Gibbs-Sobolev Potentials: Exponentially Integrable Potentials with Cameron-Martin Derivatives of Finite Moments

We work in the settings of The Real Numbers: Standing Notation and Background and Gaussian Analysis Relative to a Diagonal Gaussian Reference Measure with Noise Weights: Standing Notation, with the Lebesgue spaces $L^{p}(\gamma_{c})$ of Gaussian Analysis Relative to a Diagonal Gaussian Reference Measure with Noise Weights: Standing Notation §lebesgue, the space $\mathbb{D}^{1,2}_{c}$ of Cameron-Martin Partial Derivatives on the Gaussian Sobolev Space with the Variances as Weights, and the Cameron-Martin partial derivatives $\partial_{k}F$ of its elements; $\exp$ is the exponential function.

(Gibbs-Sobolev potential)¶ Let $\beta,q\in\mathbb{R}$ satisfy $0<\beta$ and $1<q$, and write $q'=q/(q-1)=1+1/(q-1)$, so that $1<q'$. A Borel function $V:X\to\mathbb{R}$ is a Gibbs-Sobolev potential at temperature $\beta$ with exponent $q$ if the following three conditions hold.

(a) (Exponential integrability)¶ The function $x\mapsto\exp(-qV(x)/\beta)$ is integrable with respect to $\gamma_{c}$.

(b) (Cameron-Martin derivatives)¶ $\int_{X}V^{2}\,d\gamma_{c}<\infty$, and the class of $V$ in $L^{2}(\gamma_{c})$ lies in $\mathbb{D}^{1,2}_{c}$.

(c) (Moments of the derivatives)¶ For every $k\in\mathbb{N}$, $\partial_{k}V\in L^{2q'}(\gamma_{c})$.