Capacity of shrinking condensers in the plane

We show that the capacity of a class of plane condensers is comparable to the capacity of corresponding "dyadic condensers". As an application, we show that for plane condensers in that class the capacity blows up as the distance between the plates shrinks, but there can be no asymptotic estimate of the blow-up

Stability of isometric maps in the Heisenberg group

In this paper we prove some approximation results for biLipschitz maps in the Heisenberg group. Namely, we show that a biLipschitz map with biLipschitz constant close to one can be pointwise approximated, quantitatively on any fixed ball, by an isometry. This leds to an approximation in BMO norm for the map's Pansu derivative. We also prove that a global quasigeodesic can be approximated by a geodesic in any fixed segment

From Hankel operators to Carleson measures in a quaternionic variable

We introduce and study Hankel operators defined on the Hardy space of regular functions of a quaternionic variable. Theorems analogous to those of Nehari anc C. Fefferman are proved.Comment: 19 page

The orthogonal projection on slice functions on the quaternionic sphere

We study the $L^p$ norm of the orthogonal projection from the space of quaternion valued $L^2$ functions to the closed subspace of slice $L^2$ functions.Comment: 6 page

Some Hilbert spaces related with the Dirichlet space

We study the reproducing kernel Hilbert space with kernel kd , where d is a positive integer and k is the reproducing kernel of the analytic Dirichlet space

Second order Riesz transforms on multiply-connected Lie groups and processes with jumps

We study a class of combinations of second order Riesz transforms on Lie groups that are multiply connected, composed of a discrete abelian component and a compact connected component. We prove sharp $L^{p}$ estimates for these operators, therefore generalising previous results. We construct stochastic integrals with jump components adapted to functions defined on our semi-discrete set. We show that these second order Riesz transforms applied to a function may be written as conditional expectation of a simple transformation of a stochastic integral associated with the function. The analysis shows that Ito integrals for the discrete component must be written in an augmented discrete tangent plane of dimension twice larger than expected, and in a suitably chosen discrete coordinate system. Those artifacts are related to the difficulties that arise due to the discrete component, where derivatives of functions are no longer local. Previous representations of Riesz transforms through stochastic integrals in this direction do not consider discrete components and jump processes

Carleson Measures for the Drury-Arveson Hardy space and other Besov-Sobolev spaces on Complex Balls

We characterize the Carleson measures for the Drury-Arveson Hardy space and other Hilbert spaces of analytic functions of several complex variables. This provides sharp estimates for Drury's generalization of Von Neumann's inequality. The characterization is in terms of a geometric condition, the "split tree condition", which reflects the nonisotropic geometry underlying the Drury-Arveson Hardy space