Some Calculations of Orlicz Cohomology and Poincar\'e--Sobolev--Orlicz Inequalities

TL;DR

This paper computes Orlicz cohomology for fundamental Riemannian manifolds and explores its connection to Poincaré--Sobolev--Orlicz inequalities, advancing understanding of geometric analysis in these contexts.

Contribution

It provides explicit calculations of Orlicz cohomology on key manifolds and discusses their relationship with Poincaré--Sobolev--Orlicz inequalities, a novel link in geometric analysis.

Findings

Calculated Orlicz cohomology for the real line, hyperbolic plane, and ball.

Established relationships between Orlicz cohomology and Poincaré--Sobolev--Orlicz inequalities.

Enhanced understanding of geometric inequalities in Riemannian manifolds.

Abstract

We carry out calculations of Orlicz cohomology for some basic Riemannian manifolds (the real line, the hyperbolic plane, the ball). Relationship between Orlicz cohomology and Poincar\'e--Sobolev--Orlicz-type inequalities is discussed.

Equations

$$\psi (s)=\inf \{t>0:\phi (t)>s\},s>0,$$

$$\Phi (x)=\underset{0}{\overset{|x|}{\int }}\phi (t)dt,\Psi (x)=\underset{0}{\overset{|x|}{\int }}\psi (t)dt$$

$$\Psi (y)=\sup \{x|y|-\Phi (x):x\ge 0\},y\in \mathbb{R}.$$

$${\rho }_{\Phi }(f):={\int }_{\Omega }\Phi (f)d\mu .$$

$$L^{\Phi }=L^{\Phi }(\Omega )=L^{\Phi }(\Omega ,\Sigma ,\mu )=\{f:\Omega \to \mathbb{R}\text{leavevmode}\text{measurable}\text{leavevmode}:{\rho }_{\Phi }(af)<\infty \text{leavevmode}\text{for some a>0}\}$$

$$\|f\|_{\Phi}=\|f\|_{L^{\Phi}(\Omega)}=\sup\biggl{\{}\biggl{|}\int_{\Omega}fg\,d\mu\biggr{|}:\rho_{\Psi}(g)\leq 1\biggr{\}}$$

$$\|f\|_{(\Phi)}=\|f\|_{L^{(\Phi)}(\Omega)}=\inf\biggl{\{}K>0\,:\,\rho_{\Phi}\biggl{(}\frac{f}{K}\biggr{)}\leq 1\biggr{\}}.$$

$$0\to T^k\to H^k\to {\overline{H}}^k\to 0.$$

$$\Vert \eta {\Vert }_{F^{k-1}}\le C_k\Vert \theta {\Vert }_{F^k}.$$

$$\Vert \xi -\zeta {\Vert }_{F^{k-1}}\le C_k^{{}^{\prime }}\Vert d\xi {\Vert }_{F^k}.$$

$${\rho }_{\Phi }(\omega ):={\int }_X\Phi (|\omega (x)|)d{\mu }_X.$$

$${\rho }_{\Phi }(\alpha \omega )<\infty \text{leavevmode}\text{for some}\text{leavevmode}\alpha >0.$$

$$\|\omega\|_{(\Phi)}=\inf\biggl{\{}K>0\,:\,\rho_{\Phi}\biggl{(}\frac{\omega}{K}\biggr{)}\leq 1\biggr{\}}.$$

$$\|\omega\|_{\Phi}=\sup\biggl{\{}\biggl{|}\int_{X}(\omega(x),\theta(x))\,d\mu_{X}\biggr{|}:\rho_{\Psi}(\theta)\leq 1\biggr{\}}$$

$$\underset{U}{\int }\omega \wedge du=(-1{)}^{j+1}\underset{U}{\int }\theta \wedge u$$

$${\Omega }_{{\Phi }_I,{\Phi }_{II}}^k(X)=\left\{\omega \in L^{{\Phi }_I}(X,{\Lambda }^k):d\omega \in L^{{\Phi }_{II}}(X,{\Lambda }^{k+1})\right\}.$$

$$\Vert \omega {\Vert }_{({\Phi }_I),({\Phi }_{II})}=\Vert \omega {\Vert }_{({\Phi }_I)}+\Vert d\omega {\Vert }_{({\Phi }_{II})}.$$

$$Z_{{\Phi }_{II}}^k(X)=\{\omega \in L^{{\Phi }_{II}}(X,{\Lambda }^k):d\omega =0\};$$

$$B_{{\Phi }_I,{\Phi }_{II}}^k(X)=\{\omega \in L^{{\Phi }_{II}}(X,{\Lambda }^k):\omega =d\beta \text{leavevmode}\text{for some}\text{leavevmode}\beta \in L^{{\Phi }_I}(X,{\Lambda }^{k-1})\}.$$

$$H_{{\Phi }_I,{\Phi }_{II}}^k(X):=Z_{{\Phi }_{II}}^k(X)/B_{{\Phi }_I,{\Phi }_{II}}^k(X)$$

