Aalto: Weak $L^\infty$ and BMO in Metric Spaces

Aalto, Daniel:
Weak $L^\infty$ and BMO in Metric Spaces
Bollettino dell'Unione Matematica Italiana Serie 9 5 (2012), fasc. n.2, p. 369-385, (English)
pdf (314 Kb), djvu (138 Kb). | MR 2977254 | Zbl 1256.46013

Sunto

Bennett, DeVore and Sharpley introduced the space weak $L^{\infty}$ in 1981 and studied its relationship with functions of bounded mean oscillation. Here we characterize the weak $L^{\infty}$ in measure spaces without using the decreasing rearrangement of a function. Instead, we use exponential estimates for the distribution function. In addition, we consider a localized version of the characterization that leads to a new characterization of BMO.

Referenze Bibliografiche

[1] D. AALTO - J. KINNUNEN, The discrete maximal operator in metric spaces, J. Anal. Math., 111 (2010). | fulltext (doi) | MR 2747071 | Zbl 1210.42029

[2] J. BASTERO - M. MILMAN - B. F. J. RUIZ, A note on $L(\infty ,q)$ spaces and Sobolev embeddings, Indiana Univ. Math. J., 52, 5 (2003), 1215-1230. | fulltext (doi) | MR 2010324 | Zbl 1098.46023

[3] C. BENNETT - R. A. DEVORE - R. SHARPLEY, Weak-$L^{\infty}$ and BMO, Ann. of Math. (2), 113 (1981), 601-611. | fulltext (doi) | MR 621018

[4] C. BENNETT - R. SHARPLEY, Interpolation of operators, vol. 129 of Pure and Applied Mathematics, Academic Press Inc. (Boston, MA, 1988). | MR 928802 | Zbl 0647.46057

[5] F. CHIARENZA - M. FRASCA, Morrey spaces and Hardy-Littlewood maximal function, Rend. Mat. Appl. (7), 7 (1987), 3-4 (1988), 273-279. | MR 985999 | Zbl 0717.42023

[6] R. R. COIFMAN - G. WEISS, Analyse harmonique non-commutative sur certains espaces homogènes, Lecture Notes in Mathematics, 242 (Springer-Verlag, Berlin, 1971), Étude de certaines intégrales singulières. | MR 499948 | Zbl 0224.43006

[7] J. HEINONEN, Lectures on analysis on metric spaces, Universitext (Springer-Verlag, New York, 2001). | fulltext (doi) | MR 1800917 | Zbl 0985.46008

[8] A. KORENOVSKII, Mean oscillations and equimeasurable rearrangements of functions, vol. 4 of Lecture Notes of the Unione Matematica Italiana, (Springer, Berlin, 2007). | fulltext (doi) | MR 2363526 | Zbl 1133.42035

[9] P. MACMANUS - C. PÉREZ, Generalized Poincaré inequalities: sharp self-improving properties, Internat. Math. Res. Notices, 2 (1998), 101-116. | fulltext (doi) | MR 1604816

[10] J. MALÝ - L. PICK, An elementary proof of sharp Sobolev embeddings, Proc. Amer. Math. Soc., 130 (2002), 555-563. | fulltext (doi) | MR 1862137 | Zbl 0990.46022

[11] J. MATEU - P. MATTILA - A. NICOLAU - J. OROBITG, BMO for nondoubling measures, Duke Math. J., 102, 3 (2000), 533-565. | fulltext (doi) | MR 1756109

[12] M. MILMAN - E. PUSTYLNIK, On sharp higher order Sobolev embeddings, Commun. Contemp. Math., 6, 3 (2004), 495-511. | fulltext (doi) | MR 2068850 | Zbl 1108.46029

[13] N. M. RIVIÈRE, Interpolation à la Marcinkiewicz, Rev. Un. Mat. Argentina, 25 (1970/71), 363-377. Collection of articles dedicated to Alberto González Domínguez on his sixty-fifth birthday. | MR 370169

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Referenze Bibliografiche