A sharp rearrangement inequality for the fractional maximal operator

A. Cianchi; R. Kerman; B. Opic; L. Pick

ArticleOriginal scientific text

Title

A sharp rearrangement inequality for the fractional maximal operator

Authors ¹, ², ³, ⁴

Affiliations

Istituto di Matematica, Facoltà di Architettura, Università di Firenze, Via dell'Agnolo 14, 50122 Firenze, Italy
Department of Mathematics, Brock University, St. Catharines, Ontario, Canada
Mathematical Institute, Czech Academy of Sciences, Žitná 25, 115 67 Praha 1, Czech Republic
Department of Mathematical Analysis, Faculty of Mathematics and Physics, Charles University, Sokolovská 83, 186 75 Praha 8, Czech Republic

Abstract

We prove a sharp pointwise estimate of the nonincreasing rearrangement of the fractional maximal function of ⨍,

M_{γ} ⨍

, by an expression involving the nonincreasing rearrangement of ⨍. This estimate is used to obtain necessary and sufficient conditions for the boundedness of

M_{γ}

between classical Lorentz spaces.

Keywords

ENG

fractional maximal operator, nonincreasing rearrangement, classical Lorentz spaces, weighted norm inequalities

[AM] M. Ariño and B. Muckenhoupt, Maximal functions on classical Lorentz spaces and Hardy's inequality with weights for nonincreasing functions, Trans. Amer. Math. Soc. 320 (1990), 727-735.
[BS] C. Bennett and R. Sharpley, Interpolation of Operators, Pure Appl. Math. 129, Academic Press, New York, 1988.
[OK] B. Opic and A. Kufner, Hardy-Type Inequalities, Pitman Res. Notes Math. Ser. 219, Longman Sci. & Tech., Harlow 1990.
[S] E. Sawyer, Boundedness of classical operators on classical Lorentz spaces, Studia Math. 96 (1990), 145-158.
[T] A. Torchinsky, Real-Variable Methods in Harmonic Analysis, Pure Appl. Math. 123, Academic Press, New York, 1986.

Title

A sharp rearrangement inequality for the fractional maximal operator

Affiliations

Abstract

Keywords

Bibliography