Conical square functions associated with Bessel, Laguerre and Schrödinger operators in UMD Banach spaces. (English) Zbl 1351.42029
Summary: In this paper we consider conical square functions in the Bessel, Laguerre and Schrödinger settings where the functions take values in UMD Banach spaces. Following a recent paper of T. Hytönen et al. [J. Anal. Math. 106, 317–351 (2008; Zbl 1165.46015)], in order to define our conical square functions, we use \(\gamma\)-radonifying operators. We obtain new equivalent norms in the Lebesgue-Bochner spaces \(L^p((0, \infty), \mathbb{B})\) and \(L^p(\mathbb{R}^n, \mathbb{B})\), \(1 < p < \infty\), in terms of our square functions, provided that \(\mathbb{B}\) is a UMD Banach space. Our results can be seen as Banach valued versions of known scalar results for square functions.
MSC:
| 42B99 | Harmonic analysis in several variables |
| 46E40 | Spaces of vector- and operator-valued functions |
Keywords:
conical square functions; vector-valued harmonic analysis; UMD Banach spaces; Lebesgue-Bochner spaces; Bessel operators; Laguerre operators; Schrödinger operatorsCitations:
Zbl 1165.46015References:
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