Sobolev Spaces $W^{1,p}(\mathbb {R}^{n},\gamma )$ Weighted by the Gaussian Normal Distribution $\gamma (x):=\frac {1}{\sqrt {\pi }^{n}}\exp (-|x|^{2})$ and the Spectral Theory

Abstract

In the spectral theory it does make a difference, whether we consider differential operators on bounded or unbounded domains. In order to treat eigenvalue problems on the whole Euclidean space, we construct Sobolev spaces over $\mathbb {R}^{n}$, which are weighted by the Gaussian normal distribution. By the methods presented in Chapters 2, 8, and 10 of the treatise F. Sauvigny: Partial Differential Equations 1 and 2, Springer Universitext (2012), we can prove an analogue of the Sobolev embedding theorem and a Rellich selection theorem for the Sobolev spaces $W_{0}^{1,p}(\mathbb {R}^{n},\gamma )$ weighted by γ- with vanishing values towards infinity. We achieve these specific results for our entire Sobolev spaces $W^{1,p}(\mathbb {R}^{n},\gamma )$, since we concentrate on the Gaussian normal distribution γ as our weight function. Even our notion of the weighted partial derivative depends on this weight function. Within the so-called Gauß–Rellich space $W_{0}^{1,2}(\mathbb {R}^{n},\gamma )$ we shall investigate the discrete spectrum of weighted elliptic operators over $\mathbb {R}^{n}$ by spectral methods. There we rely on the treatise F. Sauvigny: Spektraltheorie selbstadjungierter Operatoren im Hilbertraum und elliptischer Differentialoperatoren, Springer Spektrum (2019). By reflection methods, we solve eigenvalue problems for elliptic differential operators on the sectorial domain $\mathbb {R}_{+}^{n}$ under vanishing and mixed boundary conditions.

1 Introduction

Usually one considers Lebesgue and Sobolev spaces over bounded domains in $\mathbb {R}^{n}$ and derives well-known embedding theorems there. Here we refer to the chapter Sobolev spaces in the classical book [2] by D. Gilbarg and N. Trudinger. Furthermore, we recommend the Springer–Lehrbuch [4] by J. Jost in this context. Sobolev spaces over bounded domains in $\mathbb {R}^{n}$ endowed with a weight function have originally been considered in the monograph [5] by A. Kufner. Weighted Sobolev spaces over unbounded domains as well are treated in the book by H. Triebel [11]. The Gaussian normal distribution does neither appear explicitly as weight function there nor in the Springer–Grundlehren [6] on Sobolev spaces by V. Maz’ya. Our subsequent methods of proof are tailored for the density function γ, and the results later rely decisively on this density.

In order to define Sobolev spaces over the whole $\mathbb {R}^{n}$, we equip the Euclidean space with the Gaussian normal distribution

$$ \begin{array}{@{}rcl@{}} \gamma(x)&:=&\frac{1}{\sqrt{\pi}^{n}}\exp(-|x|^{2})=\frac{1}{\sqrt{\pi}}\exp(-{x_{1}^{2}})\cdot\ldots\cdot\frac{1}{\sqrt{\pi}}\exp(-{x_{n}^{2}})=\gamma_{1}(x_{1})\cdot\ldots\cdot \gamma_{n}(x_{n}),\\ &&x=(x_{1},\ldots,x_{n})\in\mathbb{R}^{n}\quad\text{with the contributions}\\ \gamma_{i}(x_{i})&:=&\frac{1}{\sqrt{\pi}}\exp(-{x_{i}^{2}}),\quad x_{i}\in\mathbb{R}\quad\text{ for } i=1,\ldots,n. \end{array} $$

Thus we receive weighted Lebesgue spaces $L^{p}(\mathbb {R}^{n},\gamma )$ in Section 2, which share the usual properties with the Lebesgue spaces over bounded domains. We consider the pull-back of the Sobolev space W^1,p(Ω) over the cube

$$ {\Omega}:=\left\{y=(y_{1},\ldots,y_{n})\in\mathbb{R}^{n}\colon 0<y_{i}<1\quad\text{for } i=1,\ldots,n\right\} $$

under the Gaussian diffeomorphism

$$ y={\Gamma}(x)=({\Gamma}_{1}(x_{1}),\ldots,{\Gamma}_{n}(x_{n}))\colon\mathbb{R}^{n}\to {\Omega}. $$

Here the function Γ contains the Gaussian error function in their components

$$ {\Gamma}_{i}(x_{i}):= \frac{1}{\sqrt{\pi}}{\int}_{-\infty}^{x_{i}}\exp(-{t_{i}^{2}})dt_{i} ={\int}_{-\infty}^{x_{i}}\gamma_{i}(t_{i})dt_{i},\quad x_{i}\in\mathbb{R}\quad\text{for }~i=1,\ldots,n $$

with ${\Gamma }(0,\ldots ,0)=(\frac 12,\ldots ,\frac 12)=:y_{0}$ and possesses the Jacobian J_Γ(x) = γ(x), $x\in \mathbb {R}^{n}$. We denote the inverse mapping by $x={\Theta }(y)=(\theta _{1}(y_{1}),\ldots ,\theta _{n}(y_{n}))\colon {\Omega }\to \mathbb {R}^{n}$, which we address as inverse Gaussian diffeomorphism.

For an arbitrary test function $\varphi \in C_{0}^{\infty }(\mathbb {R}^{n})$ we define the weighted partial derivative

$$ \delta_{i}\varphi(x):= \frac{d}{dy_{i}}\left( \varphi\circ{{\Theta}}(y)\right)\|_{y={\Gamma}(x)}=\frac{d}{dy_{i}}\hat\varphi(y)\|_{y={\Gamma}(x)},\quad x\in \mathbb{R}^{n} \text{with} i=1,\ldots,n, $$

(1.1)

where we introduce the lifted test function $\hat \varphi (y):=\varphi \circ {\Theta }(y)$, y ∈Ω of the class $C_{0}^{\infty }({\Omega })$. The weighted weak derivative $\delta ^{e_{i}}f(x)$ in direction of $e_{i}=(\delta _{1i},\ldots ,\delta _{ni})\in \mathbb {R}^{n}$ is represented by the bounded linear functional over $L^{\frac {p}{p-1}}(\mathbb {R}^{n},\gamma )$ for $1<p<+\infty$ on the right-hand side

$$ {\int}_{\mathbb{R}^{n}}[\delta^{e_{i}}f(x)\cdot \varphi(x)]\gamma(x)dx = -{\int}_{\mathbb{R}^{n}}[f(x)\cdot \delta_{i} \varphi(x)] \gamma(x)dx,\quad \forall \varphi \in C_{0}^{\infty}(\mathbb{R}^{n}). $$

In Section 3 we obtain with

$$ W^{1,p}(\mathbb{R}^{n},\gamma):=\left\{f\in L^{p}(\mathbb{R}^{n},\gamma)~|~\delta^{e_{i}}f(x)\in L^{p}(\mathbb{R}^{n},\gamma),~ i=1,\ldots,n\right\} $$

the weighted Sobolev space normed by

$$ \|f\|_{1,p,\gamma}:=\|f\|_{L^{p}(\mathbb{R}^{n},\gamma)}+{\sum}_{i=1}^{n} \|\delta^{e_{i}}f\|_{L^{p}(\mathbb{R}^{n},\gamma)}. $$

When we lift an arbitrary function $f\in L^{p}(\mathbb {R}^{n},\gamma )$ to the function

$$ \hat f:=f\circ {\Theta}(y),\quad y\in{\Omega}~\text{ of the class }~L^{p}({\Omega}), $$

we define the difference quotients of the lifted function $\hat f$ in direction of e_i by

$$ \widehat{\Delta}_{i,\varepsilon}\hat f(y):=\frac{\hat f[y+\varepsilon e_{i}]-\hat f[y]}{\varepsilon}\quad \text{ for }~y \in{\Omega},~0<y_{i}+\varepsilon<1,~i=1,\ldots,n. $$

(1.2)

For a function $f\in L^{p}(\mathbb {R}^{n},\gamma )$ being given, we define the weighted difference quotients in direction of e_i as follows:

$$ \begin{array}{@{}rcl@{}} {\Delta}_{i,\varepsilon} f(x)&:=&\frac{f\circ{\Theta}[{\Gamma}(x)+\varepsilon e_{i}]-f(x)}{\varepsilon}=\frac{\hat f[{\Gamma}(x)+\varepsilon e_{i}]-\hat f[{\Gamma}(x)]}{\varepsilon}\\ &=&\left( \widehat{\Delta}_{i,\varepsilon}\hat f(y)\right)\|_{y={\Gamma}(x)}\quad\text{ \!\!\!\!\!for\! }~x\!\in \mathbb{\!R}^{n},\!~0<{\Gamma}_{i}(x)+\varepsilon<1,~i=1,\ldots,n. \end{array} $$

(1.3)

When these weighted difference quotients (1.3) are bounded in the $L^{p}(\mathbb {R}^{n},\gamma )$-norm, we shall find a null-sequence {ε_k}_{k= 1,2,3,…} such that the functions ${\Delta }_{i,\varepsilon _{k}}f$ weakly converge in $L^{p}(\mathbb {R}^{n},\gamma )$ to the weighted weak derivative $\delta ^{e_{i}}f(x)$ for $k\to \infty$ with i = 1,…,n.

Closing the set of test functions $C_{0}^{\infty }(\mathbb {R}^{n})$ with respect to the ∥⋅∥_1,p,γ-norm, we obtain the Sobolev space $W_{0}^{1,p}(\mathbb {R}^{n},\gamma )\subset W^{1,p}(\mathbb {R}^{n},\gamma )$. Within this Sobolev space we establish an analogue of the Sobolev embedding theorem in Section 4. We address $W_{0}^{1,2}(\mathbb {R}^{n},\gamma )$ as the Gauß–Rellich space and derive the Rellich selection theorem there.

In Section 5 we construct a complete orthonormal system, with respect to the inner product in the Hilbert space $L^{2}(\mathbb {R}^{n},\gamma )$, of eigenfunctions $f_{i}\in W_{0}^{1,2}(\mathbb {R}^{n},\gamma )~(i\in \mathbb N)$ for n ≥ 3. Here we introduce a symmetric, uniformly elliptic coefficient matrix generated by $a_{ij}(x)\in L^{\infty }(\mathbb {R}^{n})$ for i,j = 1,…,n, and we consider their associate bilinear form

$$ \mathcal E (f,g):={\int}_{\mathbb{R}^{n}}\left\{\sum\limits_{i,j=1}^{n} a_{ij}(x)\delta^{e_{i}}f(x)\delta^{e_{j}} g(x)\right\}\gamma(x) dx,\quad f,g\in W^{1,2}_{0}(\mathbb{R}^{n},\gamma). $$

Then these functions f_i satisfy the weighted weak eigenvalue equations

$$ \mathcal E(f_{i},\varphi)=\lambda_{i}(f_{i},\varphi)_{L^{2}(\mathbb{R}^{n},\gamma)}\quad\text{ for all }~\varphi\in C^{\infty}_{0}(\mathbb{R}^{n})~\text{ and all }~i\in\mathbb{N} $$

with their associate eigenvalues

$$ 0<\lambda_{1}\le \lambda_{2}\le\cdots\le\lambda_{j}~\to~+\infty\quad (j\to\infty). $$

In Section 6 and Section 7 we give spectral-theoretic applications for elliptic differential operators on the sectorial domain $\mathbb {R}^{n}_{+}$ under mixed and vanishing boundary conditions by reflection methods.

2 Weighted Lebesgue Spaces $L^{p}(\mathbb {R}^{n},\gamma )$ with the Exponents $1\le p \le +\infty$

The Lebesgue integral and Lebesgue spaces are introduced within Chapters IV and V of the treatise [3] by J. Jost: Postmodern Analysis in a concise and elegant way. In our considerations here, we are starting with the following linear space of integrable functions, which are absolutely integrable with respect to the improper Riemannian integral on the entire $\mathbb {R}^{n}$ being weighted by γ:

$$ M(\mathbb{R}^{n},\gamma):=\left\{f=f(x)\colon\mathbb{R}^{n}\to\mathbb{R}\in C^{0}(\mathbb{R}^{n})~\left|~{\int}_{\mathbb{R}^{n}}|f(x)| \gamma(x)dx<+\infty\right.\right\}. $$

On $M(\mathbb {R}^{n},\gamma )$ we define the basic Daniell integral in the sense of [7, Kap. VIII, Section 1]:

$$ I_{\gamma}(f):= {\int}_{\mathbb{R}^{n}}f(x)\gamma(x)dx,\quad f\in M(\mathbb{R}^{n},\gamma). $$

By the general procedure from [7, Kap. VIII, Section 2], we continue this Daniell integral to the associate Lebesgue integral. At first, we define the class of monotonically increasing approximative functions

$$ V^{+}(\mathbb{R}^{n},\gamma):=\left\{f\colon\mathbb{R}^{n}\to\mathbb{R}\cup\{+\infty\}~|~M(\mathbb{R}^{n},\gamma)\ni f_{k}\le f_{k+1},~f_{k}\to f (k\to\infty)\right\} $$

(2.1)

with the increasing integral

$$ (-\infty,+\infty] \ni I^{+}_{\gamma}(f):=\lim_{k\to\infty}I_{\gamma}(f_{k}),\quad f\equiv\{f_{k}\}_{k\in\mathbb{N}}\in V^{+}(\mathbb{R}^{n},\gamma). $$

(2.2)

Secondly, we define the class of monotonically decreasing approximative functions

$$ V^{-}(\mathbb{R}^{n},\gamma):=\left\{f\colon\mathbb{R}^{n}\to\mathbb{R}\cup\{-\infty\}~|~M(\mathbb{R}^{n},\gamma)\ni f_{k}\ge f_{k+1},~f_{k}\to f (k\to\infty)\right\} $$

(2.3)

with the decreasing integral

$$ [-\infty,+\infty) \ni I^{-}_{\gamma}(f):=\lim_{k\to\infty}I_{\gamma}(f_{k}),\quad f\equiv\{f_{k}\}_{k\in\mathbb N}\in V^{-}(\mathbb{R}^{n},\gamma). $$

(2.4)

For an arbitrary function

$$ f=f(x)\colon \mathbb{R}^{n}\to \overline{\mathbb{R}}:=\mathbb{R}\cup\{\pm\infty\} $$

(2.5)

we define the upper Daniell integral by

$$ I^{+}(f,\gamma):=\inf\left\{I^{+}_{\gamma}(h)~|~h\in V^{+}(\mathbb{R}^{n},\gamma)~\text{ with }~h\ge f \right\} $$

(2.6)

and the lower Daniell integral by

$$ I^{-}(f,\gamma):=\sup\left\{I^{-}_{\gamma}(g)~|~g\in V^{-}(\mathbb{R}^{n},\gamma)~\text{ with }~g\le f \right\}, $$

satisfying I⁻(f,γ) ≤ I⁺(f,γ) for all functions (2.5). A function f from (2.5) belongs to the class $L(\mathbb {R}^{n},\gamma )$ of weighted Lebesgue integrable functions, if the condition

$$ I^{-}(f,\gamma)= I^{+}(f,\gamma) $$

holds true. In this case, we define by

$$ I(f,\gamma):=I^{-}(f,\gamma)= I^{+}(f,\gamma)=:{\int}_{\mathbb{R}^{n}}f(x)\gamma(x)dx,\quad f\in L(\mathbb{R}^{n},\gamma) $$

(2.7)

the weighted Lebesgue integral. The identity on the right-hand side is justified, since we have continued the weighted absolute Riemann integral to the weighted Lebesgue integral. In particular, the usual convergence theorems of the Lebesgue theory are valid.

