The Sobolev--Poincar\'e Inequality and the $L_{q,p}$-Cohomology of Twisted Cylinders

TL;DR

This paper proves a new vanishing theorem for the $L_{q,p}$-cohomology of twisted cylinders, extending previous results for warped cylinders, using advanced Sobolev--Poincaré inequality techniques.

Contribution

It introduces a vanishing result for $L_{q,p}$-cohomology of twisted cylinders, generalizing known results for warped cylinders with novel methods.

Findings

Vanishing of $L_{q,p}$-cohomology for twisted cylinders.

Extension of Sobolev--Poincaré inequality methods.

New results even for warped cylinders.

Abstract

We establish a vanishing result for the $L_{q,p}$-cohomology ($q\ge p$) of a twisted cylinder, which is a generalization of a warped cylinder. The result is new even for warped cylinders. We base on the methods for proving the $(p,q)$ Sobolev--Poincar\'e inequality developed by L.~Shartser.

Equations

$$g:=g_X+h^2(x,y)g_Y,$$

$$g=d{t}^2+h^2(t)g_N,$$

$${\int }_M\theta \wedge \omega =(-1{)}^{k+1}{\int }_M\varphi \wedge d\omega$$

$${\Omega }_{q,p}^k(M)=\left\{\omega \in L^q(M,{\Lambda }^k)|d\omega \in L^p(M,{\Lambda }^{k+1})\right\}.$$

$$\Vert \omega {\Vert }_{q,p}={\left(\Vert \omega {\Vert }_{L^q(M,{\Lambda }^k)}^2+\Vert d\omega {\Vert }_{L^p(M,{\Lambda }^{k+1})}^2\right)}^{1/2}.$$

$$B_{q,p}^k(M)\subset {\overline{B}}_{q,p}^k(M)\subset Z_p^k(M)={\overline{Z}}_p^k(M)\subset L^p(M,{\Lambda }^k).$$

$$H_{q,p}^k(M):=Z_p^k(M)/B_{q,p}^k(M),$$

$${\overline{H}}_{q,p}^k(M):=Z_p^k(M)/{\overline{B}}_{q,p}^k(M).$$

$$C^{\infty }{L}^p(M,{\Lambda }^k):=C^{\infty }(M,{\Lambda }^k)\cap L^p(M,{\Lambda }^k);$$

$$C^{\infty }{L}^p(M,{\Lambda }^k,\sigma ):=C^{\infty }(M,{\Lambda }^k)\cap L^p(M,{\Lambda }^k,\sigma );$$

$$C^{\infty }{\Omega }_{q,p}^k(M):=C^{\infty }(M,{\Lambda }^k)\cap {\Omega }_{q,p}^k(M);$$

$$C^{\infty }{H}_{q,p}^k(M):=\frac{C^{\infty }(M,{\Lambda }^k)\cap Z_p^k(M)}{C^{\infty }(M,{\Lambda }^k)\cap B_{q,p}^k(M)};$$

$$C^{\infty }{\overline{H}}_{q,p}^k(M):=\frac{C^{\infty }(M,{\Lambda }^k)\cap Z_p^k(M)}{C^{\infty }(M,{\Lambda }^k)\cap {\overline{B}}_{q,p}^k(M)}.$$

$$|\omega(t,x)|_{C_{a,b}^{h}N}=\bigl{[}h^{-2k}(t,x)|\omega_{A}(t,x)|_{N}^{2}+h^{-2(k+1)}(t,x)|\omega_{B}(t,x)|_{N}^{2}\bigr{]}^{1/2}$$

$$\|\omega\|_{L^{p}(C_{a,b}^{h}N,\Lambda^{k})}\\ =\left[\!\int_{a}^{b}\int_{N}\bigl{(}h^{2(\frac{n}{p}-k)}(t,x)|\omega_{A}(t,x)|_{N}^{2}\!+h^{2(\frac{n}{p}-k+1)}(t,x)|\omega_{B}(t,x)|_{N}^{2}\bigr{)}^{\frac{p}{2}}dxdt\!\right]^{\frac{1}{p}}.$$

