Boundedness of non-local operators with spatially dependent coefficients and Lp -estimates for non-local equations

Hongjie Dong, Pilgyu Jung, Doyoon Kim

Research output: Contribution to journalArticlepeer-review

3 Citations (Scopus)

Abstract

We prove the boundedness of the non-local operator Lau(x)=∫Rd(u(x+y)-u(x)-χα(y)(∇u(x),y))a(x,y)dy|y|d+αfrom Hp,wα(Rd) to Lp,w(Rd) for the whole range of p∈ (1 , ∞) , where w is a Muckenhoupt weight. The coefficient a(x, y) is bounded, merely measurable in y, and Hölder continuous in x with an arbitrarily small exponent. We extend the previous results by removing the largeness assumption on p as well as considering weighted spaces with Muckenhoupt weights. Using the boundedness result, we prove the unique solvability in Lp spaces of the corresponding parabolic and elliptic non-local equations.

Original languageEnglish
Article number62
JournalCalculus of Variations and Partial Differential Equations
Volume62
Issue number2
DOIs
Publication statusPublished - 2023 Mar

Bibliographical note

Funding Information:
H. Dong was partially supported by a Simons fellowship Grant No. 007638, the NSF under agreement DMS-2055244, and the Charles Simonyi Endowment at the Institute for Advanced Study.

Funding Information:
P. Jung and D. Kim were supported by the National Research Foundation of Korea (NRF) grant funded by the Korea government (MSIT) (2019R1A2C1084683).

Publisher Copyright:
© 2022, The Author(s), under exclusive licence to Springer-Verlag GmbH Germany, part of Springer Nature.

ASJC Scopus subject areas

  • Analysis
  • Applied Mathematics

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