Singular integral operators in Dunkl setting (Q2905188)

The paper deals with vector-valued inequalities in a setting of weighted Lebesgue measures \(\mu_k\) invariant under the action of a reflection group \(G\).NEWLINENEWLINEThe authors first present a theorem for Banach-valued singular integral operators. More precisely, let \(\mathfrak B_1, \mathfrak B_2\) be two Banach spaces and let \(T: L^r_k(\mathbb R^N, \mathfrak B_1) \to L^r_k(\mathbb R^N, \mathfrak B_2)\) be a bounded operator for some \(1<r\leqslant+\infty\), where \(L^r_k(\mathbb R^N, \mathfrak B)\) denotes the Bochner space with underlying measure \(\mu_k\). Suppose that for every compactly supported \(f \in L^\infty(\mathbb R^N, \mathfrak B_1)\) NEWLINE\[NEWLINET(f)(x)=\int_{\mathbb R^N}\mathcal K(x,y)f(y)d\mu_k(y),NEWLINE\]NEWLINE with \(g.x \notin \text{supp}(f)\) for all \(g \in G\). If the kernel \(\mathcal K\) satisfies the following Hörmander type conditions NEWLINE\[NEWLINE\int_{\mathrm{min }_{g \in G}|g.x-y|>2|y-y_0|}\|\mathcal K(x,y)-\mathcal K(x,y_0)\|d\mu_k(x)\leqslant C, \quad y,y_0 \in \mathbb R^N,NEWLINE\]NEWLINE NEWLINE\[NEWLINE\int_{\mathrm{min }_{g \in G}|g.x-y|>2|x-x_0|}\|\mathcal K(x,y)-\mathcal K(x_0,y)\|d\mu_k(y)\leqslant C, \quad x,x_0 \in \mathbb R^N,NEWLINE\]NEWLINE with \(\mathopen\|\cdot\mathclose\|\) the usual norm of \(\mathcal L(\mathfrak B_1,\mathfrak B_2)\), then \(T\) can be extended to a bounded operator from \(L^p_k(\mathbb R^N, \mathfrak B_1)\) to \(L^p_k(\mathbb R^N, \mathfrak B_2)\) for all \(1<p<+\infty\).NEWLINENEWLINEThe proof of this theorem is mainly based on an appropriate Calderón-Zygmund decomposition.NEWLINENEWLINEThe theorem is then exploited to deduce some maximal inequalities in the setting of Dunkl operators, which are roughly speaking deformations of the usual derivatives by reflections. Following some ideas presented in [\textit{L. Grafakos}, Classical Fourier Analysis. Graduate Texts in Mathematics 249. New York, NY: Springer (2008; Zbl 1220.42001)] and using a generalized Poisson kernel, the authors prove on one hand Fefferman-Stein type inequalities for the Dunkl maximal operator. On the other hand, they introduce a Dunkl type \(g\)-function and deduce from their main theorem some estimates for it.