Schauder estimates for drifted fractional operators in the supercritical case

Abstract

We consider a non-local operator $L_{ α}$ which is the sum of a fractional Laplacian $\triangle^{α/2} $, $α\in (0,1)$, plus a first order term which is measurable in the time variable and locally $β$-Hölder continuous in the space variables. Importantly, the fractional Laplacian $Δ^{ α/2} $ does not dominate the first order term. We show that global parabolic Schauder estimates hold even in this case under the natural condition $α+ β>1$. 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^{α/2} $ with other operators of $α$-stable type which are somehow close, including the relativistic $α$-stable operator. Moreover, when $α\in (1/2,1)$, we can prove Schauder estimates for more general $α$-stable type operators like the singular cylindrical one, i.e., when $\triangle^{α/2} $ is replaced by a sum of one dimensional fractional Laplacians $\sum_{k=1}^d (\partial_{x_k x_k}^2 )^{α/2}$.