zbMATH — the first resource for mathematics

Riesz rising sun lemma for several variables and the John-Nirenberg inequality. (English. Russian original) Zbl 1094.42021
Math. Notes 77, No. 1, 48-60 (2005); translation from Mat. Zametki 77, No. 1, 53-66 (2005).
Summary: We obtain a multidimensional analog of the well-known Riesz rising sun lemma. We prove a more precise version of this lemma for space dimension \(d = 2\). We use these lemmas to establish an anisotropic analog of the John-Nirenberg inequality for functions of bounded mean oscillation with an exact constant in the exponent. Earlier, this exact constant was only known in the one-dimensional case.

42B25 Maximal functions, Littlewood-Paley theory
42C20 Other transformations of harmonic type
Full Text: DOI
[1] A. P. Calderón and A. Zygmund, ”On the existence of certain singular integrals,” Acta Math., 88 (1952), 85–139. · Zbl 0047.10201 · doi:10.1007/BF02392130
[2] J. Garnett, Bounded Analytic Functions, Academic Press, New York, 1981. · Zbl 0469.30024
[3] F. Riesz, ”Sur un theoreme de maximum de MM. Hardy et Littlewood,” J. London Math. Soc., 7 (1932), 10–13. · Zbl 0003.39201 · doi:10.1112/jlms/s1-7.1.10
[4] G. Hardy, D. Littlewood, and G. Pólya, Inequalities, Cambridge Univ. Press, Cambridge, 1952.
[5] I. Klemes, ”A mean oscillation inequality,” Proc. Amer. Math. Soc., 93 (1985), no. 3, 497–500. · Zbl 0572.46025 · doi:10.1090/S0002-9939-1985-0774010-0
[6] B. Jessen, J. Marcinkiewicz, and A. Zygmund, ”Note on the differentiability of multiple integrals,” Fund. Math., 25 (1935), 217–234. · Zbl 0012.05901
[7] M. Gusman, Differentiation of Integrals in \(\mathbb{R}\)n, Springer-Verlag, Heidelberg, 1975.
[8] E. Stein, Singular Integrals and Differentiability Properties of Functions, Princeton Univ. Press, Princeton (USA), 1970. · Zbl 0207.13501
[9] F. John and L. Nirenberg, ”On functions of bounded mean oscillation,” Comm. Pure Appl. Math., 14 (1961), 415–426. · Zbl 0102.04302 · doi:10.1002/cpa.3160140317
[10] A. A. Korenovskii, ”On the relationship between mean oscillations and exact exponents of summability of functions,” Mat. Sb. [Math. USSR-Sb.], 181 (1990), no. 12, 1721–1727.
[11] A. M. Garsia and E. Rodemich, ”Monotonicity of certain functionals under rearrangements,” Ann. Inst. Fourier, Grenoble, 24 (1974), no. 2, 67–116. · Zbl 0274.26006
[12] C. Bennett, R. A. De Vore, and R. Sharpley, ”Weak-Land BMO,” Ann. Math., 113 (1981), no. 2, 601–611. · Zbl 0465.42015 · doi:10.2307/2006999
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.