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Harmonic Analysis

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We construct a decomposition of the identity operator on a Riemannian manifold M as   a sum of smooth orthogonal projections subordinate to an open cover of M. This extends a decomposition on the real line by smooth orthogonal projection due to Coifman and Meyer (C. R. Acad. Sci. Paris, Sér. I Math., 312(3), 259–261 1991) and Auscher, Weiss, Wickerhauser (1992), and a similar decomposition when M is the sphere by Bownik and Dziedziul (Const. Approx., 41, 23–48 2015).

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 < ∞.
This is made possible thanks to smooth orthogonal projection decomposition of the identity operator on L 2 (M ), which was recently proven by the authors in [3]. We also show a characterization of Triebel-Lizorkin  and Besov 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 manifolds M with bounded geometry.

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License:
Creative Commons: by 4.0 open in new tab
CC BY
Attribution

Details

Year of publication:
2021
Verification date:
2021-04-26
Dataset language:
English
Fields of science:
  • mathematics (Natural sciences)
DOI:
DOI ID 10.34808/b84g-e471 open in new tab
Verified by:
Gdańsk University of Technology

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