## Banach Journal of Mathematical Analysis

### Square functions and spectral multipliers for Bessel operators in UMD spaces

#### 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 $L^{p}((0,\infty),\mathbb{B})$, where $\mathbb{B}$ is a UMD Banach space. As special cases, we study square functions defined by fractional derivatives of the Poisson semigroup for the Bessel operator $\Delta_{\lambda}=-x^{-\lambda}\frac{d}{dx}x^{2\lambda}\frac{d}{dx}x^{-\lambda}$, $\lambda\gt 0$. We characterize the UMD property for a Banach space $\mathbb{B}$ by using $L^{p}((0,\infty),\mathbb{B})$-boundedness properties of $g$-functions defined by Bessel–Poisson semigroups. As a by-product, we prove that the fact that the imaginary power $\Delta_{\lambda}^{i\omega}$, $\omega\in\mathbb{R}\setminus\{0\}$, of the Bessel operator $\Delta_{\lambda}$ is bounded in $L^{p}((0,\infty),\mathbb{B})$, $1\lt p\lt \infty$, characterizes the UMD property for the Banach space $\mathbb{B}$. As applications of our results for square functions, we establish the boundedness in $L^{p}((0,\infty),\mathbb{B})$ of spectral multipliers $m(\Delta_{\lambda})$ of Bessel operators defined by functions $m$ which are holomorphic in sectors $\Sigma_{\vartheta}$.

#### Article information

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

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

https://projecteuclid.org/euclid.bjma/1459772693

Digital Object Identifier
doi:10.1215/17358787-3495627

Mathematical Reviews number (MathSciNet)
MR3481108

Zentralblatt MATH identifier
1338.42021

#### 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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