Nelson's Exponential Integrability for the Wick-Ordered $Phi^4$ Potential on the Torus of Dimension at Most Two

In the settings of The Real Numbers: Standing Notation and Background, The Free Field on the Torus as the Gaussian Reference Measure, with Square-Integrable White Noise: Standing Notation and Gaussian Analysis Relative to a Diagonal Gaussian Reference Measure with Noise Weights: Standing Notation, the last read with the space $X=H^{-m}(\mathbb{T}^{n})$, the orthonormal basis $(e_{j})_{j\in\mathbb{N}}$, the noise weights $a$ and the variance sequence $c$ of The Free Field on the Torus as the Gaussian Reference Measure, with Square-Integrable White Noise: Standing Notation §gaussian, so that $\gamma_{c}$ is the free field, with the Lebesgue spaces $L^{p}(\gamma_{c})$ and norms $\lVert\cdot\rVert_{p}$ of Gaussian Analysis Relative to a Diagonal Gaussian Reference Measure with Noise Weights: Standing Notation §lebesgue, suppose $n\le2$, and let $\varkappa,\mathfrak{m}\in\mathbb{R}$ with $0<\varkappa$. Let $V_{N}$ ($N\in\mathbb{N}$) be the Galerkin Wick-ordered $\phi^{4}$ potentials and $V$ the Wick-ordered $\phi^{4}$ potential with coupling $\varkappa$ and mass $\mathfrak{m}$, and let $t\in\mathbb{R}$ be positive; $\exp$ is the exponential function.

1. (Uniform bound)¶ There is $E_{t}\in\mathbb{R}$ with $\int_{X}\exp(-tV_{N})\,d\gamma_{c}\le E_{t}$ for every $N\in\mathbb{N}$.

2. (Integrability)¶ The function $\exp(-tV)$ is integrable with respect to $\gamma_{c}$.

3. (Convergence)¶ For every real $p\ge1$, the classes of $\exp(-tV)$ and of $\exp(-tV_{N})$ ($N\in\mathbb{N}$) lie in $L^{p}(\gamma_{c})$, and $\lVert\exp(-tV_{N})-\exp(-tV)\rVert_{p}\to0$ as $N\to\infty$.