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Projecting Lipschitz Functions Onto Spaces of Polynomials

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Abstract

The Banach space \(\mathcal {P}({}^2X)\) of 2-homogeneous polynomials on the Banach space X can be naturally embedded in the Banach space \({\mathrm{Lip}_0}(B_X)\) of real-valued Lipschitz functions on \(B_X\) that vanish at 0. We investigate whether \(\mathcal {P}({}^2X)\) is a complemented subspace of \({\mathrm{Lip}_0}(B_X)\). This line of research can be considered as a polynomial counterpart to a classical result by Joram Lindenstrauss, asserting that \(\mathcal {P}({}^1X)=X^*\) is complemented in \({\mathrm{Lip}_0}(B_X)\) for every Banach space X. Our main result asserts that \(\mathcal {P}({}^2X)\) is not complemented in \({\mathrm{Lip}_0}(B_X)\) for every Banach space X with non-trivial type.

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Notes

  1. We don’t use the standard notation \(\omega \) for a modulus of continuity as it would conflict with \(\omega \in \mathcal {O}_n\).

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We wish to thank the anonymous referee for carefully reading our manuscript and for the helpful report.

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Authors and Affiliations

  1. Department of Mathematics, Faculty of Electrical Engineering, Czech Technical University in Prague, Technická 2, 16627, Prague 6, Czech Republic

    Petr Hájek & Tommaso Russo

  2. Institute of Mathematics, Czech Academy of Sciences, Žitná 25, 11567, Prague 1, Czech Republic

    Tommaso Russo

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  1. Petr Hájek
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Correspondence to Tommaso Russo.

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Research of P. Hájek was supported in part by OPVVV CAAS CZ.02.1.01/0.0/0.0/16_019/0000778. Research of T. Russo was supported by the GAČR project 20-22230L; RVO: 67985840 and by Gruppo Nazionale per l’Analisi Matematica, la Probabilità e le loro Applicazioni (GNAMPA) of Istituto Nazionale di Alta Matematica (INdAM), Italy.

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Hájek, P., Russo, T. Projecting Lipschitz Functions Onto Spaces of Polynomials. Mediterr. J. Math. 19, 190 (2022). https://doi.org/10.1007/s00009-022-02075-6

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