Wavelet Characterization of Inhomogeneous Lipschitz Spaces on Spaces of Homogeneous Type and Its Applications

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.