$$f_{k,p}(t)=\min_{x\in N}\bigl{\{}h^{\frac{n}{p}-k}(t,x)\bigr{\}}$$

$$F_{k,p}(t)=\max_{x\in N}\bigl{\{}h^{\frac{n}{p}-k}(t,x)\bigr{\}}.$$

$$\psi {}_y^{\ast }\omega (x,t)={\left(\psi {}_y^{\ast }\omega \right)}_0(x,t)+dt\wedge {\left(\psi {}_y^{\ast }\omega \right)}_1(x,t),$$

$$K_y:{\Omega }_{loc}^k(D)\to {\Omega }_{loc}^{k-1}(D)$$

$$K_y\omega (x):={\int }_0^1{\left(\psi {}_y^{\ast }\omega \right)}_1(t)dt$$

$$C(k,p,q,n,\beta ):={\int }_0^1\underset{z\in D}{\sup }\Vert \beta (x)1_{tx+(1-t)D}(z){\Vert }_{L^q(D,dx)}{t}^k(1-t{)}^{-n/p}dt<\infty$$

$${\Vert \beta (x){\Vert \frac{K_{y}d\omega (x)}{\left|x-y\right|}\Vert }_{L^p(D,dy)}\Vert }_{L^q(D,dx)}\le C(k,p,q,n,\beta ){\Vert d\omega \Vert }_{L^p(D,{\Lambda }^{k+1})}.$$

$$\begin{gathered} {\Vert \beta (x){\Vert \frac{K_{y}d\omega (x)}{\left|x-y\right|}\Vert }_{L^p(D,dy)}\Vert }_{L^q(D,dx)}={\Vert \beta (x){\Vert {\int }_0^1\frac{{\left(\psi {}_y^{\ast }d\omega \right)}_1(x,t)}{\left|x-y\right|}dt\Vert }_{L^p(D,dy)}\Vert }_{L^q(D,dx)} \\ \le {\int }_0^1{\Vert \beta (x){\Vert \frac{{\left(\psi {}_y^{\ast }d\omega \right)}_1(x,t)}{\left|x-y\right|}\Vert }_{L^p(D,dy)}\Vert }_{L^q(D,dx)}dt \\ \le {\int }_0^1{\left\{{\int }_D{\beta }^q(x){\left[{\int }_D\frac{{\left|{\left(\psi {}_y^{\ast }d\omega \right)}_1(x,t)\right|}^p}{{\left|x-y\right|}^p}dy\right]}^{q/p}dx\right\}}^{1/q}dt. \end{gathered}$$

$$\left|{\left({\psi }_y^{\ast }d\omega \right)}_1(x,t)\right|\le \left|x-y\right|t^k\left|d\omega \left({\psi }_y(x,t)\right)\right|.$$

$$\begin{gathered} {\int }_0^1{\left\{{\int }_D{\beta }^q(x){\left[{\int }_D\frac{{\left|{\left({\psi }_y^{\ast }d\omega \right)}_1(x,t)\right|}^p}{{\left|x-y\right|}^p}dy\right]}^{q/p}dx\right\}}^{1/q}dt \\ \le {\int }_0^1{\left\{{\int }_D{\beta }^q(x){\left[{\int }_D{t}^{kp}{\left|d\omega \left({\psi }_y(x,t)\right)\right|}^{p}dy\right]}^{q/p}dx\right\}}^{1/q}dt \\ ={\int }_0^1{\left\{{\int }_D{\beta }^q(x){\left[{\int }_D{t}^{kp}{\left|d\omega \left(tx+(1-t)y\right)\right|}^{p}dy\right]}^{q/p}dx\right\}}^{1/q}dt:=I. \end{gathered}$$

$$I={\int }_0^1{\left\{{\int }_D{\beta }^q(x){\left[{\int }_{tx+(1-t)D}{\left|d\omega (z)\right|}^{p}dz\right]}^{q/p}dx\right\}}^{1/q}{t}^k(1-t{)}^{-n/p}dt$$

