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 $\triangle^{\alpha/2} $, $\alpha \in (0,1)$, plus a first order term which is measurable in the time variable and locally $\beta$-H\"older 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 $\triangle^{\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 $\triangle^{\alpha/2} $ is replaced by a sum of one dimensional fractional Laplacians $\sum_{k=1}^d (\partial_{x_k x_k}^2 )^{\alpha/2}$.