One-dimensional degenerate diffusion operators

TL;DR

This paper investigates the mathematical properties of a class of one-dimensional degenerate diffusion operators, focusing on their generation, sectoriality, and gradient estimates for associated semigroups and resolvents in continuous function spaces.

Contribution

It provides new results on the analytical behavior of these operators, particularly regarding their generation and gradient estimates in the space of continuous functions.

Findings

Establishment of generation properties for the operators.

Derivation of sectoriality conditions.

Gradient estimates for semigroups and resolvents.

Abstract

The aim of this paper is to present some results about generation, sectoriality and gradient estimates both for the semigroup and for the resolvent of suitable realizations of the operators [\Ab u(x)=\gamma xu"(x) + b u'(x),] with constants $\gamma >0$ and $b\geq 0$, in the space $C([0,\infty])$