$${\overline{H}}_{{\Phi }_I,{\Phi }_{II}}^k(X):=Z_{{\Phi }_{II}}^k(X)/{\overline{B}}_{{\Phi }_I,{\Phi }_{II}}^k(X)$$

$$T_{{\Phi }_I,{\Phi }_{II}}^k(X):={\overline{B}}_{{\Phi }_I,{\Phi }_{II}}^k(X)/B_{{\Phi }_I,{\Phi }_{II}}^k(X).$$

$${\Omega }_{\mathcal{F}}^k(X)={\Omega }_{{\Phi }_k,{\Phi }_{k+1}}^k(X);$$

$$0\to {\Omega }_{\mathcal{F}}^0(X)\to {\Omega }_{\mathcal{F}}^1(X)\to \cdots \to {\Omega }_{\mathcal{F}}^k(X)\to \cdots \to {\Omega }_{\mathcal{F}}^n(X)\to 0.$$

$$H_{\mathcal{F}}^k(X)=H_{{\Phi }_{k-1},{\Phi }_k}^k(X)=Z_{{\Phi }_k}^k(X)/B_{{\Phi }_{k-1},{\Phi }_k}^k;$$

$${\overline{H}}_{\mathcal{F}}^k(X)={\overline{H}}_{{\Phi }_{k-1},{\Phi }_k}^k(X)=Z_{{\Phi }_k}^k(X)/{\overline{B}}_{{\Phi }_{k-1},{\Phi }_k}^k.$$

$$\Vert \theta {\Vert }_{L^{({\Phi }_I)}}\le C\Vert \omega {\Vert }_{L^{({\Phi }_{II})}}.$$

$$\Vert \theta -\zeta {\Vert }_{L^{({\Phi }_I)}}\le C^{\prime }\Vert d\theta {\Vert }_{L^{({\Phi }_{II})}}.$$

$$H_{\mathcal{F}}^k(X)=H_{{\Phi }_I,{\Phi }_{II}}^k(X);{\overline{H}}_{\mathcal{F}}^k(X)={\overline{H}}_{{\Phi }_I,{\Phi }_{II}}^k(X).$$

$$\underset{z\in \mathbb{R}}{\inf }\Vert f-z{\Vert }_{({\Phi }_1)}\le C\Vert f^{\prime }{\Vert }_{({\Phi }_2)}$$

$$\theta (x)={\omega }_{1/2}(x)=\{\begin{array}{ll}C{e}^{-\frac{1}{1-4x^2}}& \text{mbox}if|x|\le 1/2,\\ 0& \text{mbox}if|x|>1/2.\end{array}$$

Full text

Key words and phrases: differential form, Orlicz cohomology, torsion, Poincaré–Sobolev–Orlicz inequality

Mathematics Subject Classification 2000: 58A12, 46E30, 22E25

The second author was supported by the Program of Basic Scientific Research of the Siberian Branch of the Russian Academy of Sciences.

Introduction

The article continues the study of Orlicz cohomology of Riemannian manifolds initiated in [7, 8].

Orlicz cohomology is a natural generalization of $L_{qp}$-cohomology (for a detailed discussion of $L_{qp}$-cohomology, the reader is referred, for example, to [4]).

Like Orlicz function spaces, the Orlicz spaces $L^{\Phi}$ of differential forms are a natural nonlinear generalization of the spaces $L^{p}$. Orlicz spaces of differential forms on domains in $\mathbb{R}^{n}$ were first considered by Iwaniec and Martin in [6] and then by Agarwal, Ding, and Nolder in [1]. Orlicz forms on an arbitrary Riemannian manifold were apparently first examined by Kopylov and Panenko in [7].

In [4], Gol*′*dshtein and Troyanov demonstrated close relationship between $L_{qp}$-cohomology and Sobolev-type inequalities on Riemannian manifolds and, basing on this and some “almost duality” techniques, performed calculations of $L_{qp}$-cohomology for some basic manifolds. It turns out that, with some significant corrections and sometimes under additional constraints on the $N$-functions from which the Orlicz cohomology is constructed, these methods prove to be fruitful in computing Orlicz cohomology.

The structure of the article is as follows: In Section 1, we recall the main notions and necessary properties of Orlicz function spaces. In Section 2, we recall some basic information on abstract Banach complexes. Section 3 contains defininitions concerning Orlicz spaces of differential forms on a Riemannian manifold, Orlicz cohomology, and its interpretation in terms of Poincaré–Sobolev–Orlicz inequalities (Theorems 3.3 and 3.4). Then we calculate the $L_{\Phi_{1},\Phi_{2}}$-cohomology of $\mathbb{R}$ (Section 4) the hyperbolic plane (Section 5) and the $L_{\Phi}$-cohomology of the ball (“$L^{\Phi}$-Poincaré inequality”, Section 6).