$${\left\{{\int }_D{\beta }^q(x){\left[{\int }_{tx+(1-t)D}{\left|d\omega (z)\right|}^{p}dz\right]}^{q/p}dx\right\}}^{1/q}$$

$$={\left\{{\int }_D{\beta }^q(x){\left[{\int }_D{1}_{tx+(1-t)D}(z){\left|d\omega (z)\right|}^{p}dz\right]}^{q/p}dx\right\}}^{1/q}$$

$$={\left\{{\left({\int }_D{\left[{\int }_D{\beta }^p(x)1_{tx+(1-t)D}(z){\left|d\omega (z)\right|}^{p}dz\right]}^{q/p}dx\right)}^{p/q}\right\}}^{1/p}$$

$$={\left\{{\Vert {\int }_D{\beta }^p(x)1_{tx+(1-t)D}(z){\left|d\omega (z)\right|}^{p}dz\Vert }_{L^{q/p}(D,dx)}\right\}}^{1/p}$$

$$\le {\left\{{\int }_D{\Vert {\beta }^p(x)1_{tx+(1-t)D}(z){\left|d\omega (z)\right|}^p\Vert }_{L^{q/p}(D,dx)}dz\right\}}^{1/p}$$

$$={\left\{{\int }_D{\left({\int }_D{\beta }^q(x)1_{tx+(1-t)D}(z){\left|d\omega (z)\right|}^{q}dx\right)}^{p/q}dz\right\}}^{1/p}$$

$$={\left\{{\int }_D{\left({\int }_D{\beta }^q(x)1_{tx+(1-t)D}(z)dx\right)}^{p/q}{\left|d\omega (z)\right|}^{p}dz\right\}}^{1/p}$$

$$\le {\left(\underset{z\in D}{\sup }{\int }_D{\beta }^q(x)1_{tx+(1-t)D}(z)dx\right)}^{1/q}{\left({\int }_D{\left|d\omega (z)\right|}^{p}dz\right)}^{1/p}$$

$$=\underset{z\in D}{\sup }\Vert \beta (x)1_{tx+(1-t)D}(z){\Vert }_{L^q(D,dx)}{\Vert d\omega \Vert }_{L^p(D,{\Lambda }^{k+1})}.$$

$$C(k,p,q,n,\beta )={\int }_0^1\underset{z\in D}{\sup }\Vert \beta (x)1_{tx+(1-t)D}(z){\Vert }_{L^q(D,dx)}{t}^k(1-t{)}^{-n/p}dt$$

Full text

Mathematics Subject Classification. 58A12, 46E30, 55N05.

Key words and phrases: differential form, $L_{q,p}$-cohomology, twisted cylinder, homotopy operator

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

1. Introduction

The $L_{q,p}$-cohomology $H_{q,p}^{k}(M)$ of a Riemannian manifold $(M,g)$ is, by definition, the quotient of the space of closed $p$-integrable differential $k$-forms by the exterior differentials of $q$-integrable $k$-forms. If $p=q$ then $L_{q,p}$-cohomology is usually referred to simply as $L_{p}$-cohomology and the index $p$ is used instead of $p,p$ in all the notations.

A twisted product $X\times_{h}Y$ of two Riemannian manifolds $(X,g_{X})$ and $(Y,g_{Y})$ is the direct product manifold $X\times_{g}Y$ endowed with a Riemannian metric of the form

[TABLE]

where $h:X\times Y\to\mathbb{R}$ is a smooth positive function (see [5]). If $X$ is a half-interval $[a,b)$ then the twisted product $X\times_{h}Y$ is called a twisted cylinder.

We refer to an $m$-dimensional Riemannian manifold $(M,g_{M})$ as an asymptotic twisted product (respectively, as an asymptotic twisted cylinder) if, outside an $m$-dimensional compact submanifold, it is bi-Lipschitz equivalent to a twisted product (respectively, to a twisted cylinder).

