Schauder estimates for drifted fractional operators in the supercritical case

We consider a non-local operator L_α which is the sum of a fractional Laplacian △^α/2, α∈(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^∞-norm of the first order term. In our approach we do not use the so-called extension property and we can replace △^α/2 with other operators of α-stable type which are somehow close, including the relativistic α-stable operator. Moreover, when α∈(1/2,1), we can prove Schauder estimates for more general α-stable type operators like the singular cylindrical one, i.e., when △^α/2 is replaced by a sum of one dimensional fractional Laplacians ∑_k=1 ^d(∂_xkxk ²)^α/2.