On account of Beispiel 4 in [7, Kap. V, Section 5] we see that

$$ f_{0}(x):=1,\quad x\in\mathbb{R}^{n}\quad\text{fulfills}\quad I(f_{0},\gamma)={\int}_{\mathbb{R}^{n}} 1 \cdot \gamma(x)dx=1. $$

Therefore, we have constructed a finite measure space—in contrast to the unweighted whole space $\mathbb {R}^{n}$, which is only σ-finite. As in [7, Kap. VIII, Section 3] we define those subsets as weighted measurable, whose characteristic function belong to $L(\mathbb {R}^{n},\gamma )$ and take their integral as our weighted measure. In particular, we receive null-sets and can formulate a.e.-convergence. On this basis we define the p-times weighted integrable functions as in [7, Kap. VIII, Section 6] and obtain the weighted Lebesgue spaces $L^{p}(\mathbb {R}^{n},\gamma )$ for all exponents $1\le p\le +\infty$. With the Lebesgue norm

$$ \begin{array}{@{}rcl@{}} \|f\|_{L^{p} (\mathbb{R}^{n},\gamma )}=\|f\|_{p,\gamma}&:=&\left( I(|f|^{p},\gamma)\right)^{\frac1p}=\left( {\int}_{\mathbb{R}^{n}}|f(x)|^{p}\gamma(x)dx\right)^{\frac1p}\\ &&\text{for all functions }~f\in L^{p}(\mathbb{R}^{n},\gamma) \end{array} $$

they constitute a Banach space for $1\le p<+\infty$; here we refer to [7, Kap. VIII, Section 8]. In the case $p=+\infty$ we take the essential supremum as our norm, and receive with $L^{\infty }(\mathbb {R}^{n},\gamma )$ a Banach space as well; here we refer to [8, Chapter 2, Theorem 7.10].

The bounded linear functionals on the space $L^{p}(\mathbb {R}^{n},\gamma )$ can be uniquely represented by an element $g\in L^{q}(\mathbb {R}^{n},\gamma )$ for all exponents $1<p<+\infty$ with the conjugate exponent satisfying $\frac 1p+\frac 1q=1$. Here we refer to the Riesz representation theorem from [8, Chapter 2, Theorem 8.7]. Therefore, we can formulate the weak convergence in $L^{p}(\mathbb {R}^{n},\gamma )$ as follows:

Definition 1

A sequence $\{f_{k}\}_{k=1,2,\ldots }\subset L^{p}(\mathbb {R}^{n},\gamma )$ converges weakly to an element $f\in L^{p}(\mathbb {R}^{n},\gamma )$ with $1<p<+\infty$, if the following limit relations hold true:

$$ \begin{array}{@{}rcl@{}} {\int}_{\mathbb{R}^{n}}f_{k}(x)\cdot g(x)\cdot\gamma(x)dx~&\to&~{\int}_{\mathbb{R}^{n}}f(x)\cdot g(x)\cdot\gamma(x)dx\quad(k\to\infty)\\ &&\text{for all functions }~g\in L^{q}(\mathbb{R}^{n},\gamma). \end{array} $$

Then we write formally $f_{k} \rightharpoondown f(k\to \infty )$ in $L^{p}(\mathbb {R}^{n},\gamma )$.

Parallel to our considerations above, we start with the improper Riemannian integral

$$ I_{\Omega}(\hat f):={\int}_{\Omega}\hat f(y)dy,\quad \hat f\in M({\Omega}):=\left\{\hat f\in C^{0}({\Omega})~\left|~{\int}_{\Omega}|\hat f(y)|dy<+\infty\right.\right\} $$

over the unit cube Ω. As in [7, Kap. VIII, Sections 2 and 3], we continue this functional to the Lebesgue integral

$$ I(\hat f,{\Omega})={\int}_{\Omega}\hat f(y)dy,\quad \hat f\in L({\Omega}) $$

onto the class L(Ω) of Lebesgue integrable functions over the unit cube Ω. As in [7, Kap. VIII, Section 3] we define those subsets as measurable, whose characteristic function belong to L(Ω) and take their integral as our measure. In particular, we receive null-sets and can formulate a.e.-convergence. As in [7, Kap. VIII, Section 6] we define the p-times integrable functions over Ω, and we obtain the Lebesgue spaces L^p(Ω) for all exponents $1\le p\le +\infty$. With the Lebesgue norm

$$ \|f\|_{L^{p}({\Omega})}=\|f\|_{p,{\Omega}}:=\left( I(|f|^{p},{\Omega})\right)^{\frac1p}=\left( {\int}_{\Omega}|f(x)|^{p} dx\right)^{\frac1p},\quad f\in L^{p}({\Omega}) $$

they constitute a Banach space for $1\le p<+\infty$ due to [7, Kap. VIII, Section 8]. In the case $p=+\infty$ we take the essential supremum as norm, and we receive with $L^{\infty }({\Omega })$ a Banach space as well due to [8, Chap. 2, Theorem 7.10].

Analogously to Definition 1, a sequence $\{\hat f_{k}\}_{k=1,2,\ldots }\subset L^{p}({\Omega })$ converges weakly to an element $\hat f\in L^{p}({\Omega })$, if the limit relations

$$ {\int}_{\Omega}\hat f_{k}(y)\cdot \hat g(y)(y)dy~\to~{\int}_{\Omega}\hat f(y)\cdot \hat g(y)dy\quad(k\to\infty)\quad\text{for all }~\hat g\in L^{q}({\Omega}) $$

hold true. Then we write formally

$$ \hat f_{k}~\rightharpoondown~\hat f\quad(k\to\infty)\quad\text{in}\quad L^{p}({\Omega}). $$

Here the exponent p satisfies $1<p<+\infty$ and $q\in (1,+\infty )$ denotes its conjugate.

Now we present the profound

Theorem 1 (Transformation formula for weighted integrable functions)

An arbitrary function f belongs to the class $L(\mathbb {R}^{n},\gamma )$, if and only if the associate lifted function $\hat f(y):= f\circ {\Theta }(y)$, y ∈Ω belongs to the class L(Ω). In this case we have the identity

$$ I(f,\gamma)={\int}_{\mathbb{R}^{n}}f(x) \gamma(x)dx={\int}_{\Omega}\hat f(y)dy=I(\hat f,{\Omega}). $$

(2.8)

Proof

We apply the transformation formula for multiple integrals from [7, Kap. V, Section 5, Satz 5] to the function $\hat f=\hat f(y)\in M({\Omega })$ under the Gaussian diffeomorphism ${\Gamma }\colon \mathbb {R}^{n}\to {\Omega }$. When we observe $\hat f\circ {\Gamma }(x)=f(x)$, $x\in \mathbb {R}^{n}$, we obtain the identity

$$ {\int}_{\Omega}\hat f(y)dy={\int}_{\mathbb{R}^{n}}\hat f\circ {\Gamma}(x)J_{\Gamma}(x)dx= {\int}_{\mathbb{R}^{n}}f(x)\gamma(x)dx\quad\text{for all }~\hat f\in M({\Omega}). $$

(2.9)

Starting with a function $f\in M(\mathbb {R}^{n},\gamma )$ on the right-hand side of (2.9), we use the transformation formula under the inverse Gaussian diffeomorphism ${\Theta }\colon {\Omega }\to \mathbb {R}^{n}$. Then we receive the identity

$$ {\int}_{\Omega}\hat f(y)dy={\int}_{\mathbb{R}^{n}}f(x)\gamma(x)dx\quad\text{for all }~f\in M({\Omega},\gamma). $$

(2.10)

Under the monotone approximations (2.1), (2.2) and (2.3), (2.4) the transformation formulae (2.9), (2.10) are conserved, which are immediately extended to the upper and lower Daniell integrals. Finally, we can deduce the validity of the transformation formula (2.8) from the characterization (2.7). □

Theorem 2

(The space $L^{p}(\mathbb {R}^{n},\gamma )$) We have the following properties:

• i) The Banach space $L^{p}(\mathbb {R}^{n},\gamma )$ is separable for all $1\le p<+\infty$: With respect to the ∥⋅∥_p,γ-norm, we have a dense sequence of test functions

  $$ \varphi_{k}=\varphi_{k}(x)\in C^{\infty}_{0}(\mathbb{R}^{n}),\quad k=1,2,3,\ldots $$

  within this weighted Lebesgue space.

• ii) The set of test functions $C^{\infty }_{0}(\mathbb {R}^{n})\subset L^{p}(\mathbb {R}^{n},\gamma )$ is dense within the weighted Lebesgue space for all exponents $1\le p<+\infty$.

• iii) The Banach space $L^{p}(\mathbb {R}^{n},\gamma )$ is weakly compact for all $1< p<+\infty$: Each sequence $\{f_{k}\}_{k=1,2,\ldots }\subset L^{p}(\mathbb {R}^{n},\gamma )$ satisfying ∥f_k∥_p,γ ≤ c for all $k\in \mathbb {N}$, with a constant $0<c<+\infty$, allows to select a subsequence $\{f_{k_{l}}\}_{l=1,2,\ldots }$ and an element $f\in L^{p}(\mathbb {R}^{n},\gamma )$, such that $f_{k_{l}}\rightharpoondown f~(l\to \infty )$ in $L^{p}(\mathbb {R}^{n},\gamma )$ holds true.

Proof

i) This property is established by an isometry of the weighted Lebesgue space $L^{p}(\mathbb {R}^{n},\gamma )$ and the Banach space L^p(Ω) of the lifted functions. Here we consider the linear lifting map

$$ \begin{array}{@{}rcl@{}} \widehat\quad\colon~ L^{p}(\mathbb{R}^{n},\gamma)&\to& L^{p}({\Omega})\quad\text{ defined by }\\ L^{p}(\mathbb{R}^{n},\gamma)\ni f=f(x) &\mapsto& \hat f=\hat f(y):=f\circ{\Theta}(y)\in L^{p}({\Omega}). \end{array} $$

(2.11)

From Theorem 1 we infer the identity

$$ \|f\|_{p,\gamma}=\left( {\int}_{\mathbb{R}^{n}}|f(x)|^{p}\gamma(x)dx\right)^{\frac1p}=\left( {\int}_{\Omega}|\hat f(y)|^{p} dy\right)^{\frac1p}=\|\hat f\|_{p,{\Omega}}\quad\forall f\in L^{p}(\mathbb{R}^{n},\gamma). $$

(2.12)

Consequently, the lifting map (2.11) furnishes an isometry between the Banach spaces $L^{p}(\mathbb {R}^{n},\gamma )$ and L^p(Ω). Theorem 7.14 in [8, Chapter 2, Section 7] tells us that the Lebesgue space L^p(Ω) is separable. More precisely, there exists a sequence of test functions $\{\hat \varphi _{k}(y)\}_{k=1,2,\ldots }\subset L^{p}({\Omega })$, which are dense within this Lebesgue space. Due to the identity (2.12) the pulled-back functions

$$ \varphi_{k}(x):=\hat\varphi\circ{\Gamma}(x),\quad x\in\mathbb{R}^{n},\quad k=1,2,\ldots\quad\text{in the class }~L^{p}(\mathbb{R}^{n},\gamma) $$

are dense with respect to the $L^{p}(\mathbb {R}^{n},\gamma )$-norm.

ii) This is a direct consequence of the statement i).

iii) This property can be taken from Theorem 8.9 in [8, Chapter 2, Section 8], which is valid for the Lebesgue spaces $L^{p}(\mathbb {R}^{n},\gamma )$ with $1<p<+\infty$ as well, since they are separable. □

Remark 1

Since a Friedrichs mollification procedure for the weighted Lebesgue integral seems to be impossible, we mollify the lifted function which we continue trivially outside of the unit cube before. Here we take the lifted function $\hat f=\hat f(y)\in L^{p}({\Omega })$ from (2.11) to a given function $f=f(x)\in L^{p}(\mathbb {R}^{n},\gamma )$, and we define the entire function

$$ \tilde f\colon \mathbb{R}^{n}\to \mathbb{R}\quad\text{satisfying}\quad \tilde f(y):=\hat f(y),~~y\in {\Omega}\quad\text{and}\quad \tilde f(y) :=0,~~y\in \mathbb{R}^{n}\setminus{\Omega}. $$

Now we consider the mollifier $\varrho =\varrho (y)\colon \mathbb {R}^{n}\to \mathbb {R}\in C^{\infty }_{0}(\mathbb {R}^{n})$ satisfying

$$ \varrho(y):=r\exp\left( \frac{1}{|y|^{2}-1}\right),\quad |y|<1\qquad\text{and}\qquad \varrho(y):=0,\quad |y|\ge 1, $$

where we choose the constant r > 0 such that ${\int \limits }_{\mathbb {R}^{n}}\varrho (y)dy=1$ holds true. Then we define the mollified function of the class $C^{\infty }(\mathbb {R}^{n})$ by setting

$$ \tilde f_{h}(y):=h^{-n}{\int}_{\mathbb{R}^{n}}\varrho\left( \frac{y-z}{h}\right)\tilde f(z)dz,\quad y\in\mathbb{R}^{n}. $$

Now we obtain with

$$ f_{h}(x):=\tilde f_{h}\circ {\Gamma}(x),\quad x\in\mathbb{R}^{n}\qquad\text{for all parameters }~h>0 $$

(2.13)

the regularized functions of the function $f=f(x)\in L^{p}(\mathbb {R}^{n},\gamma )$ within the class $C^{\infty }(\mathbb {R}^{n})$. On account of the isometry (2.12) we see that the regularized functions (2.13) fulfill the statements within Theorems 1.3 and 1.4 in [9, Chapter 10] due to K. Friedrichs.

