On the domain of elliptic operators defined in subsets of Wiener spaces

TL;DR

This paper characterizes the domain of elliptic operators associated with quadratic forms in subsets of Wiener spaces, focusing on the Ornstein-Uhlenbeck operator on half-spaces with explicit boundary conditions.

Contribution

It provides a complete characterization of the operator's domain in Wiener spaces, especially for half-spaces, with explicit boundary conditions involving the Hausdorff-Gauss measure.

Findings

Explicit domain description for Ornstein-Uhlenbeck operator on half-spaces.

Boundary conditions involve the Hausdorff-Gauss surface measure.

Results extend understanding of elliptic operators in infinite-dimensional spaces.

Abstract

Let $X$ be a separable Banach space endowed with a non-degenerate centered Gaussian measure $\mu$. The associated Cameron-Martin space is denoted by $H$. Consider two sufficiently regular convex functions $U:X\rightarrow\mathbb{R}$ and $G:X\rightarrow \mathbb{R}$. We let $\nu=e^{-U}\mu$ and $\Omega=G^{-1}(-\infty,0]$. In this paper we are interested in the domain of the the self-adjoint operator associated with the quadratic form \begin{gather} (\psi,\varphi)\mapsto \int_\Omega\langle\nabla_H\psi,\nabla_H\varphi\rangle_Hd\nu\qquad\psi,\varphi\in W^{1,2}(\Omega,\nu).\qquad\qquad (\star) \end{gather} In particular we obtain a complete characterization of the Ornstein-Uhlenbeck operator on half-spaces, namely if $U\equiv 0$ and $G$ is an affine function, then the domain of the operator defined via $(\star)$ is the space \[\{u\in W^{2,2}(\Omega,\mu)\,|\, \langle\nabla_H u(x),\nabla_H G(x)\rangle_H=0\text{ for }\rho\text{-a.e. }x\in G^{-1}(0)\},\] where $\rho$ is the Feyel-de La Pradelle Hausdorff-Gauss surface measure.

Equations

$$(\psi ,\phi )\mapsto {\int }_{\Omega }{⟨{\nabla }_H\psi ,{\nabla }_H\phi ⟩}_{H}d\nu \psi ,\phi \in W^{1,2}(\Omega ,\nu ),$$

$$\left\{u\in W^{2,2}(\Omega ,\mu )|{⟨{\nabla }_{H}u(x),{\nabla }_{H}G(x)⟩}_H=0\text{ for }\rho \text{-a.e. }x\in G^{-1}(0)\right\},$$

$$(\psi ,\phi )\mapsto {\int }_{\Omega }{⟨{\nabla }_H\psi ,{\nabla }_H\phi ⟩}_{H}d\nu \psi ,\phi \in W^{1,2}(\Omega ,\nu ),$$

$${\left\{\begin{array}[]{ll}dY(t,x)-Y(t,x)dt-\nabla_{H}U(Y(t,x))dt+N_{\Omega}(Y(t,x))dt\ni dW(t,x);\\ Y(0,x)=x,\end{array}\right.}$$

$$L_{\nu ,\Omega }u(\xi )=\Delta u(\xi )-⟨\nabla U(\xi )+\xi ,\nabla u(\xi )⟩u\in C_b^2(\Omega ),$$

$$\displaystyle D(L_{\nu,\Omega})={\left\{\begin{array}[]{ll}{\left\{u\in W^{2,2}(\mathbb{R}^{n},\nu)\,\middle|\,{\left\langle\nabla U+\xi,\nabla u\right\rangle}\in\operatorname{L}^{2}(\mathbb{R}^{n},\nu)\right\}},&\Omega=\mathbb{R}^{n};\\ {\left\{u\in W^{2,2}(\Omega,\nu)\,\middle|\,{\left\langle\nabla U+\xi,\nabla u\right\rangle}\in\operatorname{L}^{2}(\Omega,\nu),\ \partial u/\partial n=0\text{ at }\partial\Omega\right\}},&\Omega\neq\mathbb{R}^{n},\end{array}\right.}$$

$$T_{t}f(x)={\int }_{X}f\left(e^{-t}x+\sqrt{1-e^{-2t}}y\right)d\mu (y),$$

$$\underset{t\to 0}{\lim }$$

$$\nu :=e^{-U}\mu .$$

$$\displaystyle W_{U}^{2,2}(\Omega,\nu)=\bigg{\{}u\in W^{2,2}(\Omega,\nu)\,\bigg{|}\,\int_{\Omega}{\left\langle\nabla_{H}^{2}U\nabla_{H}u,\nabla_{H}u\right\rangle}d\nu<+\infty\bigg{\}},$$

$${\Vert u\Vert }_{W_U^{2,2}(\Omega ,\nu )}^2={\Vert u\Vert }_{W^{2,2}(\Omega ,\nu )}^2+{\int }_{\Omega }{⟨{\nabla }_H^{2}U{\nabla }_{H}u,{\nabla }_{H}u⟩}_{H}d\nu .$$

$$\displaystyle W_{U,N}^{2,2}(\Omega,\nu)=\bigg{\{}u\in W^{2,2}(\Omega,\nu)\,\bigg{|}\,\int_{\Omega}{\left\langle\nabla_{H}^{2}U\nabla_{H}u,\nabla_{H}u\right\rangle}d\nu<+\infty,\phantom{aaaaaaaaaaaaaaaaaaaaaaa}$$

$$\displaystyle\phantom{aaaaaaaaaaaaaaaaaa}{\left\langle\operatorname{\operatorname{Tr}}(\nabla_{H}u),\operatorname{\operatorname{Tr}}(\nabla_{H}G)\right\rangle}_{H}=0\text{ $\rho$-a.e. in }G^{-1}(0)\bigg{\}}$$

$${\Vert u\Vert }_{D(L_{\nu ,\Omega })}^2:={\Vert u\Vert }_{L^2(X,\nu )}^2+{\Vert L_{\nu ,\Omega }u\Vert }_{L^2(X,\nu )}^2.$$

$${\Vert u\Vert }_{D(L_{\nu ,X})}\le {\Vert u\Vert }_{W_U^{2,2}(X,\nu )}\le 2\sqrt{2}{\Vert u\Vert }_{D(L_{\nu ,X})}.$$

$$L_{\nu ,X}u=\sum _{n=1}^{+\infty }\left({\partial }_{nn}u-{\partial }_{n}u{\partial }_{n}U-{\partial }_{n}u{\hat{h}}_n\right),$$

$$x^{\ast }\left(\mathrm{Tr}({\nabla }_{H}u)(x)\right)={⟨\mathrm{Tr}({\nabla }_{H}u)(x),h_{x^{\ast }}⟩}_H=0$$

$$x^{\ast }(h)={⟨h_{x^{\ast }},h⟩}_H\text{ for every }h\in H.$$

$$\mathcal{Z}(\Omega )=\left\{u\in {\mathrm{FC}}_b^2(\Omega )|x^{\ast }\left(\mathrm{Tr}({\nabla }_{H}u)(x)\right)=0\text{ for }\rho \text{-a.e. }x\in G^{-1}(0)\right\},$$

$${\Vert u\Vert }_{D(L_{\nu ,\Omega })}\le {\Vert u\Vert }_{W_U^{2,2}(\Omega ,\nu )}\le 2\sqrt{2}{\Vert u\Vert }_{D(L_{\nu ,\Omega })}.$$

$$L_{\nu ,\Omega }u=\sum _{n=1}^{+\infty }\left({\partial }_{nn}u-{\partial }_{n}u{\partial }_{n}U-{\partial }_{n}u{\hat{h}}_n\right),$$

$${\Vert u\Vert }_{D(L_{\nu ,\Omega })}\le {\Vert u\Vert }_{W_U^{2,2}(\Omega ,\nu )}\le 2\sqrt{2}{\Vert u\Vert }_{D(L_{\nu ,\Omega })}.$$

$$L_{\nu ,\Omega }u=\sum _{n=1}^{+\infty }\left({\partial }_{nn}u-{\partial }_{n}u{\partial }_{n}U-{\partial }_{n}u{\hat{h}}_n\right),$$

$$D(L_{\mu ,\Omega })=\left\{u\in W^{2,2}(\Omega ,\mu )|x^{\ast }\left(\mathrm{Tr}({\nabla }_{H}u)(x)\right)=0\text{ for }\rho \text{-a.e. }x\in G^{-1}(0)\right\},$$

$${\Vert u\Vert }_{D(L_{\mu ,\Omega })}\le {\Vert u\Vert }_{W^{2,2}(\Omega ,\mu )}\le 2\sqrt{2}{\Vert u\Vert }_{D(L_{\mu ,\Omega })}.$$

$$L_{\mu ,\Omega }u=\sum _{n=1}^{+\infty }\left({\partial }_{nn}u-{\partial }_{n}u{\hat{h}}_n\right),$$

$$\mathcal{Z}(\Omega )=\left\{u\in {\mathrm{FC}}_b^2(\Omega )|x^{\ast }\left(\mathrm{Tr}({\nabla }_{H}u)(x)\right)=0\text{ for }\rho \text{-a.e. }x\in G^{-1}(0)\right\}$$

$$\displaystyle\mathcal{Z}(\Omega,H):={\left\{\Phi:\Omega\rightarrow H\,\middle|\,\begin{array}[]{c}\text{there exists $n\in\mathbb{N}$ and ${\left\{h_{1},\ldots,h_{n}\right\}}\subseteq H$}\\ \text{such that $\Phi=\sum_{i=1}^{n}\varphi_{i}h_{i}$ for some $n\in\mathbb{N}$,}\\ \text{and $\varphi_{i}\in\operatorname{\mathscr{FC}}_{b}^{2}(\Omega)$ for $i=1,\ldots,n$.}\\ \text{In addition ${\left\langle\Phi,\operatorname{\operatorname{Tr}}(\nabla_{H}G)\right\rangle}_{H}=0$ $\rho$-a.e. in $G^{-1}(0)$.}\end{array}\right\}}.$$

$${\Vert \Phi \Vert }_{Z_U^{1,2}(\Omega ,\nu ;H)}^2:={\Vert \Phi \Vert }_{W^{1,2}(\Omega ,\nu ;H)}^2+{\int }_{\Omega }{⟨{\nabla }_H^{2}U\Phi ,\Phi ⟩}_{H}d\nu +{\int }_{G^{-1}(0)}{⟨\mathrm{Tr}({\nabla }_H^{2}G)\Phi ,\Phi ⟩}_H\frac{e^{-\mathrm{Tr}(U)}}{{\left|\mathrm{Tr}({\nabla }_{H}G)\right|}_H}d\rho .$$

$$\lambda u(x)-L_{\nu ,\Omega }u(x)=f(x)\mu \text{-a.e. }x\in \Omega ,$$

Full text

[TABLE]

and we give sharp embedding results for it. In particular we obtain a characterization of the domain of the Ornstein–Uhlenbeck operator on half-spaces, namely if $U\equiv 0$ and $G$ is an affine function, then the domain of the operator defined via (0.1) is the space

[TABLE]

where $\rho$ is the Feyel–de La Pradelle Hausdorff–Gauss surface measure.

1. Introduction

Let $X$ be a separable Banach space with norm ${\left\|\cdot\right\|}_{X}$, endowed with a non-degenerate centered Gaussian measure $\mu$. The associated Cameron–Martin space is denoted by $H$, its inner product by ${\left\langle\cdot,\cdot\right\rangle}_{H}$ and its norm by ${\left|\cdot\right|}_{H}$. The spaces $W^{k,p}(X,\mu)$ for $p\geq 1$ and $k\in\mathbb{N}$ are the classical Sobolev spaces of the Malliavin calculus (see [8, Chapter 5]).

