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Boundedness of non-local operators with spatially dependent coefficients and \(L_p\)-estimates for non-local equations

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Abstract

We prove the boundedness of the non-local operator

$$\begin{aligned} \mathcal {L}^a u(x)=\int _{\mathbb {R}^d} \left( u(x+y)-u(x)-\chi _\alpha (y)\big (\nabla u(x),y\big )\right) a(x,y)\frac{dy}{|y|^{d+\alpha }} \end{aligned}$$

from \(H_{p,w}^\alpha (\mathbb {R}^d)\) to \(L_{p,w}(\mathbb {R}^d)\) for the whole range of \(p \in (1,\infty )\), 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 \(L_p\) spaces of the corresponding parabolic and elliptic non-local equations.

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Acknowledgements

The authors would like to thank the referee for his/her careful reading and many useful comments.

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Authors and Affiliations

  1. Division of Applied Mathematics, Brown University, 182 George Street, Providence, RI, 02912, USA

    Hongjie Dong

  2. Department of Mathematics, Korea University, 145 Anam-ro, Seongbuk-gu, Seoul, 02841, Republic of Korea

    Pilgyu Jung & Doyoon Kim

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  1. Hongjie Dong
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  2. Pilgyu Jung
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  3. Doyoon Kim
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Correspondence to Pilgyu Jung.

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Communicated by A. Mondino.

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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.

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

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Dong, H., Jung, P. & Kim, D. Boundedness of non-local operators with spatially dependent coefficients and \(L_p\)-estimates for non-local equations. Calc. Var. 62, 62 (2023). https://doi.org/10.1007/s00526-022-02392-4

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