Sharp Logarithmic Inequalities for Two Hardy-type Operators

• Autorzy: Osękowski A.

Identyfikatory

DOI

10.4064/ba8039-12-2015

• Języki publikacji: EN

Abstrakty

EN

For any locally integrable f on Rⁿ, we consider the operators S and T which average f over balls of radius |x| and center 0 and x, respectively: [WZÓR] for x ∈ Rⁿ. The purpose of the paper is to establish sharp localized LlogL estimates for S and T. The proof rests on a corresponding one-weight estimate for a martingale maximal function, a result which is of independent interest.

Słowa kluczowe

EN

Hardy–Littlewood operator; LlogL inequality; martingale; best constant

• Wydawca: Instytut Matematyczny PAN
• Czasopismo: Bulletin of the Polish Academy of Sciences. Mathematics
• Rocznik: 2015
• Tom: Vol. 63, no. 3
• Strony: 237--247
• Opis fizyczny: Bibliogr. 5 poz.

Twórcy

autor

Osękowski A.

• Department of Mathematics, Informatics and Mechanics, University of Warsaw, Banacha 2, 02-097 Warszawa, Poland

Bibliografia

• [1] M. Christ and L. Grafakos, Best constants for two nonconvolution inequalities, Proc. Amer. Math. Soc. 123 (1995), 1687–1693.
• [2] J. L. Doob, Stochastic Processes, Wiley, New York, 1953.
• [3] D. Gilat, The best bound in the LlogL inequality of Hardy and Littlewood and its martingale counterpart, Proc. Amer. Math. Soc. 97 (1986), 429–436.
• [4] S. E. Graversen and G. Peskir, Optimal stopping in the LlogL-inequality of Hardy and Littlewood, Bull. London Math. Soc. 30 (1998), 171–181.
• [5] G. Hardy, J. Littlewood, and G. Pólya, Inequalities, The University Press, Cambridge, 1959.