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.

Supported by the Australian Research Council Laureate Fellowship FL190100081 and by the Australian Future Fellowship FT230100333.