Schauder estimates for drifted fractional operators in the supercritical case

We consider a non-local operator Lα 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 Δα/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.