The aim of this paper is to study the domain of the self-adjoint operator $L_{\nu,\Omega}$ associated with the quadratic form

[TABLE]

where $\Omega$ is a convex subset of $X$, $\nu:=e^{-U}\mu$ and $U:X\rightarrow\mathbb{R}$ is a convex function, $\nabla_{H}\psi$ is the gradient along $H$ of $\psi$ and $W^{1,2}(\Omega,\nu)$ is the Sobolev space on $\Omega$ associated to the measure $\nu$ (see Section 2). These operators arise in Kolmogorov equations in Hilbert spaces corresponding to stochastic variational inequalities with reflection, such as

[TABLE]

where $N_{\Omega}$ is the normal cone to $\Omega$ and $W(t,\cdot)$ is a $X$-valued cylindrical Wiener process (here $X$ is a Hilbert space). This is because, at least formally, the transition semigroup $T(t)f(x):=\mathbb{E}[f(Y(t,x))]$ is generated by $L_{\nu,\Omega}$.

In the case of the standard Gaussian measure in a convex subset $\Omega\subseteq\mathbb{R}^{n}$ with sufficiently regular boundary, the operator $L_{\nu,\Omega}$ reads as

[TABLE]

so that, if $U$ is sufficiently regular, $L_{\nu,\Omega}$ is an elliptic operator with possibly unbounded coefficients, and its domain in $\operatorname{L}^{2}(\Omega,\nu)$ is

[TABLE]

where $\partial/\partial n$ is the exterior normal derivative at the boundary of $\partial\Omega$ (see [15] and [31]). In the infinite dimensional case there is a characterization for the Ornstein–Uhlenbeck operator, when $\Omega$ is the whole space and $U\equiv 0$ (see [8, Section 5.6]). In this case the operator $L_{\mu,X}$ is the infinitesimal generator of the Ornstein–Uhlenbeck semigroup

[TABLE]

in $\operatorname{L}^{2}(X,\mu)$ and its domain is $D(L_{\mu,X})=W^{2,2}(X,\mu)$. Further results were obtained in [12], assuming $U$ has $H$-Lipschitz gradient, and $\Omega$ is the whole space. In this case too the domain is $D(L_{\nu,X})=W^{2,2}(X,\nu)$. We want to point out that in [33] the authors study in detail the case of non-symmetric Ornstein–Uhlenbeck operators on the whole space.

This paper is a first attempt to give a characterization of the domain of $L_{\nu,\Omega}$ in a more general setting. In order to state the main results of this paper we need some hypotheses on the set $\Omega$ and on the weighted measure $\nu$.

Throughout the paper we take $\Omega=G^{-1}(-\infty,0]$, where $G$ satisfies the following assumptions.

Hypothesis 1.1**.**

Let $G:X\rightarrow\mathbb{R}$ be a version of a function belonging to $W^{3,q}(X,\mu)$ for every $q>1$. We fix a version of $\nabla_{H}G$ and a version of $\nabla_{H}^{2}G$ such that

1. (1)

$G$ is convex and, for every $q>1$, the functions $G$ is $(3,q)$-precise (see Section 2.2); 2. (2)

for every $q>1$, the functions $\nabla_{H}G$ and $\nabla_{H}^{2}G$ are $(2,q)$-precise and $(1,q)$-precise, respectively (see Section 2.2); 3. (3)

$\mu(G^{-1}(-\infty,0])>0$ and $G^{-1}(-\infty,0]$ is closed; 4. (4)

${\left|\nabla_{H}G\right|}_{H}^{-1}\in\operatorname{L}^{q}(G^{-1}(-\infty,0],\mu)$ for every $q>1$; 5. (5)

for $\rho$-a.e. $x\in G^{-1}(0)$, $G$ is twice differentiable along $H$ at $x$, i.e., for $\rho$-a.e. $x\in G^{-1}(0)$ there exists $\nabla_{H}G(x)\in H$ and a Hilbert-Schmidt operator $\nabla_{H}^{2}G(x)$ such that

[TABLE]

Here $\rho$ is the Feyel–de La Pradelle Hausdorff–Gauss surface measure (see [23]); 6. (6)

${\left|\nabla_{H}G(x)\right|}_{H}\neq 0$ for $\rho$-a.e $x\in G^{-1}(0)$.

Hypotheses 1.1(1)-(4) are taken from [13] and [11] in order to define traces of Sobolev functions on level sets of $G$ and to get maximal Sobolev regularity estimates for elliptic equations associated to the operator $L_{\nu,\Omega}$. In particular, Hypothesis 1.1(3) implies that the distance function $d_{H}(\cdot,\Omega)$ introduced in Section 5 enjoys good properties. Hypotheses 1.1(5)-(6) allow us to prove Lemma 4.3 which is generalization of a classical result in differential geometry (see [29], [6] and [10]).

Hypothesis 1.2**.**

$U:X\rightarrow\mathbb{R}\cup{\left\{+\infty\right\}}$ is a proper, convex, lower semicontinuous and twice continuously differentiable along $H$ function belonging to $W^{2,t}(X,\mu)$ for some $t>3$ (see Section 2 for the definition of differentiability along $H$). We set

[TABLE]

The assumption $t>3$ may sound strange, but it is helpful to define the weighted Sobolev spaces $W^{1,2}(X,\nu)$. Indeed, let us observe that, by [1, Lemma 7.5], $e^{-U}$ belongs to $W^{1,r}(X,\mu)$ for every $r<t$. Thus if $U$ satisfies Hypothesis 1.2, then it satisfies [21, Hypothesis 1.1]; namely $e^{-U}\in W^{1,s}(X,\mu)$ for some $s>1$ and $U\in W^{1,r}(X,\mu)$ for some $r>s^{\prime}$. Then following [21] it is possible to define the space $W^{1,2}(X,\nu)$ as the domain of the closure of the gradient operator along $H$ (see Section 2 for an in-depth discussion).

From here on, we will denote by $\operatorname{\operatorname{Tr}}$ the trace operator acting on Sobolev functions (see Section 2.6), by $\rho$ the Feyel–de La Pradelle Hausdorff–Gauss surface measure (see [22]) and by $\operatorname{\mathscr{FC}}^{2}_{b}(\Omega)$ the space of the restriction to $\Omega$ of cylindrical twice differentiable functions on $X$ with bounded derivatives (see Section 2.2). We remark that, by [24, Theorem 3.1(2)], $\langle\nabla^{2}_{H}U(x)h,h\rangle_{H}\geq 0$, for $\mu$-a.e. $x\in X$ and every $h\in H$. An important space in our investigation is

[TABLE]

endowed with the norm

[TABLE]

We remark that $W_{U}^{2,2}(\Omega,\nu)$ is a Hilbert space. We will also study the following subspace of $W^{2,2}_{U}(\Omega,\nu)$

[TABLE]

endowed with the norm (1.3).

Our main results are the following characterizations of the domain of the self-adjoint operator $L_{\nu,\Omega}$ when $\Omega$ is the whole space or a half-space. We recall that by ${\left\|\cdot\right\|}_{D(L_{\nu,\Omega})}$ we denote the graph norm, i.e. for $u\in D(L_{\nu,\Omega})$

[TABLE]

Theorem 1.3**.**

Assume that Hypothesis 1.2 holds and that $\operatorname{\mathscr{FC}}_{b}^{2}(X)$ is dense in $W^{2,2}_{U}(X,\nu)$. Then $D(L_{\nu,X})=W^{2,2}_{U}(X,\nu)$. Moreover, for every $u\in D(L_{\nu,X})$, it holds

[TABLE]

and fixed any orthornomal basis $\{h_{n}\,|\,n\in\mathbb{N}\}$ of $H$

[TABLE]

where the series converges in $\operatorname{L}^{2}(X,\nu)$ (See Section 2 for the definition of the $\widehat{\cdot}$ operator).

We remark that if the weight $U$ is such that $\nabla_{H}U$ is Lipschitz continuous, or more generally $H$-Lipschitz (see Section 2), then $\operatorname{\mathscr{FC}}^{2}_{b}(X)$ is dense in $W^{2,2}_{U}(X,\nu)$, so that the assumption of Theorem 1.3 is satisfied (see Corollary 6.2).

When $G=x^{*}-r$ where $x^{*}\in X^{*}\smallsetminus{\left\{0\right\}}$ and $r\in\mathbb{R}$, i.e. if $\Omega$ is a half-space, we want to remark that the Neumann boundary condition: ${\left\langle\operatorname{\operatorname{Tr}}(\nabla_{H}u),\operatorname{\operatorname{Tr}}(\nabla_{H}G)\right\rangle}_{H}=0$ for $\rho$-a.e. $x\in G^{-1}(0)$, read

[TABLE]

for $\rho$-a.e. $x\in G^{-1}(0)$, where $h_{x^{*}}$ is the unique vector of $H$ such that

[TABLE]

Such an element exists since $x^{*}$ is a continuous linear functional on $H$.

Theorem 1.4**.**

Assume that Hypothesis 1.2 holds and $G$ is an affine function, namely $G=x^{*}-r$ where $x^{*}\in X^{*}\smallsetminus{\left\{0\right\}}$ and $r\in\mathbb{R}$. If the space

[TABLE]

where $h_{x^{*}}$ is defined in (1.4), is dense in the space of $W^{2,2}_{U,N}(\Omega,\nu)$, then $D(L_{\nu,\Omega})=W^{2,2}_{U,N}(\Omega,\nu)$. Moreover, for every $u\in D(L_{\nu,\Omega})$, it holds

[TABLE]

and fixed any orthornomal basis $\{h_{n}\,|\,n\in\mathbb{N}\}$ of $H$

[TABLE]

where the series converges in $\operatorname{L}^{2}(\Omega,\nu)$ (See Section 2 for the definition of the $\widehat{\cdot}$ operator).

We remark that showing the density of $\mathcal{Z}(\Omega)$ in $W^{2,2}_{U,N}(\Omega)$ is not an easy task. This difficulty can be overcome if $\Omega$ belongs to the class of Neumann extension domains.

Definition 1.5**.**

Let $Z_{U}^{2,2}(X,\nu)$ be the completion of the space $\operatorname{\mathscr{FC}}^{2}_{b}(X)$ with respect to the norm defined in (1.3). We say that $\Omega$ is a Neumann extension domain if there exists a linear operator $E^{\mathcal{N}}$ from $W^{2,2}_{U,N}(\Omega,\nu)$ into $Z_{U}^{2,2}(X,\nu)$ such that for every $\phi\in W^{2,2}_{U,N}(\Omega,\nu)$

1. (1)

$E^{\mathcal{N}}\phi(x)=\phi(x)$for $\nu$-a.e $x\in\Omega$; 2. (2)

there is $K>0$, independent of $\phi$, such that $\|E^{\mathcal{N}}\phi\|_{Z^{2,2}_{U}(X,\nu)}\leq K\|\phi\|_{W^{2,2}_{U}(\Omega,\nu)}$.

The operator $E^{\mathcal{N}}$ is called Neumann extension operator.

Theorem 1.6**.**

Assume that Hypothesis 1.2 holds and that $\Omega$ is a Neumann extension domain satisfying Hypothesis 1.1. Then $D(L_{\nu,\Omega})=W^{2,2}_{U,N}(\Omega,\nu)$. Moreover, for every $u\in D(L_{\nu,\Omega})$, it holds

[TABLE]

and fixed any orthornomal basis $\{h_{n}\,|\,n\in\mathbb{N}\}$ of $H$

[TABLE]

where the series converges in $\operatorname{L}^{2}(\Omega,\nu)$ (See Section 2 for the definition of the $\widehat{\cdot}$ operator).

The characterization of Neumann extension domains is an open problem in Wiener space theory. The only known results are mainly negative (see [7]), but if $\Omega$ is a half-space and $U\equiv 0$, it is known that an extension operator can be constructed (see [7]). Since we were unable to find explicit computations in the literature, we made them in Lemma 7.1. Applying Theorems 1.4, 1.6 and Lemma 7.1 we get the following characterization of the domain of the Ornstein–Uhlenbeck operator on half-spaces, i.e. $U\equiv 0$ and $G$ is an affine function.

