Banach Journal of Mathematical Analysis

Square functions and spectral multipliers for Bessel operators in UMD spaces

Jorge J. Betancor, Alejandro J. Castro, and L. Rodríguez-Mesa

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Abstract

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.

Article information

Source
Banach J. Math. Anal., Volume 10, Number 2 (2016), 338-384.

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

Permanent link to this document
https://projecteuclid.org/euclid.bjma/1459772693

Digital Object Identifier
doi:10.1215/17358787-3495627

Mathematical Reviews number (MathSciNet)
MR3481108

Zentralblatt MATH identifier
1338.42021

Subjects
Primary: 42A25
Secondary: 42B20: Singular and oscillatory integrals (Calderón-Zygmund, etc.) 43A15: $L^p$-spaces and other function spaces on groups, semigroups, etc. 46B20: Geometry and structure of normed linear spaces 46E40: Spaces of vector- and operator-valued functions 47D03: Groups and semigroups of linear operators {For nonlinear operators, see 47H20; see also 20M20}

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

Citation

Betancor, Jorge J.; Castro, Alejandro J.; Rodríguez-Mesa, L. Square functions and spectral multipliers for Bessel operators in UMD spaces. Banach J. Math. Anal. 10 (2016), no. 2, 338--384. doi:10.1215/17358787-3495627. https://projecteuclid.org/euclid.bjma/1459772693


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