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Article Dans Une Revue Journal of Functional Analysis Année : 2020

Schauder estimates for drifted fractional operators in the supercritical case

Résumé

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 (∂ 2 x k x k) α/2.
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Dates et versions

hal-02018619 , version 1 (13-02-2019)

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Paul-Éric Chaudru de Raynal, Stéphane Menozzi, Enrico Priola. Schauder estimates for drifted fractional operators in the supercritical case. Journal of Functional Analysis, 2020, 278 (8), pp.108425. ⟨10.1016/j.jfa.2019.108425⟩. ⟨hal-02018619⟩
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