Schauder estimates for drifted fractional operators in the supercritical case
Section snippets
Statement of the problem and main results
We are interested in establishing global Schauder estimates for the following parabolic integro-partial differential equation (IPDE): where is a fixed final time horizon and the dimension .
The operator can be the fractional Laplacian , i.e., for regular functions , or a more general symmetric non-local α-stable operator with symbol comparable
Frozen semigroup and associated smoothing effects
The key idea in our approach consists in considering a suitable proxy IPDE, for which we have good controls along which to expand a solution to (1.1). Under (A), which involves potentially unbounded drifts, we will use for the proxy IPDE a non zero first order term which involves a flow associated with the drift coefficient F (which in the current setting exists from the Peano theorem). This flow is, for given freezing parameters , defined as:
Existence result
We point out here that the classical continuity method, which is direct from the a priori estimate, and which was successfully used in [28] to establish existence in the elliptic setting, does not work here for . The key point is that when one tries to write: where then, the product has under (A) a Hölder-regularity of order , since . Therefore, we cannot in this framework readily apply
Proof of Proposition 2 for stable like operators close to ,
In other words, the Lévy measure ν in (1.9) rewrites:
We have to prove (1.21) for all , and ; this will rely on global estimates on the derivatives of , , which can be deduced from the work of Kolokoltsov [20]. First by [20, formula (2.38) in Proposition 2.6] we know that there exists (where η denotes the non degeneracy constant associated with the spectral measure in (1.15)) such that
Acknowledgements
For the first author, this work has been partially supported by the ANR project ANR-15-IDEX-02. For the second author, the article was prepared within the framework of the HSE University Basic Research Program. The third author has been partially supported by the grant 346300 for IMPAN from the Simons Foundation and the matching 2015-2019 Polish MNiSW fund.
We would eventually like to thank the anonymous referees for their careful reading and comments.
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