Parseval Wavelet Frames on Riemannian Manifold

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.