1. $N$-Functions and Orlicz Function Spaces

Definition 1.1**.**

A nonnegative function $\Phi:\mathbb{R}\to\mathbb{R}$ is called an $N$-function if

(i)* $\Phi$ is even and convex;*

(ii)* $\Phi(x)=0\Longleftrightarrow x=0$;*

(iii)* $\lim\limits_{x\to 0}\frac{\Phi(x)}{x}=0;\quad\lim\limits_{x\to\infty}\frac{\Phi(x)}{x}=\infty$.*

An $N$-function $\Phi$ has left and right derivatives (which can differ only on an at most countable set, see, for instance, [10, Theorem 1, p. 7]). The left derivative $\varphi$ of $\Phi$ is left continuous, nondecreasing on $(0,\infty)$, and such that $0<\varphi(t)<\infty$ for $t>0$, $\varphi(0)=0$, $\lim\limits_{t\to\infty}\varphi(t)=\infty$. The function

[TABLE]

is called the left inverse of $\varphi$.

The functions $\Phi,\Psi$ given by

[TABLE]

are called complementary $N$-functions.

The $N$-function $\Psi$ complementary to an $N$-function $\Phi$ can also be expressed as

[TABLE]

Throughout the article, given an $N$-function $\Phi:\mathbb{R}\to[0,\infty)$, we denote by $\Phi^{-1}$ its “positive” inverse $\Phi^{-1}:[0,\infty)\to[0,\infty)$.

$N$-functions are classified in accordance with their growth rates as follows:

Definition 1.2**.**

An $N$-function $\Phi$ is said to satisfy the $\Delta_{2}$-condition (for all $x$), which is written as $\Phi\in\Delta_{2}$ if there exists a constant $K>2$ such that $\Phi(2x)\leq K\Phi(x)$ for all $x\geq 0$; $\Phi$ is said to satisfy the $\nabla_{2}$-condition (for all $x$), which is denoted symbolically as $\Phi\in\nabla_{2}$, if there is a constant $c>1$ such that $\Phi(x)\leq\frac{1}{2c}\Phi(cx)$ for all $x\geq 0$.

It is not hard to see that an $N$-function $\Phi$ satisfies the the $\nabla_{2}$-condition if and only if its dual $N$-function satisfies the $\Delta_{2}$-condition.

Henceforth, let $\Phi$ be an $N$-function and let $(\Omega,\Sigma,\mu)$ be a measure space.

Definition 1.3**.**

Given a measurable function $f:\Omega\to\mathbb{R}$, we put

[TABLE]

Definition 1.4**.**

The linear space

[TABLE]

is called an Orlicz space on $(\Omega,\Sigma,\mu)$.

Let $\Psi$ be the complementary $N$-function to $\Phi$.

Below we as usual identify two functions equal outside a set of measure zero.

If $f\in L^{\Phi}$ then the functional $\|\cdot\|_{\Phi}$ (called the Orlicz norm) defined by

[TABLE]

is a seminorm. It becomes a norm if $\mu$ satisfies the finite subset property (see [10, p. 59]): if $A\in\Sigma$ and $\mu(A)>0$ then there exists $B\in\Sigma$, $B\subset A$, such that $0<\mu(B)<\infty$.

The equivalent gauge (or Luxemburg) norm of a function $f\in L^{\Phi}$ is defined by the formula

[TABLE]

This is a norm without any constraint on the measure $\mu$ (see [10, p. 54, Theorem 3]).

2. Banach Complexes

Like in the case of $L_{q,p}$-cohomology, treated in [4], we apply some abstract facts about Banach complexes to the Orlicz cohomology of Rimennian manifolds.

In this section, we recall some definitions and assertions about abstract Banach complexes given in [4].

Definition 2.1**.**

A Banach complex is a sequence $F^{*}=\{F^{k},d_{k}\}_{k\in\mathbb{N}}$ where $F^{k}$ is a Banach space and $d=d^{k}:F^{k}\rightarrow F^{k+1}$ is a bounded operator with $d^{k+1}\circ d^{k}=0$.

Definition 2.2**.**

Given a Banach complex $\{F^{k},d\}$, introduce the vector spaces:

• •

$Z^{k}:=\ker(d:F^{k}\rightarrow F^{k+1})$* (a closed subspace of $F^{k}$);*

• •

$B^{k}:=$Im$(d:F^{k-1}\rightarrow F^{k})\subset Z^{k}$;

• •

$H^{k}(F^{*}):=Z^{k}/B^{k}$* is the cohomology of the complex $F^{*}=\{F^{k},d\}$;*

• •

$\overline{H}^{k}(F^{*}):=Z^{k}/\overline{B}^{k}$* is the reduced cohomology of the complex $F^{*}$;*

• •

$T^{k}(F^{*}):=\overline{B}^{k}/B^{k}=H^{k}/\overline{H}^{k}$* is the torsion of the complex $F^{*}$.*

As was observed in [4], the following easy assertion holds:

1. (a)

$\overline{H}^{k},Z^{k}$ and $\overline{B}^{k}$ are Banach spaces; 2. (b)

The natural (quotient) topology on $T^{k}:=\overline{B}^{k}/B^{k}$ is coarse (any closed set is either empty or $T^{k}$); 3. (c)

there is a natural exact sequence

[TABLE]

Lemma 2.3**.**

[4, Lemma 4.4]** For any Banach complex $\{F^{k},d\}$, the following are equivalent:

1. (i)

$T^{k}=0$; 2. (ii)

