Fredholm alternative for a general class of nonlocal operators

We develop a Fredholm alternative for a fractional elliptic operator 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 [Tru73]. We build 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 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.