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In this paper we establish a T1 criterion for the boundedness ofHermite-Calderon-Zygmund operators on the BMO_H(R^n) space naturally associatedto the Hermite operator H. We apply this criterion in a systematic way to provethe boundedness on BMO_H(R^n) of certain harmonic analysis operators related toH (Riesz transforms, maximal operators, Littlewood-Paley g-functions andvariation operators).

A T1 criterion for Hermite-Calderon-Zygmund operators on the BMO_H(R^n) space and applications