Wavelet Characterization of Inhomogeneous Lipschitz Spaces on Spaces of Homogeneous Type and Its Applications
Fan Wang
TL;DR
This work addresses the problem of characterizing the inhomogeneous Lipschitz space $\mathrm{lip}_{\theta}(\mathcal{X})$ on spaces of homogeneous type via a wavelet Carlson/sequences framework. The authors develop a wavelet-based Carleson-sequence characterization by decomposing functions into wavelet coefficients $\langle f, \phi_\alpha^0 \rangle$ and $\langle f, \psi_\beta^{k+1} \rangle$, with a norm $\|f\|_{\ast}$ that captures the Lipschitz semi-norm and a corresponding Carleson condition. A key novelty is that the characterization holds without relying on reverse-doubling or quasi-metric assumptions; exponential decay of the wavelets is employed to control cross-terms and establish the Lip–Carleson equivalence. As applications, the paper derives geometric characterizations for upper bounds, lower bounds, and Ahlfors regularity, including that $\mathrm{lip}_{\theta}(\mathcal{X})\subset L^{\infty}(\mathcal{X})$ if and only if $\mu(B(x,1))$ is uniformly bounded and that, on Ahlfors-regular spaces, $\mathrm{lip}_{\theta}(\mathcal{X})=C^{\theta\omega}(\mathcal{X})$ with equivalent norms. Overall, the results extend Lipschitz-space theory from Euclidean settings to general spaces of homogeneous type and provide robust tools for Hardy–Lipschitz duality and related analysis on these spaces.
Abstract
In this article, the author establishes a wavelet characterization of inhomogeneous Lipschitz space $\mathrm{lip}_θ(\mathcal{X})$ via Carlson sequence, where $\mathcal{X}$ is a space of homogeneous type introduced by R. R. Coifman and G. Weiss. As applications, characterizations of several geometric conditions on $\mathcal{X}$, involving the upper bound, the lower bound, and the Ahlfors regular condition, are obtained.