Theorem 1.7**.**

Assume that Hypothesis 1.2 holds and $G$ is an affine function, namely $G(x)=x^{*}(x)-r$ with $x^{*}\in X^{*}\smallsetminus{\left\{0\right\}}$ and $r\in\mathbb{R}$. Then

[TABLE]

where $h_{x^{*}}$ is defined in (1.4). Moreover, for every $u\in D(L_{\mu,\Omega})$, it holds

[TABLE]

and fixed any orthornomal basis $\{h_{n}\,|\,n\in\mathbb{N}\}$ of $H$

[TABLE]

where the series converges in $\operatorname{L}^{2}(\Omega,\mu)$ (See Section 2 for the definition of the $\widehat{\cdot}$ operator). In addition the space

[TABLE]

is dense in $D(L_{\mu,\Omega})$ with respect to the graph norm.

The paper is organized as follows: in Section 2 we recall some basic definitions and we fix the notations. Section 3 is dedicated to the study of the second order analysis of the Moreau–Yosida approximations along $H$, that are used to prove Theorems 1.3. In section 4 we will introduce the divergence operator $\operatorname{\operatorname{div}}_{\nu,\Omega}$ as minus the formal adjoint of the gradient operator along $H$ and investigate its properties. Namely, consider the space

[TABLE]

For every $\Phi\in\mathcal{Z}(\Omega,H)$ put

[TABLE]

Let $Z_{U}^{1,2}(\Omega,\nu;H)$ be the completion of the space $\mathcal{Z}(\Omega,H)$ with respect to the norm defined in (1.10). As usual the elements of $Z_{U}^{1,2}(\Omega,\nu;H)$ can be identified as equivalence classes of vector fields with respect to the $\nu$-a.e. equivalence relation. It is easy to see that $Z_{U}^{1,2}(\Omega,\nu;H)$ is a Hilbert space. In Proposition 4.4 we will prove that the space $Z_{U}^{1,2}(\Omega,\nu;H)$ is contained in the domain of the divergence operator $\operatorname{\operatorname{div}}_{\nu,\Omega}$ in $\operatorname{L}^{2}$ and $\operatorname{\operatorname{div}}_{\nu,\Omega}\Phi\in\operatorname{L}^{2}(\Omega,\nu)$ for every $\Phi\in Z_{U}^{1,2}(X,\nu;H)$. Furthermore an explicit formula for the calculation of $\operatorname{\operatorname{div}}_{\nu,\Omega}$ is given by (4.16).

We remark that without loss of generality we can assume that the sequence ${\left\{h_{1},\ldots,h_{n}\right\}}$ in (1.9) is a sequence of orthonormal elements of $H$ (indeed, it is enough to apply the Gram-Schmidt procedure). Moreover, we stress that the boundary integral in (1.10) in general cannot be estimated by the $W^{1,2}$-norm of $\Phi$. This fact depends not only from the presence of the second order derivatives of $G$, but also from the trace theory in infinite dimensions. Indeed, as shown in [21] the trace of $f\in W^{1,p}(\Omega)$ belongs to $\operatorname{L}^{q}(G^{-1}(0),e^{-U}\rho)$ for any $q\in[1,p(t-2)(t-1)^{-1}]$, where $t$ is the number fixed in Hypothesis 1.2. In particular if $p=2$ then we do not know if the trace operator is continuous in $\operatorname{L}^{2}(G^{-1}(0),e^{-U}\rho)$.

In Section 5 we obtain maximal Sobolev regularity estimates for the weak solution of the problem

[TABLE]

where $\lambda>0$, and $f\in\operatorname{L}^{2}(\Omega,\nu)$. We say that $u\in W^{1,2}(\Omega,\nu)$ is a weak solution of problem (1.11) if

[TABLE]

Notice that the unique weak solution $u$ of problem (1.11) satisfies $u=R(\lambda,L_{\nu,\Omega})f$, where $R(\lambda,L_{\nu,\Omega})$ is the resolvent of $L_{\nu,\Omega}$. We recall that results about existence, uniqueness and regularity of the weak solution of problem (5.1), in domains with sufficiently regular boundary, are known in the finite dimensional case (see the classical books [25] and [28] for a bounded $\Omega$ and [6], [15], [32], [16] and [17] for an unbounded $\Omega$). If $X$ is infinite dimensional maximal Sobolev regularity results are known when $X$ is a separable Hilbert space. See for example [2] and [3] where $U\equiv 0$ and [19] where $U$ is bounded from below. When $\Omega=X$ more results are known, see for example [14], [34] and [30] if $X$ is finite dimensional, [18] if $X$ is a Hilbert space and [12] if $X$ is a separable Banach space. If $X$ is general separable Banach space and $\Omega\varsubsetneq X$, then the only results regarding maximal Sobolev regularity are the one contained in [10], where the second named author studied problem (5.1) when $U\equiv 0$, namely when $L_{\nu,\Omega}$ is the Ornstein–Uhlenbeck operator on $\Omega$, and in [11], where the second and third named authors studied the general case.

In Section 6 we prove Theorems 1.3, 1.4 and 1.6 and some related corollaries. Finally, in Section 7 we provide some examples to which our results can be applied. In particular we study the case when $\Omega$ is the unit ball of a Hilbert space and we prove Theorem 1.7.

2. Notation and preliminaries

We will denote by $X^{*}$ the topological dual of $X$. We recall that $X^{*}\subseteq\operatorname{L}^{2}(X,\mu)$. The linear operator $R_{\mu}:X^{*}\rightarrow(X^{*})^{\prime}$

[TABLE]

is called the covariance operator of $\mu$. Since $X$ is separable, then it is actually possible to prove that $R_{\mu}:X^{*}\rightarrow X$ (see [8, Theorem 3.2.3]). We denote by $X^{*}_{\mu}$ the closure of $X^{*}$ in $\operatorname{L}^{2}(X,\mu)$. The covariance operator $R_{\mu}$ can be extended by continuity to the space $X^{*}_{\mu}$, still by formula (2.1). By [8, Lemma 2.4.1] for every $h\in H$ there exists a unique $g\in X^{*}_{\mu}$ with $h=R_{\mu}g$, in this case we set

[TABLE]

Throughout the paper we fix an orthonormal basis ${\left\{e_{i}\right\}}_{i\in\mathbb{N}}$ of $H$ such that $\widehat{e}_{i}$ belongs to $X^{*}$, for every $i\in\mathbb{N}$. Such basis exists by [8, Corollary 3.2.8(ii)].

2.1. Differentiability along $H$

We say that a function $f:X\rightarrow\mathbb{R}$ is differentiable along $H$ at $x$ if there exists $v\in H$ such that

[TABLE]

uniformly with respect to $h\in H$, with ${\left|h\right|}_{H}=1$. In this case, the vector $v\in H$ is unique and we set $\nabla_{H}f(x):=v$. Moreover, for every $k\in\mathbb{N}$ the derivative of $f$ in the direction of $e_{k}$ exists and it is given by

[TABLE]

We denote by $\mathcal{H}_{2}$ the space of the Hilbert–Schmidt operators in $H$, that is the space of the bounded linear operators $A:H\rightarrow H$ such that ${\left\|A\right\|}_{\mathcal{H}_{2}}^{2}=\sum_{i}{\left|Ae_{i}\right|}^{2}_{H}$ is finite (see [20]). We say that a function $f:X\rightarrow\mathbb{R}$ is twice differentiable along $H$ at $x$ if it is differentiable along $H$ at $x$ and there exists $A\in\mathcal{H}_{2}$ such that

[TABLE]

uniformly with respect to $h\in H$, with ${\left|h\right|}_{H}=1$. In this case the operator $A$ is unique and we set $\nabla_{H}^{2}f(x):=A$. Moreover, for every $i,j\in\mathbb{N}$ we set

[TABLE]

2.2. Special classes of functions

For $k\in\mathbb{N}\cup{\left\{\infty\right\}}$, we denote by $\operatorname{\mathscr{FC}}^{k}(X)$ ($\operatorname{\mathscr{FC}}_{b}^{k}(X)$ respectively) the space of the cylindrical function of the type $f(x)=\varphi(x^{*}_{1}(x),\ldots,x^{*}_{n}(x))$ where $\varphi\in\operatorname{\mathscr{C}}^{k}(\mathbb{R}^{n})$ ($\varphi\in\operatorname{\mathscr{C}}^{k}_{b}(\mathbb{R}^{n})$, respectively) and $x^{*}_{1},\ldots,x^{*}_{n}\in X^{*}$, for some $n\in\mathbb{N}$. We remark that $\operatorname{\mathscr{FC}}^{\infty}_{b}(X)$ is dense in $\operatorname{L}^{p}(X,\nu)$ for all $p\geq 1$ (see [21, Proposition 3.6]). We recall that if $f\in\operatorname{\mathscr{FC}}^{2}(X)$, then $\partial_{ij}f(x)=\partial_{ji}f(x)$ for every $i,j\in\mathbb{N}$ and $x\in X$.

If $Y$ is a Banach space, a function $F:X\rightarrow Y$ is said to be $H$-Lipschitz if there exists a positive constant $C$ such that

[TABLE]

for every $h\in H$ and $\mu$-a.e. $x\in X$ (see [8, Section 4.5 and Section 5.11]). We denote with $[F]_{H\text{-Lip}}$ the best constant $C$ appearing in (2.3).

A function $F:X\rightarrow\mathbb{R}$ is said to be $H$-continuous, if $\lim_{{\left|h\right|}_{H}\rightarrow 0}F(x+h)=F(x)$, for $\mu$-a.e. $x\in X$.

2.3. Sobolev spaces

The Gaussian Sobolev spaces $W^{1,p}(X,\mu)$ and $W^{2,p}(X,\mu)$, with $p\geq 1$, are the completions of the smooth cylindrical functions $\operatorname{\mathscr{FC}}_{b}^{\infty}(X)$ in the norms

[TABLE]

Such spaces can be identified with subspaces of $\operatorname{L}^{p}(X,\mu)$ and the (generalized) gradient and Hessian along $H$, $\nabla_{H}f$ and $\nabla_{H}^{2}f$, are well defined and belong to $\operatorname{L}^{p}(X,\mu;H)$ and $\operatorname{L}^{p}(X,\mu;\mathcal{H}_{2})$, respectively. The spaces $W^{1,p}(X,\mu;H)$ are defined in a similar way, replacing smooth cylindrical functions with $H$-valued smooth cylindrical functions (i.e. the linear span of the functions $x\mapsto f(x)h$, where $f$ is a smooth cylindrical function and $h\in H$). For more information see [8, Section 5.2].

Now we consider $\nabla_{H}:\operatorname{\mathscr{FC}}^{\infty}_{b}(X)\rightarrow\operatorname{L}^{p}(X,\nu;H)$. This operator is closable in $\operatorname{L}^{p}(X,\nu)$ whenever $p>\frac{t-1}{t-2}$ (see [21, Definition 4.3]). For such $p$ we denote by $W^{1,p}(X,\nu)$ the domain of its closure in $\operatorname{L}^{p}(X,\nu)$. In the same way the operator $(\nabla_{H},\nabla^{2}_{H}):\operatorname{\mathscr{FC}}^{\infty}_{b}(X)\rightarrow\operatorname{L}^{p}(X,\nu;H)\times\operatorname{L}^{p}(X,\nu;\mathcal{H}_{2})$ is closable in $\operatorname{L}^{p}(X,\nu)$, whenever $p>\frac{t-1}{t-2}$ (see [12, Proposition 2.1]). For such $p$ we denote by $W^{2,p}(X,\nu)$ the domain of its closure in $\operatorname{L}^{p}(X,\nu)$. The spaces $W^{1,p}(X,\nu;H)$ are defined in a similar way, replacing smooth cylindrical functions with $H$-valued smooth cylindrical functions.

We want to point out that if Hypothesis 1.2 holds, then $\frac{t-1}{t-2}<2$. In particular the above arguments allows us to define the Sobolev spaces $W^{1,2}(X,\nu)$ and $W^{2,2}(X,\nu)$.

We shall use the integration by parts formula (see [21, Lemma 4.1]) for $\varphi\in W^{1,p}(X,\nu)$ with $p>\frac{t-1}{t-2}$:

[TABLE]

where $\widehat{e}_{k}$ is defined in formula (2.2). Finally, we recall that if $U$ satisfies Hypothesis 1.2 then for every $u\in\operatorname{\mathscr{FC}}^{2}_{b}(X)$

[TABLE]

where the series converges in $\operatorname{L}^{2}(X,\nu)$ (see [21, Proposition 5.3]).