3 The Difference Quotients Δ_i,ε and Weak Derivatives $\delta ^{e_{i}}$ in the Sobolev Space $W^{1,p}(\mathbb {R}^{n},\gamma )$

As in [9, Chapter 10, Section 1] we shall define the weak weighted partial derivatives in the direction of e_i for suitable functions $f\in L^{p}(\mathbb {R}^{n},\gamma )$ with $1<p<+\infty$ and i = 1,…,n. Therefore, we consider the linear functionals

$$ A_{f,i}(\varphi):={\int}_{\mathbb{R}^{n}}f(x)\delta_{i}\varphi(x)\gamma(x)dx\quad\forall\varphi=\varphi(x)\in C^{\infty}_{0}(\mathbb{R}^{n}) $$

using the weighted partial derivatives (1.1). If this functional A_f,i is bounded with respect to the $L^{q}(\mathbb {R}^{n},\gamma )$-norm for the conjugate exponent $q=\frac {p}{p-1}$, we can continue this functional from the dense space of test functions $C^{\infty }_{0}(\mathbb {R}^{n})$ onto the Banach space $L^{q}(\mathbb {R}^{n},\gamma )$. The Riesz representation theorem from [8, Chapter 2, Theorem 8.7] uniquely yields the existence of a function $g_{i}\in L^{p}(\mathbb {R}^{n},\gamma )$, such that the following identity holds true:

$$ A_{f,i}(\varphi)={\int}_{\mathbb{R}^{n}}g_{i}(x)\varphi(x)\gamma(x)dx\quad\forall\varphi=\varphi(x)\in C^{\infty}_{0}(\mathbb{R}^{n}). $$

These functions g_i represent the weak weighted derivatives in the direction of e_i.

Definition 2

For $1<p<+\infty$ and $f\in L^{p}(\mathbb {R}^{n},\gamma )$, the weak weighted derivative $\delta ^{e_{i}}f(x)\in L^{p}(\mathbb {R}^{n},\gamma )$ in the direction of e_i satisfies the integral identity

$$ {\int}_{\mathbb{R}^{n}}\delta^{e_{i}}f(x)\varphi(x)\gamma(x)dx=-{\int}_{\mathbb{R}^{n}}f(x)\delta_{i}\varphi(x)\gamma(x)dx\quad\forall\varphi=\varphi(x)\in C^{\infty}_{0}(\mathbb{R}^{n}) $$

with i = 1,…,n.

Proposition 1

The function $f=f(x)\in L^{p}(\mathbb {R}^{n},\gamma )$ possesses the weak weighted partial derivative $\delta ^{e_{i}}f(x)\in L^{p}(\mathbb {R}^{n},\gamma )$, if and only if the lifted function $\hat f=\hat f(x)\in L^{p}({\Omega })$ from (2.11) possesses the weak partial derivatives $D^{e_{i}}\hat f(y)\in L^{p}({\Omega })$ satisfying

$$ {\int}_{\Omega} D^{e_{i}}\hat f(y)\hat\varphi(y)dy=-{\int}_{\Omega}\hat f(y)\frac{\partial}{\partial y_{i}}\hat \varphi(y)dy\quad\forall\hat\varphi=\hat\varphi(y)\in C^{\infty}_{0}({\Omega}) $$

(3.1)

for i = 1,…,n. In this case we have the identities

$$ \delta^{e_{i}}f(x)=D^{e_{i}}\hat f(y)\|_{y={\Gamma}(x)}\quad \text{a.e. }~x\in \mathbb{R}^{n}. $$

(3.2)

Proof

We transform the identity (3.1) under the Gaussian diffeomorphism ${\Gamma }={\Gamma }(x)\colon \mathbb {R}^{n}\to {\Omega }$ with their Jacobian J_Γ(x) = γ(x), $x\in \mathbb {R}^{n}$. Thus we receive

$$ \begin{array}{@{}rcl@{}} {\int}_{\mathbb{R}^{n}} D^{e_{i}}\hat f\circ{\Gamma}(x)\cdot\hat\varphi\circ{\Gamma}(x)\gamma(x)dx&=&-{\int}_{\mathbb{R}^{n}}\hat f\circ{\Gamma}(x)\cdot\frac{\partial}{\partial y_{i}}\hat\varphi\circ{\Gamma}(x)\gamma(x)dx\\ &&\text{for all test functions }~\hat\varphi(y)\in C^{\infty}_{0}({\Omega}). \end{array} $$

Inserting the pulled-back function $\varphi (x):=\hat \varphi \circ {\Gamma }(x)$, $x\in \mathbb {R}^{n}$ of the class $C^{\infty }_{0}(\mathbb {R}^{n})$ and remembering $\hat f\circ {\Gamma }(x)=f(x)$, $x\in \mathbb {R}^{n}$ from (2.11), we arrive at the identity

$$ {\int}_{\mathbb{R}^{n}} D^{e_{i}}\hat f(y)\|_{y={\Gamma}(x)}\varphi(x)\gamma(x)dx=-{\int}_{\mathbb{R}^{n}}f(x) \delta_{i}\varphi(x)\gamma(x)dx\quad\forall\varphi(x)\in C^{\infty}_{0}(\mathbb{R}^{n}). $$

(3.3)

Application of the inverse Gaussian diffeomorphism ${\Theta }={\Theta }(y)\colon {\Omega }\to \mathbb {R}^{n}$ yields the reverse implication, and consequently we have an equivalence. The statement (3.3) implies the identity (3.2). □

Definition 3

For $1<p<+\infty$ we define the entire Sobolev space

$$ W^{1,p}(\mathbb{R}^{n},\gamma):=\left\{f\in L^{p}(\mathbb{R}^{n},\gamma)~|~\delta^{e_{i}}f(x)\in L^{p}(\mathbb{R}^{n},\gamma)~\text{exist for }i=1,\ldots,n\right\} $$

with the entire Sobolev norm

$$ \begin{array}{@{}rcl@{}} \|f\|_{W^{1,p}(\mathbb{R}^{n},\gamma)} &=& \|f\|_{1,p,\gamma}:=\|f\|_{p,\gamma}+{\sum}_{i=1}^{n} \|\delta^{e_{i}}f\|_{p,\gamma}\\ &=& \|f\|_{L^{p}(\mathbb{R}^{n},\gamma)}+{\sum}_{i=1}^{n} \|\delta^{e_{i}}f\|_{L^{p}(\mathbb{R}^{n},\gamma)}. \end{array} $$

(3.4)

Remark 2

Proposition 1 implies that $f\in W^{1,p}(\mathbb {R}^{n},\gamma )$ holds true for a function $f\in L^{p}(\mathbb {R}^{n},\gamma )$, if and only if their lifted function $\hat f\in L^{p}({\Omega })$ satisfies $\hat f\in W^{1,p}({\Omega })$. Within the latter space, we have the familiar Sobolev norm

$$ \|\hat f\|_{W^{1,p}({\Omega})}=\|\hat f\|_{1,p,{\Omega}}:=\|\hat f\|_{p,{\Omega}}+{\sum}_{i=1}^{n} \|D^{e_{i}}\hat f\|_{p,{\Omega}} = \|\hat f\|_{L^{p}({\Omega})}+{\sum}_{i=1}^{n} \|D^{e_{i}}\hat f\|_{L^{p}({\Omega})} $$

for arbitrary elements $\hat f\in W^{1,p}({\Omega })$.

Theorem 3 (Characterization of the entire Sobolev spaces $W^{1,p}(\mathbb {R}^{n},\gamma )$)

When the exponent $1<p<+\infty$ and the function $f\in L^{p}(\mathbb {R}^{n},\gamma )$ are given, the following properties are equivalent:

• i) We have $f\in W^{1,p}(\mathbb {R}^{n},\gamma )$.

• ii) There exists a constant $C\in [0,+\infty )$, such that all open bounded sets $B\subset \mathbb {R}^{n}$ fulfill the estimate

  $$ \|{\Delta}_{i,\epsilon}f\|_{L^{p}(B,\gamma)} := \left( {\int}_{B}|{\Delta}_{i,\epsilon} f(x)|^{p}\gamma(x)dx\right)^{\frac1p} \le C $$

  for all indices i ∈{1,…,n} and all parameters

  $$ \varepsilon \in\mathbb{R} \quad \text{satisfying}\quad 0<|\varepsilon|<\text{dist}\left( {\Gamma}(B),\mathbb{R}^{n}\setminus{\Omega}\right). $$

  (3.5)

  Here the weighted difference quotients Δ_i,𝜖 are given in (1.3).

Proof

The condition ii) for f yields for the lifted function $\hat f$ the equivalent

Condition $\widehat ii$: There exists a constant $C\in [0,+\infty )$, such that all open bounded sets $B\subset \mathbb {R}^{n}$ fulfill the estimate

$$ \|\widehat {\Delta}_{i,\epsilon} \hat f\|_{L^{p}({\Gamma}(B))}:=\left( {\int}_{\Gamma(B)}|\widehat {\Delta}_{i,\epsilon}\hat f(y)|^{p} dy\right)^{\frac1p} \le C $$

for all indices i ∈{1,…,n} and all parameters ε from (3.5).

We show this equivalence via Theorem 1 under the transformation Γ: B →Γ(B) applied to the difference quotient from (1.3), which represents the lifting of the usual difference quotient $\widehat {\Delta }_{i,\epsilon }$ from (1.2).

Now the criterion $\widehat {ii}$ is valid for the lifted function $\hat f\in L^{p}({\Omega })$ exactly if $\hat f\in W^{1,p}({\Omega })$ holds true, on account of Theorem 1.8 within [9, Chapter 10]. Here we notice that the open sets Γ(B) ⊂Ω exhaust the unit cube Ω. Due to Remark 2 the inclusion $\hat f\in W^{1,p}({\Omega })$ holds true if and only if $f\in W^{1,p}(\mathbb {R}^{n},\gamma )$ is satisfied. Thus we receive the equivalence between the conditions i) and ii).□

Remark 3

On account of Theorem 3 and the statement on the weak compactness in Theorem 2 iii), each Sobolev function $f\in W^{1,p}(\mathbb {R}^{n},\gamma )$ possesses a weakly convergent sequence of difference quotients $\{\triangle _{i,\varepsilon _{k}}f\}_{k=1,2,\ldots }$ with ε_k ↓ 0, such that

$$ \triangle_{i,\varepsilon_{k}}f\rightharpoondown \delta^{e_{i}}f \qquad\text{in }~L^{p}(\mathbb{R}^{n},\gamma),\quad i=1,\ldots,n. $$

(3.6)

In the proof of Theorem 1.8 within [9, Chapter 10], this statement is elaborated for the corresponding lifted functions. With the aid of Theorem 1 we obtain the weak convergence property (3.6). This relation explains the notion weak weighted partial derivative within the entire Sobolev space $W^{1,p}(\mathbb {R}^{n},\gamma )$. These derivatives differ substantially from the usual weak derivatives within the space $W^{1,p}_{loc}(\mathbb {R}^{n})$, which are not utilized in our investigation.

