TY - JOUR

T1 - γ-Radonifying operators and UMD-valued Littlewood-Paley-Stein functions in the Hermite setting on BMO and Hardy spaces

AU - Betancor, J. J.

AU - Castro, A. J.

AU - Curbelo, J.

AU - Fariña, J. C.

AU - Rodríguez-Mesa, L.

N1 - Funding Information:
✩ The authors are partially supported by MTM2010/17974. The second author is also supported by a FPU grant from the Government of Spain. The third author is partially supported by MINECO: ICMAT Severo Ochoa project SEV-2011-0087. * Corresponding author. E-mail addresses: jbetanco@ull.es (J.J. Betancor), ajcastro@ull.es (A.J. Castro), jezabel.curbelo@icmat.es (J. Curbelo), jcfarina@ull.es (J.C. Fariña), lrguez@ull.es (L. Rodríguez-Mesa).

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

UR - http://www.scopus.com/inward/record.url?scp=84868214861&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84868214861&partnerID=8YFLogxK

U2 - 10.1016/j.jfa.2012.09.010

DO - 10.1016/j.jfa.2012.09.010

M3 - Article

AN - SCOPUS:84868214861

VL - 263

SP - 3804

EP - 3856

JO - Journal of Functional Analysis

JF - Journal of Functional Analysis

SN - 0022-1236

IS - 12

ER -