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Denote by $(t, x_{1}, \ldots, x_{d-1})$ the coordinates of $x \in \mathbb{R}^{d}$ and set $$\mathbb{R}^{d}_{+}=\left\{t, x_{1}, \ldots, x_{d-1} \in \mathbb{R}^{d}|t > 0\right\}. $$ Write $\theta$ fo …

Let $\mu$ be a finite, finitely additive measure defined on the Borel $\sigma$-algebra of a real separable Hilbert space $\mathcal{H}$ with dual $\mathcal{H}^{*}$. Write $L^{p}(\mathcal{H},\mu)$ for t …

Let $\mathcal{H}$ be a separable Hilbert space equipped with a strongly continuous unitary representation of a locally compact group $G$. Denote by $\mathcal{L}^{\infty}(H)$ the space of the bounded …

In Bogachev's book "Gaussian Measures" (Example 3.8.13) sufficient conditions for the (uniform) tightness of a sequence of centered Borel Gaussian probability measures on a separable Hilbert spaces in …

Do there exist versions of the Nash-Moser inverse function theorem applicable to spaces of unbounded smooth functions on unbounded domains in $\mathbb{R}^n$? Any reference would be appreciated but her …

Let $M$ be a closed Riemannian manifold and denote by $H^s(M), \, s\in \mathbb{R} $ the standard Sobolev spaces on $M$ defined using powers of $1+\triangle$. Let $J_n: \mathcal{D}'(M)\rightarrow \mat …

In white noise analysis, one starts with a real Gelfand triple $\mathcal{N}\subset \mathcal{H} \subset \mathcal{N}^{*}$ and produces out of it, using complexifications along the way, the complex spa …

(I'm reposting here this question from MSE as it didn't receive any answer for two weeks.) Consider a sequence of finite lattices in $\mathbb{R}^n$ defined by $$L_k= [-k,k]^n \cap 2^{-k}\cdot \mathbb{ …

Let $\mu_C$ be a centered Gaussian probability Borel measure on a real separable Hilbert space $\mathcal{H}$ with covariance operator $C$. Denote the ball with radius $r$ in $\mathcal{H}$ centered at …