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Parseval Wavelet Frames on Riemannian Manifold

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Abstract

We construct Parseval wavelet frames in \(L^2(M)\) for a general Riemannian manifold M and we show the existence of wavelet unconditional frames in \(L^p(M)\) for \(1< p <\infty \). This is made possible thanks to smooth orthogonal projection decomposition of the identity operator on \(L^2(M)\), which was recently proven by Bownik et al. (Potential Anal 54:41–94, 2021). We also show a characterization of Triebel–Lizorkin \({\mathbf {F}}_{p,q}^s(M)\) and Besov \({\mathbf {B}}_{p,q}^s(M)\) spaces on compact manifolds in terms of magnitudes of coefficients of Parseval wavelet frames. We achieve this by showing that Hestenes operators are bounded on \({\mathbf {F}}_{p,q}^s(M)\) and \({\mathbf {B}}_{p,q}^s(M)\) spaces on manifolds M with bounded geometry.

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Authors and Affiliations

  1. Department of Mathematics, University of Oregon, Eugene, OR, 97403–1222, USA

    Marcin Bownik

  2. Faculty of Applied Mathematics, Gdańsk University of Technology, ul. G. Narutowicza 11/12, 80-952, Gdańsk, Poland

    Karol Dziedziul

  3. Institute of Mathematics, Polish Academy of Sciences, ul. Abrahama 18, 80–825, Sopot, Poland

    Anna Kamont

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  1. Marcin Bownik
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  2. Karol Dziedziul
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  3. Anna Kamont
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Correspondence to Marcin Bownik.

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Marcin Bownik was partially supported by the NSF grant DMS-1956395. The authors are grateful to Isaak Pesenson for useful comments on Lemma 2.1.

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Bownik, M., Dziedziul, K. & Kamont, A. Parseval Wavelet Frames on Riemannian Manifold. J Geom Anal 32, 4 (2022). https://doi.org/10.1007/s12220-021-00742-w

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