$\dim T_{k}<\infty$; 3. (iii)

$H^{k}$* is a Banach space;* 4. (iv)

$B^{k}\subset F^{k}$* is closed.*

Lemma 2.4**.**

[4, Proposition 4.5]** The following are equivalent:

1. (i)

$H^{k}=0$; 2. (ii)

The operator $d_{k-1}:F^{k-1}/Z^{k-1}\rightarrow Z^{k}$ admits a bounded inverse $d_{k-1}^{-1}$; 3. (iii)

There exists a constant $C_{k}$ such that for any  $\theta\in Z^{k}$ there is an element $\eta\in F^{k-1}$ with $d\eta=\theta$ and

[TABLE]

Lemma 2.5**.**

[4, Propositions 4.6 and 4.7]** The following conditions (i) and (ii) are equivalent:

1. (i)

$T^{k}=0$. 2. (ii)

The operator $d_{k-1}:F^{k-1}/Z^{k-1}\rightarrow B^{k}$ admits a bounded inverse $d_{k-1}^{-1}$.

Any of these conditions implies

1. (iii)

There exists a constant $C_{k}^{{}^{\prime}}$ such that for any $\xi\in F^{k-1}$ there is an element $\zeta\in Z^{k-1}$ such that

[TABLE]

Moreover, if $F^{k-1}$ is a reflexive Banach space then conditions (i)-(iii) are equivalent.

3. Orlicz Spaces of Differential Forms and Orlicz Cohomology

Let $X$ be a Riemannian manifold of dimension $n$. Given $x\in X$, denote by $(\omega(x),\theta(x))$ the scalar product of exterior $k$-forms $\omega(x)$ and $\theta(x)$ on $T_{x}X$. This gives a function $x\mapsto(\omega(x),\theta(x))$ on $X$.

Let $\Phi:\mathbb{R}\to\mathbb{R}$ and $\Psi:\mathbb{R}\to\mathbb{R}$ be two complementary $N$-functions. Given a measurable $k$-form $\omega$, we put

[TABLE]

Here $d\mu_{X}$ stands for the volume element of the Riemannian manifold $X$. We will identify $k$-forms differing on a set of measure zero.

Given a (not necessarily orientable) Riemannian manifold $X$, introduce the space $L^{\Phi}(X,\Lambda^{k})$ as the class of all measurable $k$-forms $\omega$ satisfying the condition

[TABLE]

As in the case of Orlicz function spaces, the space $L^{\Phi}(X,\Lambda^{k})$ is endowed with two equivalent norms: the gauge norm

[TABLE]

and the Orlicz norm ($\Psi$ is the complementary $N$-function to $\Phi$):

[TABLE]

As in the case of function spaces, it can be proved that $L^{\Phi}(X,\Lambda^{k})$ endowed with one of these norms is a Banach space.

Obviously, the gauge norm of a $k$-form $\omega$ is nothing but the gauge norm of its modulus function $|\omega|$. The same holds for the Orlicz norm ([7, Lemma 2.1]).

Unless otherwise specified, we endow the $L^{\Phi}$ spaces with the gauge norms; the quotient (semi)norm on each of the cohomology spaces to be defined below depends on the choice of the norms on $L^{\Phi_{I}}$ and $L^{\Phi_{II}}$ but the resulting topology does not.

Definition 3.1**.**

A form $\theta\in L_{1,\mathop{\rm loc}}^{j+1}(X)$ is called the (weak) differential $d\omega$ of $\omega\in L_{1,\mathop{\rm loc}}^{j}(X)$ if

[TABLE]

for every orientable domain $U\subset\mathop{Int}X$ and every form $u\in D^{n-j-1}(X)$ having support in $U$.

Let $\Phi_{I}$ and $\Phi_{II}$ be $N$-functions. For $0\leq k\leq n$, put

[TABLE]

This is a Banach space with the norm

[TABLE]

Consider also the spaces

[TABLE]

Denote by $\overline{B}_{\Phi_{I},\Phi_{II}}^{k}(X)$ the closure of $B_{\Phi_{I},\Phi_{II}}^{k}(X)$ in $L^{\Phi_{II}}(X,\Lambda^{k})$.

Definition 3.2**.**

The quotient spaces

[TABLE]

and

[TABLE]

are called the $k$th $L_{\Phi_{I},\Phi_{II}}$-cohomology and the $k$th reduced $L_{\Phi_{I},\Phi_{II}}$-cohomology of the Riemannian manifold $X$, the latter cohomology being a Banach space. Define the $L_{\Phi_{I},\Phi_{II}}$-torsion as

[TABLE]

The torsion $T_{\Phi_{I},\Phi_{II}}^{k}(X)$ can be either $\{0\}$ or infinite-dimensional. In fact, if $\dim T_{\Phi_{I},\Phi_{II}}^{k}(X)<\infty$ then $B_{\Phi_{I},\Phi_{II}}^{k}(X)$ is closed, hence $T_{\Phi_{I},\Phi_{II}}^{k}(X)=\{0\}$. In particular, if $\dim T_{\Phi_{I},\Phi_{II}}^{k}(X)\neq 0$ then $\dim H_{\Phi_{I},\Phi_{II}}^{k}(X)=\infty$.

