Annales Academiæ Scientiarum Fennicæ
Mathematica
Volumen 41, 2016, 491-501

MARCINKIEWICZ SPACES, GARSIA-RODEMICH SPACES AND THE SCALE OF JOHN-NIRENBERG SELF IMPROVING INEQUALITIES

Mario Milman

CONICET, Instituto Argentino de Matemática
Saavedra 15, 1083 Buenos Aires, Argentina; mario.milman 'at' gmail.com

Abstract. We extend to n-dimensions a characterization of the Marcinkiewicz L(p,∞) spaces first obtained by Garsia-Rodemich in the one dimensional case. This leads to a new proof of the John-Nirenberg self-improving inequalities. We also show a related result that provides still a new characterization of the L(p,∞) spaces in terms of distribution functions, reflects the self-improving inequalities directly, and also characterizes L(∞,∞), the rearrangement invariant hull of BMO. We show an application to the study of tensor products with L(∞,∞) spaces, which complements the classical work of O'Neil [19] and the more recent work of Astashkin [2].

2010 Mathematics Subject Classification: Primary 42B35, 46E30.

Key words: John-Nirenberg inequality, rearrangement, BMO.

Reference to this article: M. Milman: Marcinkiewicz spaces, Garsia-Rodemich spaces and the scale of John-Nirenberg self improving inequalities. Ann. Acad. Sci. Fenn. Math. 41 (2016), 491-501.

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doi:10.5186/aasfm.2016.4129