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 a(ij) are measurable functions of (t, x(1)) except for a(11), which is a measurable function of either t or x(1). 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.