If $\Phi_{I}=\Phi_{II}=\Phi$ then we use the notations $\Omega_{\Phi}^{k}(X)$, $H_{\Phi}^{k}(X)$, and $\overline{H}_{\Phi}^{k}(X)$ instead of $\Omega_{\Phi,\Phi}^{k}(X)$, $H_{\Phi,\Phi}^{k}(X)$, and $\overline{H}_{\Phi,\Phi}^{k}(X)$ respectively. Thus, the $L_{\Phi}$-cohomology $H_{\Phi}^{k}(X)$ (respectively, the reduced $L_{\Phi}$-cohomology $\overline{H}_{\Phi}^{k}(X)$) is the $k$th cohomology (respectively, the $k$th reduced cohomology) of the cochain complex $\{\Omega_{\Phi}^{*}(X),d\}$.

In [4], Gol*′*dshtein and Troyanov realized the $k$th $L_{q,p}$-cohomology as the $k$th cohomology of some Banach complex. Here we apply this approach to $L_{\Phi_{I},\Phi_{II}}$-cohomology.

Fix an $(n+1)$-tuple of $N$-functions $\mathcal{F}=\{\Phi_{0},\Phi_{1},\dots,\Phi_{n}\}$ and put

[TABLE]

Since the weak exterior differential is a bounded operator $d:\Omega_{\mathcal{F}}^{k}(X)\to\Omega_{\mathcal{F}}^{k+1}(X)$, we obtain a Banach complex

[TABLE]

The $L_{\mathcal{F}}$-cohomology $H^{k}_{\mathcal{F}}(X)$ (respectively, the reduced $L_{\mathcal{F}}$-cohomology ${\overline{H}}^{k}_{\mathcal{F}}(X)$) of $X$ is the $k$th cohomology (respectively, the $k$th reduced cohomology) of the Banach complex $(\Omega_{\mathcal{F}}^{*},d)$.

The above-defined cohomology spaces $H^{k}_{\mathcal{F}}(X)$ and $\overline{H}^{k}_{\mathcal{F}}(X)$ in fact depend only on $\Phi_{k-1}$ and $\Phi_{k}$:

[TABLE]

The results on abstract Banach complexes by Gol*′*dhstein and Troyanov enable us to interprete Orlicz cohomology in terms of a Poincaré–Sobolev–Orlicz type inequality for differential forms on a Riemannian manifold $X$:

Theorem 3.3**.**

$H_{\Phi_{I},\Phi_{II}}^{k}(X)=0$* if and only if there exists a constant $C<\infty$ such that for any closed differential form $\omega\in L^{\Phi_{II}}(X,\Lambda^{k})$ there exists a differential form $\theta\in L^{\Phi_{I}}(X,\Lambda^{k-1})$ such that $d\theta=\omega$ and*

[TABLE]

This result is an immediate consequence of Lemma 2.4.

Theorem 3.4**.**

(A) If $T_{\Phi_{II}}^{k}(X)=0$ then there exists a constant $C^{\prime}$ such that for any differential form $\theta\in\Omega^{k-1}_{\Phi_{I},\Phi_{II}}(X)$ there exists a closed form $\zeta\in Z_{\Phi_{I}}^{k-1}(X)$ such that

[TABLE]

(B) Conversely, if $\Phi_{II}\in\Delta_{2}\cap\nabla_{2}$ and there exists a constant $C^{\prime}$ such that for any form $\theta\in\Omega^{k-1}_{\Phi_{I},\Phi_{II}}(X)$ there exists $\zeta\in Z_{\Phi_{I}}^{k-1}(X)$ such that (3.1) holds then $T_{\Phi_{I},\Phi_{II}}(X)=0$.

Proof.

Considering the Banach complex $\Omega^{*}_{\mathcal{F}}$ with $\mathcal{F}=\{\Phi_{II},\dots,\Phi_{II},\Phi_{I},\dots,\Phi_{I}\}$, where $\Phi_{II}$ changes to $\Phi_{I}$ at the $k$th position, we get

[TABLE]

Since $\Phi_{I}\in\Delta_{2}\cap\nabla_{2}$, the Banach space $\Omega^{k-1}_{\Phi_{II},\Phi_{II}}(X)$ is reflexive. Theorem 3.4 now stems from Lemma 2.5. ∎

4. The $L_{\Phi_{1},\Phi_{2}}$-Cohomology of $\mathbb{R}$

Let $\Phi_{1}$ and $\Phi_{2}$ be $N$-functions.

Proposition 4.1**.**

$T_{\Phi_{1},\Phi_{2}}^{1}(\mathbb{R})\neq 0$.

Proof.

Suppose on the contrary that $T_{\Phi_{1},\Phi_{2}}^{1}(\mathbb{R})=0$. In accordance with Theorem 3.4, then there is a Sobolev inequality for functions on $\mathbb{R}$

[TABLE]

for some real positive constant $C$.