2.4. Capacity

Let $L_{p}$ be the infinitesimal generator of the Ornstein–Uhlenbeck semigroup $T(t)$ in $\operatorname{L}^{p}(X,\mu)$, where

[TABLE]

For $k=1,2,3$, we define the $C_{k,p}$-capacity of an open set $A\subseteq X$ as

[TABLE]

For a general Borel set $B\subseteq X$ we let $C_{k,p}(B)=\inf{\left\{C_{k,p}(A)\,\middle|\,B\subseteq A\text{ open}\right\}}$. By $f\in W^{k,p}(X,\mu)$ we mean an equivalence class of functions and we call every element “version”. For any $f\in W^{k,p}(X,\mu)$ there exists a version $\overline{f}$ of $f$ which is Borel measurable and $C_{k,p}$-quasicontinuous, i.e. for every $\varepsilon>0$ there exists an open set $A\subseteq X$ such that $C_{k,p}(A)\leq\varepsilon$ and $\overline{f}_{|_{X\smallsetminus A}}$ is continuous. Furthermore, for every $r>0$

[TABLE]

See [8, Theorem 5.9.6]. Such a version is called a $(k,p)$-precise version of $f$. Two precise versions of the same $f$ coincide outside sets with null $C_{k,p}$-capacity. All our results will be independent on our choice of a precise version of $G$ in Hypothesis 1.1. With obvious modification the same definition can be adapted to functions belonging to $W^{k,p}(X,\mu;H)$ and $W^{k,p}(X,\mu;\mathcal{H}_{2})$.

2.5. Sobolev spaces on sublevel sets

The proof of the results stated in this subsection can be found in [13] and [21]. Let $G$ be a function satisfying Hypothesis 1.1. We are interested in Sobolev spaces on sublevel sets of $G$.

For $k\in\mathbb{N}\cup{\left\{\infty\right\}}$, we denote by $\operatorname{\mathscr{FC}}^{k}_{b}(\Omega)$ the space of the restriction to $\Omega$ of functions in $\operatorname{\mathscr{FC}}^{k}_{b}(X)$. For any $p\geq 1$, the spaces $W^{1,p}(\Omega,\mu)$ and $W^{2,p}(\Omega,\mu)$ are defined as the domain of the closure of the operators $\nabla_{H}:\operatorname{\mathscr{FC}}_{b}^{\infty}(\Omega)\rightarrow\operatorname{L}^{p}(\Omega,\mu;H)$ and $(\nabla_{H},\nabla_{H}^{2}):\operatorname{\mathscr{FC}}^{\infty}_{b}(\Omega)\rightarrow\operatorname{L}^{p}(\Omega,\mu;H)\times\operatorname{L}^{p}(\Omega,\mu;\mathcal{H}_{2})$, respectively. See [13, Lemma 2.2] and [10, Proposition 1].

We recall that $\nabla_{H}:\operatorname{\mathscr{FC}}^{\infty}_{b}(\Omega)\rightarrow\operatorname{L}^{p}(\Omega,\nu;H)$ and $(\nabla_{H},\nabla^{2}_{H}):\operatorname{\mathscr{FC}}_{b}^{\infty}(\Omega)\rightarrow\operatorname{L}^{p}(\Omega,\nu;H)\times\operatorname{L}^{p}(\Omega,\nu;\mathcal{H}_{2})$ are closable operators in $\operatorname{L}^{p}(\Omega,\nu)$, whenever $p>\frac{t-1}{t-2}$ (see [21, Proposition 6.1] and [11, Proposition 2.2]). For such values of $p$ we denote by $W^{1,p}(\Omega,\nu)$ the domain of its closure in $\operatorname{L}^{p}(\Omega,\nu)$ and we will still denote by $\nabla_{H}$ the closure operator. The space $W^{2,p}(\Omega,\nu)$ is defined in the same way.

Finally we want to remark that if Hypotheses 1.1 and 1.2 hold, then $\frac{t-1}{t-2}<2$. In particular the Sobolev spaces $W^{1,2}(\Omega,\nu)$ and $W^{2,2}(\Omega,\nu)$ are well defined.

2.6. Traces of Sobolev functions

By $\rho$ we indicate the Feyel–de La Pradelle Hausdorff–Gauss surface measure. For a comprehensive treatment of surface measures in infinite dimensional Banach spaces with Gaussian measures we refer to [23], [22] and [13].

Traces of Sobolev functions in infinite dimensional Banach spaces have been studied in [13] in the Gaussian case and in [21] in the weighted Gaussian case. We stress that in [13] the definition of Sobolev Spaces is different with respect to the our one, but these two definitions coincide in the case of Gaussian measure. Assume that Hypotheses 1.1 and 1.2 hold and let $p>\frac{t-1}{t-2}$. If $\varphi\in W^{1,p}(\Omega,\nu)$ we define the trace of $\varphi$ on $G^{-1}(0)$ as follows:

[TABLE]

and it is possible to prove that ${\rm Tr}\varphi\in\operatorname{L}^{q}(G^{-1}(0),e^{-U}\rho)$ for any $q\in[1,p(t-2)(t-1)^{-1}]$, where $t$ is the real number fixed in Hypothesis 1.2. Here, $(\varphi_{n})_{n\in\mathbb{N}}$ is any sequence in $\operatorname{\operatorname{Lip}}_{b}(\Omega)$, the space of bounded and Lipschitz functions on $\Omega$, which converges in $W^{1,p}(\Omega,\nu)$ to $\varphi$. The definition does not depend on the choice of the sequence $(\varphi_{n})_{n\in\mathbb{N}}$ in $\operatorname{\operatorname{Lip}}_{b}(\Omega)$ approximating $\varphi$ in $W^{1,p}(\Omega,\nu)$ (see [21, Proposition 7.1]). In addition the following result holds.

Proposition 2.1**.**

Assume that Hypotheses 1.1 and 1.2 hold. Then the operator $\operatorname{\operatorname{Tr}}:W^{1,p}(\Omega,\nu)\rightarrow\operatorname{L}^{q}(G^{-1}(0),e^{-U}\rho)$ is continuous for every $p>\frac{t-1}{t-2}$ and $q\in{\left[1,p\frac{t-2}{t-1}\right]}$. Moreover, if $U\equiv 0$, then the trace operator is continuous from $W^{1,p}(\Omega,\mu)$ to $\operatorname{L}^{q}(G^{-1}(0),\rho)$ for every $p>1$ and $q\in[1,p)$ (see [13, Corollary 4.2] and [21, Corollary 7.3]).

We will still denote by $\operatorname{\operatorname{Tr}}\Psi=\sum_{n=1}^{+\infty}(\operatorname{\operatorname{Tr}}\psi_{n})e_{n}$ if $\Psi\in W^{1,p}(\Omega,\nu;H)$, for $p>\frac{t-1}{t-2}$, and $\psi_{n}={\left\langle\Psi,e_{n}\right\rangle}_{H}$. The main result of [21] is the following integration by parts formula.

Theorem 2.2**.**

Assume that Hypotheses 1.1 and 1.2 hold and let $p>\frac{t-1}{t-2}$. For every $\varphi\in W^{1,p}(\Omega,\nu)$ and $k\in\mathbb{N}$ we have

[TABLE]

Another important result, that we will use in this paper, is the following (see [13, Proposition 4.8] and [21, Proposition 7.5]).

Proposition 2.3**.**

Assume that Hypotheses 1.1 and 1.2 hold and let $p>\frac{t-1}{t-2}$. Then for every $\varphi\in W^{1,p}(\Omega,\nu)$, the trace of $\operatorname{\operatorname{Tr}}(\varphi)$ at $G^{-1}(0)$ coincides $\rho$-a.e. with the restriction to $G^{-1}(0)$ of any precise version $\widetilde{\varphi}$ of $\varphi$.

2.7. The spaces $W^{2,2}_{U,N}$ and $Z^{2,2}_{U,N}$

We recall the definition of the space $W^{2,2}_{U}$ and $W^{2,2}_{U,N}$.

[TABLE]

endowed with the norm

[TABLE]

We consider the space $W^{2,2}_{U}(\Omega,\nu)$

[TABLE]

endowed with the norm (2.5).

We denote by $Z_{U}^{2,2}(X,\nu)$ be the completion of the space $\operatorname{\mathscr{FC}}^{2}_{b}(X)$ with respect to the norm defined in (2.5) and by $Z^{2,2}_{U,N}(\Omega,\nu)$ the completion of the space

[TABLE]

with respect to the norm (2.5).

3. Second-order analysis of the Moreau–Yosida approximations along $H$

We start this section by recalling the definition of the subdifferential of a convex semicontinuous function. If $f:X\rightarrow\mathbb{R}$ is a proper, convex and lower semicontinuous function, we denote by $\operatorname{\operatorname{dom}}(f)$ the domain of $f$, namely $\operatorname{\operatorname{dom}}(f):={\left\{x\in X\,\middle|\,f(x)<+\infty\right\}}$, and by $\partial f(x)$ the subdifferential of $f$ at the point $x$, i.e.

[TABLE]

For a classical treatment of subdifferentials of convex functions we refer to [37] and [4].

We recall that for $\alpha>0$ the Moreau–Yosida approximation along $H$ of a proper convex and lower semicontinuous function $f:X\rightarrow\mathbb{R}\cup{\left\{+\infty\right\}}$ is

[TABLE]

See [12, Section 3] and [11, Section 4] for more details and [9] and [5, Section 12.4] for a treatment of the classical Moreau–Yosida approximations in Hilbert spaces, which are different from the ones defined in (3.1). Second-order analysis of the classical Moreau–Yosida approximations have been studied in various papers, e.g. [38], [36] and [35].

In the following proposition we recall some results contained in [12, Section 3] and in [11, Section 4].

Proposition 3.1**.**

Let $x\in X$, $\alpha>0$ and $f:X\rightarrow\mathbb{R}\cup{\left\{+\infty\right\}}$ be a proper convex and lower semicontinuous function. The following properties hold:

1. (1)

the function $g_{\alpha,x}:H\rightarrow\mathbb{R}$ defined as $g_{\alpha,x}(h):=f(x+h)+\frac{1}{2\alpha}{\left|h\right|}^{2}_{H}$, has a unique global minimum point $P(x,\alpha)\in H$. Moreover $P(x,\alpha)\rightarrow 0$ in $H$ as $\alpha$ goes to zero; 2. (2)

$f_{\alpha}(x)\nearrow f(x)$* as $\alpha\rightarrow 0^{+}$. In particular $f_{\alpha}(x)\leq f(x)$ for every $\alpha>0$ and $x\in X$;* 3. (3)

for $p\in H$, we have $p=P(x,\alpha)$ if, and only if, $f(x+p)\leq f(x+h)+\frac{1}{\alpha}{\left\langle p,h-p\right\rangle}_{H}$, for every $h\in H$; 4. (4)

the function $P_{x,\alpha}:H\rightarrow H$ defined as $P_{x,\alpha}(h):=P(x+h,\alpha)$ is Lipschitz continuous, with Lipschitz constant less than or equal to $1$; 5. (5)

$f_{\alpha}$* is differentiable along $H$ at every point $x\in X$. In addition, for every $x\in X$, we have $\nabla_{H}f_{\alpha}(x)=-\alpha^{-1}P(x,\alpha)$;* 6. (6)

$f_{\alpha}$* belongs to $W^{2,p}(X,\mu)$, whenever $f\in\operatorname{L}^{p}(X,\mu)$ for some $1\leq p<+\infty$;* 7. (7)

let $x\in\operatorname{\operatorname{dom}}(f)$ and assume that $f$ belongs to $W^{1,p}(X,\mu)$ for some $p>1$. If we define $F:H\rightarrow\mathbb{R}$ as $F(h):=f(x+h)$, then $F$ is proper convex and lower semicontinuous function. Moreover, $\nabla_{H}f(x)\in\partial F(0)$ and $\nabla_{H}f_{\alpha}(x)\in\partial F(P(x,\alpha))$; 8. (8)

let $x\in\operatorname{\operatorname{dom}}(f)$ and assume that $f$ belongs to $W^{1,p}(X,\mu)$ for some $p>1$. Then $\nabla_{H}f_{\alpha}(x)$ converges to $\nabla_{H}f(x)$ as $\alpha$ goes to zero.

