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Muckenhoupt class weight decomposition and BMO distance to bounded functions. (English) Zbl 1425.42025

Summary: We study the connection between the Muckenhoupt \(A_p\) weights and bounded mean oscillation (BMO) for general bases for \(\mathbb{R}^d\). New classes of bases are introduced that allow for several deep results on the Muckenhoupt weights-BMO connection to hold in a very general form. The John-Nirenberg type inequality and its consequences are valid for the new class of Calderón-Zygmund bases which includes cubes in \(\mathbb{R}^d\), but also the basis of rectangles in \(\mathbb{R}^d\). Of particular interest to us is the Garnett-Jones theorem on the BMO distance, which is valid for cubes. We prove that the theorem is equivalent to the newly introduced \(A_2\)-decomposition property of bases. Several sufficient conditions for the theorem to hold are analysed as well. However, the question whether the theorem fully holds for rectangles remains open.

MSC:

42B25 Maximal functions, Littlewood-Paley theory
42B35 Function spaces arising in harmonic analysis
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[1] 1S.-Y. A.Chang and R.Fefferman, Some recent developments in Fourier analysis and H^p-theory on product domains, Bull. Amer. Math. Soc. (N.S.)12(1) (1985), 1-43. · Zbl 0557.42007
[2] 2R. R.Coifman and R.Rochberg, Another characterization of BMO, Proc. Amer. Math. Soc.79(2) (1980), 249-254. · Zbl 0432.42016
[3] 3D. V.Cruz-Uribe, J. M.Martell and C.Pérez, Weights, extrapolation and the theory of Rubio de Francia, Operator Theory: Advances and Applications, Volume 215 (Birkhäuser/Springer Basel, Basel, 2011). · Zbl 1234.46003
[4] 4J.Duoandikoetxea, F. J.Martín Reyes and S.Ombrosi, On the A_∞ conditions for general bases, Math. Z.282(3-4) (2016), 955-972. · Zbl 1343.42034
[5] 5J.García-Cuerva and J. L.Rubio de Francia, Weighted norm inequalities and related topics, North-Holland Mathematics Studies, Volume 116 (North-Holland, Amsterdam, 1985). · Zbl 0578.46046
[6] 6J. B.Garnett and P. W.Jones, The distance in BMO to L^∞, Ann. of Math. (2)108(2) (1978), 373-393. · Zbl 0358.26010
[7] 7L.Grafakos, Modern Fourier analysis, Graduate Texts in Mathematics, 3rd edn, Volume 250 (Springer, New York, 2014). · Zbl 1304.42002
[8] 8C.Heil and A. M.Powell, Gabor Schauder bases and the Balian-Low theorem, J. Math. Phys.47(11) (2006), 113506, 21. · Zbl 1112.42004
[9] 9T.Hytönen and C.Pérez, Sharp weighted bounds involving A_∞, Anal. PDE6(4) (2013), 777-818. · Zbl 1283.42032
[10] 10B.Jawerth, Weighted inequalities for maximal operators: linearization, localization and factorization, Amer. J. Math.108(2) (1986), 361-414. · Zbl 0608.42012
[11] 11P. W.Jones, Factorization of A_p weights, Ann. of Math. (2)111(3) (1980), 511-530. · Zbl 0493.42030
[12] 12A. A.Korenovskyy, A. K.Lerner and A. M.Stokolos, On a multidimensional form of F. Riesz’s ‘rising sun’ lemma, Proc. Amer. Math. Soc.133(5) (2005), 1437-1440. · Zbl 1070.42012
[13] 13M.Nielsen and H.Šikić, Schauder bases of integer translates, Appl. Comput. Harmon. Anal.23(2) (2007), 259-262. · Zbl 1137.42009
[14] 14M.Nielsen and H.Šikić, Quasi-greedy systems of integer translates, J. Approx. Theory155(1) (2008), 43-51. · Zbl 1157.42011
[15] 15M.Nielsen and H.Šikić, On stability of Schauder bases of integer translates, J. Funct. Anal.266(4) (2014), 2281-2293. · Zbl 1308.46019
[16] 16C.Pérez, Weighted norm inequalities for general maximal operators, Publ. Mat.35(1) (1991), 169-186. · Zbl 0722.42011
[17] 17J. L.Rubio de Francia, Factorization theory and A_p weights, Amer. J. Math.106(3) (1984), 533-547. · Zbl 0558.42012
[18] 18F.Soria, A remark on A_1-weights for the strong maximal function, Proc. Amer. Math. Soc.100(1) (1987), 46-48. · Zbl 0665.42022
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