Definition 4

For a function $f\in W^{1,p}(\mathbb {R}^{n},\gamma )$ with $1<p<+\infty$, we define their weighted weak gradient

$$ \delta f(x):=\left( \delta^{e_{1}}f(x),\ldots,\delta^{e_{n}}f(x)\right)\in L^{p}(\mathbb{R}^{n},\gamma)\times \cdots\times L^{p}(\mathbb{R}^{n},\gamma). $$

Theorem 4 (Differentiation within the entire Sobolev space $W^{1,p}(\mathbb {R}^{n},\gamma )$)

For arbitrary exponents $1<p<+\infty$ we have the subsequent calculus rules:

• i) Weak product rule: For Sobolev functions $f,g\in W^{1,p}(\mathbb {R}^{n},\gamma )\cap L^{\infty }(\mathbb {R}^{n},\gamma )$, we have the property $h:=fg\in W^{1,p}(\mathbb {R}^{n},\gamma )\cap L^{\infty }(\mathbb {R}^{n},\gamma )$ and the formula

  $$ \delta^{e_{i}} h=f \delta^{e_{i}} g+g\delta^{e_{i}}f\quad\text{ for }~ i=1,\ldots,n. $$

• ii) Weak chain rule: For the Sobolev function $f\in W^{1,p}(\mathbb {R}^{n},\gamma )\cap L^{\infty }(\mathbb {R}^{n},\gamma )$ and the differentiable function $\omega : \mathbb {R}\rightarrow \mathbb {R} \in C^{1} (\mathbb {R})$, their composition l := ω ∘ f belongs to the Sobolev class $W^{1,p}(\mathbb {R}^{n},\gamma )\cap L^{\infty }(\mathbb {R}^{n},\gamma )$, and we have the formula

  $$ \delta^{e_{i}}l(x)=\omega^{\prime}(f(x)) \delta^{e_{i}} f(x)\quad\text{ for a.e. }~x\in\mathbb{R}^{n}\quad\text{and}\quad i=1,\ldots,n. $$

• iii) Lattice property: With the Sobolev function $f\in W^{1,p}(\mathbb {R}^{n},\gamma )$ their absolute function |f|(x) := |f(x)|, $x\in \mathbb {R}^{n}$ a.e. belongs to the Sobolev space $W^{1,p}(\mathbb {R}^{n},\gamma )$, and we have the following formula

  $$ \delta |f|=\left\{\begin{array}{ll} \delta f&\quad \text{if }~f>0, \\ 0&\quad \text{if }~f=0,\\ -\delta f &\quad \text{if }~f<0, \end{array}\right. $$

  (3.7)

Proof

i) We lift the functions $f(x),g(x)\in W^{1,p}(\mathbb {R}^{n},\gamma )\cap L^{\infty }(\mathbb {R}^{n},\gamma )$ to

$$ \hat f(y):=f \circ {\Theta}(y) \in W^{1,p}({\Omega})\cap L^{\infty}({\Omega}),\quad \hat g(y):=g\circ {\Theta}(y)\in W^{1,p}({\Omega})\cap L^{\infty}({\Omega}) $$

(3.8)

and receive the representations

$$ f(x):=\hat f\circ {\Gamma}(x),\quad g(x):=\hat g\circ {\Gamma}(x)\quad\text{and}\quad h(x)=f(x)\cdot g(x)=(\hat f\cdot \hat g)\circ {\Gamma}(x). $$

(3.9)

With the aid of Theorem 1.9 from [9, Chapter 10] and our Proposition 1, we calculate

$$ \begin{array}{@{}rcl@{}} \delta^{e_{i}} h(x)&=&D^{e_{i}}(\hat f\cdot \hat g)\circ {\Gamma}(x)=[(D^{e_{i}}\hat f)\cdot\hat g+\hat f \cdot (D^{e_{i}}\hat g)]\circ {\Gamma}(x)\\ & =& [(D^{e_{i}}\hat f)\circ {\Gamma}(x)]\cdot[\hat g\circ {\Gamma}(x)]+[\hat f\circ {\Gamma}(x)]\cdot [D^{e_{i}}\hat g\circ {\Gamma}(x)]\\ & =& \delta^{e_{i}} f(x)\cdot g(x)+ f(x) \cdot \delta^{e_{i}}g(x)\quad\text{ for a.e. }~x\in\mathbb{R}^{n}\quad\text{and}\quad i=1,\ldots,n. \end{array} $$

Thus we have established the weak weighted product rule.

ii) We take the functions f and $\hat f$ out of (3.8), (3.9) and consider the composition

$$ l(x)=\omega\circ f=(\omega\circ \hat f)\circ {\Gamma}(x)\quad\text{for a.e. }~x\in\mathbb{R}^{n}. $$

We apply the weak chain rule from Theorem 1.10 in [9, Chapter 10] to the composition $\omega \circ \hat f\in W^{1,p}({\Omega })$, and we obtain for i = 1,…,n via Proposition 1 the identities

$$ \begin{array}{@{}rcl@{}} \delta^{e_{i}} l(x) &=& D^{e_{i}}(\omega\circ \hat f)\circ {\Gamma}(x)=\left[\omega^{\prime}(\hat f(y))\cdot D^{e_{i}}\hat f(y)\right]\|_{y={\Gamma}(x)}\\ &=&\omega^{\prime}(f(x))\cdot\delta^{e_{i}} f(x)\quad\text{for a.e. }~x\in\mathbb{R}^{n}. \end{array} $$

Thus we have established the weak weighted chain rule.

iii) Again we take the functions f and $\hat f$ out of (3.8), (3.9), and we consider the absolute function

$$ |f|(x)=|f(x)|=|\hat f\circ {\Gamma}(x)|=|\hat f|\circ {\Gamma}(x)\quad\text{with }~|\hat f|(y):=|\hat f(y)|,~y\in{\Omega}. $$

With the aid of Theorem 1.11 from [9, Chapter 10] we see that $|\hat f|(y)$ lies within W^1,p(Ω). Furthermore, the weak gradient

$$ D \hat f:=(D^{e_{1}}\hat f,\ldots,D^{e_{n}}\hat f)\in L^{p}({\Omega})\times\cdots\times L^{p}({\Omega}) $$

fulfills the identities

$$ D|\hat f|=\left\{\begin{array}{ll} D \hat f& \quad \text{if }~\hat f>0, \\ 0& \quad \text{if }~\hat f=0, \\ -D \hat f & \quad \text{if }~\hat f<0. \end{array}\right. $$

Therefore, we obtain via Proposition 1 the identities

$$ \delta |f|(x)=D|\hat f|\circ {\Gamma}(x)=\left\{\begin{array}{ll} D \hat f\circ {\Gamma}(x)=\delta f(x)& \quad \text{if }~\hat f\circ {\Gamma}(x)=f(x)>0, \\ 0&\quad \text{if }~\hat f\circ {\Gamma}(x)=f(x)=0, \\ -D \hat f\circ {\Gamma}(x)=-\delta f(x) & \quad \text{if }~\hat f\circ {\Gamma}(x)=f(x)<0. \end{array}\right. $$

(3.10)

The statement (3.10) yields the formula (3.7) for the weak weighted gradient of our absolute function |f|(x).□

In the subsequent calculations, it is convenient to parametrize the axis going through an arbitrary point $x=(x_{1},\ldots ,x_{n})\in \mathbb {R}^{n}$ in the direction of e_i by the setting

$$ x|^{t_{i}}:=(x_{1},\ldots,x_{n-1},t_{i},x_{n+1},\ldots,x_{n}),\quad -\infty<t_{i}<+\infty\quad\text{for }~i=1,\ldots,n, $$

and we observe $\{x|^{t_{i}}\}|_{t_{i}=x_{i}}=x$. Furthermore, we abbreviate

$$ x|^{t_{1},t_{i}}:= (t_{1},x_{2},\ldots,x_{i-1},t_{i},x_{i+1},\ldots, x_{n}),\quad -\infty<t_{1},t_{i}<+\infty\quad \text{for }~i=2,\ldots,n. $$

We proceed correspondingly until we arrive at

$$ x|^{t_{1},\ldots,t_{n}}:=(t_{1},\ldots,t_{n}),\quad -\infty<t_{1},\ldots,t_{n}<+\infty. $$

Thus we can distinguish between the variables of integration t_i and the parameters x_i appearing in the respective integrals. Finally, we define the i th reduced vector

$$ x|^{i}:=(x_{1},\ldots,x_{i-1},x_{i+1},\ldots,x_{n})\in\mathbb{R}^{n-1}\quad\text{for }~i=1,\ldots,n. $$

With the symbol dx|ⁱ = dx₁…dx_i− 1dx_i+ 1…dx_n we indicate the differentials for the corresponding (n − 1)-dimensional Lebesgue integration over $\mathbb {R}^{n-1}$.

Now we can prove the enlightening

Theorem 5 (Differentiation a.e. in the space $W^{1,p}(\mathbb {R}^{n},\gamma )\cap C^{1}(\mathbb {R}^{n})$)

With $1<p<+\infty$ we have for the function $f\in W^{1,p}(\mathbb {R}^{n},\gamma )\cap C^{1}(\mathbb {R}^{n})$ the pointwise derivative

$$ \delta^{e_{i}} f(x)=f_{x_{i}}(x)\exp({x_{i}^{2}})\quad\text{for a.e. }~x \in \mathbb{R}^{n}\quad\text{and}\quad i=1,\ldots,n. $$

(3.11)

Here we denote by $f_{x_{i}}(x)$ the partial derivative of the function $f=f(x)\in C^{1}(\mathbb {R}^{n})$ for i = 1,…,n.

Proof

For arbitrary $x\in \mathbb {R}^{n}$ and i ∈{1,…,n}, we immediately comprehend the following identity on the axis through x in direction of e_i:

$$ \begin{array}{@{}rcl@{}} \gamma(x|^{t_{i}})&=&\gamma_{1}(x_{1})\cdot{\ldots} \cdot\gamma_{i-1}(x_{i-1})\cdot\gamma_{i}(t_{i})\cdot\gamma_{i+1}(x_{i+1})\cdot\ldots\cdot\gamma_{n}(x_{n})\\ &=&\gamma_{i}(t_{i})\cdot \gamma(x|^{i}) \quad\text{for all }~-\infty<t_{i}<+\infty\qquad \text{with}\\ \gamma(x|^{i})&:=&\gamma_{1}(x_{1})\cdot\ldots\cdot\gamma_{i-1}(x_{i-1})\cdot\gamma_{i+1}(x_{i+1})\cdot\ldots\cdot\gamma_{n}(x_{n}). \end{array} $$

(3.12)

With an arbitrary function $\varphi =\varphi (x)\in C^{\infty }_{0}(\mathbb {R}^{n})$, we calculate the crucial identity

$$ \begin{array}{@{}rcl@{}} \delta_{i} \varphi(x|^{t_{i}})\cdot\gamma_{i}(t_{i}) &=& \frac{d}{d y_{i}}\hat\varphi(y)\|_{y={\Gamma}(x|^{t_{i}})}\cdot\gamma_{i}(t_{i})\\ & =&\frac{d}{d y_{i}}\left( \varphi\circ{\Theta}(y)\right)\|_{y={\Gamma}(x|^{t_{i}})}\cdot\frac{d}{dt_{i}}{\Gamma}_{i}(t_{i})\\ & =&\frac{d}{d t_{i}}\left( \varphi\circ{\Theta}\circ{\Gamma}(x|^{t_{i}})\right)\\ & =& \frac{d}{d t_{i}}\varphi(x|^{t_{i}}),\quad\!\!\!\!\! -\infty\!<\!t_{i}\!<\!+\infty~\text{ for all }~x\in \mathbb{R}^{n}~\text{ and }~i=1,\ldots,n.\qquad \end{array} $$

(3.13)

With the aid of (3.12) and (3.13), we determine the weak weighted partial derivatives for the given function $f\in W^{1,p}(\mathbb {R}^{n},\gamma )\cap C^{1}(\mathbb {R}^{n})$ as follows:

$$ \begin{array}{@{}rcl@{}} -{\int}_{\mathbb{R}^{n}}[f(x)\cdot \delta_{i} \varphi(x)]\gamma(x)dx\!&=&-{\int}_{\mathbb{R}^{n-1}}\left\{{\int}_{-\infty}^{+\infty}f(x|^{t_{i}})\delta_{i} \varphi(x|^{t_{i}}) \gamma(x|^{t_{i}})dt_{i}\right\}d x|^{i}\\ &=&-{\int}_{\mathbb{R}^{n-1}}\left\{{\int}_{-\infty}^{+\infty}f(x|^{t_{i}})\delta_{i} \varphi(x|^{t_{i}}) \gamma_{i}(t_{i})dt_{i}\right\}\gamma(x|^{i})d x|^{i}\\ &=&-{\int}_{\mathbb{R}^{n-1}}\left\{{\int}_{-\infty}^{+\infty}f(x|^{t_{i}})\frac{d}{d t_{i}}\varphi(x|^{t_{i}}) dt_{i}\right\}\gamma(x|^{i})d x|^{i}\\ & =&{\int}_{\mathbb{R}^{n-1}}\left\{{\int}_{-\infty}^{+\infty}\frac{d}{d t_{i}} f(x|^{t_{i}})\cdot \varphi(x|^{t_{i}}) dt_{i}\right\}\gamma(x|^{i})d x|^{i}\\ & =&{\int}_{\mathbb{R}^{n-1}}\!\!\left\{{\int}_{-\infty}^{+\infty}\!f_{x_{i}}(x|^{t_{i}}){\gamma_{i}}^{-1}(t_{i})\cdot \varphi(x|^{t_{i}}) \gamma_{i}(t_{i})dt_{i}\!\right\}\!\gamma(x|^{i})d x|^{i}\\ & =&{\int}_{\mathbb{R}^{n}} f_{x_{i}}(x)\gamma_{i}^{-1}(x_{i})\cdot \varphi(x)\gamma(x)d x\\ & =&{\int}_{\mathbb{R}^{n}} \left[f_{x_{i}}(x)\exp({x_{i}^{2}})\right]\cdot \varphi(x)\gamma(x)d x\quad \forall \varphi\in C_{0}^{\infty}(\mathbb{R}^{n}). \end{array} $$

Here the partial integration is justified, since $f\in C^{1}(\mathbb {R}^{n})$ and $\varphi \in C_{0}^{\infty }(\mathbb {R}^{n})$ is valid. This yields the stated identity (3.11). □

4 Embedding of $W^{1,p}_{0}(\mathbb {R}^{n},\gamma )$ into $L^{\frac {np}{n-p}}(\mathbb {R}^{n},\gamma )$

We begin our considerations with the instructive

Theorem 6 (Density property for the space $W^{1,p}(\mathbb {R}^{n},\gamma )$)

With the exponent $1\le p<+\infty$ being chosen, the subspace $W^{k,p}(\mathbb {R}^{n},\gamma )\cap C^{\infty }({\Omega })$ is dense in the Sobolev space $W^{k,p}(\mathbb {R}^{n},\gamma )$.

