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.

Sharp endpoint estimates for some operators associated with the Laplacian with drift in Euclidean space

Let $v \ne 0$ be a vector in ${\mathbb {R}}^n$ . Consider the Laplacian on ${\mathbb {R}}^n$ with drift $\Delta _{v} = \Delta + 2v\cdot \nabla $ and the measure $d\mu (x) = e^{2 \langle v, x \rangle } dx$ , with respect to which $\Delta _{v}$ is self-adjoint. This measure has exponential growth with respect to the Euclidean distance. We study weak type $(1, 1)$ and other sharp endpoint estimates for the Riesz transforms of any order, and also for the vertical and horizontal Littlewood–Paley–Stein functions associated with the heat and the Poisson semigroups.

Generalized Riesz potential spaces and their characterization viawavelet-type transform

We introduce a wavelet-type transform generated by the so-called beta-semigroup, which is a natural generalization of the Gauss-Weierstrass and Poisson semigroups associated to the Laplace-Bessel convolution. By making use of this wavelet-type transform we obtain new explicit inversion formulas for the generalized Riesz potentials and a new characterization of the generalized Riesz potential spaces. We show that the usage of the concept beta-semigroup gives rise to minimize the number of conditions on wavelet measure, no matter how big the order of the generalized Riesz potentials is.