Time Fractional Parabolic Equations with Measurable Coefficients and Embeddings for Fractional Parabolic Sobolev Spaces

We consider time fractional parabolic equations in divergence and non-divergence form when the leading coefficients aij are measurable functions of (t,x1) except for a11, which is a measurable function of either t or x1. We obtain the solvability in Sobolev spaces of the equations in the whole space, on a half space, and on a partially bounded domain. The proofs use a level set argument, a scaling argument, and embeddings in fractional parabolic Sobolev spaces for which we give a direct and elementary proof.