Proof

Let $f=f(x)\in W^{k,p}(\mathbb {R}^{n},\gamma )$ be given. Now we lift this function to the function $\hat f=\hat f(y)\in W^{k,p}({\Omega })$ due to (2.11) and observe $f(x)=\hat f \circ {\Gamma }(x)$, $x\in \mathbb {R}^{n}$. Formulae (3.4), (3.2), and (2.12) yield the coincidence of their Sobolev norms:

$$ \|f\|_{1,p,\gamma}=\|f\|_{p,\gamma}+\sum\limits_{i=1}^{n} \|\delta^{e_{i}}f\|_{p,\gamma}=\|\hat f\|_{p,{\Omega}}+\sum\limits_{i=1}^{n} \|D^{e_{i}}\hat f\|_{p,{\Omega}}=\|\hat f\|_{1,p,{\Omega}}. $$

(4.1)

Due to Theorem 1.6 from [9, Chapter 10] of Meyers and Serrin we can find a sequence

$$ \left\{\hat f_{k}=\hat f_{k}(y)\right\}_{k=1,2,\ldots}\subset W^{1,p}({\Omega}) \cap C^{\infty}({\Omega})\quad\text{satisfying }~\|\hat f-\hat f_{k}\|_{1,p,{\Omega}}\to 0~~(k\to\infty). $$

For the pulled-back sequence

$$ f_{k}(x):=\hat f_{k}\circ {\Gamma}(x)\in W^{1,p}(\mathbb{R}^{n},\gamma)\cap C^{\infty}({\Omega}),\quad k=1,2,3,\ldots $$

we easily deduce $\|f- f_{k}\|_{1,p,\gamma }\to 0~(k\to \infty )$ from the isometry (4.1) above.

Therefore, the linear space $W^{k,p}(\mathbb {R}^{n},\gamma )\cap C^{\infty }({\Omega })$ is dense within the entire Sobolev space $W^{k,p}(\mathbb {R}^{n},\gamma )$. □

Let us introduce the entire Sobolev spaces $W^{1,p}_{0}(\mathbb {R}^{n},\gamma )$, whose functions are vanishing towards infinity. More precisely, we formulate

Definition 5

For $1\le p<+\infty$ we define the Sobolev space

$$ \begin{array}{@{}rcl@{}} W^{1,p}_{0}(\mathbb{R}^{n},\gamma)&:=&\!\left\{f\in W^{1,p}(\mathbb{R}^{n},\gamma)~|~\text{There exists a sequence of functions}\right.\\ &&\left.\!\{\varphi_{k}=\varphi_{k}(x)\}_{k=1,2,\ldots}\!\subset\! C^{\infty}_{0}(\mathbb{R}^{n})~\text{ satisfying }~\|f-\varphi_{k}\|_{1,p,\gamma}\!\to\! 0~\!(k\!\to\!\infty)\right\}\!. \end{array} $$

For this space $W^{1,p}_{0}(\mathbb {R}^{n},\gamma )$ we establish the Sobolev embedding theorem, which we prove directly by the fundamental method of L. Nirenberg (see [9, Chapter 10, Theorem 2.1]) in our weighted situation.

Theorem 7 (Embedding result for the space $W^{1,p}_{0}(\mathbb {R}^{n},\gamma )$)

Let the dimension $n\in \mathbb {N}$ with n ≥ 3 and the exponent 1 ≤ p < n be given. Then the Sobolev space

$$ W_{0}^{1,p}(\mathbb{R}^{n},\gamma)\subset L^{\frac{np}{n-p}}(\mathbb{R}^{n},\gamma) $$

is continuously embedded into the specified Lebesgue space: This means that the following estimate

$$ \|f\|_{L^{\frac{np}{n-p}}(\mathbb{R}^{n},\gamma)}\le C\|\delta f\|_{L^{p}(\mathbb{R}^{n},\gamma)}\quad\text{for all }~f\in W_{0}^{1,p}(\mathbb{R}^{n},\gamma) $$

(4.2)

holds true with the constant $C=C(n,p)\in (0,+\infty )$ from (4.8).

Proof

1.) Because of Definition 5 it suffices to prove the inequality (4.2) for all $f\in C_{0}^{\infty }({\Omega })$. In this context we need the generalized weighted Hölder inequality, which can easily be deduced from the weighted Hölder inequality within the spaces $L^{p}(\mathbb {R},\gamma _{i}(t_{i}))$ with i = 1,…,n by an induction.

For the integer $m \in \mathbb {N}$ with m ≥ 2 we choose the exponents $p_{1},\ldots ,p_{m}\in (1,\infty )$ satisfying $p_{1}^{-1}+\cdots +p_{m}^{-1}=1$ and select an index i ∈{1,…,n}. For all functions $f_{j}=f_{j}(t_{i})\in {C_{0}^{0}}(\mathbb {R})$ with j = 1,…,m, the following inequality holds true:

$$ \begin{array}{@{}rcl@{}} &&{\int}_{\mathbb{R}} f_{1}(t_{i}){\ldots} f_{m}(t_{i})\gamma_{i}(t_{i})dt_{i}\\ && \le \|f_{1}(t_{i})\|_{L^{p_{1}}(\mathbb{R},\gamma_{i}(t_{i}))}\cdot\ldots\cdot\|f_{m} (t_{i})\|_{L^{p_{m}}(\mathbb{R},\gamma_{i}(t_{i}) )}\\ && = \left( {\int}_{-\infty}^{+\infty}|f_{1}(t_{i})|^{p_{1}}\gamma_{i}(t_{i})dt_{i}\right)^{\frac{1}{p_{1}}}\cdot {\ldots} \cdot \left( {\int}_{-\infty}^{+\infty}|f_{m}(t_{i})|^{p_{m}}\gamma_{i}(t_{i})dt_{i}\right)^{\frac{1}{p_{m}}}. \end{array} $$

(4.3)

Now we specialize this inequality (4.3) to the exponents p₁ = ⋯ = p_m = n − 1 and m = n − 1 ≥ 2, since n ≥ 3 holds true for the spatial dimension. Thus we receive the weighted m-th Hölder inequality

$$ \begin{array}{@{}rcl@{}} &&{\int}_{\mathbb{R}} f_{1}(t_{i}){\ldots} f_{n-1}(t_{i})\gamma_{i}(t_{i})dt_{i}\\ &&\le \|f_{1}(t_{i})\|_{L^{n-1}(\mathbb{R},\gamma_{i}(t_{i}))}\cdot\ldots\cdot\|f_{n-1} (t_{i})\|_{L^{n-1}(\mathbb{R},\gamma_{i}(t_{i}))}\\ &&= \left( {\int}_{-\infty}^{+\infty}|f_{1}(t_{i})|^{n-1}\gamma_{i}(t_{i})dt_{i}\right)^{\frac{1}{n-1}}\cdot \ldots\cdot \left( {\int}_{-\infty}^{+\infty}|f_{n-1}(t_{i})|^{n-1}\gamma_{i}(t_{i})dt_{i}\right)^{\frac{1}{n-1}}\\ &&\qquad\text{for all }f_{1}=f_{1}(t_{i}),\ldots,f_{n-1}=f_{n-1}(t_{i})\in {C_{0}^{0}}(\mathbb{R})~\text{ and }~i=1,\ldots,n. \end{array} $$

(4.4)

2.) At first, we deduce the estimate (4.2) in the case p = 1. We note that $f\in C_{0}^{\infty }({\Omega })$ holds true, and we use the crucial identities (3.13). Thus we receive for all $x\in {\mathbb {R}}^{n}$ and i ∈{1,…,n} the following representation

$$ f(x)={\int}_{-\infty}^{x_{i}}\frac{d}{d t_{i}}f(x|^{t_{i}})dt_{i}={\int}_{-\infty}^{x_{i}}\delta_{i}f(x|^{t_{i}})\cdot\gamma_{i}(t_{i})dt_{i}. $$

For i = 1,…,n this implies the estimate

$$ |f(x)|\le {\int}_{-\infty}^{x_{i}}|\delta_{i} f(x|^{t_{i}})|\cdot \gamma_{i}(t_{i})dt_{i} \le {\int}_{-\infty}^{+\infty}|\delta_{i} f(x|^{t_{i}})| \cdot \gamma_{i}(t_{i})dt_{i},\quad x\in\mathbb{R}^{n}. $$

Consequently, we receive the estimate

$$ |f(x)|^{\frac{n}{n-1}}\le {\prod}_{i=1}^{n}\left( {\int}_{-\infty}^{+\infty}|\delta_{i}f(x|^{t_{i}})|\gamma_{i}(t_{i}) dt_{i}\right)^{\frac{1}{n-1}},\quad x\in\mathbb{R}^{n}. $$

(4.5)

With the aid of the weighted m th Hölder inequality (4.4) for the index i = 1, we integrate the inequality (4.5) with respect to the variable x₁ via the weighted differential γ₁(t₁)dt₁. Thus we receive

$$ \begin{array}{@{}rcl@{}} {\int}_{-\infty}^{+\infty}\!|f(x|^{t_{1}})|^{\frac{n}{n-1}} \gamma_{1}(t_{1})dt_{1}\!\!& \le& \!\left( {\int}_{-\infty}^{+\infty}|\delta^{e_{1}}f(x|^{t_{1}})|\gamma_{1}(t_{1})dt_{1}\right)^{\frac{1}{n-1}}\\ &&\!\times {\int}_{-\infty}^{+\infty}{\prod}_{i=2}^{n}\left( {\int}_{-\infty}^{+\infty}|\delta^{e_{i}}f(x|^{t_{1},t_{i}})|dt_{i}\right)^{\frac{1}{n-1}}\gamma_{1}(t_{1})dt_{1} \\ &\le& \!\left( {\int}_{-\infty}^{+\infty}|\delta^{e_{1}}f(x|^{t_{1}})|\gamma_{1}(t_{1})dt_{1}\right)^{\frac{1}{n-1}}\\ &&\!\times {\prod}_{i=2}^{n} \!\left( {\int}_{-\infty}^{+\infty}{\int}_{-\infty}^{+\infty}\!|\delta^{e_{i}}f(x|^{t_{1},t_{i}})\gamma_{1}(t_{1})dt_{1}\gamma_{i}(t_{i}) dt_{i}\!\right)^{\!\frac{1}{n-1}}.\qquad \end{array} $$

(4.6)

A similar weighted integration of (4.6) over the variable x₂ yields

$$ \begin{array}{@{}rcl@{}} &&{\int}_{-\infty}^{+\infty}{\int}_{-\infty}^{+\infty}|f(x|^{t_{1},t_{2}})|^{\frac{n}{n-1}}\gamma_{1}(t_{1})dt_{1}\gamma_{2}(t_{2})dt_{2}\\ &&\le\left( {\int}_{-\infty}^{+\infty}{\int}_{-\infty}^{+\infty}|\delta^{e_{1}}f(x|^{t_{1},t_{2}})|\gamma_{1}(t_{1})dt_{1}\gamma_{2}(t_{2})dt_{2}\right)^{\frac{1}{n-1}}\\ &&\quad\times \left( {\int}_{-\infty}^{+\infty}{\int}_{-\infty}^{+\infty}|\delta^{e_{2}}f(x|^{t_{1},t_{2}})\gamma_{1}(t_{1})dx_{1}\gamma_{2}(t_{2}) dt_{2}\right)^{\frac{1}{n-1}}\\ &&\quad\times {\prod}_{i=3}^{n} \left( {\int}_{-\infty}^{+\infty}{\int}_{-\infty}^{+\infty}{\int}_{-\infty}^{+\infty}|\delta^{e_{i}}f(x|^{t_{1},t_{2},t_{i}})\gamma_{1}(t_{1})dt_{1}\gamma_{2}(t_{2})dt_{2} \gamma_{i}(t_{i})d t_{i}\right)^{\frac{1}{n-1}}. \end{array} $$

Proceeding successively with the weighted integration over x₃,…,x_n, we arrive at the following inequality

$$ \begin{array}{@{}rcl@{}} &&{\int}_{\mathbb{R}^{n}}|f(x|^{t_{1},\ldots,t_{n}})|^{\frac{n}{n-1}}\gamma_{1}(t_{1})dt_{1}\ldots\gamma_{n}(t_{n})dt_{n}\\ &&\le\left( {\prod}_{i=1}^{n}{\int}_{{\mathbb{R}}^{n}}|\delta^{e_{i}}f(x|^{t_{1},\ldots,t_{n}})|\gamma_{1}(t_{1})dt_{1}\ldots\gamma_{n}(t_{n})dt_{n} \right)^{\frac{1}{n-1}}. \end{array} $$

Finally, we obtain

$$ \begin{array}{@{}rcl@{}} \|f\|_{\frac{n}{n-1},\gamma} &=& \|f\|_{L^{\frac{n}{n-1}}(\mathbb{R}^{n},\gamma)} \le \left( {\prod}_{i=1}^{n}{\int}_{\mathbb{R}^{n}}|\delta^{e_{i}}f(x)|\gamma(x)dx\right)^{\frac{1}{n}}\\ &\le& \frac{1}{n}{\int}_{\mathbb{R}^{n}}\left( {\sum}_{i=1}^{n}|\delta^{e_{i}}f(x)|\right)\gamma(x) dx \le \frac{1}{\sqrt{n}}{\int}_{\mathbb{R}^{n}} |\delta f(x)| \gamma(x) dx\\ &=&\frac{1}{\sqrt{n}}\|\delta f(x)\|_{L^{1}(\mathbb{R}^{n},\gamma)}=\frac{1}{\sqrt{n}}\|\delta f(x)\|_{1,\gamma} \quad \text{for all }~f\in C_{0}^{\infty}({\Omega}). \end{array} $$

