In this paper, we consider square functions (also called Littlewood–Paley -functions) associated to Hankel convolutions acting on functions in the Bochner–Lebesgue space , where is a UMD Banach space. As special cases, we study square functions defined by fractional derivatives of the Poisson semigroup for the Bessel operator , . We characterize the UMD property for a Banach space by using -boundedness properties of -functions defined by Bessel–Poisson semigroups. As a by-product, we prove that the fact that the imaginary power , , of the Bessel operator is bounded in , , characterizes the UMD property for the Banach space . As applications of our results for square functions, we establish the boundedness in of spectral multipliers of Bessel operators defined by functions which are holomorphic in sectors .
"Square functions and spectral multipliers for Bessel operators in UMD spaces." Banach J. Math. Anal. 10 (2) 338 - 384, April 2016. https://doi.org/10.1215/17358787-3495627