Consider the function

[TABLE]

Here the constant $C$ is chosen so that

[TABLE]

Now, consider the family of smooth functions with compact support $\{f_{a}:\mathbb{R}\to\mathbb{R}:a>0\}$, where

[TABLE]

(we owe this construction to [2, pp. 8–9]). Then $f_{a}(x)=1$ if $x\in[1,a]$, $f_{a}(x)=0$ if $x\not\in[0,a+1]$, and $\|f_{a}^{\prime}\|_{L^{\infty}}=:L<\infty$. Clearly, $\|f_{a}-z\|_{(\Phi_{1})}$ is finite only for $z=0$. Estimate the Orlicz norms involved in (4.1). We have

[TABLE]

If $\rho_{\Phi_{1}}\left(\frac{\Phi_{1}}{K}\right)\leq 1$ then $(a-1)\Phi_{1}\left(\frac{1}{K}\right)\leq 1$, which is equivalent to

[TABLE]

Hence,

[TABLE]

On the other hand,

[TABLE]

We have

[TABLE]

Put $\mathcal{M}_{f^{\prime},a}=\left\{K:\rho_{\Phi_{2}}\left(\frac{f^{\prime}_{a}}{K}\right)\leq 1\right\}$. We have shown that if $K\geq\frac{L}{\Phi_{2}^{-1}\left(\frac{1}{2}\right)}$ then $K\in\mathcal{M}_{f^{\prime},a}$. Therefore,

[TABLE]

Thus,

[TABLE]

The obtained contradiction proves the proposition. ∎

Corollary 4.2**.**

If $\Phi_{1}$ and $\Phi_{2}$ are $N$-functions then the space $H^{1}_{\Phi_{1},\Phi_{2}}(\mathbb{R})$ is not separated; in particular, $H^{1}_{\Phi_{1},\Phi_{2}}(\mathbb{R})\neq 0$.

Proposition 4.3**.**

If $\Phi_{1}$ and $\Phi_{2}$ are $N$-functions and $\Phi_{2}\in\Delta_{2}$ then $\overline{H}^{1}_{\Phi_{1},\Phi_{2}}(\mathbb{R})=0$.

Proof.

Let $\omega=a(x)dx\in L^{\Phi_{2}}(\mathbb{R})$. For each $n$, put

[TABLE]

If $C_{m}=0$ then put $\lambda_{m}(x)\equiv 0$ for all $x\in\mathbb{R}$. If $C_{m}\neq 0$ then put

[TABLE]

where $\varepsilon_{m}=t_{m}/m$ and $t_{m}$ is the only root of the equation

[TABLE]

(The function $t\mapsto\Phi_{2}(t)/t$ is strictly increasing; see, for example, [9]). We obviously have

[TABLE]

Compute the norm $\|\lambda_{m}\|_{(\Phi_{2})}$. We have

[TABLE]

Thus,

[TABLE]

Here $\Phi_{2}^{-1}$ stands for the inverse function to $\Phi_{2}:[0,\infty)\to[0,\infty)$. Hence, $\|\lambda_{m}\|_{(\Phi_{2})}=\frac{\varepsilon_{m}}{\Phi_{2}^{-1}(\varepsilon_{m}/|C_{m}|)}$. By the choice of $\varepsilon_{m}$,

[TABLE]

and so $\|\lambda_{m}\|_{(\Phi_{2})}=\frac{1}{m}$.

Let $b_{m}(x):=\int_{-\infty}^{x}\left(\chi_{[-m,m]}(t)a(t)-\lambda_{m}(t)\right)dt$. Since $b_{m}$ has compact support, $b_{m}\in L^{\Phi_{1}}(\mathbb{R})$ for each $m$. Furthermore, $\|db_{m}-\omega\|_{(\Phi_{2})}\leq\|a\|_{L^{(\Phi_{2})}(\mathbb{R}\setminus[-m,m])}+\|\lambda_{m}\|_{L^{(\Phi_{2})}(\mathbb{R})}\to 0$ as $m\to\infty$ since for $\Phi_{2}\in\Delta_{2}$ all functions in $L^{\Phi_{2}}$ have absolutely continuous norm ([9, Theorem 10.3]). Thus, $\overline{H}^{1}_{\Phi_{1},\Phi_{2}}(\mathbb{R})=0$. ∎

All the results of this section are also valid for the half-line $\mathbb{R}_{+}$ (with similar proofs).

5. The $L_{\Phi_{1},\Phi_{2}}$-Cohomology of the Hyperbolic Plane

We will need the following Orlicz versions of Propositions 8.3 and 8.4 in [4], which are proved in absolutely the same manner:

Proposition 5.1**.**

Let $M$ be a complete manifold of dimension $n$ and let $(\Phi_{1},\Psi_{1})$ and $(\Phi_{2},\Psi_{2})$ be two pairs of complementary Orlicz functions. Suppose that $\alpha\in Z_{\Phi_{2}}^{k}(X)$ and there exists a smooth closed $(n-k)$-form $\gamma$ such that $\gamma\in Z_{\Psi_{1}}^{n-k}(X)$, $\gamma\wedge\alpha\in L^{1}(X,\Lambda^{n})$, and

[TABLE]

then $\alpha\notin B_{\Phi_{1},\Phi_{2}}^{k}(X)$. In particular, $H_{\Phi_{1},\Phi_{2}}^{k}(X)\neq 0$.

