Schauder estimates for drifted fractional operators in the supercritical case

Abstract

We consider a non-local operator $L_{\alpha }$ which is the sum of a fractional Laplacian ${△}^{\alpha /2}$, $\alpha \in (0,1)$, plus a first order term which is measurable in the time variable and locally $\beta$-Hölder continuous in the space variables. Importantly, the fractional Laplacian ${\Delta }^{\alpha /2}$ does not dominate the first order term. We show that global parabolic Schauder estimates hold even in this case under the natural condition $\alpha +\beta >1$. Thus, the constant appearing in the Schauder estimates is in fact independent of the $L^{\infty }$-norm of the first order term. In our approach we do not use the so-called extension property and we can replace ${△}^{\alpha /2}$ with other operators of $\alpha$-stable type which are somehow close, including the relativistic $\alpha$-stable operator. Moreover, when $\alpha \in (1/2,1)$, we can prove Schauder estimates for more general $\alpha$-stable type operators like the singular cylindrical one, i.e., when ${△}^{\alpha /2}$ is replaced by a sum of one dimensional fractional Laplacians ${\sum }_{k=1}^d({\partial }_{x_k{x}_k}^2{)}^{\alpha /2}$.