOUNDED LOWER OSCILLATION

Part of: Harmonic analysis in several variables

Published online by Cambridge University Press:  28 May 2012

ADAM OSȨKOWSKI
ADAM OSȨKOWSKI*
Affiliation:
Department of Mathematics, Informatics and Mechanics, University of Warsaw, Banacha 2, 02-097 Warsaw, Poland (email: ados@mimuw.edu.pl)
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Abstract

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Let f:ℝ→ℝ be a locally integrable function of bounded lower oscillation. The paper contains the proofs of sharp strong-type, weak-type and exponential estimates for the mean oscillation of f. In particular, this yields the precise value of the norm of the embedding BLO⊂BMOp, 1≤p<∞. Higher-dimensional analogues for anisotropic BLO spaces are also established.

Keywords

BMOBLOBellman functionbest constants

MSC classification

Secondary: 42B37: Harmonic analysis and PDE
Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 87 , Issue 1 , February 2013 , pp. 68 - 81
Copyright
Copyright © Australian Mathematical Publishing Association Inc. 2012

References

[1]Coifman, R. R. and Rochberg, R., ‘Another characterization of BMO’, Proc. Amer. Math. Soc. 79 (1980), 249–254.CrossRefGoogle Scholar
[2]Dindoš, M. and Wall, T., ‘The sharp A p constant for weights in a reverse-Hölder class’, Rev. Mat. Iberoam. 25(2) (2009), 559–594.CrossRefGoogle Scholar
[3]Fefferman, C., ‘Characterizations of bounded mean oscillation’, Bull. Amer. Math. Soc. 77 (1971), 587–588.CrossRefGoogle Scholar
[4]Garcia-Cuerva, J. and Rubio de Francia, J. L., Weighted Norm Inequalities and Related Topics (North-Holland, Amsterdam–New York–Oxford, 1985).Google Scholar
[5]Garnett, J. and Jones, P., ‘The distance in BMO to L ∞’, Ann. of Math. (2) 108 (1978), 373–393.CrossRefGoogle Scholar
[6]John, F. and Nirenberg, L., ‘On functions of bounded mean oscillation’, Comm. Pure Appl. Math. 14 (1961), 415–426.CrossRefGoogle Scholar
[7]Jones, P., ‘Factorization of A p weights’, Ann. of Math. (2) 111 (1980), 511–530.CrossRefGoogle Scholar
[8]Korenovskii, A. A., Mean Oscillations and Equimeasurable Rearrangements of Functions, Lecture Notes of the Unione Matematica Italiana, 4 (Springer, Berlin; UMI, Bologna, 2007).CrossRefGoogle Scholar
[9]Nazarov, F. L. and Treil, S. R., ‘The hunt for a Bellman function: applications to estimates for singular integral operators and to other classical problems of harmonic analysis’, Algebra i Analiz 8(5) (1996), 32–162; translation in St. Petersburg Math. J. 8(5) (1997), 721–824 (in Russian).Google Scholar
[10]Nazarov, F. L., Treil, S. R. and Volberg, A., ‘Bellman function in stochastic control and harmonic analysis’, in: Systems, Approximation, Singular Integral Operators, and Related Topics (Bordeaux, 2000), Operator Theory: Advances and Applications, 129 (Birkhäuser, Basel, 2001), pp. 393–423.CrossRefGoogle Scholar
[11]Slavin, L. and Vasyunin, V., ‘Sharp results in the integral-form John–Nirenberg inequality’, Trans. Amer. Math. Soc. 363 (2011), 4135–4169.CrossRefGoogle Scholar
[12]Slavin, L. and Volberg, A., ‘Bellman function and the H 1−BMO duality’, in: Harmonic Analysis, Partial Differential Equations, and Related Topics, Contemporary Mathematics, 428 (American Mathematical Society, Providence, RI, 2007), pp. 113–126.CrossRefGoogle Scholar
[13]Vasyunin, V., ‘The exact constant in the inverse Hölder inequality for Muckenhoupt weights’, Algebra i Analiz 15(1) (2003), 73–117; translation in St. Petersburg Math. J. 15(1) (2004), 49–79 (in Russian. Russian summary).Google Scholar
[14]Vasyunin, V., ‘Mutual estimates for L p-norms and the Bellman function’, Zap. Nauchn. Sem. S.-Peterburg. Otdel. Mat. Inst. Steklov. (POMI) 355 (2008); Issledovaniya po Lineinym Operatoram i Teorii Funktsii. 36, 81–138, 237–238; translation in J. Math. Sci. (N.Y.) 156(5) (2009), 766–798 (in Russian).Google Scholar
[15]Wittwer, J., ‘Survey article: a user’s guide to Bellman functions’, Rocky Mountain J. Math. 41(3) (2011), 631–661.CrossRefGoogle Scholar