Geodesic
A generalization of John--Nirenberg's inequailty
Trudy Instituta matematiki, Tome 22 (2014) no. 2, pp. 63-73.

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In work the generalization $BMO_\varphi$ of space of $BMO$ for functions on space of gomogeneous type that defined by integral $\varphi$-oscillations is studied. The analog of John–Nirenberg's inequality for functions from these classes is proved. As a corollary we prove coincidence of the classes $BMO_\varphi$ for rather wide class of functions $\varphi$. Furthermore, generalizations of Kampanato–Meyers's and Spanne's theorems are obtained. 
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A. I. Porabkovich; R. V. Shanin. A generalization of John--Nirenberg's inequailty. Trudy Instituta matematiki, Tome 22 (2014) no. 2, pp. 63-73. http://geodesic.mathdoc.fr/item/TIMB_2014_22_2_a6/

[1] Heinonen J., Lectures on analysis on metric spaces, Springer-Verlag, New York–Berlin–Heidelberg, 2001 | MR | Zbl

[2] John F., Nirenberg L., “On functions of bounded mean oscillation”, Comm. Pure Appl. Math., 14:2 (1961), 415–426 | DOI | MR | Zbl

[3] Spanne S., “Some function spaces defined using the mean oscillation over cubes”, Ann. Scuola norm. super. Pisa, 19:4 (1965), 593–608 | MR | Zbl

[4] Campanato C., “Proprieta di holderianita di alcune classi di funzioni”, Ann. Sc. Norm. Super. Pisa Cl. Sci., 17:1–2 (1963), 175–188 | MR | Zbl

[5] Meyers N. J., “Mean oscillation over cubes and Hölder continuity”, Proc. Amer. Math. Soc., 15:5 (1964), 717–721 | MR | Zbl

[6] Long R., Yang L., “BMO functions in spaces of homogeneous type”, Sci. Sinica, Ser. A, 27:7 (1984), 695–708 | MR | Zbl

[7] Logunov A. A., Slavin L., Stolyarov D. M., Vasyunin V., Zatitskiy P. B., Weak integral conditions for BMO, 2013, arXiv: 1309.6780v1 | MR

[8] Aimar H., “Singular integrals and approximate identities on spaces of homogeneous type”, Trans. Amer. Math. Soc., 292 (1985), 135–153 | DOI | MR | Zbl

[9] Ivanishko I. A., Krotov V. G., “Obobschennoe neravenstvo Puankare–Soboleva na metricheskikh prostranstvakh”, Trudy Instituta matematiki NAN Belarusi, 14:1 (2006), 51–61

[10] Korenovskii A., Mean oscillations and equimeasurable rearrangements of functions, Springer-Verlag, Berlin–Heidelberg, 2007 | MR | Zbl

[11] Ivanishko I. A., Krotov V. G., Porabkovich A. I., “Obobschenie teoremy Kampanato–Meiersa”, Trudy Instituta matematiki NAN Belarusi, 20:2 (2012), 30–35

[12] Gorka P., “Campanato theorem on metric measure spaces”, Ann. Acad. Sci. Fenn., 34 (2009), 523–528 | MR | Zbl