Proposition 5.2**.**

Let $M$ be a complete manifold of dimension $n$ and let $(\Phi_{1},\Psi_{1})$ and $(\Phi_{2},\Psi_{2})$ be two pairs of complementary Orlicz functions. Suppose that $\alpha\in Z_{\Phi_{2}}^{k}(X)$ and there exists a smooth closed $(n-k)$-form $\gamma\in Z_{\Psi_{1}}^{n-k}(X)\cap Z_{\Psi_{2}}^{n-k}(X)$ such that

[TABLE]

then $\alpha\notin\overline{B}_{\Phi_{1},\Phi_{2}}^{k}(X)$. In particular, $\overline{H}_{\Phi_{1},\Phi_{2}}^{k}(X)\neq 0$.

The hyperbolic plane $\mathbb{H}^{2}$ is the Riemannian manifold that can be modelled as the space $\mathbb{R}^{2}$ endowed with the Riemannian metric

[TABLE]

For an $N$-function $\Phi$, introduce the condition

[TABLE]

(The upper integration limit $1$ can be replaced by any positive number.)

Theorem 5.3**.**

If $\Phi_{1}$ and $\Phi_{2}$ are $N$-functions such that their complementary $N$-functions $\Psi_{1}$ and $\Psi_{2}$ and the function $\Phi_{2}$ satisfy condition $(A)$ then

[TABLE]

We will need the following lemma, which is in fact Lemma 10.2 in [4]:

Lemma 5.4**.**

There exist two smooth functions $f$ and $g$ on $\mathbb{H}^{2}$ such that

(1)* $f$ and $g$ are nonnegative;*

(2)* $f(y,z)=g(y,z)=0$ if $z\leq 0$ or $|y|\geq 1$;*

(3)* $df$ and $dg\in L^{r}(\mathbb{H}^{2},\Lambda^{1})$ for any $1<r\leq\infty$;*

(4)* the support of $df\wedge dg$ is contained in $\{(y,z):|y|\leq 1\,,\,0\leq z\leq 1\}$;*

(5)* $df\wedge dg\geq 0$;*

(6)* $\int_{\mathbb{H}^{2}}df\wedge dg=1$;*

(7)* $\frac{\partial f}{\partial y}$ and $\frac{\partial g}{\partial y}\in L^{\infty}(\mathbb{H}^{2})$;*

(8)* $\frac{\partial f}{\partial z}$ and $\frac{\partial g}{\partial z}$ have compact support.*

We will also need the following generalization of item (3) above:

Lemma 5.5**.**

If $\Phi$ is an $N$-function satisfying condition $(A)$ then $df,\,dg\in L^{\Phi}(\mathbb{H}^{2},\Lambda^{1})$.

Proof.

Recall the construction of the functions $f$ and $g$ of [4, Lemma 10.2].

Choose smooth functions $h_{1}$, $h_{2}$, and $k:\mathbb{R}\rightarrow\mathbb{R}$ with the following properties:

(1) $h_{1}$, $h_{2}$, and $k$ are nonnegative;

(2) $h_{i}(y)=0$ if $|y|\geq 1$;

(3) $h^{\prime}_{1}(y)h_{2}(y)\geq 0$ and $h_{1}(y)h^{\prime}_{2}(y)\leq 0$ for all $y$;

(4) the support of the function $h^{\prime}_{1}(y)h_{2}(y)-h_{1}(y)h^{\prime}_{2}(y)$ is not empty;

(5) $k^{\prime}(z)\geq 0$ for all $z$;

(6) $k(z)=1$ if $z\geq 1$ and $k(z)=0$ if $z\leq 0$.

Then $f$ and $g$ are defined as $f(y,z):=h_{1}(y)k(z)$ and $g(y,z):=h_{2}(y)k(z)$ respectively.

We will now prove that $df\in L^{\Phi}$ by modifying the argument of the proof of [4, Lemma 10.2].

Indeed,

[TABLE]

The first summand $h_{1}(y)k^{\prime}(z)dz$ has compact support, and the second summand $k(z)h^{\prime}_{1}(y)dy$ is zero outside the infinite rectangle $Q=\{|y|\leq 1\,;z\geq 0\}$.

Choose $D<\infty$ such that $|k(z)h^{\prime}_{1}(y)|\leq D$ on $Q$. We have

[TABLE]

Since the area element of $\mathbb{H}^{2}$ is $dA=e^{z}dydz$, for any $a>0$ we infer

[TABLE]

Putting $aDe^{-z}=v$ in the last integral, we get

[TABLE]

Thus, $\rho_{\Phi}(ak(z)h^{\prime}_{1}(y)\,dy)<\infty$ for any $a$. Consequently, $k(z)h^{\prime}_{1}v\in L^{\Phi}(\mathbb{H}^{2},\Lambda^{1})$. Thus, $df=h_{1}(y)k^{\prime}(z)dz+k(z)h^{\prime}_{1}(y)dy$ also lies in $L^{\Phi}(\mathbb{H}^{2},\Lambda^{k})$.