The last property we need is the convergence of the second-order derivative along $H$.

Proposition 3.2**.**

Let $f\in W^{2,p}(X,\mu)$ for some $p>1$ and $\alpha>0$. Assume that $f$ is twice differentiable along $H$ at every point $x\in\operatorname{\operatorname{dom}}f$. Then for every $x\in dom(f)$ there exists $\nabla_{H}^{2}f_{\alpha}(x)$, and $\nabla_{H}^{2}f_{\alpha}(x)$ converges to $\nabla_{H}^{2}f(x)$ as $\alpha$ goes to zero.

Proof.

By Proposition 3.1(7) we get $\nabla_{H}f_{\alpha}(x)=\nabla_{H}f(x+P(x,\alpha))$. We can differentiate along $H$ since $P(x,\alpha)$ admits a $H$-gradient (it is $H$-Lipschitz).

[TABLE]

If we let $\alpha\rightarrow 0$ then, by 3.1(8), we get $\lim_{\alpha\rightarrow 0}\nabla_{H}^{2}f_{\alpha}(x)=\nabla_{H}^{2}f(x)$. ∎

4. The divergence operator

We start this section by recalling the definition of divergence, see [8, Section 5.8] for the case $\Omega=X$. For every measurable map $\Phi:\Omega\rightarrow X$ and for every $f\in\operatorname{\mathscr{FC}}^{\infty}_{b}(\Omega)$ we define

[TABLE]

Definition 4.1**.**

Let $\Phi\in\operatorname{L}^{1}(\Omega,\nu;X)$ be a vector field. We say that $\Phi$ admits divergence if there exists a function $g\in\operatorname{L}^{1}(\Omega,\nu)$ such that

[TABLE]

for every $f\in\operatorname{\mathscr{FC}}_{b}^{\infty}(\Omega)$, where $\partial_{\Phi}f$ has been defined in (4.1). If such a function $g$ exists, then we set $\operatorname{\operatorname{div}}_{\nu,\Omega}\Phi:=g$. Observe that, when $\operatorname{\operatorname{div}}_{\nu,\Omega}\Phi$ exists, it is unique by the density of $\operatorname{\mathscr{FC}}_{b}^{\infty}(\Omega)$ in $\operatorname{L}^{p}(\Omega,\nu)$ (see [21]). We denote by $D(\operatorname{\operatorname{div}}_{\nu,\Omega})$ the domain of $\operatorname{\operatorname{div}}_{\nu,\Omega}$ in $\operatorname{L}^{1}(\Omega,\nu;X)$. Lastly, we observe that if $\Phi\in\operatorname{L}^{1}(\Omega,\nu;H)$, then $\partial_{\Phi}f(x)={\left\langle\nabla_{H}f(x),\Phi(x)\right\rangle}_{H}$ for $x\in\Omega$. In this case (4.2) becomes

[TABLE]

We remark that in $\operatorname{L}^{2}$-setting, the divergence operator $\operatorname{\operatorname{div}}_{\nu,\Omega}$ is $-\nabla_{H}^{*}$, the $\operatorname{L}^{2}$-adjoint of the the gradient along $H$ operator. Indeed, for any $\Phi\in\operatorname{L}^{2}(\Omega,\nu;H)$ and any $f\in W^{1,2}(\Omega,\nu)$ we get

[TABLE]

The following two technical lemmata are crucial to show Theorems 1.3 and 1.4. In particular, the second one is a generalization of a well known result in differential geometry, see [29], [6] and [10].

Lemma 4.2**.**

If Hypothesis 1.2 holds, then

[TABLE]

If $\Omega\subsetneq X$, let Hypotheses 1.1 and 1.2 hold true, and let $f,g\in\operatorname{\mathscr{FC}}^{2}_{b}(\Omega)$ and $h,k\in H$. Then

[TABLE]

Proof.

We will only prove (4.5), since the proof of (4.4) is essentially the same. We will use Theorem 2.2 several times. We have

[TABLE]

∎

Lemma 4.3**.**

Assume Hypotheses 1.1. Let $\Phi\in\mathcal{Z}(\Omega,H)$ the space defined in (1.9). Then for $\rho$-a.e. $x\in G^{-1}(0)$

[TABLE]

Proof.

The proof is rather long and it will be split into various steps. Let $\{h_{i}\}_{i\in\mathbb{N}}$ be the orthonormal basis of $H$ associated with $\Phi$ given by the definition of the space $\mathcal{Z}(\Omega,H)$. By Hypothesis 1.1, Proposition 2.3 and the very definition of $\mathcal{Z}(\Omega,H)$ the set

[TABLE]

has full $\rho$ measure. We will prove that (4.6) holds for every point $x_{0}$ belonging to $A$. By (4.10) we have $\nabla_{H}G(x_{0})\neq 0$, so there exists $n_{0}\in\mathbb{N}$ such that

[TABLE]

Without loss of generality, we can assume that $n_{0}=1$. By the very definition of the space $\mathcal{Z}(\Omega,H)$ there exist $K(\Phi)>0$, $k\in\mathbb{N}$ and $(\varphi_{i})_{i=1}^{k}\subseteq\operatorname{\mathscr{FC}}^{2}_{b}(\Omega)$ such that for every $k_{1},k_{2}\in H$ it holds

[TABLE]

and $\Phi(x)=\sum_{i=1}^{k}\varphi_{i}(x)h_{i}$. For $i>k$ we set $\varphi_{i}(x)\equiv 0$.

1. Step 1:

Let us consider the space

[TABLE]

endowed with the Hilbert space norm ${\left|\overline{h}\right|}_{h_{1}^{\perp}}=\sum_{i=2}^{+\infty}{\left\langle\overline{h},h_{i}\right\rangle}_{H}^{2}$. We denote its inner product by ${\left\langle\cdot,\cdot\right\rangle}_{h_{1}^{\perp}}$ and recall that $\{h_{i}\}_{i\geq 2}$ is an orthonormal basis for $h_{1}^{\perp}$ and $H=h_{1}^{\perp}\oplus\operatorname{\operatorname{span}}{\left\{h_{1}\right\}}$. We want to apply the implicit function theorem to a function defined on $h_{1}^{\perp}\oplus\mathbb{R}$. Let $G_{x_{0}}:h_{1}^{\perp}\oplus\mathbb{R}\rightarrow\mathbb{R}$ be the function defined as

[TABLE]

Observe that $G_{x_{0}}((0,0))=G(x_{0})=0$ and

[TABLE]

where $D_{2}$ is the derivative with respect the second variable. Since (1.1) implies that $G_{x_{0}}$ is Fréchet differentiable at [math], applying the implicit function theorem, see [29, Theorem 5.9], we get an open neighborhood $U_{0}\subseteq h_{1}^{\perp}$ of the origin and a continuously Fréchet differentiable function $g_{x_{0}}:U_{0}\rightarrow\mathbb{R}$ such that for every $\overline{h}\in U_{0}$ we have

[TABLE]

Moreover, the function $g_{x_{0}}:U_{0}\rightarrow\mathbb{R}$ satisfying (4.12) is uniquely determined. Without loss of generality we may assume that $U_{0}$ is an open ball centered at the origin of radius $R$. We remark that (4.12) implies that for every $\overline{h}\in h_{1}^{\perp}$

[TABLE] 2. Step 2:

We denote by $D_{h_{1}^{\perp}}g_{x_{0}}(0)$ the Fréchet derivative of $g_{x_{0}}$ at the origin. For $t>0$ sufficiently small and by (4.13), for any $i\geq 2$ we get

[TABLE]

Letting $t$ go to zero, for any $i\geq 2$ we get

[TABLE] 3. Step 3:

The vector field $\Phi_{x_{0}}(\overline{h})=\sum_{i=2}^{+\infty}\varphi_{i}(x_{0}+\overline{h})h_{i}$ is defined from $h_{1}^{\perp}$ to itself. Let $\delta$ be a positive real number which satisfies

[TABLE]

where $K(\Phi)$ has been introduced in (4.11). We consider the complete metric space $\operatorname{\mathscr{C}}_{b}([-\delta,\delta],\overline{U_{0}})$, i.e. the set

[TABLE]

endowed with the complete metric $d(f,g):=\sup_{t\in[-\delta,\delta]}{\left|f(t)-g(t)\right|}_{h_{1}^{\perp}}$. Let $\Gamma:\operatorname{\mathscr{C}}_{b}([-\delta,\delta],\overline{U_{0}})\rightarrow\operatorname{\mathscr{C}}_{b}([-\delta,\delta],h_{1}^{\perp})$ be the function defined as follows:

[TABLE]

for any $t\in[-\delta,\delta]$. The integral in (4.15) should be understood in the Bochner sense. We look for a fixed point of $\Gamma$ in $\operatorname{\mathscr{C}}_{b}([-\delta,\delta],\overline{U_{0}})$. We want to use Banach fixed-point theorem, so

[TABLE]

Therefore $\Gamma$ is a contraction in $\operatorname{\mathscr{C}}_{b}([-\delta,\delta],\overline{U_{0}})$. We claim that $\Gamma$ maps $\operatorname{\mathscr{C}}_{b}([-\delta,\delta],\overline{U_{0}})$ into itself. The continuity of $\Gamma(\gamma)(t)$ is clear, and

[TABLE]

By the Banach fixed-point theorem there exists a unique fixed point $\gamma_{x_{0}}\in\operatorname{\mathscr{C}}_{b}([-\delta,\delta],\overline{U_{0}})$ of $\Gamma$. We remark that $\gamma_{x_{0}}(0)=0$ and that, up to replace $\delta>0$ with a smaller one, we can assume that $\gamma_{x_{o}}([-\delta,\delta])\subseteq U_{0}$. 4. Step 4:

We consider the function $\psi_{x_{0}}:U_{0}\rightarrow H$, defined as $\psi_{x_{0}}(\overline{h})=\overline{h}+g_{x_{0}}(\overline{h})h_{1}$. We now want to evaluate the function $\sigma_{x_{0}}:(-\delta,\delta)\rightarrow H$ defined as

[TABLE]

and its derivative at the origin. Observe that

[TABLE]

so $\sigma_{x_{0}}(0)=0$. Furthermore

[TABLE]

by (4.10) and (4.14) we get

[TABLE]

We finally claim that for every $t\in(-\delta,\delta)$ we have $G(x_{0}+\sigma_{x_{0}}(t))=0$. Indeed, recalling that $\Gamma(\gamma_{x_{0}})(t)\in U_{0}$ and (4.13), we get

[TABLE] 5. Step 5:

Now We are able to prove (4.6). Indeed, from (1.2), (4.1), $\sigma_{x_{0}}(0)=0$ and $\sigma_{x_{0}}^{\prime}(0)=\Phi(x_{0})$ we deduce that

[TABLE]

Then, we have

[TABLE]

∎

In the next theorem we prove that the space $Z^{1,2}_{U}(\Omega,\nu;H)$ is contained in the domain of the divergence, where $Z^{1,2}_{U}(\Omega,\nu;H)$ is the completion of the space $\mathcal{Z}(\Omega,H)$ with respect to the norm defined in (1.10).

Theorem 4.4**.**

Assume that either Hypotheses 1.1 and 1.2 hold or Hypothesis 1.2 holds and $\Omega$ is the whole space. Every vector field $\Phi\in Z^{1,2}_{U}(\Omega,\nu;H)$ has a divergence $\operatorname{\operatorname{div}}_{\nu,\Omega}\Phi\in L^{2}(\Omega,\nu)$ and for every $f\in W^{1,2}(\Omega,\nu)$, the following equality holds:

[TABLE]

Furthermore, if $\varphi_{n}={\left\langle\Phi,h_{n}\right\rangle}_{H}$ for every $n\in\mathbb{N}$ where $(h_{n})_{n\in\mathbb{N}}$ is an orthonormal basis of $H$, then

[TABLE]

where the series converges in $\operatorname{L}^{2}(\Omega,\nu)$. In addition ${\left\|\operatorname{\operatorname{div}}_{\nu}\Phi\right\|}_{\operatorname{L}^{2}(\Omega,\nu)}\leq{\left\|\Phi\right\|}_{Z_{U}^{1,2}(\Omega,\nu;H)}$.