In this paper, we prove some vanishing results for the $L_{q,p}$-cohomology of twisted cylinders $[a,b)\times_{h}N$ for a positive smooth function $h:[a,b)\times N\to\mathbb{R}$ in the case where the base $N$ is a closed manifold and $p\geq q>1$, $\frac{1}{p}-\frac{1}{q}<\frac{q-1}{q(\dim N+1)}$.

If in (1) the function $h$ depends only on $x$ then we obtain the familiar notion of a warped product (see [1]). Twisted products were the object of recent investigations [4, 6, 8, 9, 10, 16, 20]. The $L_{q,p}$-cohomology of warped cylinders $[a,b)\times_{h}N$, i.e., of product manifolds $[a,b)\times N$ endowed with a warped product metric

[TABLE]

where $g_{N}$ is the Riemannian metric of $N$ and $h:[a,b)\to\mathbb{R}$ is a positive smooth function, was studied by Gol*′dshtein, Kuz′minov, and Shvedov [11], Kuz′*minov and Shvedov [18, 19] (for $p=q$), and Kopylov [17] for $p,q\in[1,\infty)$, $\frac{1}{p}-\frac{1}{q}<\frac{1}{\dim\,N+1}$.

The main result of the paper (Theorem 7.1) states that the $L_{q,p}$-cohomology $H^{k}_{q,p}(C_{a,b}^{h}N)$ of the twisted cylinder $C_{a,b}^{h}N$ with $q\geq p\geq 1$ and $\frac{1}{p}-\frac{1}{q}<\frac{q-1}{q(\dim N+1)}$ is zero provided that the de Rham cohomology $H^{k}_{\mathrm{DR}}(N)$ of the base $N$ is trivial and some integral conditions on the twisting function involving $p$, $q$ and an auxiliary parameter $\overline{p}$ are fulfilled.

The paper is organized as follows: In Sec. 2, we recall some basic definitions concerning the $L_{q,p}$-cohomology of Riemannian manifolds. Sec. 3 describes the representations of differential forms on a twisted cylinder obtained in [10] and analogous to the representations of forms on a warped product proposed by Gol*′dshtein, Kuz′*minov, and Shvedov in [12]. In Sec. 4, we develop a version of the weighted Sobolev–Poncaré inequality for convex sets in $\mathbb{R}^{n}$ by introducing a homotopy operator and consider some of its consequences; the exposition is based on the ideas of Shartser suggested in [21] and [22]. In Sec. 5, we consider a new homotopy operator $A_{\alpha}$ on differential forms defined on a convex domain in $\mathbb{R}^{n}$ and show that it guarantees the fulfillment of an inequality of Sobolev–Poincaré-type for $q\geq p\geq 1$ and $\frac{1}{p}-\frac{1}{q}<\frac{1}{n}$. In Sec. 6, using the ideas of Shartser’s article [22], we “glue” local homotopy operators on a twisted cylinder to obtain a global homotopy operator. In Sec. 7, we use this global homotopy operator for proving our above-metioned main result on the triviality of the $L_{q,p}$-cohomology of a twisted cylinder (Theorem 7.1), and in Sec. 8, we extend this theorem to asymptotic twisted cylinders (Theorem 8.2). Sec. 9 contains some examples.

2. Basic Definitions

We recall the main definitions and notations.

Below we tacitly assume all manifolds to be oriented.

Let $M$ be a smooth oriented Riemannian manifold. Denote by $\mathcal{D}^{k}(M):=C_{0}^{\infty}(M,\Lambda^{k})$ the space of all smooth differential $k$-forms with compact support contained in $M\setminus\partial M$ denote by $L_{loc}^{1}(M,\Lambda^{k})$ the space of locally integrable differential forms.

Denote by $L^{p}(M,\Lambda^{k})$ the Banach space of locally integrable differential $k$-forms endowed with the norm $\|\theta\|_{L^{p}(M,\Lambda^{k})}:=\left(\int_{M}|\theta|^{p}dx\right)^{\frac{1}{p}}<\infty$ (as usual, we identify forms coinciding outside a set of measure zero). Of course, we can add a positive (smooth) weight $\sigma:M\to\mathbb{R}$ and thus integrate $|\theta|^{p}\sigma^{p}$ to obtain the weighted $L^{p}$-space $L^{p}(M,\Lambda^{k},\sigma)$.

