Fredholm alternative for a general class of nonlocal operators

TL;DR

This paper establishes a Fredholm alternative for a broad class of nonlocal fractional elliptic operators built on fractional gradients, accommodating variable coefficients and non-homogeneous structures.

Contribution

It introduces a functional analytic framework for a general class of nonlocal operators with variable, possibly unbounded or discontinuous coefficients, extending classical elliptic theory.

Findings

Developed a Fredholm alternative for fractional elliptic operators.

Constructed a functional framework for nonlocal operators with variable coefficients.

Analyzed the properties of associated functional spaces.

Abstract

We develop a Fredholm alternative for a fractional elliptic operator~$\mathcal{L}$ of mixed order built on the notion of fractional gradient. This operator constitutes the nonlocal extension of the classical second order elliptic operators with measurable coefficients treated by Neil Trudinger in~\cite{trudinger}. We build~$\mathcal{L}$ by weighing the order~$s$ of the fractional gradient over a measure (which can be either continuous, or discrete, or of mixed type). The coefficients of~$\mathcal{L}$ may also depend on~$s$, giving this operator a possibly non-homogeneous structure with variable exponent. These coefficients can also be either unbounded, or discontinuous, or both. A suitable functional analytic framework is introduced and investigated and our main results strongly rely on some custom analysis of appropriate functional spaces.