The lemma is proved. ∎

Proof of Theorem 5.3. Take the functions $f$ and $g$ on $\mathbb{H}^{2}$ defined in Lemma 5.4 and consider the $1$-forms $\alpha=df$ and $\gamma=dg$ on $\mathbb{H}^{2}$. Obviously, $d\alpha=d\gamma=0$. By Lemmas 5.4 and 5.5, $\alpha\in L^{\Phi}$ for any $N$-function $\Phi$ such that $\int_{0}^{1}\Phi(v)/v^{2}\,dv<\infty$ and $\gamma$ is smooth and $\gamma\in L^{\Psi_{1}}\cap L^{\Psi_{2}}$ if $\int_{0}^{1}\Psi_{1}(v)/v^{2}\,dv<\infty$ and $\int_{0}^{1}\Psi_{2}(v)/v^{2}\,dv<\infty$.

Since $\int_{\mathbb{H}^{2}}\alpha\wedge\gamma\neq 0$, Proposition 5.1 shows that $\alpha\not\in\overline{B}_{\Phi_{1},\Phi_{2}}^{1}(\mathbb{H}^{2})$.

Now, using the isometry group of $\mathbb{H}^{2}$, we obtain an infinite family of linearly independent classes in $\overline{H}_{\Phi_{1},\Phi_{2}}^{1}(\mathbb{H}^{2})$. ∎

6. The $L_{\Phi}$-Cohomology of the Ball

In this section, we prove the “$L^{\Phi}$-Poincaré lemma”, i.e., the vanishing of the $L^{\Phi}$-cohomology of the unit ball $\mathbb{B}^{n}\subset\mathbb{R}^{n}$.

Since $\mathbb{B}^{n}$ has finite volume, $H_{\Phi_{1},\Phi_{2}}^{0}(\mathbb{B}^{n})=\overline{H}_{\Phi_{1},\Phi_{2}}^{0}(\mathbb{B}^{n})=\mathbb{R}$ for any $N$-functions $\Phi_{1}$ and $\Phi_{2}$.

For the case of $L^{p}$ spaces, Gol*′dshtein, Kuz′minov, and Shvedov proved the vanishing of the $L^{p}$-cohomology of the ball in [3, Lemma 3.2]; for $p\neq q$, Gol′*dshtein and Troyanov found necessary and sufficient conditions on $p$ and $q$ for the vanishing of the $L^{q,p}$-cohomology of $\mathbb{B}^{n}$. Their proof is based on the following fact, established by Iwaniec and Lutoborski in [5]:

Proposition 6.1**.**

For any bounded convex domain $U\subset\mathbb{R}^{n}$ and any $k=1,2,\dots,n$, there exists an operator

[TABLE]

with the following properties:

1. (a)

$T(d\theta)+dT\theta=\theta$* (in the sense of currents);* 2. (b)

$\displaystyle\left|T\theta(x)\right|\leq C\int_{U}\frac{|\theta(y)|}{\;\>|y-x|^{n-1}}dy$.

∎

We prove

Corollary 6.2**.**

If $\Phi$ is an $N$-function then the operator $T$ maps $L^{\Phi}(U,\Lambda^{k})$ continuously into $L^{\Phi}(U,\Lambda^{k-1})$.

Proof.

The following Orlicz space version of Young’s inequality for convolution holds (see the proof of Corollary 7 in [10, pp. 230–231]111Though it is required in Corollary 7 in [10, pp. 230–231] that $\Phi\in\Delta_{2}$, the proof of Young’s inequality works for general $N$-functions $\Phi$.: If $f\in L^{\Phi}$ and $g\in L^{1}$ then $f*g\in L^{\Phi}$ and

[TABLE]

Applying this inequality to $f=|\theta|$ and $g(x)=|x|^{1-n}$, we obtain the corollary from Proposition 6.1. In the Orlicz norms, the norm of the operator $T$ is bounded by $\|g\|_{1}$. ∎

Corollary 6.3**.**

The operator $T:\Omega_{\Phi}(U,\Lambda^{k})\to\Omega_{\Phi}(U,\Lambda^{k-1})$ is bounded and $Td\omega+dT\omega=\omega$ for any $\omega\in\Omega_{\Phi}^{k}(U)$.

Corollary 6.3 gives the following

Theorem 6.4**.**

If $\Phi$ is an $N$-function then $H_{\Phi}^{k}(\mathbb{B}^{n})=0$ for all $k=1,...,n$.

Proof.

Let $\omega\in Z_{\Phi}^{k}(\mathbb{B}^{n})$. By Corollary 6.3, $T\omega\in L^{\Phi}(\mathbb{B}^{n},\Lambda^{k+1})$. Since $\omega=dT\omega+Td\omega=d(T\omega)$, we conclude that $[\omega]=[dT\omega]=0\in H_{\Phi}^{k}(\mathbb{B}^{n})$ and so $H_{\Phi}^{k}(\mathbb{B}^{n})=0$. ∎