Definition 2.1**.**

We call a differential $(k+1)$-form $\theta\in L^{1}_{loc}(M,\Lambda^{k+1})$ the weak exterior derivative (or differential) of a differential $k$-form $\phi\in L^{1}_{\mathrm{loc}}(M,\Lambda^{k})$ and write $d\phi=\theta$ if

[TABLE]

for any $\omega\in\mathcal{D}^{n-k}(M)$.

Remark 2.2*.*

Note that the orientability of $M$ is not substantial in this definition since one may take integrals over orientable domains on $M$ instead of integrals over $M$.

We then introduce an analog of Sobolev spaces for differential $k$-forms, i.e., the space of $q$-integrable forms with $p$-integrable weak exterior derivative:

[TABLE]

This is a Banach space for the graph norm

[TABLE]

The space $\Omega_{q,p}^{k}(M)$ is a reflexive Banach space for any $1<q,p<\infty$. This can be proved using standard arguments of functional analysis.

We now define our basic ingredients (for three parameters $r,q,p$).

Definition 2.3**.**

Put

(a) $Z_{p,r}^{k}(M)=\mathrm{Ker}[d:\Omega_{p,r}^{k}(M)\to L^{r}(M,\Lambda^{k+1})]$.

(b) $B_{q,p}^{k}(M)=\mathrm{Im}[d:\Omega_{q,p}^{k-1}(M)\to L^{p}(M,\Lambda^{k})]$.

The subspace $Z_{p,r}^{k}(M)$ does not depend on $r$ and is a closed subspace in $L^{p}(M,\Lambda^{k})$ (see Lemma [14, Lemma 2.4(i)]). This allows us to use the notation $Z_{p}^{k}(M)$ for all $Z_{p,r}^{k}(M)$. Note that $Z_{p}^{k}(M)\subset L^{p}(M,\Lambda^{k})$ is always a closed subspace but that is in general not true for $B_{q,p}^{k}(M)$. Denote by $\overline{B}_{q,p}^{k}(M)$ its closure in the $L^{p}$-topology. Observe also that since $d\circ d=0$, one has $\overline{B}_{q,p}^{k}(M)\subset Z_{p}^{k}(M)$. Thus,

[TABLE]

Definition 2.4**.**

Suppose that $1\leq q,p\leq\infty$. The $L_{q,p}$-cohomology of $(M,g)$ is defined as the quotient

[TABLE]

and the reduced $L_{q,p}$-cohomology of $(M,g)$ is, by definition, the space

[TABLE]

Since $B_{p,q}^{k}$ is not always closed, the $L_{p}$-cohomology is in general a (non-Hausdorff) semi-normed space, while the reduced $L_{p}$-cohomology is a Banach space.

Below $|X|$ stands for the volume of a Riemannian manifold $(X,g)$.

It follows from the results of [13] that, under suitable assumptions on $p,q$, the $L_{q,p}$-cohomology of a Riemannian manifold $M$ can be expressed in terms of smooth forms.

Let $C^{\infty}(M,\Lambda^{k})$ be the space of smooth $k$-forms on $M$.

Introduce the notations:

[TABLE]

Theorem 2.5**.**

[13, Theorem 12.5 and 12.8, Corollary 12.9].* Let $(M,g)$ be a $n$-dimensional Riemannian manifold and suppose the fulfillment of one of the following conditions:*

$\bullet$* $p,q\in(1,\infty)$ and $\frac{1}{p}-\frac{1}{q}\leq\frac{1}{n}$;*

$\bullet$* $p,q\in[1,\infty)$ and $\frac{1}{p}-\frac{1}{q}<\frac{1}{n}$.*

Then the cohomology $H^{*}_{q,p}(M)$ can be represented by smooth forms, and thus $H^{*}_{q,p}(M)=C^{\infty}H_{q,p}^{*}(M)$.

