Fractional Sobolev spaces via interpolation, and applications to mixed local-nonlocal operators

In this note, we present a well-known connection between the Sobolev-Slobodeckij spaces, also known as Fractional Sobolev spaces, and interpolation theory. We show how Sobolev spaces can be equivalently characterized as real and complex interpolation spaces between Lebesgue spaces and integer-order Sobolev spaces. We also state a spectral theorem for the so-called mixed local-nonlocal operators, and show how interpolation theory leads to its proof. This note is intended for early-career researchers, and aims to provide a concise and accessible introduction to the subject.