(4.7)

3.) We now consider the case 1 < p < n. Here we insert |f|^α with a suitable exponent α > 1 into the estimate (4.7). This is possible on account of Theorem 4, when we utilize the differentiation rules given there. Therefore, Hölder’s inequality yields the following relation, under the condition p^− 1 + q^− 1 = 1 for the conjugate exponent $q=\frac {p}{p-1}$:

$$ \begin{array}{@{}rcl@{}} \||f|^{\alpha}\|_{\frac{n}{n-1},\gamma}&\le& \frac{1}{\sqrt{n}}{\int}_{\mathbb{R}^{n}}\|\delta |f|^{\alpha}(x)\| \gamma(x)dx\\ &=&\frac{\alpha}{\sqrt{n}}{\int}_{\mathbb{R}^{n}}|f(x)|^{\alpha-1}|\delta f(x)|\gamma(x)dx \le \frac{\alpha}{\sqrt{n}}\||f|^{\alpha-1}\|_{q,\gamma} \cdot \|\delta f\|_{p,\gamma}. \end{array} $$

This implies the estimate

$$ \|f\|_{\frac{\alpha n}{n-1},\gamma}^{\alpha} \le \frac{\alpha}{\sqrt{n}}\|f\|_{(\alpha-1)q,\gamma}^{\alpha-1} \cdot \|\delta f\|_{p,\gamma}. $$

Choosing the exponent $\alpha :=\frac {(n-1)p}{n-p}=\frac {np-p}{n-p}>1$, we infer

$$ \frac{\alpha n}{n-1}=(\alpha-1)q=\frac{np}{n-p}. $$

Finally, we arrive at the inequality

$$ \|f\|_{\frac{np}{n-p},\gamma}\le\frac{\alpha}{\sqrt{n}} \|\delta f\|_{p,\gamma}\qquad\text{for all }~f\in C_{0}^{\infty}(\mathbb{R}^{n}). $$

With the constant

$$ C:=\frac{np-p}{\sqrt{n}(n-p)} $$

(4.8)

the statement (4.2) follows. □

5 Spectral Theory of Weighted Elliptic Operators on the Gauß–Rellich space $W^{1,2}_{0}(\mathbb {R}^{n},\gamma )$

At first, we formulate our

Definition 6

We introduce the Gauß–Rellich space $W^{1,2}_{0}(\mathbb {R}^{n},\gamma )$ with the norm

$$ \|f\|_{W^{1,2}(\mathbb{R}^{n},\gamma)}:=\|f\|_{L^{2}(\mathbb{R}^{n},\gamma)}+{\sum}_{i=1}^{n} \|\delta^{e_{i}}f\|_{L^{2}(\mathbb{R}^{n},\gamma)}. $$

Definition 7

We prescribe the coefficient functions $a_{ij}=a_{ij}(x)\in L^{\infty }(\mathbb {R}^{n})$ for i,j = 1,…,n with an ellipticity constant $M_{0}\in [1,+\infty )$, such that the following conditions hold true:

$$ \begin{array}{@{}rcl@{}} &&\text{For a.e. }~x\in\mathbb{R}^{n}~\text{ we have }~a_{ij}(x)=a_{ji}(x)\quad(i,j=1,\ldots,n)\quad\text{and}\\ &&\frac{1}{M_{0}}|\xi|^{2}\le {\sum}_{i,j=1}^{n} a_{ij}(x)\xi_{i}\xi_{j}\le M_{0}|\xi|^{2}\quad \forall\xi=(\xi_{1},\ldots,\xi_{n})\in\mathbb{R}^{n}. \end{array} $$

(5.1)

Then we introduce the weighted Dirichlet integral

$$ \mathcal E (f):={\int}_{\mathbb{R}^{n}}\left\{{\sum}_{i,j=1}^{n} a_{ij}(x)\delta^{e_{i}}f(x)\delta^{e_{j}} f(x)\right\}\gamma(x)dx,\quad f\in W^{1,2}_{0}(\mathbb{R}^{n},\gamma) $$

(5.2)

with the associate weighted bilinear form

$$ \mathcal E (f,g):={\int}_{\mathbb{R}^{n}}\left\{{\sum}_{i,j=1}^{n} a_{ij}(x)\delta^{e_{i}}f(x)\delta^{e_{j}} g(x)\right\}\gamma(x) dx\quad\forall f,g\in W^{1,2}_{0}(\mathbb{R}^{n},\gamma). $$

(5.3)

With the constant

$$ {\Lambda}_{n}:=\left( \frac{\sqrt n(n-2)}{2n-2}\right)^{2} $$

our Theorem 7 above yields the following estimate:

$$ \begin{array}{@{}rcl@{}} \mathcal E(f) &=& {\int}_{\mathbb{R}^{n}}\left\{{\sum}_{i,j=1}^{n} a_{ij}(x)\delta^{e_{i}}f(x)\delta^{e_{j}} f(x)\right\}\gamma(x)dx\\ &\ge& \frac{1}{M_{0}}{\int}_{\mathbb{R}^{n}}\left\{{\sum}_{i=1}^{n}|\delta^{e_{i}}f(x)|^{2}\right\}\gamma(x)dx=\frac{1}{M_{0}}{\int}_{\mathbb{R}^{n}}|\delta f(x)|^{2}\gamma(x)dx\\ &=&\frac{1}{M_{0}}\|\delta f\|_{L^{2}(\mathbb{R}^{n},\gamma)}^{2}\ge\frac{{\Lambda}_{n}}{M_{0}}\|f\|_{L^{\frac{2n}{n-2}}(\mathbb{R}^{n},\gamma)}^{2}\ge\frac{{\Lambda}_{n}}{M_{0}}\|f\|_{L^{2}(\mathbb{R}^{n},\gamma)}^{2}\\ &=&\frac{{\Lambda}_{n}}{M_{0}}{\int}_{\mathbb{R}^{n}} |f(x)|^{2}\gamma(x)dx\quad \forall f\in W^{1,2}_{0}(\mathbb{R}^{n},\gamma). \end{array} $$

(5.4)

Now we establish a version of Rellich’s selection theorem within our entire Sobolev space $W^{1,2}_{0}(\mathbb {R}^{n},\gamma )$. Since this proof requires Friedrichs mollifiers, the transition to the lifted functions seems to be natural. Here we refer to the Remark 1 from above.

Theorem 8 (Compactness of the unit sphere in $L^{2}(\mathbb {R}^{n},\gamma )\subset W^{1,2}_{0}(\mathbb {R}^{n},\gamma )$)

For the spatial dimension n ≥ 3 being given, let the elements

$$ f_{k}\in W^{1,2}_{0}(\mathbb{R}^{n},\gamma)\quad \text{ with }~\|f_{k}\|_{L^{2}(\mathbb{R}^{n},\gamma)}=1~\text{ for }~k=1,2,3,\ldots $$

denote a sequence on the unit sphere of the Lebesgue space $L^{2}(\mathbb {R}^{n},\gamma )$, whose energy is bounded from above by

$$ \sup_{k\in\mathbb{N}}\mathcal E(f_{k})<+\infty. $$

(5.5)

Then we can select a subsequence $\{f_{k_{l}}\}_{l=1,2,\ldots }$ of this sequence {f_k}_k= 1,2,… and a limit element $f\in W^{1,2}_{0}(\mathbb {R}^{n},\gamma )$ with $\|f\|_{L^{2}(\mathbb {R}^{n},\gamma )}=1$, such that the following convergence properties hold true:

$$ \delta f_{k_{l}} \rightharpoondown \delta f~~(l\to \infty)\quad\text{ in }~L^{2}(\mathbb{R}^{n},\gamma)\quad\text{ and }\quad \|f-f_{k_{l}}\|_{L^{2}(\mathbb{R}^{n},\gamma)}~\to~0~~(l\to\infty). $$

(5.6)

Proof

1.) With the aid of the condition (5.5) and the estimate (5.4), we deduce

$$ \sup_{k\in\mathbb{N}}\|\delta f_{k}\|_{L^{2}(\mathbb{R}^{n},\gamma)}^{2} < +\infty. $$

(5.7)

For each component of {δf_k}_k= 1,2,…, our Theorem 2 iii) allows the transition to a subsequence, which is weakly convergent in $L^{2}(\mathbb {R}^{n},\gamma )$. Thus we receive a subsequence $\{f_{k_{l}}\}_{l=1,2,\ldots } \subset \{f_{k}\}_{k=1,2,\ldots }$ such that

$$ \delta f_{k_{l}}~\rightharpoondown~\delta f~~(l\to \infty)\quad\text{ in }~L^{2}(\mathbb{R}^{n},\gamma)\quad\text{ with }\quad \delta f\in L^{2}(\mathbb{R}^{n},\gamma)\times \cdots\times L^{2}(\mathbb{R}^{n},\gamma) $$

holds true. We assume that we have already performed this transition to a subsequence, without indicating this subsequence any more, and we continue with the sequence {f_k}_k= 1,2,… now.

2.) As in (2.11) we lift the sequence $\{f_{k}\}_{k=1,2,\ldots }\subset W^{1,2}_{0}(\mathbb {R}^{n},\gamma )$ to the sequence $\{\hat f_{k}\}_{k=1,2,\ldots } \subset W^{1,2}_{0}({\Omega })$. When we denote the weak gradient within the Sobolev space W^1,2(Ω) by D, the condition (5.7) is transformed via (3.2) and (2.12) into the following estimate for Dirichlet’s integral of the lifted functions:

$$ \sup_{k\in\mathbb{N}} \|D \hat f_{k}\|_{L^{2}({\Omega})}^{2} < +\infty. $$

(5.8)

From (5.5), (5.4), and (2.12) we infer the estimate

$$ \sup_{k\in\mathbb N} \|\hat f_{k}\|_{L^{2}({\Omega})}^{2} = \sup_{k\in\mathbb N} \|f_{k}\|_{L^{2}(\mathbb{R}^{n},\gamma)}^{2}<+\infty. $$

(5.9)

Now we invoke Theorem 2.3 from [9, Chapter 10] for p = 2 = q, namely the selection theorem by F. Rellich in the Sobolev space $W^{1,2}_{0}({\Omega })$. Then the properties (5.8), (5.9) imply the possibility to select a subsequence $\{\hat f_{k_{l}}\}_{l=1,2,\ldots }\subset \{\hat f_{k}\}_{k=1,2,\ldots }$ and the existence of a function $\hat f\in L^{2}({\Omega })$ with their pulled-back function $f\in L^{2}(\mathbb {R}^{n},\gamma )$, such that

$$ \|f_{k_{l}}- f\|_{L^{2}(\mathbb{R}^{n},\gamma)}=\|\hat f_{k_{l}}-\hat f\|_{L^{2}({\Omega})}\to 0\quad (l\to\infty) $$

holds true, where we utilize the identity (2.12) again. Thus we have found a subsequence with the properties (5.6) stated above. □

With the aid of the spectral theorem for selfadjoint operators on the Hilbert space $L^{2}(\mathbb {R}^{n},\gamma )$, we establish

Theorem 9 (Elliptic operators on $W^{1,2}_{0}(\mathbb {R}^{n},\gamma )$ with discrete spectrum)

For the spatial dimension n ≥ 3 being given, there exists a complete orthonormal system $f_{i}\in W_{0}^{1,2}(\mathbb {R}^{n},\gamma )$ with $i\in \mathbb N$, satisfying

$$ (f_{i},f_{j})_{L^{2}(\mathbb{R}^{n},\gamma)}=\delta_{ij}\quad\text{ for }~i,j=1,2,3,\ldots, $$

(5.10)

and the associate eigenvalues

$$ 0<\lambda_{1}\le \lambda_{2}\le\cdots\le\lambda_{j} \to +\infty\quad (j\to\infty), $$

(5.11)

such that the weak eigenvalue equations

$$ \mathcal E(f_{i},\varphi)=\lambda_{i}(f_{i},\varphi)_{L^{2}(\mathbb{R}^{n},\gamma)}\quad\text{ for all }~\varphi\in C_{0}^{\infty}(\mathbb{R}^{n})~\text{ and }~i\in\mathbb N $$

(5.12)

are valid.

