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.

我们在\(L^2(M)\) 中为一般黎曼流形M构造 Parseval 小波框架，并且我们证明了在\(L^p(M)\) 中存在小波无条件框架对于\(1< p <\ infty \)。这要归功于对\(L^2(M)\)上的恒等运算符的平滑正交投影分解，最近 Bownik 等人证明了这一点。（潜在的肛门 54：41-94，2021 年）。我们还展示了 Triebel–Lizorkin \({\mathbf {F}}_{p,q}^s(M)\)和 Besov \({\mathbf {B}}_{p,q}^s (M)\)在 Parseval 小波框架系数大小方面的紧凑流形上的空间。我们通过证明 Hestenes 算子有界来实现这一点\（{\ mathbf {F}} _ {P，Q} ^ S（M）\）和\（{\ mathbf {B}} _ {P，Q} ^ S（M）\）流形上的空间中号与有界几何。