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On the John-Nirenberg inequality. (English) Zbl 1503.30128

Summary: We present a version of the John-Nirenberg inequality for a sub-class of BMO by estimating the corresponding mean oscillating distribution function via dyadic decomposition. The dominating functions are of the form of decreasing step functions which are finer than the classical exponential functions and might be much more efficient for some sophisticated analysis. We also prove that the modified BMO-norm is equivalent to the classical BMO-norm under the convexity assumption.

MSC:

30H35 BMO-spaces
42B35 Function spaces arising in harmonic analysis
46E30 Spaces of measurable functions (\(L^p\)-spaces, Orlicz spaces, Köthe function spaces, Lorentz spaces, rearrangement invariant spaces, ideal spaces, etc.)
49K20 Optimality conditions for problems involving partial differential equations
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References:

[1] John, F.; Nirenberg, L., On functions of bounded mean oscillation, Commun. Pure Appl. Math., 14, 415-426 (1961) · Zbl 0102.04302 · doi:10.1002/cpa.3160140317
[2] Pak, H.-C., Existence of solutions for a nonlinear elliptic equation with general flux term, Fixed Point Theory Appl. (2011) · Zbl 1221.35153 · doi:10.1155/2011/496417
[3] Pak, H.-C.; Park, Y.-J., Trace operator and a nonlinear boundary value problem in a new space, Bound. Value Probl. (2014) · Zbl 1304.35294 · doi:10.1186/s13661-014-0153-z
[4] Pak, H.-C.; Park, Y.-J., Spectrum and singular integrals on a new weighted function space, Acta Math. Sin. Engl. Ser., 34, 11, 1692-1702 (2018) · Zbl 1411.46026 · doi:10.1007/s10114-018-7043-8
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