Proof.

We prove the theorem assuming Hypotheses 1.1 and 1.2 hold, since the case when Hypothesis 1.2 holds and $\Omega$ is the whole space can be proved in a similar way. We start with a preliminary computation. Let $\Phi\in\mathcal{Z}(\Omega,H)$, so there exists an orthonormal basis ${\left\{h_{i}\right\}}_{i\in\mathbb{N}}$ of $H$ such that $\Phi=\sum_{i=1}^{n}\varphi_{i}h_{i}$ for some $n\in\mathbb{N}$ and $\varphi_{i}\in\operatorname{\mathscr{FC}}^{2}_{b}(\Omega)$ for every $i=1,\ldots,n$. In addition ${\left\langle\Phi(x),\nabla_{H}G(x)\right\rangle}_{H}=0$ for $\rho$-a.e $x\in G^{-1}(0)$. By the integration by parts formula if $f\in\operatorname{\mathscr{FC}}^{\infty}_{b}(\Omega)$ we have

[TABLE]

So we have

[TABLE]

We recall the definition of the trace operator for nuclear operators $A$. Let $x\in\Omega$ and let $\{h_{n}\}_{n\in\mathbb{N}}$ be an orthonormal basis of $H$; we say that $A$ is a trace class operator if $\sum_{n=1}^{\infty}\langle(A^{*}A)^{1/2}h_{n},h_{n}\rangle_{H}$ is finite, and we set $\text{trace}_{H}(A):=\sum_{n=1}^{\infty}\langle(A^{*}A)^{1/2}h_{n},h_{n}\rangle_{H}$. In particular, $(\nabla_{H}\Phi)^{2}$ is a trace class operator and $\text{trace}_{H}(\nabla_{H}\Phi(x)^{2})\leq{\left\|\nabla_{H}\Phi(x)\right\|}^{2}_{\mathcal{H}_{2}}$ (see [8, Appendix A.2]). By Lemmata 4.2 and 4.3

[TABLE]

Let $(\Phi^{n})_{n\in\mathbb{N}}\subseteq\mathcal{Z}(\Omega,H)$ be a sequence of vector fields which converges to $\Phi$ in $Z_{U}^{1,2}(\Omega,\nu;H)$. By (4.22), $(\operatorname{\operatorname{div}}_{\nu,\Omega}\Phi^{n})$ is a Cauchy sequence in $\operatorname{L}^{2}(\Omega,\nu)$ and therefore it converges to an element of $\operatorname{L}^{2}(\Omega,\nu)$ which we denote by $\operatorname{\operatorname{div}}_{\nu,\Omega}\Phi$. By formula (4.21), it is easily seen that $\operatorname{\operatorname{div}}_{\nu,\Omega}\Phi$ satisfies (4.3). Finally, by a standard approximation argument we can conclude that $\operatorname{\operatorname{div}}_{\nu,\Omega}\Phi$ fulfills (4.3) also for every $f\in W^{1,2}(\Omega,\nu)$. ∎

We say that a subspace $S$ of $W^{1,2}(\Omega,\nu;H)$, endowed with a Banach norm ${\left\|\cdot\right\|}_{S}$, is a Neumann extension subspace if any $\Phi\in S$ satisfies $\langle\Phi,\nabla_{H}G\rangle_{H}=0$ $\rho$-a.e. on $G^{-1}(0)$ and it admits a continuous linear extension operator, i,e., if there exists a linear operator $E_{S}:S\rightarrow Z_{U}^{1,2}(X,\nu,H)$ such that for every $\Phi\in S$

1. (1)

$E_{S}\Phi(x)=\Phi(x)$ and $\nabla_{H}E_{S}\Phi(x)=\nabla_{H}\Phi(x)$ for $\mu$-a.e $x\in\Omega$; 2. (2)

there is $K_{S}>0$, independent of $\Phi$, such that ${\left\|E_{S}\Phi\right\|}_{Z_{U}^{1,2}(X,\mu;H)}\leq K_{S}{\left\|\Phi\right\|}_{S}$.

As a corollary of Theorem 4.4 we get the following.

Corollary 4.5**.**

Assume that Hypotheses 1.1 and 1.2 hold and let $S$ be a Neumann extension subspace with norm ${\left\|\cdot\right\|}_{S}$. Every field $\Phi\in S$ has a divergence $\operatorname{\operatorname{div}}_{\nu,\Omega}\Phi\in\operatorname{L}^{2}(\Omega,\nu)$ and for every $f\in W^{1,2}(\Omega,\nu)$, the following equality holds:

[TABLE]

Furthermore, if $\varphi_{n}={\left\langle\Phi,h_{n}\right\rangle}_{H}$ for every $n\in\mathbb{N}$, where $(h_{n})_{n\in\mathbb{N}}$ is an orthonormal basis of $H$, then

[TABLE]

where the series converges in $\operatorname{L}^{2}(\Omega,\nu)$. In addition, ${\left\|\operatorname{\operatorname{div}}_{\nu}\Phi\right\|}_{\operatorname{L}^{2}(\Omega,\nu)}\leq K_{S}{\left\|\Phi\right\|}_{S}$.

Proof.

Let us consider the divergence $\operatorname{\operatorname{div}}_{\nu,X}E_{S}\Phi$ (Theorem 4.4). For $\nu$-a.e. every $x\in\Omega$ let

[TABLE]

We have that

[TABLE]

where $E_{S}\varphi_{n}:=\langle E_{S}\Phi,h_{n}\rangle_{H}$. Since the right hand side of (4.23) converges to zero (the series converges to $\operatorname{\operatorname{div}}_{\nu,X}E_{S}\Phi$) we get that $(D_{k})_{k\in\mathbb{N}}$ is a Cauchy sequence in $\operatorname{L}^{2}(\Omega,\nu)$. We denote by $D_{\infty}\Phi$ the limit of $D_{n}$ in $L^{2}(\Omega,\nu)$ and we observe that for every $f\in W^{1,2}(\Omega,\nu)$

[TABLE]

We remark that $\rho$-a.e we have

[TABLE]

and

[TABLE]

Therefore, by the Lebesgue’s dominated convergence theorem and the continuity of the trace operator (Proposition 2.1) we get $\int_{\Omega}{\left\langle\nabla_{H}f,\Phi\right\rangle}_{H}d\nu=-\int_{\Omega}fD_{\infty}\Phi d\nu$ for any $f\in W^{1,2}(\Omega,\nu)$. This means that $\operatorname{\operatorname{div}}_{\nu,\Omega}\Phi$ exists and $\operatorname{\operatorname{div}}_{\nu,\Omega}\Phi=D_{\infty}\Phi$. Moreover

[TABLE]

∎

Remark 4.6**.**

The subspace of the vector fields $\Phi\in Z_{U}^{1,2}(\Omega,\nu;H)$ such that the extension

[TABLE]

belongs to $Z_{U}^{1,2}(X,\nu;H)$ satisfies the hypotheses of Corollary 4.5.

5. Maximal Sobolev regularity

This Section is devoted to the the study of maximal Sobolev regularity for the equation

[TABLE]

where $\lambda>0$, and $f\in\operatorname{L}^{2}(\Omega,\nu)$, since a part of the proofs of Theorems 1.3, 1.4 and 1.6 relies on them. The results of this section are sharper than the results contained in [12] and [11].

Our main result is the following theorem.

Theorem 5.1**.**

Assume that Hypotheses 1.1 and 1.2 hold. For every $\lambda>0$ and $f\in\operatorname{L}^{2}(\Omega,\nu)$ problem (5.1) has a unique weak solution $u\in W^{2,2}_{U}(\Omega,\nu)$. In addition the following hold

[TABLE]

In particular $u\in W^{2,2}_{U,N}(\Omega,\nu)$.

We split the proof of Theorem 5.1 into two parts: in the Section 5.1 we study the case of $\Omega=X$ and $U$ with $H$-Lipschitz gradient, in Section 5.2 we use the results of Section 5.1 to prove Theorem 5.1.

5.1. $\Omega$ is the whole space

We start this subsection assuming the following hypothesis on the weight:

Hypothesis 5.2**.**

Let $U:X\rightarrow\mathbb{R}$ be a function satisfying Hypothesis 1.2. Assume that $U$ is differentiable along $H$ at every point $x\in X$, and $\nabla_{H}U$ is $H$-Lipschitz.

We remark that every convex function in $\operatorname{\mathscr{FC}}^{2}_{b}(X)$ and every continuous linear functional $x^{*}\in X^{*}$ satisfy Hypothesis 5.2.

We will recall some results about maximal Sobolev regularity contained in [12]. Let us consider the problem

[TABLE]

where $\lambda>0$, $f\in\operatorname{L}^{2}(X,\nu)$, and $L_{\nu}:=L_{\nu,X}$. A function $u\in D(L_{\nu})$ of problem (5.5) is said to be a strong solution if there exists a sequence $\{u_{n}\}_{n\in\mathbb{N}}\subseteq\operatorname{\mathscr{FC}}^{3}_{b}(X)$ such that $u_{n}$ converges to $u$ in $\operatorname{L}^{2}(X,\nu)$ and

[TABLE]

Moreover a sequence ${\left\{u_{n}\right\}}_{n\in\mathbb{N}}\subseteq\operatorname{\mathscr{FC}}^{3}_{b}(X)$ satisfying the above conditions is called a strong solution sequence for $u$. The following proposition is borrowed from [12, Proposition 5.8].

Theorem 5.3**.**

Assume that Hypothesis 5.2 holds. For every $\lambda>0$ and $f\in\operatorname{L}^{2}(X,\nu)$, there exists a unique strong solution of equation (5.5). Such strong solution is also a weak solution of problem (5.5). In addition, if ${\left\{u_{n}\right\}}_{n\in\mathbb{N}}\subseteq\operatorname{\mathscr{FC}}^{3}_{b}(X)$ is a strong solution sequence for $u$, then $(u_{n})$ converges to $u$ in $W^{2,2}(X,\nu)$.

When $U$ satisfies Hypothesis 5.2 we have the following regularity result.

Theorem 5.4**.**

Let $U$ be a function satisfying Hypothesis 5.2, let $\lambda>0$, $f\in\operatorname{L}^{2}(X,\nu)$, and let $u$ be the strong solution of equation (5.5). Then $u\in W^{2,2}_{U}(X,\nu)$ and

[TABLE]

The difference between Theorem 5.4 and the results of [12] is that estimate (5.7) is sharper, since it contains the integral $\int_{X}{\left\langle\nabla_{H}^{2}U\nabla_{H}u,\nabla_{H}u\right\rangle}_{H}d\nu$. We stress that, even if $\nabla_{H}U$ is $H$-Lipschitz, which means that $\nabla^{2}_{H}U$ is essentially bounded, we can not use the second inequality in (5.6) to estimate (5.7). Indeed, (5.7) is independent of $\lambda$, while (5.6) does not.

Proof.

The proof of (5.6) can be found in [12, Theorem 5.10]. By Proposition 5.3 there exists a sequence ${\left\{u_{n}\right\}}_{n\in\mathbb{N}}\subseteq\operatorname{\mathscr{FC}}^{3}_{b}(X)$ and a function $u\in W^{1,2}(X,\nu)$ such that $u_{n}$ converges to $u$ in $\operatorname{L}^{2}(X,\nu)$ and

[TABLE]

Let $f_{n}:=\lambda u_{n}-L_{\nu}u_{n}$. Using formula (2.4), we differentiate the equality $\lambda u_{n}-L_{\nu}u_{n}=f_{n}$ with respect to the $e_{j}$ direction, multiply the result by $\partial_{j}u$, sum over $j$ and finally integrate over $X$ with respect to $\nu$. Then we obtain

[TABLE]

By Fatou’s Lemma and recalling that $u_{n}$ and $f_{n}$ converge to $u$ and $f$ in $\operatorname{L}^{2}(X,\nu)$, respectively, we get

[TABLE]

Using inequalities (5.6) we get

[TABLE]

∎

We will not give the prove of the following theorem, since it can be easily deduced using the results of [12] and the arguments in the proof of Theorem 5.1.

