The characterization of the Carleson measures for analytic Besov spaces: a simple proof

We give a simple proof of the characterization of the Carleson measures for the weighted analytic Besov spaces. Such characterization provides some information on the radial variation of an analytic Besov function.Comment: 12 page

Potential Theory on Trees, Graphs and Ahlfors Regular Metric Spaces

We investigate connections between potential theories on a Ahlfors-regular metric space X, on a graph G associated with X, and on the tree T obtained by removing the "horizontal edges" in G. Applications to the calculation of set capacity are given.Comment: 45 pages; presentation improved based on referee comment

Geometric Hardy inequalities for the sub-elliptic Laplacian on convex domains in the Heisenberg group

We prove geometric $L^p$ versions of Hardy's inequality for the sub-elliptic Laplacian on convex domains $\Omega$ in the Heisenberg group $\mathbb{H}^n$, where convex is meant in the Euclidean sense. When $p=2$ and $\Omega$ is the half-space given by $\langle \xi, \nu\rangle > d$ this generalizes an inequality previously obtained by Luan and Yang. For such $p$ and $\Omega$ the inequality is sharp and takes the form \begin{equation} \int_\Omega |\nabla_{\mathbb{H}^n}u|^2 \, d\xi \geq \frac{1}{4}\int_{\Omega} \sum_{i=1}^n\frac{\langle X_i(\xi), \nu\rangle^2+\langle Y_i(\xi), \nu\rangle^2}{\textrm{dist}(\xi, \partial \Omega)^2}|u|^2\, d\xi, \end{equation} where $\textrm{dist}(\, \cdot\,, \partial \Omega)$ denotes the Euclidean distance from $\partial \Omega$.Comment: 14 page

Sharp LpL^p estimates for second order Riesz transforms on multiply–connected Lie groups

We study a class of combinations of second order Riesz transforms on Lie groups $G = G_x \times G_y$ that are multiply connected, composed of a discrete abelian component $G_x$ and a compact connected component $G_y$ . We prove sharp $L^p$ estimates for these operators, therefore generalizing previous results [13][4]. The proof uses stochastic integrals with jump components adapted to functions defined on the semi-discrete set $G = G_x \times G_y$ . The analysis shows that Itô 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

Some notions of subharmonicity over the quaternions

This works introduces several notions of subharmonicity for real-valued functions of one quaternionic variable. These notions are related to the theory of slice regular quaternionic functions introduced by Gentili and Struppa in 2006. The interesting properties of these new classes of functions are studied and applied to construct the analogs of Green's functions.Comment: 16 page

Equilibrium measures on trees

We give a characterization of equilibrium measures for p-capacities on the boundary of an infinite tree of arbitrary (finite) local degree. For p= 2 , this provides, in the special case of trees, a converse to a theorem of Benjamini and Schramm, which interpretes the equilibrium measure of a planar graph’s boundary in terms of square tilings of cylinders

Ahlfors regular spaces have regular subspaces of any dimension

We characterize Q-dimensional Ahlfors regular spaces among trees’ boundaries and show how to construct, for each 0 < α < Q, an α-regular subspace. As an application, we give an alternative simple proof of the existence of α-regular subspaces of a Q-dimensional complete Ahlfors regular metric space (X, ρ), which was proved in [8]

Ahlfors regular spaces have regular subspaces of any dimension

We characterize Q-dimensional Ahlfors regular spaces among trees' boundaries and show how to construct, for each 0 < alpha < Q, an alpha-regular subspace. As an application, we give an alternative simple proof of the existence of alpha-regular subspaces of a Q-dimensional complete Ahlfors regular metric space (X, rho), which was proved in [8]

Two-weight dyadic Hardy inequalities

We present various results concerning the two-weight Hardy inequality on infinite trees. Our main aim is to survey known characterizations (and proofs) for trace measures, as well as to provide some new ones. Also for some of the known characterizations we provide here new proofs. In particular, we obtain a new characterization in terms of a reverse Hölder inequality for trace measures, and one based on the well-known Muckenhoupt–Wheeden–Wolff inequality, of which we here give a new probabilistic proof. We provide a new direct proof for the so-called isocapacitary characterization and a new simple proof, based on a monotonicity argument, for the so-called mass-energy characterization. Furthermore, we introduce a conformally invariant version of the two-weight Hardy inequality, characterize the compactness of the Hardy operator, provide a list of open problems, and suggest some possible lines of future research