Proof

We start with the weighted elliptic differential operator in divergence form

$$ \mathcal L f(x):=-{\sum}_{j,k=1}^{n} \delta^{e_{k}}\left( a_{kj}(x)\delta^{e_{j}}f(x)\right),\quad f\in {C^{2}_{0}}(\mathbb{R}^{n}). $$

(5.13)

As in Theorem I.2.2 of the treatise [10] Spektraltheorie selbstadjungierter Operatoren im Hilbertraum und elliptischer Differentialoperatoren we continue the operator $\mathcal L$ to a selfadjoint linear operator $L \colon \mathcal D_{L}\to L^{2}(\mathbb {R}^{n},\gamma )$ on the dense linear subset $\mathcal D_{L}:=W^{1,2}_{0}(\mathbb {R}^{n})$ within the Hilbert space $L^{2}(\mathbb {R}^{n},\gamma )$, such that the identity

$$ (Lf,g)_{L^{2}(\mathbb{R}^{n},\gamma)}=\mathcal E(f,g)=(f,Lg)_{L^{2}(\mathbb{R}^{n},\gamma)}\quad\text{ for all }~f,g\in \mathcal D_{L}=W^{1,2}_{0}(\mathbb{R}^{n}) $$

is valid. To this operator L we apply the spectral theorem for selfadjoint operators from [10, Theorem I.12.1]. Thus we obtain a resolution of the identity or a spectral family $\{E_{\lambda }\}_{\lambda \in \mathbb {R}}$, such that the spectral representation

$$ L f={\int}_{-\infty}^{+\infty} \lambda dE(\lambda)\quad \text{ for all }~f\in \mathcal D_{L}=W^{1,2}_{0}(\mathbb{R}^{n}) $$

is valid. From [10, Proposition I.12.3] and our Theorem 8 above we see that the operator $L\colon W^{1,2}_{0}(\mathbb {R}^{n})\to L^{2}(\mathbb {R}^{n},\gamma )$ solely possesses a discrete spectrum. Therefore, we receive a complete system of eigenfunctions $\{f_{i}\}_{i\in \mathbb N}\subset W^{1,2}_{0}(\mathbb {R}^{n})$ satisfying (5.10) with their associate eigenvalues (5.11), such that the weak eigenvalue equations (5.12) are fulfilled. □

Remark 4

We could prove Theorem 9 by R. Courant’s direct variational method as well. Here we have to minimize the weighted Dirichlet integral over the unit sphere in $L^{2}(\mathbb {R}^{n},\gamma )$ as follows:

$$ \mathcal E(f)~\to~ \text{Minimum},\quad f\in W^{1,2}_{0}(\mathbb{R}^{n},\gamma)\quad\text{ with }~(f,f)_{L^{2}(\mathbb{R}^{n},\gamma)}=1. $$

Now we need a lower semicontinuity result for the weighted Dirichlet integral (5.2) under weak convergence and Theorem 8 from above. This method is elaborated in [10, Theorem I.3.4] and supplies certain minimum properties for the respective eigenfunctions. The spectral theorem for selfadjoint operators, however, solely deals with the weak identities and is independent of an eventual variational structure.

6 Least Eigenvalues for Elliptic Operators on $\mathbb {R}_{+}^{n}$ with Mixed Boundary Conditions

For the dual numbers ρ₁,…,ρ_n ∈{0,1} we define the sectorial domains

$$ {\Omega}_{\rho_{1},\ldots,\rho_{n}}:=\left\{(x_{1},\ldots, x_{n})\in\mathbb{R}^{n}~|~(-1)^{\rho_{1}}x_{1}>0,\ldots,(-1)^{\rho_{n}} x_{n}>0 \right\}. $$

With the positive continuum $\mathbb {R}_{+}:=(0,+\infty )$ we notice that

$$ {\Omega}_{0,\ldots,0}=\mathbb{R}_{+}\times\cdots\times\mathbb{R}_{+} $$

holds true. The union of the sectorial domains ${\Omega }_{\rho _{1},\ldots ,\rho _{n}}$, ρ₁,…,ρ_n ∈{0,1} represents the entire $\mathbb {R}^{n}$ apart from the planes x_j = 0 for j = 1,…,n.

Let us reflect the function f exactly ρ_j ∈{0,1} times at the plane x_j = 0 for j = 1,…,n as follows:

$$ \begin{array}{@{}rcl@{}} {\Sigma}_{\rho_{1},\ldots,\rho_{n}}\colon W_{0}^{1,2}(\mathbb{R}^{n},\gamma) &\to& W_{0}^{1,2}(\mathbb{R}^{n},\gamma)\quad\text{defined by}\\ {\Sigma}_{\rho_{1},\ldots,\rho_{n}}f(x_{1},\ldots,x_{n})&:=& f\left( (-1)^{\rho_{1}}x_{1},\ldots,(-1)^{\rho_{n}} x_{n}\right)\\ &&\text{for almost all }~ x=(x_{1},\ldots,x_{n})\in\mathbb{R}^{n}. \end{array} $$

(6.1)

We introduce the basic reflections at the planes x_j = 0 by

$$ \begin{array}{@{}rcl@{}} {\Sigma}_{j}&:=&{\Sigma}_{\delta_{1j},\ldots,\delta_{jj},\ldots,\delta_{nj}}\colon L^{2}(\mathbb{R}^{n},\gamma)\to L^{2}(\mathbb{R}^{n},\gamma)\quad\text{satisfying}\\ {{\Sigma}_{j}^{2}}&=&{\Sigma}_{j}\circ{\Sigma}_{j}=E\quad \text{ for }~j=1,\ldots,n\quad\text{and}\\ {\Sigma}_{j}\circ {\Sigma}_{k}&=& {\Sigma}_{k}\circ {\Sigma}_{j}\quad\text{ for }~j,k=1,\ldots,n. \end{array} $$

(6.2)

Thus we obtain the compositions

$$ \begin{array}{@{}rcl@{}} {\Sigma}_{\rho_{1},\ldots,\rho_{n}}f(x) &=& {\Sigma}_{n}^{\rho_{n}}\circ\cdots\circ{\Sigma}_{1}^{\rho_{1}}f(x),\quad x\in\mathbb{R}^{n}~\text{ a.e.}\\ &&\text{for all }~f\in W_{0}^{1,2}(\mathbb{R}^{n},\gamma)\quad \text{and}\quad \rho_{1},\ldots,\rho_{n}\in\{0,1\}. \end{array} $$

(6.3)

Each basic reflection Σ_j yields an isometry from the Lebesgue space $L^{2}(\mathbb {R}^{n},\gamma )$ into itself for j = 1,…,n, and consequently represents a bounded linear operator there.

Definition 8

We introduce the symmetric Sobolev space

$$ W_{0,{\Sigma}}^{1,2}(\mathbb{R}^{n},\gamma):=\left\{f\in W_{0}^{1,2}(\mathbb{R}^{n},\gamma)\colon {\Sigma}_{j} f=E f~\text{ for }~ j=1,\ldots,n\right\}, $$

whose functions are symmetric with respect to the reflections Σ_j for j = 1,…,n.

Remark 5

As an intersection of the kernels from the bounded linear operators

$$ (E-{\Sigma}_{j})\colon L^{2}(\mathbb{R}^{n},\gamma)\to L^{2}(\mathbb{R}^{n},\gamma),\quad j=1,\ldots,n, $$

the symmetric Sobolev space $W_{0,{\Sigma }}^{1,2}(\mathbb {R}^{n},\gamma )$ constitutes a nonvoid linear subspace of the entire Sobolev space $W_{0}^{1,2}(\mathbb {R}^{n},\gamma )$, which is closed with respect to the $L^{2}(\mathbb {R}^{n},\gamma )$-convergence.

Definition 9

When we denote by $\tilde f=f|_{\mathbb {R}_{+}^{n}}$ the restriction of a function $f\colon \mathbb {R}^{n} \to \mathbb {R}$ onto the sectorial domain $\mathbb {R}^{n}_{+}$, we define the sectorial Sobolev space with mixed boundary conditions

$$ W_{0,{\Sigma}}^{1,2}(\mathbb{R}^{n}_{+},\gamma):=\left\{\tilde f=f|_{\mathbb{R}_{+}^{n}} \colon f\in W_{0,{\Sigma}}^{1,2}(\mathbb{R}^{n},\gamma)\right\}. $$

Remark 6

We interpret the reflection conditions on the finite boundary of $\mathbb {R}^{n}_{+}$ as weak Neumann conditions. However, these functions are vanishing towards the infinite boundary of the sectorial domain and yield weak Dirichlet conditions there. Thus we obtain mixed boundary conditions within the space $W_{0,{\Sigma }}^{1,2}(\mathbb {R}^{n}_{+},\gamma )$.

In a unique way, we can continue an arbitrary function $\tilde f\in W_{0,{\Sigma }}^{1,2}(\mathbb {R}^{n}_{+},\gamma )$ to an entire function $f\in W_{0,{\Sigma }}^{1,2}(\mathbb {R}^{n},\gamma )$ by the reflections (6.1) at the planes x_j = 0 for j = 1,…,n.

Definition 10

The weighted elliptic operators from (5.1) and (5.13) are called admissible, if their coefficient functions a_ij = a_ij(x) for i,j = 1,…,n are situated within the linear space

$$ L^{\infty}_{\Sigma}(\mathbb{R}^{n}):=\left\{a(x)\in L^{\infty}(\mathbb{R}^{n})\colon {\Sigma}_{j} a(x)=Ea(x),~x\in\mathbb{R}^{n}~\text{ a.e.};~j=1,\ldots,n\right\}. $$

If the coefficients additionally vanish outside of the diagonal, that means

$$ \begin{array}{@{}rcl@{}} && a_{ii}\in L^{\infty}_{\Sigma}(\mathbb{R}^{n})\quad\text{and}\quad \frac{1}{M_{0}}\le a_{ii}(x)\le M_{0},\qquad x\in\mathbb{R}^{n}_{+}~\text{ a.e.}\quad\text{for }~i=1,\ldots,n\\ \text{and }&& a_{ij}(x)=a_{ii}(x)\delta_{ij},\qquad x\in\mathbb{R}^{n}~\text{ a.e.}\quad\text{for }~i,j=1,\ldots,n, \end{array} $$

we address this as a reflective operator.

Definition 11

With an admissible operator from Definition 10 we associate the sectorial energy form

$$ \begin{array}{@{}rcl@{}} \mathcal E (f,g;\rho_{1},\ldots,\rho_{n})&:=&{\int}_{{\Omega}_{\rho_{1},\ldots,\rho_{n}}}\left\{{\sum}_{i,j=1}^{n} a_{ij}(x)\delta^{e_{i}}f(x)\delta^{e_{j}}g(x)\right\}\gamma(x)dx\\ &&\text{for all }~f,g\in W^{1,2,}_{0}(\mathbb{R}^{n},\gamma)~\text{ and }~\rho_{1},\ldots,\rho_{n}\in\{0,1\}. \end{array} $$

(6.4)

Now we consider reflective operators with their weighted bilinear form (5.3) and the sectorial energy form

$$ \begin{array}{@{}rcl@{}} \mathcal E(f,g;\rho_{1},\ldots,\rho_{n})&=&{\int}_{{\Omega}_{\rho_{1},\ldots,\rho_{n}}}\left\{{\sum}_{i=1}^{n} a_{ii}(x)\delta^{e_{i}}f(x)\delta^{e_{i}} g(x)\right\}\gamma(x)dx\\ &&\text{for all }~f,g\in W^{1,2}_{0,{\Sigma}}(\mathbb{R}^{n},\gamma). \end{array} $$

(6.5)

We shall verify the following identities

$$ \begin{array}{@{}rcl@{}} \mathcal E (f,g;\rho_{1},\ldots,\rho_{n})&=&\mathcal E (f,g;0,\ldots,0)=:\mathcal E (f,g;\mathbb{R}_{+}^{n})\\ &&\text{for all }~f,g\in W^{1,2}_{0,{\Sigma}}(\mathbb{R}^{n},\gamma)~\text{ and }~\rho_{1},\ldots,\rho_{n}\in\{0,1\}. \end{array} $$

(6.6)

Here we utilize that the functions $f,g\in W^{1,2}_{0,{\Sigma }}(\mathbb {R}^{n},\gamma )$ on the sectorial domain ${\Omega }_{\rho _{1},\ldots ,\rho _{n}}$ can be gained by the reflection ${\Sigma }_{\rho _{1},\ldots ,\rho _{n}}$from the domain $\mathbb {R}_{+}^{n}$. These reflections are composed as in (6.3) by the basic reflections (6.2). Under each basic reflection the weighted derivatives within the sectorial energy form (6.5) are antisymmetric, however, they appear quadratically. The other terms in the sectorial energy forms (6.5) are symmetric with respect to the basic reflections. Thus we can easily verify the identities (6.6) by the transformation formula for multiple integrals.

In a similar way we comprehend the identities

$$ \begin{array}{@{}rcl@{}} (f,g)_{L^{2}({\Omega}_{\rho_{1},\ldots,\rho_{n}},\gamma)}&=&(f,g)_{L^{2}({\Omega}_{0,\ldots,0},\gamma)}=: (f,g)_{L^{2}(\mathbb{R}^{n}_{+},\gamma)} \\ &&\text{for all }~f,g\in W^{1,2}_{0,{\Sigma}}(\mathbb{R}^{n},\gamma)~\text{ and }~\rho_{1},\ldots,\rho_{n}\in\{0,1\}. \end{array} $$

(6.7)

Here we utilize the weighted inner products over the sectorial domains

$$ (f,g)_{L^{2}({\Omega}_{\rho_{1},\ldots,\rho_{n}},\gamma)}:= {\int}_{{\Omega}_{\rho_{1},\ldots,\rho_{n}}}\left\{f(x)g(x)\right\}\gamma(x) dx,\quad f,g\in L^{2}(\mathbb{R}^{n},\gamma). $$

We are now prepared to establish

Theorem 10 (Least eigenvalues for admissible elliptic operators on $\mathbb {R}_{+}^{n}$ under mixed boundary conditions)

For the spatial dimension n ≥ 3 let a reflective elliptic operator from Definition 10 be given. Then an eigenfunction $\tilde f$ in the sectorial Sobolev space $W_{0,{\Sigma }}^{1,2}(\mathbb {R}_{+}^{n},\gamma )$ from Definition 9 to the least eigenvalue

$$ \tilde \lambda:=\inf\left\{\frac{\mathcal E (\tilde f,\tilde f;\mathbb{R}_{+}^{n})}{(\tilde f,\tilde f)_{L^{2}(\mathbb{R}^{n}_{+},\gamma)}}~\|~ {\tilde f\in W_{0,{\Sigma}}^{1,2}(\mathbb{R}_{+}^{n},\gamma)\setminus\{0\}}\right\}>0, $$

(6.8)

exists, which satisfies the weak eigenvalue equation

$$ \mathcal E(\tilde f,\tilde \phi;\mathbb{R}^{n}_{+})=\tilde\lambda(\tilde f,\tilde \phi)_{L^{2}(\mathbb{R}_{+}^{n},\gamma)}\quad\text{ for all }~\tilde \phi\in W_{0,{\Sigma}}^{1,2}(\mathbb{R}_{+}^{n},\gamma) $$

(6.9)

over the sectorial domain $\mathbb {R}^{n}_{+}$ under mixed boundary conditions.