Theorem 5.5**.**

Assume Hypothesis 1.2 holds. Let $\lambda>0$, $f\in\operatorname{L}^{2}(X,\nu)$, and let $u$ be the strong solution of equation (5.5). Then $u\in W^{2,2}_{U}(X,\nu)$ and

[TABLE]

5.2. The general case

Assume that Hypotheses 1.1 and 1.2 hold. Let $x\in X$ and let $\mathcal{C}\subseteq X$ be a Borel set. We define

[TABLE]

$d_{H}$ can be seen as a distance function from $\mathcal{C}$ along $H$. This function has been already considered in [27], [39], [8, Example 5.4.10], [26], and [11]. For $\alpha\in(0,1]$ let $U_{\alpha}$ be the Moreau–Yosida approximation along $H$ of the weight $U$ defined in Section 3.

We approach the problem in $\Omega$ by penalized problems in the whole space $X$, replacing $U$ by

[TABLE]

for $\alpha\in(0,1]$. Namely for $\alpha\in(0,1]$, we consider the problem

[TABLE]

where $\lambda>0$, $f\in\operatorname{L}^{2}(X,\nu_{\alpha})$, $\nu_{\alpha}=e^{-V_{\alpha}}\mu$ and $L_{\nu_{\alpha}}:=L_{\nu_{\alpha},X}$. The first result we need to recall is [11, Proposition 5.2].

Proposition 5.6**.**

Assume that Hypotheses 1.1 and 1.2 hold and let $\alpha\in(0,1]$. Then the following properties hold:

1. (1)

$V_{\alpha}$* is a convex and $H$-continuous function;* 2. (2)

$V_{\alpha}$* is differentiable along $H$ for $\mu$-a.e. $x\in X$, and $\nabla_{H}V_{\alpha}$ $H$-Lipschitz;* 3. (3)

$e^{-V_{\alpha}}\in W^{1,p}(X,\mu)$, for every $p\geq 1$; 4. (4)

$V_{\alpha}\in W^{2,t}(X,\mu)$, where $t$ is given by Hypothesis 1.2; 5. (5)

\lim_{\alpha\rightarrow 0^{+}}V_{\alpha}(x)={\left\{\begin{array}[]{ll}U(x)&x\in\Omega;\\ +\infty&x\notin\Omega.\end{array}\right.}**

By Proposition 5.6 we can apply Theorem 5.4 to problem (5.8) and get the following maximal Sobolev regularity result (see also [11, Theorem 5.3]).

Theorem 5.7**.**

Assume Hypotheses 1.1 and 1.2 hold and let $\alpha\in(0,1]$, $\lambda>0$ and $f\in\operatorname{L}^{2}(X,\nu_{\alpha})$. Equation (5.8) has a unique weak solution $u_{\alpha}$. Moreover $u_{\alpha}\in W^{2,2}_{V_{\alpha}}(X,\nu_{\alpha})$ and

[TABLE]

In addition, for every $\alpha\in(0,1]$, there exists a sequence $\{u_{\alpha}^{(n)}\}_{n\in\mathbb{N}}\subseteq\operatorname{\mathscr{FC}}^{3}_{b}(X)$ such that $u_{\alpha}^{(n)}$ converges to $u_{\alpha}$ in $W^{2,2}(X,\nu_{\alpha})$ and $\lambda u_{\alpha}^{(n)}-L_{\nu_{\alpha}}u_{\alpha}^{(n)}$ converges to $f$ in $\operatorname{L}^{2}(X,\nu_{\alpha})$.

We are now ready to prove Theorem 5.1.

Proof of Theorem 5.1.

The Neumann condition (5.2) and estimates (5.3) have been proved in [11, Theorems 1.3 and 1.4]. Hence, it remains to prove (5.4). Let $f\in\operatorname{\mathscr{FC}}^{\infty}_{b}(X)$. By Theorem 5.7, for every $\alpha\in(0,1]$ the equation (5.8) has a unique weak solution $u_{\alpha}\in W^{2,2}(X,\nu_{\alpha})$ such that inequalities (5.9) and (5.10) hold. Moreover, for every $\varphi\in\operatorname{\mathscr{FC}}^{\infty}_{b}(X)$ we have

[TABLE]

By Proposition 5.6 and Proposition 3.1(2) we have

[TABLE]

and so the inclusion $W^{2,2}(\Omega,\nu_{\alpha})\subseteq W^{2,2}(\Omega,\nu)$ follows, for every $\alpha\in(0,1]$.

Let ${\left\{\alpha_{n}\right\}}_{n\in\mathbb{N}}$ be a sequence converging to zero such that $0<\alpha_{n}\leq 1$ for every $n\in\mathbb{N}$. By inequalities (5.9) and (5.10) the sequence ${\left\{u_{\alpha_{n}}\,\middle|\,n\in\mathbb{N}\right\}}$ is bounded in $W^{2,2}(\Omega,\nu)$. By weak compactness there exists a subsequence, that we will still denote by ${\left\{\alpha_{n}\right\}}_{n\in\mathbb{N}}$, such that $u_{\alpha_{n}}$ weakly converges to an element $u\in W^{2,2}(\Omega,\nu)$. Without loss of generality we can assume that $u_{\alpha_{n}}$, $\nabla_{H}u_{\alpha_{n}}$ and $\nabla_{H}^{2}u_{\alpha_{n}}$ converge pointwise $\mu$-a.e. respectively to $u$, $\nabla_{H}u$ and $\nabla_{H}^{2}u$. By Fatou’s lemma and inequality (5.10) we get

[TABLE]

Finally, if $f\in\operatorname{L}^{2}(\Omega,\nu)$, a standard density argument gives us the assertions of our theorem. ∎

6. Proof of the main results and some corollaries

Theorems 1.3, 1.4 and 1.6 are consequence of the following result.

Theorem 6.1**.**

Assume that either Hypotheses 1.1 and 1.2 hold or Hypothesis 1.2 holds and $\Omega$ is the whole space. Then $Z_{U,N}^{2,2}(\Omega,\nu)\subseteq D(L_{\nu,\Omega})\subseteq W_{U,N}^{2,2}(\Omega,\nu)$. Furthermore if we denote with ${\left\|\cdot\right\|}_{D(L_{\nu,\Omega})}$ the graph norm in $D(L_{\nu,\Omega})$, i.e. for $u\in D(L_{\nu,\Omega})$

[TABLE]

then for $u\in Z_{U}^{2,2}(\Omega,\nu)$ and $v\in D(L_{\nu,\Omega})$ it holds that

[TABLE]

Proof.

We prove the theorem assuming Hypotheses 1.1 and 1.2 hold, since in the case when Hypothesis 1.2 holds and $\Omega$ is the whole space the proof can be obtained in a similar way using Theorem 5.5.

Let $u\in D(L_{\nu,\Omega})$. Hence, $\lambda u-L_{\nu}u\in\operatorname{L}^{2}(\Omega,\nu)$, for every $\lambda\in(0,1)$, and by Theorem 5.1 we get $u\in W^{2,2}_{U,N}(\Omega,\nu)$. Moreover

[TABLE]

Letting $\lambda\rightarrow 1^{-}$ in inequality (6.4) we get ${\left\|u\right\|}_{W^{2,2}_{U}(\Omega,\nu)}\leq 2\sqrt{2}{\left\|u\right\|}_{D(L_{\nu,\Omega})}$.

Assume that $u\in Z^{2,2}_{U,N}(\Omega,\nu)$. Proposition 4.4 implies that $\operatorname{\operatorname{div}}_{\nu,\Omega}\nabla_{H}u\in\operatorname{L}^{2}(\Omega,\nu)$ and

[TABLE]

for every $f\in\operatorname{\mathscr{FC}}^{\infty}_{b}(\Omega)$. Then we have $u\in D(L_{\nu,\Omega})$ and $L_{\nu,\Omega}u=\operatorname{\operatorname{div}}_{\nu,\Omega}\nabla_{H}u$. By Proposition 4.4 we have

[TABLE]

for every $u\in Z^{2,2}_{U,N}(\Omega,\nu)$. ∎

We can actually simplify the statement of Theorem 1.3 when $\nabla_{H}U$ is $H$-Lipschitz and $\Omega=X$. Indeed, let us observe that if $\nabla_{H}U$ is $H$-Lipschitz then the function $x\mapsto\|\nabla_{H}^{2}U(x)\|_{\mathcal{H}_{2}}$ is essentially bounded (see [8, Theorem 5.11.2(ii)]). So $W^{2,2}(X,\nu)$ is isomorphic to $W^{2,2}_{U}(X,\nu)$, with

[TABLE]

In particular if $\nabla_{H}U$ is $H$-Lipschitz, then $\operatorname{\mathscr{FC}}^{2}_{b}(X)$ is dense in $W^{2,2}_{U}(X,\nu)$.

Corollary 6.2**.**

Assume Hypothesis 1.2 holds and $\nabla_{H}U$ is $H$-Lipschitz. Then $D(L_{\nu,X})=W^{2,2}(X,\nu)$. Moreover, for every $u\in D(L_{\nu,X})$, it holds $L_{\nu,X}u=\operatorname{\operatorname{div}}_{\nu,X}\nabla_{H}u$ and

[TABLE]

The same holds true, with obvious modifications, when $\Omega$ is a Neumann extension domain.

This result has been already proved in [12, Theorem 6.2].

7. Examples

We conclude the paper by presenting some examples. In Subsection 7.1 we study in detail the case when $\Omega$ is the ball sphere of a Hilbet space and we show that, in this case, the spaces $\mathcal{Z}(\Omega,H)$ is non-trivial, namely it is infinite dimensional, but the space $\mathcal{Z}(\Omega)$ contains only the constant functions. In Subsection 7.2 we prove Theorem 1.7 giving a characterization of the domain of the Ornstein–Uhlenbeck operator on half-spaces.

7.1. The unit sphere of a Hilbert space

Let $X$ be a separable Hilbert space, with norm ${\left\|\cdot\right\|}_{X}$ and inner product $(\cdot,\cdot)_{X}$, and let $\mu$ be a centered non-degenerate Gaussian measure on $X$. Let $\{h_{n}\}_{n\in\mathbb{N}}$ be an orthonormal basis of $X$ which consists of eigenvector of the covariance operator $Q$, i.e. $Qh_{n}=\lambda_{i}h_{n}$, it is known that an orthonormal basis of the Cameron–Martin space $H$ is $\{\sqrt{\lambda_{n}}h_{n}\}_{n\in\mathbb{N}}$ (see [8]).

Consider $G(x)=(x,x)_{X}-1$, for any $x\in X$, then

[TABLE]

Clearly, $G(x)=0$ if and only if $x\in\operatorname{\mathcal{S}}_{X}$ the unit sphere of $X$. Moreover, easy computations show that $\partial_{h}G(x)=2(x,h)_{X}$ for any $x\in X$ and any $h\in H$. Hence, if , we have

[TABLE]

for any $x\in X$, and so $|\nabla_{H}G(x)|^{2}_{H}=4\|Q^{1/2}x\|_{X}^{2}=4\sum_{n=1}^{\infty}\lambda_{n}(x,h_{n})_{X}^{2}$. So $\nabla_{H}G(x)=0$ if, and only if, $x=0$. Finally $G$ satisfies Hypothesis 1.1(4)-(5) (see [11]) and $\partial_{n,m}G(x)=2\lambda_{n}\delta_{n,m}$.

As an admissible weight we can take $U(x):=\Phi(\|x\|_{X}^{2})$, where $\Phi:\mathbb{R}\longrightarrow\mathbb{R}$ is a $C^{2}$ convex function which satisfies

[TABLE]

for some positive integer $k$. It is easy to prove that $U$ is convex and satisfies the Hypothesis 1.2.

