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Discrete Bilinear Operators and Commutators

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Abstract

We discuss boundedness properties of certain classes of discrete bilinear operators that are similar to those of the continuous bilinear pseudodifferential operators with symbols in the Hörmander classes \(BS^{\omega }_{1, 0}\). In particular, we investigate their relation to discrete analogs of the bilinear Calderón–Zygmund singular integral operators and show compactness of their commutators.

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Notes

  1. Recall that \(H^s({\mathbb {T}}^d) = L^2_s({\mathbb {T}}^d)\), where \(L^2_s({\mathbb {T}}^d)\) is the Bessel potential space; see also [4].

  2. By using duality, we can take the mixed-Lebesgue \(\ell ^r_j \ell _k^{p'}\ell ^{q'}_{\ell }\)-norm in any order, and thus, it suffices to assume that the minimum mixed-Lebesgue norm is finite to guarantee the boundedness of the operator \(\mathcal T_{\Theta }: \ell ^p(\mathbb Z^d)\times \ell ^q(\mathbb Z^d)\rightarrow \ell ^r(\mathbb Z^d)\).

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Acknowledgements

T.O. was supported by the European Research Council (Grant No. 864138 “SingStochDispDyn”).

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Authors and Affiliations

  1. School of Mathematics, The University of Edinburgh, James Clerk Maxwell Building, The King’s Buildings, Peter Guthrie Tait Road, Edinburgh, EH9 3FD, UK

    Árpád Bényi

  2. The Maxwell Institute for the Mathematical Sciences, James Clerk Maxwell Building, The King’s Buildings, Peter Guthrie Tait Road, Edinburgh, EH9 3FD, UK

    Tadahiro Oh

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  1. Árpád Bényi
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  2. Tadahiro Oh
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Correspondence to Árpád Bényi.

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Bényi, Á., Oh, T. Discrete Bilinear Operators and Commutators. J Geom Anal 33, 102 (2023). https://doi.org/10.1007/s12220-022-01136-2

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