A sharp rearrangement inequality for the fractional maximal operator

• Autorzy: A. Cianchi , R. Kerman , B. Opic , L. Pick
• Języki publikacji: EN

Abstrakty

EN

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.

Słowa kluczowe

EN

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

Kategorie tematyczne

• 47B38: Operators on function spaces (general)
• 47G10: Integral operators
• 46E30: Spaces of measurable functions ( L p -spaces, Orlicz spaces, K\"othe function spaces, Lorentz spaces, rearrangement invariant spaces, ideal spaces, etc.)
• 42B25: Maximal functions, Littlewood-Paley theory
• 26D10: Inequalities involving derivatives and differential and integral operators

• Wydawca: Institute of Mathematics Polish Academy of Sciences
• Czasopismo: Studia Mathematica
• Rocznik: 2000
• Tom: 138
• Numer: 3
• Strony: 277-284

Daty

wydano

2000

otrzymano

1999-05-24

poprawiono

2000-01-03

Twórcy

autor

A. Cianchi

• Istituto di Matematica, Facoltà di Architettura, Università di Firenze, Via dell'Agnolo 14, 50122 Firenze, Italy

autor

R. Kerman

• Department of Mathematics, Brock University, St. Catharines, Ontario, Canada

autor

B. Opic

• Mathematical Institute, Czech Academy of Sciences, Žitná 25, 115 67 Praha 1, Czech Republic

autor

L. Pick

• Department of Mathematical Analysis, Faculty of Mathematics and Physics, Charles University, Sokolovská 83, 186 75 Praha 8, Czech Republic

Bibliografia

• [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.