Weighted variational inequalities for the fractional Dunkl heat semigroup

Abstract

We investigate the convergence properties of the family of operators $${\mathcal{T}}_{\mathbf{N}}f(x)=\sum _{j=N_1}^{N_2}{v}_j(e^{-a_{j+1}(-{\Delta }_k{)}^s}f(x)-e^{-a_j(-{\Delta }_k{)}^s}f(x)),x\in {\mathbb{R}}^d,$$ where $\{e^{-t(-{\Delta }_k{)}^s}{\}}_{t>0}$ denotes the fractional heat semigroup generated by the Dunkl Laplacian ${\Delta }_k$. Here $\mathbf{N}=(N_1,N_2)\in {\mathbb{Z}}^2$, $N_1<N_2$, the coefficients $\{v_j{\}}_{j\in \mathbb{Z}}$ form a bounded sequence of real numbers, and $\{a_j{\}}_{j\in \mathbb{Z}}$ is a monotone increasing sequence of reals. The primary objective of this work is to establish boundedness results for these differential transform operators on weighted $L^p({\mathbb{R}}^d,d{\mu }_k)$ spaces as well as on Dunkl $\mathrm{BMO}({\mathbb{R}}^d)$ spaces. We also establish analogous boundedness properties for the associated maximal operator $${\mathcal{T}}^{\ast }f(x)=\underset{\mathbf{N}}{\sup }|{\mathcal{T}}_{\mathbf{N}}f(x)|$$ and study the pointwise convergence of the corresponding series. In addition, we prove that, for compactly supported functions, the maximal differential transform operator ${\mathcal{T}}^{\ast }$ exhibits local behaviour comparable to that of classical singular integral operators.

37 pages