In this paper we characterise the pointwise size and regularity estimates for the Dunkl Riesz transform kernel involving both the Euclidean metric and the Dunkl metric, where the two metrics are not equivalent. We further establish a suitable version
of the pointwise lower bound via the Euclidean metric and then characterise boundedness of commutator of the Dunkl Riesz transform via the BMO space associated with the Euclidean metric and the Dunkl measure. This shows that BMO space via the Euclidean metric is the suitable one associated to the Dunkl setting but not the one via the Dunkl metric.
Let $M = mathbb R^m sharp mathcal R^n$ be a non-doubling manifold with two ends $mathbb R^m sharp mathcal R^n$, $m > n ge 3$. Let $Delta$ be the Laplace--Beltrami operator which is non-negative self-adjoint on $L^2(M)$. Then $Delta$ and its square ro
ot $sqrt{Delta}$ generate the semigroups $e^{-tDelta}$ and $e^{-tsqrt{Delta}}$ on $L^2(M)$, respectively. We give testing conditions for the two weight inequality for the Poisson semigroup $e^{-tsqrt{Delta}}$ to hold in this setting. In particular, we prove that for a measure $mu$ on $M_{+}:=Mtimes (0,infty)$ and $sigma$ on $M$: $$ |mathsf{P}_sigma(f)|_{L^2(M_{+};mu)} lesssim |f|_{L^2(M;sigma)}, $$ with $mathsf{P}_sigma(f)(x,t):= int_M mathsf{P}_t(x,y)f(y) ,dsigma(y)$ if and only if testing conditions hold for the Poisson semigroup and its adjoint. Further, the norm of the operator is shown to be equivalent to the best constant in these testing conditions.
In this paper, we prove a $Tb$ theorem on product spaces $Bbb R^ntimes Bbb R^m$, where $b(x_1,x_2)=b_1(x_1)b_2(x_2)$, $b_1$ and $b_2$ are para-accretive functions on $Bbb R^n$ and $Bbb R^m$, respectively.