Proof

We solve the variational problem

$$ \mathcal E(f,f)~\to~\text{Minimum},\quad f\in W^{1,2}_{0,{\Sigma}}(\mathbb{R}^{n},\gamma)\quad\text{ with }~(f,f)_{L^{2}(\mathbb{R}^{n},\gamma)}=1. $$

(6.10)

This is possible with the aid of Theorem 8 under the observation that $W^{1,2}_{0,{\Sigma }}(\mathbb {R}^{n},\gamma )$ represents a closed subspace of $W^{1,2}_{0}(\mathbb {R}^{n},\gamma )$ with respect to $L^{2}(\mathbb {R}^{n},\gamma )$-convergence. On account of the identities (6.6), we observe

$$ \mathcal E (f,f)=2^{n} \mathcal E (\tilde f,\tilde f;\mathbb{R}_{+}^{n})\quad \text{ with }~\tilde f=f|_{\mathbb{R}_{+}^{n}}~\text{ for all }~f\in W^{1,2}_{0,{\Sigma}}(\mathbb{R}^{n},\gamma). $$

(6.11)

Furthermore, the identities (6.7) yield

$$ (f,f)_{L^{2}(\mathbb{R}^{n},\gamma)}=2^{n}(\tilde f,\tilde f)_{L^{2}(\mathbb{R}^{n}_{+},\gamma)}\quad\text{ with }~\tilde f=f|_{\mathbb{R}_{+}^{n}}~\text{ for all }~f\in W^{1,2}_{0,{\Sigma}}(\mathbb{R}^{n},\gamma). $$

(6.12)

With the aid of (6.11) and (6.12) we comprehend the invariance for the Raleigh quotient (6.8) as follows:

$$ \tilde \lambda=\inf\left\{\frac{\mathcal E (f,f)}{(f,f)_{L^{2}(\mathbb{R}^{n},\gamma)}} ~\|~ f\in W_{0,{\Sigma}}^{1,2}(\mathbb{R}^{n},\gamma)\setminus\{0\}\right\} >0. $$

The variational problem (6.10) is equivalent to the following problem

$$ \mathcal E(\tilde f,\tilde f;\mathbb{R}^{n}_{+})~\to~\text{Minimum},\quad \tilde f\in W^{1,2}_{0,{\Sigma}}(\mathbb{R}_{+}^{n},\gamma)\quad\text{with }~(\tilde f,\tilde f)_{L^{2}(\mathbb{R}_{+}^{n},\gamma)}=1. $$

The identities (6.11) and (6.12), which say that these variational problems only differ by the dilation factor $2^{-\frac n2}$, imply this fact. For the solution $f\in W^{1,2}_{0,{\Sigma }}(\mathbb {R}^{n},\gamma )$ of the variational problem (6.10) we obtain the weak eigenvalue equation

$$ \mathcal E(f,\phi)=\tilde \lambda(f,\phi)_{L^{2}(\mathbb{R}^{n},\gamma)}\quad\text{ for all }~\phi\in W^{1,2}_{0,{\Sigma}}(\mathbb{R}^{n},\gamma) $$

(6.13)

in the usual way. When we specialize the equation (6.13) to the sectorial domain $\mathbb {R}^{n}_{+}$ for the function $\tilde f=f|_{\mathbb {R}_{+}^{n}}$, we obtain the weak eigenvalue equation (6.9). □

7 Spectral Theory for Elliptic Operators on $\mathbb {R}_{+}^{n}$ with Vanishing Boundary Conditions

For the dual numbers ρ₁,…,ρ_n ∈{0,1} we define the reflections

$$ \begin{array}{@{}rcl@{}} {\Sigma}^{\prime}_{\rho_{1},\ldots,\rho_{n}}&\colon& W_{0}^{1,2}(\mathbb{R}^{n},\gamma)\to W_{0}^{1,2}(\mathbb{R}^{n},\gamma)\quad\text{defined by}\\ {\Sigma}^{\prime}_{\rho_{1},\ldots,\rho_{n}}f(x_{1},\ldots,x_{n})&:=&(-1)^{\rho_{1}+\cdots+\rho_{n}} f\left( (-1)^{\rho_{1}}x_{1},\ldots,(-1)^{\rho_{n}} x_{n}\right)\\ &=&(-{\Sigma}_{n})^{\rho_{n}}\circ\cdots\circ(-{\Sigma}_{1})^{\rho_{1}}f(x_{1},\ldots,x_{n})\\ &&\text{ for a.e. }~x=(x_{1},\ldots,x_{n})\in\mathbb{R}^{n}. \end{array} $$

(7.1)

These reflections are composed of ρ_j uneven basic reflections −Σ_j at the planes x_j = 0 for j = 1,…,n which evidently commute.

Definition 12

We introduce the antisymmetric Sobolev space

$$ W_{0,{\Sigma}^{\prime}}^{1,2}(\mathbb{R}^{n},\gamma):=\left\{f\in W_{0}^{1,2}(\mathbb{R}^{n},\gamma)\colon -{\Sigma}_{j}f=Ef ~\text{ for }~ j=1,\ldots,n\right\}, $$

whose functions are symmetric with respect to the uneven basic reflections −Σ_j.

Remark 7

As an intersection of the kernels from the bounded linear operators

$$ (E+{\Sigma}_{j})\colon L^{2}(\mathbb{R}^{n},\gamma)~ \to~L^{2}(\mathbb{R}^{n},\gamma),\quad j=1,\ldots,n, $$

the antisymmetric Sobolev space $W_{0,{\Sigma }^{\prime }}^{1,2}(\mathbb {R}^{n},\gamma )$ constitutes a nonvoid linear subspace of the entire Sobolev space $W_{0}^{1,2}(\mathbb {R}^{n},\gamma )$, which is closed under $L^{2}(\mathbb {R}^{n},\gamma )$-convergence.

Definition 13

Let us define the sectorial Sobolev space with vanishing boundary conditions

$$ W_{0}^{1,2}(\mathbb{R}^{n}_{+},\gamma):=\left\{\tilde f=f|_{\mathbb{R}_{+}^{n}} \colon f\in W_{0,{\Sigma}^{\prime}}^{1,2}(\mathbb{R}^{n},\gamma)\right\}. $$

Remark 8

We interpret the reflection condition on the finite boundary of $\mathbb {R}^{n}_{+}$ as weak Dirichlet boundary condition zero. Furthermore, these functions are vanishing towards the infinite boundary of the sectorial domain. Thus we obtain vanishing boundary conditions within the space $W_{0}^{1,2}(\mathbb {R}^{n}_{+},\gamma )$. In a unique way, we can continue an arbitary function $\tilde f\in W_{0}^{1,2}(\mathbb {R}^{n}_{+},\gamma )$ to an entire function $f\in W_{0,{\Sigma }^{\prime }}^{1,2}(\mathbb {R}^{n},\gamma )$ by the reflections (7.1) at the planes x_j = 0 for j = 1,…,n.

We consider admissible elliptic operators in Definition 10 and verify the following identities for their sectorial energy forms of Definition 11:

$$ \begin{array}{@{}rcl@{}} \mathcal E (f,g;\rho_{1},\ldots,\rho_{n})&=&\mathcal E (f,g;0,\ldots,0)=:\mathcal E (f,g;\mathbb{R}_{+}^{n})\\ &&\text{for all }~f,g\in W^{1,2}_{0,{\Sigma}^{\prime}}(\mathbb{R}^{n},\gamma)~\text{ and }~\rho_{1},\ldots,\rho_{n}\in\{0,1\}. \end{array} $$

(7.2)

Here we utilize that the functions $f,g\in W^{1,2}_{0,{\Sigma }^{\prime }}(\mathbb {R}^{n},\gamma )$ on the sectorial domain ${\Omega }_{\rho _{1},\ldots ,\rho _{n}}$ can be gained by the reflection ${\Sigma }^{\prime }_{\rho _{1},\ldots ,\rho _{n}}$ from the domain $\mathbb {R}_{+}^{n}$. These reflections are composed as in (7.1) by uneven basic reflections −Σ_j, j = 1,…,n. Under each uneven basic reflection the weighted derivatives within the sectorial energy form (6.4) are invariant. The other terms in the sectorial energy forms (6.4) are symmetric with respect to the basic reflections. Thus we can easily verify the identities (7.2) by the transformation formula for multiple integrals.

Furthermore, we comprehend the following identities in a similar way:

$$ \begin{array}{@{}rcl@{}} (f,g)_{L^{2}({\Omega}_{\rho_{1},\ldots,\rho_{n}},\gamma)}&=& (f,g)_{L^{2}({\Omega}_{0,\ldots,0},\gamma)}=: (f,g)_{L^{2}(\mathbb{R}^{n}_{+},\gamma)} \\ &&\text{for all }~f,g\in W^{1,2}_{0,{\Sigma}^{\prime}}(\mathbb{R}^{n},\gamma)~\text{ and }~\rho_{1},\ldots,\rho_{n}\in\{0,1\}. \end{array} $$

(7.3)

Theorem 11 (Elliptic operators on $W^{1,2}_{0}(\mathbb {R}^{n}_{+},\gamma )$ with discrete spectrum)

For the spatial dimension n ≥ 3 we consider an admissible differential operator from Definition 10. Then there exists an orthonormal system $\tilde f_{i}\in W_{0}^{1,2}(\mathbb {R}^{n}_{+},\gamma )~(i\in \mathbb N)$ of eigenfunctions, which satisfies

$$ (\tilde f_{i},\tilde f_{j})_{L^{2}(\mathbb{R}^{n}_{+},\gamma)}=\delta_{ij}\quad\text{ for }~i,j=1,2,3,\ldots $$

(7.4)

and is complete in the space $L^{2}(\mathbb {R}^{n}_{+},\gamma )$. For the associate eigenvalues

$$ 0<\tilde\lambda_{1}\le \tilde\lambda_{2}\le\cdots\le\tilde \lambda_{j}~\to~+\infty\quad (j\to\infty) $$

(7.5)

we have the weak eigenvalue equations

$$ \mathcal E(\tilde f_{i},\tilde \varphi;\mathbb{R}^{n}_{+})=\tilde \lambda_{i}(\tilde f_{i},\tilde \varphi)_{L^{2}(\mathbb{R}^{n}_{+},\gamma)}\quad\text{ for all }~\tilde \varphi\in C_{0}^{\infty}(\mathbb{R}^{n}_{+})~\text{ and }~i\in\mathbb N. $$

(7.6)

Proof

We start with the admissible elliptic differential operator

$$ \mathcal L \tilde f(x):=-{\sum}_{j,k=1}^{n} \delta^{e_{k}}\left( a_{kj}(x)\delta^{e_{j}}\tilde f(x)\right),\quad \tilde f\in {C^{2}_{0}}(\mathbb{R}^{n}_{+}). $$

Via (7.1) we reflect these functions to the whole $\mathbb {R}^{n}$ and extend the differential operator to the antisymmetric functions f of the class ${C^{2}_{0}}(\mathbb {R}^{n}\setminus \cup _{j=1}^{n}\{x_{j}=0\})$. As in Theorem I.2.2 of the treatise [10] we continue this operator $\mathcal L$ to a selfadjoint linear operator $L \colon \mathcal D_{L}\to L^{2}(\mathbb {R}^{n},\gamma )$ on the dense subspace $\mathcal D_{L}:=W^{1,2}_{0,{\Sigma }^{\prime }}(\mathbb {R}^{n})$ within

$$ L^{2}_{{\Sigma}^{\prime}}(\mathbb{R}^{n},\gamma):=\left\{a(x)\in L^{2}(\mathbb{R}^{n},\gamma)\colon -{\Sigma}_{j} a(x)=a(x)~ \forall x\in\mathbb{R}^{n};~j=1,\ldots,n\right\}. $$

Furthermore, we have the identity

$$ (Lf,g)_{L^{2}(\mathbb{R}^{n},\gamma)}=\mathcal E(f,g)=(f,Lg)_{L^{2}(\mathbb{R}^{n},\gamma)}\quad\text{ for all }~f,g\in \mathcal D_{L}=W^{1,2}_{0,{\Sigma}^{\prime}}(\mathbb{R}^{n}). $$

On this operator L we apply the spectral theorem for selfadjoint operators from [10, Theorem I.12.1]. Thus we obtain a spectral family $\{\widetilde {E_{\lambda }}\}_{\lambda \in \mathbb {R}}$, such that

$$ L f={\int}_{-\infty}^{+\infty} \lambda d\widetilde{E(\lambda)}\quad \text{ for all }~f\in \mathcal D_{L}=W^{1,2}_{0,{\Sigma}^{\prime}}(\mathbb{R}^{n}). $$

From [10, Proposition I.12.3] and our Theorem 8 above we see that the operator $L\colon W^{1,2}_{0,{\Sigma }^{\prime }}(\mathbb {R}^{n}) \to L^{2}(\mathbb {R}^{n},\gamma )$ solely possesses a discrete spectrum. We receive a system of eigenfunctions $\{f_{i}\}_{i\in \mathbb N}\subset W^{1,2}_{0,{\Sigma }^{\prime }}(\mathbb {R}^{n})$ satisfying (5.10), which is complete in the space $L^{2}_{{\Sigma }^{\prime }}(\mathbb {R}^{n},\gamma )$. Their associate eigenvalues $\tilde {\lambda }_{i},~i\in \mathbb N$ satisfying (7.5) fulfill the weak eigenvalue equations (5.12). Respecting the identities (7.2) and (7.3) we see that the functions $\tilde f_{i}:=f_{i}|_{\mathbb {R}^{n}_{+}},~i\in \mathbb N$ satisfy the identities (7.4) as well as (7.6) and are complete as specified above. □