Vector-valued extensions of operators related to the Ornstein-Uhlenbeck semigroup
The authors consider some operators on functions taking values on a Banach space \(X\). In this setting, the authors investigate the Riesz transforms, the Littlewood-Paley \(g\)-functions, some maximal operators like the maximal operators of Ornstein-Uhlenbeck and the corresponding Poisson semigroups, and the maximal function with respect to the Gaussian measure. Necessary and sufficient conditions on a Banach space \(X\) are found in order that the above operators be of weak type \((1,1)\) or strong type \((p,p)\) in the range \(1 < p < \infty\). These conditions on \(X\) are the same as those required for the corresponding harmonic operator: (1) the UMD-property (which means that the Hilbert transform is a bounded operator in \(L^p_X\) for some \(p\in(1,\infty)\)); (2) Lusin cotype 2 (which means boundedness of the Littlewood-Paley \(g\)-operator for functions defined in the torus); (3) the Hardy-Littlewood property (which means boundedness of the classical Hardy-Littlewood operator). Some results are new even for the classical harmonic case.
scientific article; zbMATH DE number 2103487