No abstract

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 ∆ 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 d k=1 (∂ 2x k x k ) α/2 .

No abstract