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On the local integrability condition for generalised translation-invariant systems

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Abstract

This paper considers the local integrability condition for generalised translation-invariant systems and its relation to the Calderón integrability condition, the temperateness condition and the uniform counting estimate. It is shown that sufficient and necessary conditions for satisfying the local integrability condition are closely related to lower and upper bounds on the number of lattice points that intersect with the translates of a compact set. The results are complemented by examples that illustrate the crucial interplay between the translation subgroups and the generating functions of the system.

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Acknowledgements

The author thanks José Luis Romero for useful discussions and for his help with several of the examples. Thanks also goes to Peter Kuleff and Jakob Lemvig for reading the manuscript and providing helpful comments.

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  1. Faculty of Mathematics, University of Vienna, Oskar-Morgenstern-Platz 1, 1090, Vienna, Austria

    Jordy Timo van Velthoven

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  1. Jordy Timo van Velthoven
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Correspondence to Jordy Timo van Velthoven.

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The author acknowledges support from the Austrian Science Fund (FWF): P29462-N35.

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van Velthoven, J.T. On the local integrability condition for generalised translation-invariant systems. Collect. Math. 70, 407–429 (2019). https://doi.org/10.1007/s13348-019-00238-5

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