Gaussian Dirichlet energy (${\mathcal{E}}_{\gamma }(f)=\int |f^{\prime }{|}^{2}d\gamma ={\sum }_{n\ge 1}n|c_n{|}^2$)

The energy form of the Gaussian number operator is the displayed squared derivative integral. Its form domain requires $\sum n|c_n{|}^2<\infty$, while the full operator domain requires $\sum n^2|c_n{|}^2<\infty$. It can be defined by ${\lim }_{t\downarrow 0}{t}^{-1}(\Vert f{\Vert }_2^2-⟨f,P_{t}f⟩)$ even when $Nf$ does not exist in $L^2$. This energy occurs in the Gaussian Poincaré inequality and Gaussian logarithmic Sobolev inequality.