Boundedness of differential transforms for Dunkl heat semigroups
Streszczenie
We study the boundedness and the convergence of the differential transform $$ T_{N,\kappa }f(x)=\sum _{j=N_1}^{N_2}v_j(e^{a_{j+1}\varDelta _\kappa }f(x)-e^{a_{j}\varDelta _\kappa }f(x)) $$ associated with the Dunkl heat semigroup $\{e^{t\varDelta _\kappa }f\}_{t \gt 0}$. Here $\{v_j\}_{j\in \mathbb {Z}}$ is a bounded sequence, $\{a_j\}_{j\in \mathbb {Z}}$ is an increasing sequence of positive real numbers , and $N=(N_1,N_2)\subset \mathbb {Z}^2$ with $N_1 \lt N_2$. We prove a Cotlar-type inequality for the differential maximal operator $$T_\kappa ^*f=\sup _{\substack{N=(N_1,N_2)\in \mathbb {Z}^2\\ N_1 \lt N_2}}|T_{N,\kappa }f(x)|.$$ Using this, we obtain the boundedness of $T_\kappa ^*$ on weighted $L^p$ spaces for $1 \lt p \lt \infty $. Moreover, we establish the boundedness of $T_\kappa ^*$ on the spaces of bounded mean oscillations in Dunkl settings. Finally, as an application, we show the pointwise convergence of $T_{N,\kappa }f$ as $N\rightarrow (-\infty ,\infty )$ for $f\in L^p(\mathbb R^n,d\mu _\kappa )$, $1 \lt p \lt \infty $.
