arXiv subjects

Michelangelo Cavina

Publications and source records attributed to Michelangelo Cavina.

3 recordsLinked to original sources

arXiv2023

On a quasi-additivity formula for the capacity on Ahlfors-regular spaces

In this work we prove formulas of quasi-additivity for the capacity associated to kernels of radial type in the setting of the boundary of a tree structure and in the setting of compact Ahlfors-regular spaces. We also define a notion of harmonic extension, to one additional variable, of a function defined over a compact Ahlfors-regular space, and we prove a result of tangential convergence of the harmonic extension to the values at the boundary.

math.FA↗

arXiv2023

Bellman function for Hardy's inequality over dyadic trees

In this article we use the Bellman function technique to characterize the measures for which the weighted Hardy's inequality holds on dyadic trees. We enunciate the (dual) Hardy's inequality over the dyadic tree and we use the associated "Burkholder-type" function to find the main inequality required in the proof of the result. We enunciate a "Bellman-type" function satisfying the required properties and we use the Bellman function technique to prove the main result of this article. We prove the optimality of the associated constant with an explicit example of an extremal family of maps. We also give an explicit interpretation of the corresponding Bellman function in terms of the theory of stochastic optimal control.

math.CA↗

arXiv2023

A note on a stochastic approach to Caffarelli-Silvestre Theorem

In this note we analyze the Caffarelli-Silvestre extension function using tools from the theory of stochastic analysis applied to Dirichlet problems. We use a stochastic approach to give the explicit formulation of the kernel associated to the Dirichlet problem which defines the Cafferelli-Silvestre extension function.

math.PR↗