The boundedness of variation associated with the commutators of approximate identities

In this paper, we establish $L^{p}$ L p -boundedness and endpoint estimates for variation associated with the commutators of approximate identities, which are new for variation operators. As corollaries, we obtain the corresponding boundedness results for variation associated with the commutators of heat semigroups and Poisson semigroups.

Property RD and Hypercontractivity for Orthogonal Free Quantum Groups

We prove that the twisted property RD introduced in [ 2] fails to hold for all non-Kac type, non-amenable orthogonal free quantum groups. In the Kac case we revisit property RD, proving an analogue of the $L_p-L_2$ non-commutative Khintchine inequality for free groups from [ 29]. As an application, we give new and improved hypercontractivity and ultracontractivity estimates for the generalized heat semigroups on free orthogonal quantum groups, both in the Kac and non-Kac cases.

Boundedness of Differential Transforms for Heat Semigroups Generated by Schrödinger Operators

In this paper we analyze the convergence of the following type of series $$\begin{eqnarray}T_{N}^{{\mathcal{L}}}f(x)=\mathop{\sum }_{j=N_{1}}^{N_{2}}v_{j}\big(e^{-a_{j+1}{\mathcal{L}}}f(x)-e^{-a_{j}{\mathcal{L}}}f(x)\big),\quad x\in \mathbb{R}^{n},\end{eqnarray}$$ where ${\{e^{-t{\mathcal{L}}}\}}_{t>0}$ is the heat semigroup of the operator ${\mathcal{L}}=-\unicode[STIX]{x1D6E5}+V$ with $\unicode[STIX]{x1D6E5}$ being the classical laplacian, the nonnegative potential $V$ belonging to the reverse Hölder class $RH_{q}$ with $q>n/2$ and $n\geqslant 3$ , $N=(N_{1},N_{2})\in \mathbb{Z}^{2}$ with $N_{1}<N_{2}$ , ${\{v_{j}\}}_{j\in \mathbb{Z}}$ is a bounded real sequences, and ${\{a_{j}\}}_{j\in \mathbb{Z}}$ is an increasing real sequence. Our analysis will consist in the boundedness, in $L^{p}(\mathbb{R}^{n})$ and in $BMO(\mathbb{R}^{n})$ , of the operators $T_{N}^{{\mathcal{L}}}$ and its maximal operator $T^{\ast }f(x)=\sup _{N}T_{N}^{{\mathcal{L}}}f(x)$ . It is also shown that the local size of the maximal differential transform operators (with $V=0$ ) is the same with the order of a singular integral for functions $f$ having local support. Moreover, if ${\{v_{j}\}}_{j\in \mathbb{Z}}\in \ell ^{p}(\mathbb{Z})$ , we get an intermediate size between the local size of singular integrals and Hardy–Littlewood maximal operator.