γ-Radonifying operators and UMD-valued Littlewood-Paley-Stein functions in the Hermite setting on BMO and Hardy spaces

J. J. Betancor; A. J. Castro; J. Curbelo; J. C. Fariña; L. Rodríguez-Mesa

doi:10.1016/j.jfa.2012.09.010

γ-Radonifying operators and UMD-valued Littlewood-Paley-Stein functions in the Hermite setting on BMO and Hardy spaces

J. J. Betancor, A. J. Castro, J. Curbelo, J. C. Fariña, L. Rodríguez-Mesa

Department of Mathematics

Research output: Contribution to journal › Article

4 Citations (Scopus)

Abstract

In this paper we study Littlewood-Paley-Stein functions associated with the Poisson semigroup for the Hermite operator on functions with values in a UMD Banach space B. If we denote by H the Hilbert space L ²((0, ∞), dt/t), γ(H,B) represents the space of γ-radonifying operators from H into B. We prove that the Hermite square function defines bounded operators from BMO _L(R ⁿ,B) (respectively, H _L ¹(R _n,B)) into BMO _L(R ⁿ,γ(H,B)) (respectively, H _L ¹(R ⁿ,γ(H,B))), where BMO _L and H _L ¹ denote BMO and Hardy spaces in the Hermite setting. Also, we obtain equivalent norms in BMO _L(R ⁿ,B) and H _L ¹(R _n,B) by using Littlewood-Paley-Stein functions. As a consequence of our results, we establish new characterizations of the UMD Banach spaces.

Original language	English
Pages (from-to)	3804-3856
Number of pages	53
Journal	Journal of Functional Analysis
Volume	263
Issue number	12
DOIs	https://doi.org/10.1016/j.jfa.2012.09.010
Publication status	Published - Dec 15 2012

Keywords

BMO
Hardy spaces
Hermite operator
Littlewood-Paley-Stein functions
UMD Banach spaces
γ-Radonifying operators

ASJC Scopus subject areas

Analysis

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Betancor, J. J., Castro, A. J., Curbelo, J., Fariña, J. C., & Rodríguez-Mesa, L. (2012). γ-Radonifying operators and UMD-valued Littlewood-Paley-Stein functions in the Hermite setting on BMO and Hardy spaces. Journal of Functional Analysis, 263(12), 3804-3856. https://doi.org/10.1016/j.jfa.2012.09.010

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abstract = "In this paper we study Littlewood-Paley-Stein functions associated with the Poisson semigroup for the Hermite operator on functions with values in a UMD Banach space B. If we denote by H the Hilbert space L 2((0, ∞), dt/t), γ(H,B) represents the space of γ-radonifying operators from H into B. We prove that the Hermite square function defines bounded operators from BMO L(R n,B) (respectively, H L 1(R n,B)) into BMO L(R n,γ(H,B)) (respectively, H L 1(R n,γ(H,B))), where BMO L and H L 1 denote BMO and Hardy spaces in the Hermite setting. Also, we obtain equivalent norms in BMO L(R n,B) and H L 1(R n,B) by using Littlewood-Paley-Stein functions. As a consequence of our results, we establish new characterizations of the UMD Banach spaces.",

keywords = "BMO, Hardy spaces, Hermite operator, Littlewood-Paley-Stein functions, UMD Banach spaces, γ-Radonifying operators",

author = "Betancor, {J. J.} and Castro, {A. J.} and J. Curbelo and Fari{\~n}a, {J. C.} and L. Rodr{\'i}guez-Mesa",

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AU - Betancor, J. J.

AU - Castro, A. J.

AU - Curbelo, J.

AU - Fariña, J. C.

AU - Rodríguez-Mesa, L.

PY - 2012/12/15

Y1 - 2012/12/15

N2 - In this paper we study Littlewood-Paley-Stein functions associated with the Poisson semigroup for the Hermite operator on functions with values in a UMD Banach space B. If we denote by H the Hilbert space L 2((0, ∞), dt/t), γ(H,B) represents the space of γ-radonifying operators from H into B. We prove that the Hermite square function defines bounded operators from BMO L(R n,B) (respectively, H L 1(R n,B)) into BMO L(R n,γ(H,B)) (respectively, H L 1(R n,γ(H,B))), where BMO L and H L 1 denote BMO and Hardy spaces in the Hermite setting. Also, we obtain equivalent norms in BMO L(R n,B) and H L 1(R n,B) by using Littlewood-Paley-Stein functions. As a consequence of our results, we establish new characterizations of the UMD Banach spaces.

AB - In this paper we study Littlewood-Paley-Stein functions associated with the Poisson semigroup for the Hermite operator on functions with values in a UMD Banach space B. If we denote by H the Hilbert space L 2((0, ∞), dt/t), γ(H,B) represents the space of γ-radonifying operators from H into B. We prove that the Hermite square function defines bounded operators from BMO L(R n,B) (respectively, H L 1(R n,B)) into BMO L(R n,γ(H,B)) (respectively, H L 1(R n,γ(H,B))), where BMO L and H L 1 denote BMO and Hardy spaces in the Hermite setting. Also, we obtain equivalent norms in BMO L(R n,B) and H L 1(R n,B) by using Littlewood-Paley-Stein functions. As a consequence of our results, we establish new characterizations of the UMD Banach spaces.

KW - BMO

KW - Hardy spaces

KW - Hermite operator

KW - Littlewood-Paley-Stein functions

KW - UMD Banach spaces

KW - γ-Radonifying operators

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