Let S be the Lie group RxR^n, where R acts on R^n by dilations, endowed with the left-invariant Riemannian symmetric space structure and the right Haar measure ρ, which is a Lie group of exponential growth. Hebisch and Steger in [Math. Z. 245: 37–61, 2003] proved that any integrable function on (S, ρ) admits a Calderón–Zygmund decomposition which involves a particular family of sets, called Calderón–Zygmund sets. In this paper, we show the existence of a dyadic grid in the group S, which has nice properties similar to the classical Euclidean dyadic cubes. Using the properties of the dyadic grid, we prove a Fefferman–Stein type inequality, involving the dyadic Hardy–Littlewood maximal function and the dyadic sharp function. As a consequence, we obtain a complex interpolation theorem involving the Hardy space H^1 and the space BMO introduced in [Collect. Math. 60: 277–295, 2009].

Dyadic sets, maximal functions and applications on ax + b-groups / Liu, L.; Vallarino, Maria; Yang, D.. - In: MATHEMATISCHE ZEITSCHRIFT. - ISSN 0025-5874. - STAMPA. - 270:1-2(2012), pp. 515-529. [10.1007/s00209-010-0809-z]

Dyadic sets, maximal functions and applications on ax + b-groups

VALLARINO, MARIA;
2012

Abstract

Let S be the Lie group RxR^n, where R acts on R^n by dilations, endowed with the left-invariant Riemannian symmetric space structure and the right Haar measure ρ, which is a Lie group of exponential growth. Hebisch and Steger in [Math. Z. 245: 37–61, 2003] proved that any integrable function on (S, ρ) admits a Calderón–Zygmund decomposition which involves a particular family of sets, called Calderón–Zygmund sets. In this paper, we show the existence of a dyadic grid in the group S, which has nice properties similar to the classical Euclidean dyadic cubes. Using the properties of the dyadic grid, we prove a Fefferman–Stein type inequality, involving the dyadic Hardy–Littlewood maximal function and the dyadic sharp function. As a consequence, we obtain a complex interpolation theorem involving the Hardy space H^1 and the space BMO introduced in [Collect. Math. 60: 277–295, 2009].
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Utilizza questo identificativo per citare o creare un link a questo documento: http://hdl.handle.net/11583/2498282
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