A De Giorgi-Nash type theorem for time fractional diffusion equations

We study the regularity of weak solutions to linear time fractional diffusion equations in divergence form of arbitrary time order . The coefficients are merely assumed to be bounded and measurable, and they satisfy a uniform parabolicity condition. Our main result is a De Giorgi-Nash type theorem, which gives an interior Holder estimate for bounded weak solutions in terms of the data and the -bound of the solution. The proof relies on new a priori estimates for time fractional problems and uses De Giorgi's technique and the method of non-local growth lemmas, which has been introduced recently in the context of nonlocal elliptic equations involving operators like the fractional Laplacian.