Schauder estimates for drifted fractional operators in the supercritical case

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 (∂ 2 x k x k) α/2.
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Contributor : Paul-Eric Chaudru de Raynal <>
Submitted on : Wednesday, February 13, 2019 - 10:56:35 PM
Last modification on : Friday, March 8, 2019 - 1:27:15 AM
Long-term archiving on : Tuesday, May 14, 2019 - 8:23:14 PM

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  • HAL Id : hal-02018619, version 1
  • ARXIV : 1902.02616

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Paul-Éric Chaudru de Raynal, Stephane Menozzi, Enrico Priola. Schauder estimates for drifted fractional operators in the supercritical case. 2019. ⟨hal-02018619⟩

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