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Square functions and spectral multipliers for Bessel operators in UMD spaces. (English) Zbl 1338.42021
Summary: 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 \(\Delta_{\lambda}=-x^{-\lambda}\frac{d}{dx}x^{2\lambda}\frac{d}{dx}x^{-\lambda}\), \(\lambda> 0\). 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 \(L^{p}((0,\infty),\mathbb{B})\), \(1< p< \infty\), 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 \(\Sigma_{\vartheta}\).

MSC:
42B25 Maximal functions, Littlewood-Paley theory
42B15 Multipliers for harmonic analysis in several variables
42B20 Singular and oscillatory integrals (Calderón-Zygmund, etc.)
46B20 Geometry and structure of normed linear spaces
46E40 Spaces of vector- and operator-valued functions
47D03 Groups and semigroups of linear operators
43A15 \(L^p\)-spaces and other function spaces on groups, semigroups, etc.
47H20 Semigroups of nonlinear operators
20M20 Semigroups of transformations, relations, partitions, etc.
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