Fundamental Solution of the Fractional Diffusion Equation with a Singular Drift*

Abstract

We prove pointwise upper estimates of a fundamental solution of a linear evolution equation with fractional Laplacian and with a singular drift term. Such a result has already been obtained in [Maekawa and Miura, Journal of Functional Analysis 264 (2013) 2245–2268] by generalizing the classical ideas of Carlen, Kusuoka, and Stroock. Here, we propose a different and direct method to show such type of estimates, which appears also to be more elementary.