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On Tent Spaces for the Gaussian Measure

Liliana Forzani, Roberto Scotto, Wilfredo Urbina

TL;DR

The paper generalizes classical tent-space theory to the Gaussian measure by introducing Gaussian tent spaces $T^{p,q}_{\alpha,\beta}(\gamma)$ via a Gaussian area function. It establishes an atomic decomposition for $T^{1,q}_{\alpha,\beta}(\gamma)$, develops a duality framework with Gaussian Carleson measures at the endpoint and a full duality for $1<p<\infty$, $1<q<\infty$, and proves independence of the spaces from cone aperture and cutoff positioning, yielding a canonical $T^{p,q}(\gamma)$. The work also demonstrates immersion properties into classical Gaussian function spaces and investigates open problems relating area and Carleson functionals and interpolation. These results provide a robust harmonic-analytic toolkit for Gaussian settings, with potential applications to Gaussian-singular integrals and Gaussian Hardy spaces. Overall, the paper extends tent-space methods to the Gaussian context, offering new structural and duality insights with broad applicability in analysis on Gaussian spaces.

Abstract

Following the scheme of tent spaces in classical harmonic analysis developed by R. Coifman, Y. Meyer, and E. Stein in \cite{cms}, we succeed in doing so for the Gaussian setting. In \cite{MNP}, part of this theory (an atomic decomposition) is developed for a specific tent space where functions are defined just in a proper subset of $\mathbb{R}^{n+1}_+,$ and without the use of an area function. In the present paper, using a variation of the area function considered in \cite{FSU}, we define the Gaussian area function and Gaussian tent spaces and prove both their atomic decompositions and the characterization of their dual spaces. Some applications are also considered.

On Tent Spaces for the Gaussian Measure

TL;DR

The paper generalizes classical tent-space theory to the Gaussian measure by introducing Gaussian tent spaces via a Gaussian area function. It establishes an atomic decomposition for , develops a duality framework with Gaussian Carleson measures at the endpoint and a full duality for , , and proves independence of the spaces from cone aperture and cutoff positioning, yielding a canonical . The work also demonstrates immersion properties into classical Gaussian function spaces and investigates open problems relating area and Carleson functionals and interpolation. These results provide a robust harmonic-analytic toolkit for Gaussian settings, with potential applications to Gaussian-singular integrals and Gaussian Hardy spaces. Overall, the paper extends tent-space methods to the Gaussian context, offering new structural and duality insights with broad applicability in analysis on Gaussian spaces.

Abstract

Following the scheme of tent spaces in classical harmonic analysis developed by R. Coifman, Y. Meyer, and E. Stein in \cite{cms}, we succeed in doing so for the Gaussian setting. In \cite{MNP}, part of this theory (an atomic decomposition) is developed for a specific tent space where functions are defined just in a proper subset of and without the use of an area function. In the present paper, using a variation of the area function considered in \cite{FSU}, we define the Gaussian area function and Gaussian tent spaces and prove both their atomic decompositions and the characterization of their dual spaces. Some applications are also considered.

Paper Structure

This paper contains 23 sections, 14 theorems, 171 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Area function and Tent spaces
  4. Carleson Measure
  5. Main results
  6. Atomic decomposition
  7. Duality
  8. Independence of $\alpha$ and $\beta$
  9. Conclusions and open problems
  10. Proofs of Section \ref{['prelim']}
  11. Proof of Lemma\ref{['GaussTent']}
  12. Proof of Lemma \ref{['continuity']}
  13. Proof of Lemma \ref{['Bannach']}
  14. Proof of Lemma \ref{['NOSE']}
  15. Proofs of Section \ref{['atomic']}
  16. ...and 8 more sections

Key Result

Lemma 1

Given $B\in \mathscr{B}_\beta,$ let us define $q_B$ the point in $\bar{B}$ such that $|q_B|=\inf_{y\in B}|y|,$ and assume that $\beta\ge 1$ and $|q_B|\ge \sqrt{\beta}.$ Then where $D_{\alpha,\beta}=\{(y,t)\in \mathbb{R}^{n+1}_+: \alpha t<m_\beta(y)\}.$ Moreover, where $T^\alpha(B)=\{(y,t)\in \mathbb{R}^{n+1}_+:\text{dist}(y,B^c)\ge \alpha t\}$ is the classical tent over $B.$

Figures (3)

  • Figure 1: Fabes-Forzani's cone $\widetilde{\Gamma}_\gamma(2)$ vs new cone $\Gamma_\gamma(3)$
  • Figure 2: Gaussian tent for $r_B < m(c_B)$
  • Figure 3: Gaussian tent for $r_B \ge m(c_B)$

Theorems & Definitions (36)

  • Definition 1
  • Remark 1
  • Definition 2
  • Lemma 1
  • Definition 3
  • Remark 2
  • Lemma 2
  • Definition 4
  • Remark 3
  • Lemma 3
  • ...and 26 more