Whitney coverings and the tent spaces \(T^{1,q}(\gamma)\) for the Gaussian measure
The authors introduce the notion of tent spaces associated to the Gaussian measure and prove the atomic decomposition for these spaces. This is done by introducing a technique for handling Whitney-type decompositions for the Gaussian harmonic analysis. They start by considering the admissible balls introduced by \textit{G. Mauceri} and \textit{S. Meda} [J. Funct. Anal. 252, No. 1, 278--313 (2007; Zbl 1136.46027)] in the setting of Gaussian measure and which enjoy the doubling property. They denote \(\mathcal B_\alpha\) the family of balls \(B(x,r)\) in \(\mathbb R^n\) such that \(0<r\leq \alpha\min\{1,\frac{1}{|x|}\}\) and write \(D=\{(x,t)\in \mathbb R^n\times (0,\infty): 0<t<\min\{1,\frac{1}{|x|}\}\}.\) Denoting by \(\gamma\) the Gaussian measure given by \(d\gamma(x)=(2\pi)^{-n/2}\exp(- |x|^2/2)dx\) they define, for \(\alpha\geq 1\) and \(1<q<\infty\), the Gaussian tent space \(T_\alpha^{1,q}(\gamma)\), as the completion of compactly supported continuous functions defined on \(D\) with the norm \[ \|f\|_{T^{1,q}(\gamma)}= \int_{\mathbb R^n}\left(\int\int_D \frac{\chi_{B(y,\alpha t)(x)}}{\gamma(B(y,t))} |f(y,t)|^qd\gamma(y)\frac{dt}{t}\right)^{1/q}d\gamma(x). \] A function \(a:D\to \mathbb C\) is called a \(T^{1,q}(\gamma)\) \(\alpha\)-atom if \(\text{supp}(a)\subset T_1(B)\) for some \(B\in \mathcal B_\alpha\) where \(T_1(B)\) stands for the tent with aperture \(1\) over \(B\), meaning \((y,t)\) such that \(d(y, \mathbb R^n\setminus B)\geq t\) and \(\|a\|_{L^q(D, d\gamma \frac{dt}{t})}\leq \gamma(B)^{-1/q}\). The main result establishes that, for \(\alpha>1\), all functions \(f\in T^{1,q}(\gamma)\) admit a decomposition into \(T^{1,q}(\gamma)\) \(\alpha\)-atoms, that is, there exists \((\lambda_n)\in \ell^1\) and a sequence \((a_n)\) of \(T^{1,q}(\gamma)\) \(\alpha\)-atoms such that \(f= \sum_n \lambda_n a_n\) with \(\sum_n |\lambda_n|\leq C \|f\|_{T^{1,q}(\gamma)}\). Using the atomic decomposition it is shown that \(T_{\alpha_1}^{1,q}(\gamma)=T_{\alpha_2}^{1,q}(\gamma)\) for \(1<\alpha_1<\alpha_2\).
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