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April 2016 Square functions and spectral multipliers for Bessel operators in UMD spaces
Jorge J. Betancor, Alejandro J. Castro, L. Rodríguez-Mesa
Banach J. Math. Anal. 10(2): 338-384 (April 2016). DOI: 10.1215/17358787-3495627


In this paper, we consider square functions (also called Littlewood–Paley g-functions) associated to Hankel convolutions acting on functions in the Bochner–Lebesgue space Lp((0,),B), where B is a UMD Banach space. As special cases, we study square functions defined by fractional derivatives of the Poisson semigroup for the Bessel operator Δλ=xλddxx2λddxxλ, λ>0. We characterize the UMD property for a Banach space B by using Lp((0,),B)-boundedness properties of g-functions defined by Bessel–Poisson semigroups. As a by-product, we prove that the fact that the imaginary power Δλiω, ωR{0}, of the Bessel operator Δλ is bounded in Lp((0,),B), 1<p<, characterizes the UMD property for the Banach space B. As applications of our results for square functions, we establish the boundedness in Lp((0,),B) of spectral multipliers m(Δλ) of Bessel operators defined by functions m which are holomorphic in sectors Σϑ.


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Jorge J. Betancor. Alejandro J. Castro. L. Rodríguez-Mesa. "Square functions and spectral multipliers for Bessel operators in UMD spaces." Banach J. Math. Anal. 10 (2) 338 - 384, April 2016.


Received: 10 April 2015; Accepted: 9 July 2015; Published: April 2016
First available in Project Euclid: 4 April 2016

zbMATH: 1338.42021
MathSciNet: MR3481108
Digital Object Identifier: 10.1215/17358787-3495627

Primary: 42A25
Secondary: 42B20 , 43A15 , 46B20 , 46E40 , 47D03

Keywords: $\gamma$-radonifying operator , Bessel operator , spectral multiplier , square function , UMD space

Rights: Copyright © 2016 Tusi Mathematical Research Group


Vol.10 • No. 2 • April 2016
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