More exactly, any closed form in $Z^{k}_{p}(M)$ is cohomologous to a smooth form in $L^{p}(M)$. Furthermore, if two smooth closed forms $\alpha,\beta\in C^{\infty}(M,\Lambda^{k})\cap Z^{k}_{p}(M)$ are cohomologous modulo $d\Omega_{q,p}^{k-1}(M)$ then they are cohomologous modulo $dC^{\infty}\Omega_{q,p}^{k-1}(M)$.

Similarly, any reduced cohomology class can be represented by a smooth form.

3. Differential Forms on a Twisted Cylinder

From now on, $C_{a,b}^{h}N$ is the twisted cylinder $[a,b)\times_{h}N$, that is, the product of a half-interval $[a,b)$ and a closed smooth $n$-dimensional Riemannian manifold $(N,g_{N})$ equipped with the Riemannian metric $dt^{2}+h^{2}(t,x)g_{N}$, where $h:[a,b)\times N\to\mathbb{R}$ is a smooth positive function.

Every differential form on $[a,b)\times N$ admits a unique representation of the form $\omega=\omega_{A}+dt\wedge\omega_{B}$, where the forms $\omega_{0}$ and $\omega_{1}$ do not contain $dt$ (cf. [12]). It means that $\omega_{0}$ and $\omega_{1}$ can be viewed as one-parameter families $\omega_{A}(t)$ and $\omega_{B}(t)$, $t\in I$, of differential forms on $N$.

The modulus of a form $\omega$ of degree $k$ on $C_{a,b}^{h}N$ is expressed via the moduli of $\omega_{A}(t)$ and $\omega_{B}(t)$ on $N$ as follows:

[TABLE]

Consequently,

[TABLE]

Put

[TABLE]

and

[TABLE]

4. The Weighted Sobolev–Poincare Inequality for Convex Sets in $\mathbb{R}^{n}$

Denote by $\Omega_{loc}^{*}(M)$ the space all locally integrable differential forms with locally integrable weak differential.

Suppose that $D\subset R^{n}$ is a convex set and $\psi_{y}:D\times[0,1]\to D$, $\psi_{y}(x,t):=tx+(1-t)y$, is the homotopy induced by the convex structure. For a $k$-form $\omega\in\Omega_{loc}^{k}(D)$ the pullback $\psi{}_{y}^{*}\omega$ can be written in the form

[TABLE]

where $\left(\psi{}_{y}^{*}\omega\right)_{0}$ and $\left(\psi{}_{y}^{*}\omega\right)_{1}$ do not contain $dt$.

For each $y\in D$ define a homotopy operator

[TABLE]

as follows:

[TABLE]

It is easy to see that $K_{y}$ takes smooth forms to smooth forms. It is proved in [15] that $K_{y}d\omega+dK_{y}\omega=\omega$ The following proposition is a generalization of results from [2] and Shartser’s thesis [21] (see also [22]) to the weighted case and to unbounded convex domains.

Proposition 4.1**.**

Suppose that $D$ is a convex set in $\mathbb{R}^{n}$, $q\geq p\geq 1$, and $\beta:D\to\mathbb{R}$ is a positive smooth function.

If the inequality

[TABLE]

holds then the inequality

[TABLE]

is valid for every $\omega\in\Omega_{loc}^{k}(D)$ such that $d\omega\in L^{p}(D,\Lambda^{k+1})$. Here $\mathbf{1}_{xt+(1-t)D}$ is the characteristic function of the set $xt+(1-t)D$.

Proof.

By the definition of $K_{y},$ we have

[TABLE]

As usual, we identify the tangent space to $\mathbb{R}^{n}$ at any of its points with $\mathbb{R}^{n}.$

By easy calculations,

[TABLE]

Therefore,

[TABLE]

The change of variables $z=tx+(1-t)y$ in the inner integral yields

[TABLE]

Since $D$ is convex, the set $tx+(1-t)D$ is contained in $D$ for all $x\in D$ and $t\in[0,1]$. Using Minkowski’s integral inequality, we infer

[TABLE]

The proposition follows. ∎

Estimate

[TABLE]

in particular cases.

