Weak integral conditions for BMO
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- by A. A. Logunov, L. Slavin, D. M. Stolyarov, V. Vasyunin and P. B. Zatitskiy PDF
- Proc. Amer. Math. Soc. 143 (2015), 2913-2926 Request permission
Abstract:
We study the question of how much one can weaken the defining condition of BMO. Specifically, we show that if $Q$ is a cube in $\mathbb {R}^n$ and $h:[0,\infty )\to [0,\infty )$ is such that $h(t)\to \infty$ as $t\to \infty ,$ then \[ \sup _{J~\text {subcube}~Q}\frac 1{|J|}\int _J h\big (\big |\varphi -\textstyle {\frac 1{|J|} \int _J\varphi } \big |\big )<\infty \quad \Longrightarrow \quad \varphi \in \mathrm {BMO}(Q). \] Under some additional assumptions on $h$ we obtain estimates on $\|\varphi \|_{\mathrm {BMO}}$ in terms of the supremum above. We also show that even though the limit condition on $h$ is not necessary for this implication to hold, it becomes necessary if one considers the dyadic $\mathrm {BMO}$.References
- Paata Ivanishvili, Nikolay N. Osipov, Dmitriy M. Stolyarov, Vasily I. Vasyunin, and Pavel B. Zatitskiy, On Bellman function for extremal problems in BMO, C. R. Math. Acad. Sci. Paris 350 (2012), no. 11-12, 561–564 (English, with English and French summaries). MR 2956143, DOI 10.1016/j.crma.2012.06.011
- P. Ivanishvili, N. Osipov, D. Stolyarov, V. Vasyunin, P. Zatitskiy, Bellman function for extremal problems in BMO. To appear in Transactions of the AMS, arxiv:1205.7018v3.
- F. John, Quasi-isometric mappings, Seminari 1962/63 Anal. Alg. Geom. e Topol., Vol. 2, Ist. Naz. Alta Mat., Ediz. Cremonese, Rome, 1965, pp. 462–473. MR 0190905
- F. John and L. Nirenberg, On functions of bounded mean oscillation, Comm. Pure Appl. Math. 14 (1961), 415–426. MR 131498, DOI 10.1002/cpa.3160140317
- Ivo Klemes, A mean oscillation inequality, Proc. Amer. Math. Soc. 93 (1985), no. 3, 497–500. MR 774010, DOI 10.1090/S0002-9939-1985-0774010-0
- Rui Lin Long and Lo Yang, BMO functions in spaces of homogeneous type, Sci. Sinica Ser. A 27 (1984), no. 7, 695–708. MR 764353
- Xian Liang Shi and Alberto Torchinsky, Local sharp maximal functions in spaces of homogeneous type, Sci. Sinica Ser. A 30 (1987), no. 5, 473–480. MR 1000919
- L. Slavin, The John–Nirenberg constant of $\mathrm {BMO}^p, 1\leqslant p\leqslant 2.$ Submitted.
- L. Slavin and V. Vasyunin, Sharp results in the integral-form John-Nirenberg inequality, Trans. Amer. Math. Soc. 363 (2011), no. 8, 4135–4169. MR 2792983, DOI 10.1090/S0002-9947-2011-05112-3
- Leonid Slavin and Vasily Vasyunin, Sharp $L^p$ estimates on BMO, Indiana Univ. Math. J. 61 (2012), no. 3, 1051–1110. MR 3071693, DOI 10.1512/iumj.2012.61.4651
- Jan-Olov Strömberg, Bounded mean oscillation with Orlicz norms and duality of Hardy spaces, Indiana Univ. Math. J. 28 (1979), no. 3, 511–544. MR 529683, DOI 10.1512/iumj.1979.28.28037
- Vasily Vasyunin and Alexander Volberg, Sharp constants in the classical weak form of the John-Nirenberg inequality, Proc. Lond. Math. Soc. (3) 108 (2014), no. 6, 1417–1434. MR 3218314, DOI 10.1112/plms/pdt063
Additional Information
- A. A. Logunov
- Affiliation: Chebyshev Laboratory, St. Petersburg State University, St. Petersburg, Russia
- L. Slavin
- Affiliation: Department of Mathematical Sciences, University of Cincinnati, Cincinnati, Ohio 45211-0025
- MR Author ID: 121075
- ORCID: 0000-0002-9502-8852
- D. M. Stolyarov
- Affiliation: Chebyshev Laboratory, St. Petersburg State University, St. Petersburg, Russia – and – Steklov Institute of Mathematics, St. Petersburg Branch, St. Petersburg, Russia
- MR Author ID: 895114
- V. Vasyunin
- Affiliation: Steklov Institute of Mathematics, St. Petersburg Branch, St. Petersburg, Russia
- P. B. Zatitskiy
- Affiliation: Chebyshev Laboratory, St. Petersburg State University, St. Petersburg, Russia – and – Steklov Institute of Mathematics, St. Petersburg Branch, St. Petersburg, Russia
- MR Author ID: 895184
- Received by editor(s): September 26, 2013
- Published electronically: March 6, 2015
- Additional Notes: The second author was supported by the NSF (DMS-1041763)
The third author was supported by a Rokhlin grant and the RFBR (grant 11-01-00526)
The fourth author was supported by the RFBR (grant 11-01-00584-a)
The fifth author was supported by the President of Russia grant for young researchers MK-6133.2013.1 and by the RFBR (grant 13-01-12422 ofi_m2)
This research was supported in part by the Chebyshev Laboratory (Department of Mathematics and Mechanics, St. Petersburg State University) under the RF Government grant 11.G34.31.0026, and by JSC “Gazprom Neft” - Communicated by: Alexander Iosevich
- © Copyright 2015 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 143 (2015), 2913-2926
- MSC (2010): Primary 42B35, 49K20
- DOI: https://doi.org/10.1090/S0002-9939-2015-12424-0
- MathSciNet review: 3336616