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Discrete Maximal Operators Over Surfaces of Higher Codimension

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Abstract

Integration over curved manifolds with higher codimension and, separately, discrete variants of continuous operators, have been two important, yet separate themes in harmonic analysis, discrete geometry and analytic number theory research. Here we unite these themes to study discrete analogs of operators involving higher (intermediate) codimensional integration. We consider a maximal operator that averages over triangular configurations and prove several bounds that are close to optimal. A distinct feature of our approach is the use of multilinearity to obtain non-trivial \(\ell ^1\)-estimates by a rather general idea that is likely to be applicable to other problems.

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Acknowledgements

The first author was supported in part by NSF grants DMS-1502464 and DMS-1954407. The second author thanks Towson University for sabbatical support that allowed this work to be completed. The third author was supported in part by Simons Foundation Grant #360560. Last but not least, the first two authors thank Trevor Wooley for several helpful discussions and generous advice.

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Authors and Affiliations

  1. Department of Mathematics, Purdue University, 150 N. University Street, West Lafayette, IN, 47907, USA

    Theresa C. Anderson

  2. Department of Mathematics, Towson University, 8000 York Road, Towson, MD, 21252, USA

    Angel V. Kumchev

  3. Department of Mathematics, Virginia Tech, 225 Stanger Street, Blacksburg, VA, 24061, USA

    Eyvindur A. Palsson

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  1. Theresa C. Anderson
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  2. Angel V. Kumchev
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  3. Eyvindur A. Palsson
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Correspondence to Angel V. Kumchev.

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Anderson, T.C., Kumchev, A.V. & Palsson, E.A. Discrete Maximal Operators Over Surfaces of Higher Codimension. La Matematica 1, 442–479 (2022). https://doi.org/10.1007/s44007-021-00017-4

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