On Lusin's area integrals and \(g\)-functions in certain Dunkl and Laguerre settings (Q2909663)

The paper contains a thorough analysis of square functions in certain multi-dimensional Dunkl and Laguerre frameworks. The Laguerre framework is related to the so-called Laguerre expansions of convolution type, while the Dunkl setting corresponds to choosing a group of reflections isomorphic to \(\mathbb Z^d_2\) with the Dunkl harmonic oscillator as an underlying `Laplacian'. The Dunkl setting reduces to the Laguerre framework after restricting to a space of reflection invariant functions.NEWLINENEWLINEResults obtained in the present paper and related to Laguerre expansions complete and improve those obtained earlier by the author in [\textit{T. Z. Szarek}, Acta. Math. Hungar. 131, 59--109 (2011; Zbl 1240.42135)]. Moreover, a trivial choice of the multiplicity function reduces the Dunkl setting to the framework of the classical harmonic oscillator. Thus results obtained earlier by several authors and related to square functions for Hermite function expansions are covered or even augmented. In addition, the investigation of Lusin's area type integrals constitutes an essential novelty of the paper.NEWLINENEWLINEThe Dunkl setting in the paper is as follows. The group of reflections \(G\simeq \mathbb Z^d_2\) is generated by \(\sigma_j\), \(j=1,\dots,d\), where \(\sigma_j\) means the reflection in the hyperplane orthogonal to the \(j\)th coordinate vector; the multiplicity function is identified with a multi-index \(\alpha\in[-1\slash2,\infty)^d\), the space \(\mathbb R^d\) is equipped with the measure NEWLINE\[NEWLINE dw_\alpha(x)=\prod_{j=1}^d|x_j|^{2\alpha+1}\,dx,\qquad x=(x_1,\dots,x_d)\in \mathbb R^d, NEWLINE\]NEWLINE such that the Dunkl Laplacian \(\Delta_\alpha=\sum_{j=1}^d(T^\alpha_j)^2\) and the Dunkl harmonic oscillator \(L_\alpha=-\Delta_\alpha+|x|^2\) are symmetric in \(L^2(\mathbb R^d,dw_\alpha)\); here the differential-difference operators \(T^\alpha_j\) are given by \(T^\alpha_jf(x)=\partial_jf(x)+(\alpha_j+1\slash2)\frac{f(x)-f(\sigma_jx)}{x_j}\). The main object of study are vertical and horizontal \(g\)-functions and Luzin's type area integrals based on the semigroup \(\exp(-t\mathcal L_\alpha)\), where \(\mathcal L_\alpha\) is a natural self-adjoint extension of \(L_\alpha\). The main result of the paper, Theorem 2.1, provides weighted \(L^p\) or weighted weak type \((1,1)\) estimates for a large class of weights. In the proof an argument from [\textit{A. Nowak} and \textit{K. Stempak}, SIGMA, Symmetry Integrability Geom. Methods Appl. 5, Paper 016, 12 p. (2009; Zbl 1162.42013)] is used that reduces the analysis to the context of \((\mathbb R^d_+, dw^+_\alpha)\), where \(w^+_\alpha\) is the restriction of \(w_\alpha\) to \(\mathbb R^d_+=(0,\infty)^d\). Then the theory of vector-valued Calderón-Zygmund operators is applied to prove that these auxiliary square functions are Calderón-Zygmund operators in the framework of the space of homogeneous type \((\mathbb R^d_+, dw^+_\alpha,|\cdot|)\).