Observe that

[TABLE]

In particular all the vector fields

[TABLE]

belongs to $\mathcal{Z}(\operatorname{\mathscr{B}}_{X},H)$, so the space $Z^{1,2}_{U}(\operatorname{\mathscr{B}}_{X},\nu;H)$ is infinite dimensional and contained in the domain of the divergence operator (see Theorem 4.4).

The domain of the operator $L_{\nu,\operatorname{\mathscr{B}}_{X}}$ contains the space $Z^{2,2}_{U,N}(\operatorname{\mathscr{B}}_{X},\nu)$, i.e. the completion of the space

[TABLE]

with respect to the norm

[TABLE]

We want to show that in this case the space $Z^{2,2}_{U,N}(\Omega,\nu)$ only contains the constant functions. Indeed let $u\in\mathcal{Z}(\operatorname{\mathscr{B}}_{X})$, without loss of generality assume that $u(x)=\varphi((x,h_{1})_{X},(x,h_{2})_{X})$ with $\varphi\in\operatorname{\mathscr{C}}^{2}_{b}(\mathbb{R}^{2})$. The Neumann boundary condition

[TABLE]

implies

[TABLE]

for $\rho$-a.e $x\in\operatorname{\mathcal{S}}_{X}$. So the function $\varphi$ satisfies the differential equation

[TABLE]

We want to remak that the condition $\xi\in\operatorname{\mathscr{B}}_{\mathbb{R}^{2}}$ is a consequence of the fact that, if $x\in\operatorname{\mathcal{S}}_{X}$, then the vector $(\xi_{1},\xi_{2})=((x,h_{1}),(x,h_{2}))$ belongs to the unit ball of $\mathbb{R}^{2}$. All the solutions of (7.1) are functions of the form

[TABLE]

where $g$ is a sufficiently regular function in $\mathbb{R}$. It is easy to see that if $\varphi$ is non-constant, then $\varphi$ cannot be continuous at the origin.

So Theorem 6.1 only gives us

[TABLE]

We want to remark that a positive answer to the question “Is $\operatorname{\mathscr{B}}_{X}$ a Neumann extension domain?” would allow us to apply Theorem 1.6 and get a characterization of the domain of $L_{\operatorname{\mathscr{B}}_{X},\nu}$.

7.2. The Ornstein–Uhlenbeck operator on half-spaces

In this section we give a characterization of the domain of the operator $L_{\mu,\Omega}$, where $\Omega$ is a half-space and $\mu$ is a centered non-degenerate Gaussian measure on a separable Banach space $X$. To do so we need some preliminary results, in particular a lemma about extensions of Sobolev functions and a proposition about finite dimensional approximations. We recall that $Z_{0}^{2,2}(X,\mu)=W^{2,2}(X,\mu)$ (see [8]).

Let $x^{*}\in X^{*}\smallsetminus{\left\{0\right\}}$ and $r\in\mathbb{R}$, throughout this section we set $G(x):=x^{*}(x)-r$ and $\Omega:=G^{-1}(-\infty,0]$. We recall that $x^{*}$ is a linear and continuous functional on $H$, so there exists $h_{x^{*}}\in H$ such that for every $h\in H$

[TABLE]

Finally we remind the reader that

[TABLE]

Lemma 7.1**.**

There exists a Neumann extension operator from $W_{0,N}^{2,2}(\Omega,\mu)$ to $W^{2,2}(X,\mu)$.

Proof.

We use a generalization of the reflection method, adapted to our Gaussian measure. Let $f\in\operatorname{\mathscr{FC}}_{b}^{2}(\Omega)$ and put

[TABLE]

where for every $j=1,\ldots,7$,

[TABLE]

and

[TABLE]

We start by proving that $Ef$ is well defined. Indeed for $j=1,\ldots,7$ and $x\in X$ such that $G(x)>0$ we have

[TABLE]

We point out that (7.6) are the classical conditions to prove the continuity of $Ef$ and its derivatives. (7.5) and (7.7) arise from the exponential term in (7.4), which is used to prove the continuity estimate for the extension operator.

The fact that $Ef$ belongs to $\operatorname{\mathscr{FC}}_{b}^{0}(X)$ is obvious. Fix an orthonormal basis $\{h_{i}\}_{i\in\mathbb{N}}$ of $H$ obtained by completing the set $\{h_{x^{*}}/{\left|h_{x^{*}}\right|}_{H}\}$, without loss of generality we let $h_{1}=h_{x^{*}}/{\left|h_{x^{*}}\right|}$. Let $x_{0}\in X$ such that $G(x_{0})=0$, then $G(x_{0}+th_{i})=t\delta_{1i}{\left|h_{x^{*}}\right|}_{H}$. We have for $i\neq 1$

[TABLE]

while

[TABLE]

Thus, letting $T_{j}(x):=x-(j+1)G(x)\frac{h_{x^{*}}}{{\left|h_{x^{*}}\right|}_{H}^{2}}$ and $A_{j}(x):=\exp{\left(-\frac{c_{j}G(x)+b_{j}G^{2}(x)}{2{\left|h_{x^{*}}\right|}_{H}}\right)}$

[TABLE]

In the same way it holds

[TABLE]

where

[TABLE]

So $Ef$ belongs to $\operatorname{\mathscr{FC}}_{b}^{2}(X)$ and $Ef(x)=f(x)$, $\nabla_{H}Ef(x)=\nabla_{H}f(x)$, $\nabla_{H}^{2}Ef(x)=\nabla_{H}^{2}f(x)$ for every $x\in\Omega$. Without loss of generality we can assume that there exists $n\in\mathbb{N}$ and $\psi\in\operatorname{\mathscr{C}}^{2}_{b}(\mathbb{R}^{n})$ such that for every $x\in X$

[TABLE]

We remark that

[TABLE]

So we have

[TABLE]

We remark that $d\mu_{n}(\xi)=\exp(-{\left|\xi\right|}^{2}/2{\left|h_{x^{*}}\right|}_{H})dx$. For every $j=1,\ldots,7$, consider the change of variable:

[TABLE]

We use (7.11) in the second integral of (7.8), and we get

[TABLE]

Using the definition of $a_{j},b_{j}$ and $c_{j}$ we get

[TABLE]

for some constant $C>0$. So

[TABLE]

where the constant $K>0$ depend only on $r$ and $a_{i}$ for $i=1,\ldots,7$. Using similar arguments on $\nabla_{H}Ef$ and $\nabla_{H}^{2}Ef$ we get for every $f\in\operatorname{\mathscr{FC}}^{2}_{b}(\Omega)$

[TABLE]

where $\widetilde{K}>0$ is an adequate constant independent of $f$. A standard denstity argument gives the thesis of our lemma. ∎

Using Lemma 7.1 and Theorem 1.6 we get a characterization of the domain of $L_{\mu,\Omega}$. In order to get Theorem 1.7 we need a further approximation argument.

Proposition 7.2**.**

Let $u\in W^{2,2}(\Omega,\mu)$ be such that ${\left\langle\nabla_{H}u(x),h_{x^{*}}\right\rangle}_{H}=0$ for $\rho$-a.e. $x\in G^{-1}(0)$. There exists a sequence $(u_{n})_{n\in\mathbb{N}}$ belonging to $\operatorname{\mathscr{FC}}^{2}_{b}(\Omega)$ such that

1. (1)

${\left\langle\nabla_{H}u_{n}(x),h_{x^{*}}\right\rangle}_{H}=0$* for every $n\in\mathbb{N}$ and $\rho$-a.e. $x\in G^{-1}(0)$;* 2. (2)

$(u_{n})_{n\in\mathbb{N}}$* converges to $u$ in $W^{2,2}(\Omega,\mu)$.*

Proof.

Fix an orthonormal basis $\{h_{i}\}_{i\in\mathbb{N}}$ of $H$ obtained by completing the set $\{h_{x^{*}}/{\left|h_{x^{*}}\right|}_{H}\}$, without loss of generality we let $h_{1}=h_{x^{*}}/{\left|h_{x^{*}}\right|}_{H}$. Let $u\in W^{2,2}(\Omega,\mu)$ be such that

[TABLE]

Let $Eu$ be the extension defined in Lemma 7.1. We denote with $P_{n},S_{n}:X\rightarrow H$ the functions defined as

[TABLE]

for every $x,y\in X$. We recall that $P_{n}x$ converges pointwise $\mu$-a.e. $x\in X$ to $x$ (see [8, Theorem 3.5.1]). Let

[TABLE]

by [8, Corollary 3.5.2 and Proposition 5.4.5] $u_{n}$ converges to $Eu$ in $W^{2,2}(X,\mu)$ as $n$ goes to infinity and for every $i,n\in\mathbb{N}$

[TABLE]

Observe that if $x\in G^{-1}(0)$, then for every $y\in X$ and $n\in\mathbb{N}$

[TABLE]

By (7.12) we get

[TABLE]

for $\rho$-a.e. $x\in G^{-1}(0)$.

We are almost done, but we need smoother function satisfying Proposition 7.2(1)-(2). Let $\psi_{n}(\xi):=v_{n}(\sum_{i=1}^{n}\xi_{i}h_{i})$. We remind the reader that $\psi_{n}$ belongs to $W^{2,2}(\mathbb{R}^{n},\mu\circ P_{n}^{-1})$ and

[TABLE]

Let $\mathcal{L}_{0}^{n}$ be the generator of the $m$-dimensional Ornstein–Uhlenbeck operator with homogeneous Neumann condition in $\operatorname{L}^{2}(\mathcal{O}_{n},\mu\circ P_{n}^{-1})$, where $\mathcal{O}_{n}={\left\{\xi\in\mathbb{R}^{n}\,\middle|\,\xi_{1}\leq r\right\}}$. By [30, Theorem 12.4.9] we know that the domain of $\mathcal{L}^{n}_{0}$ in $\operatorname{L}^{2}(\mathcal{O}_{n},\mu\circ P_{n}^{-1})$ is

[TABLE]

and

[TABLE]

where $R(\cdot,\mathcal{L}_{0}^{n})$ is the resolvent operator associate to $\mathcal{L}_{0}^{n}$ and $k=0,1,2$. Let $f_{n}:=\psi_{n}-\mathcal{L}_{0}^{n}\psi_{n}$, where the equality is meant in $\operatorname{L}^{2}(\mathcal{O}_{n},\mu\circ P_{n}^{-1})$. Let $(f_{n,k})_{k\in\mathbb{N}}$ be a sequence of bounded smooth function such that $f_{n,k}$ converges in $\operatorname{L}^{2}(\mathcal{O}_{n},\mu\circ P_{n}^{-1})$ to $f_{n}$ as $k$ goes to infinity. We let

[TABLE]

We recall that $\psi_{n,k}$ belongs to $D(\mathcal{L}_{n}^{0})$ and to $\operatorname{\mathscr{C}}^{2}_{b}(\mathbb{R}^{n})$ (see [30, Section 12]). Let

[TABLE]

We get that $F_{n,k}u$ belongs to $\operatorname{\mathscr{FC}}_{b}^{2}(\Omega)$ and satisfy the Neumann condition at the boundary. Let $\varepsilon>0$ and consider $n_{\varepsilon},k_{\varepsilon}\in\mathbb{N}$ such that

[TABLE]

So

[TABLE]

Thus the sequence $u_{m}:=F_{n_{m^{-1}},k_{m^{-1}}}u$ for $m\in\mathbb{N}$ is the sequence we were lookong for. ∎

As a consequence of Corollary 1.4 and Proposition 7.2, we get Theorem 1.7.

Acknowledgements**.**

The authors would like to thank Prof. Alessandra Lunardi, Prof. Diego Pallara and Prof. Leonardo Biliotti for many useful discussions and comments. The authors are members of GNAMPA of the Italian Istituto Nazionale di Alta Matematica (INdAM).

This research was partially supported by the PRIN 2015 grant: “Deterministic and stochastic evolution equations” and the GNAMPA 2017 project: “Equazioni e sistemi di equazioni di Kolmogorov in dimensione finita e non”.