Wavelet system and Muckenhoupt $\mathcal {A}_2$ condition on the Heisenberg group
Volume 158 / 2019
Colloquium Mathematicum 158 (2019), 59-76
MSC: Primary 42C15; Secondary 43A30.
DOI: 10.4064/cm7467-9-2018
Published online: 17 June 2019
Abstract
Let $\mathbb{H}^n$ denote the Heisenberg group. It is shown that under certain conditions the wavelet system $\{\psi_{j,k,l,m}:k,l\in\mathbb{Z}^n,\,j,m\in\mathbb{Z}\}$ on $\mathbb{H}^n$ arising from integer translations and nonisotropic dilations forms a Schauder basis for its closed linear span in $L^2(\mathbb{H}^n)$ if and only if the function $\sum_{r\in\mathbb{Z}}\|\widehat{\psi}(\cdot+r)\|_{\mathcal{B}_2}^2|\cdot+\,r|^n$ satisfies the Muckenhoupt $\mathcal{A}_2$ condition, where $\mathcal{B}_2$ denotes the class of Hilbert–Schmidt operators on $L^2(\mathbb{R}^n)$.