Corollary 4.2**.**

Suppose that $D$ is a convex set of finite measure in $\mathbb{R}^{n}$, $q\geq p\geq 1$, $\frac{1}{p}-\frac{1}{q}<\frac{1}{n}$, and the weight $\beta(x)\equiv 1$. Then

[TABLE]

Remark 4.3*.*

It is easy to see that the integral of the corollary exists because of the conditions imposed on $p$ and $q$.

Proof.

Using the change of variables $u=tx$, we obtain

[TABLE]

Note that $\left|tD\cap\left\{u+(1-t)D\right\}\right|\leq\left|D\right|\min(t^{n},(1-t)^{n})$. It follows that

[TABLE]

∎

Corollary 4.4**.**

Suppose that $U$ is a convex set of finite measure $|U|$ in $\mathbb{R}^{n}$, $D=[a,b)\times U$, $q\geq p\geq 1$, $\frac{1}{p}-\frac{1}{q}<\frac{q-1}{q(n+1)}$, and $\beta:[a,b)\to\mathbb{R}$ is an integrable positive function. If $\left\|\beta\right\|_{L^{q}([a,b))}<\infty$ then

[TABLE]

Proof.

If $x\in D$ then $x=(\tau,w)$, where $\tau\in[a,b)$ and $w\in U$. Using the special type of the weight $\beta(x):=\beta(\tau)$ and representing $z\in D$ as $z=(\eta,\zeta)$ with $\eta\in[a,b)$ and $\zeta\in U$, we obtain

[TABLE]

where $x=(\tau,w)$.

Using the change of variables $u=tw$ and the estimate

[TABLE]

we finally get

[TABLE]

The conditions on $p$ and $q$ imply the finiteness of the last integral. ∎

Corollary 4.4 is a key ingredient in the proof of out main result, Theorem 7.1. Unfortunately, for being able to “separate” the variable $t$, we have to impose the stronger constraint $\frac{1}{p}-\frac{1}{q}<\frac{1}{n+1}-\frac{1}{q(n+1)}$ than the condition $\frac{1}{p}-\frac{1}{q}<\frac{1}{n+1}$ given by Proposition 4.1.

5. A New Homotopy Operator for $q\geq p$.

The Case of a Convex Domain in $\mathbb{R}^{n}$

In the previous section, we considered the homotopy operator on $\Omega_{\mathrm{loc}}^{*}$ of the form

[TABLE]

for a convex set $D$ in $\mathbb{R}^{n}$. We will need to modify $A$ for obtaining some estimates.

Consider the same operator $K_{y}$ as in the previous section:

[TABLE]

Recall that $dK_{y}\omega+K_{y}d\omega=\omega$. Choose a smooth positive function $\alpha:D\to\mathbb{R}$ such that $\int_{D}\alpha(x)dx=1$ and put

[TABLE]

By a straightforward calculation,

[TABLE]

[TABLE]

[TABLE]

In particular, if $d\omega=0$ then

[TABLE]

The definition of $A_{\alpha}$ easily implies the following

Proposition 5.1**.**

The homotopy operator $A_{\alpha}$ takes smooth forms to smooth forms.

Definition 5.2**.**

Call a smooth positive function $\alpha:D\to\mathbb{R}$ an admissible weight for a convex domain $D\subset\mathbb{R}^{n}$ and $p\geq 1$ if

[TABLE]

For $p\geq 1$, we as usual put

[TABLE]

Theorem 5.3**.**

Suppose that $q\geq p\geq 1$, $D\subset\mathbb{R}^{n}$ is a convex set, $\beta:D\to\mathbb{R}$ is a positive smooth function, and $\alpha:D\to\mathbb{R}$ is an admissible weight. If

[TABLE]

[TABLE]

then for any $\omega\in C^{\infty}L^{p}(D,\Lambda^{k})$ we have

[TABLE]

where

[TABLE]

Proof.

Put $\xi:=A_{\alpha}\omega$. If $p>1$ then, by Hölder’s inequality, we infer

[TABLE]

The above estimate also obviously holds for $p=1$.

By the triangle inequality,

[TABLE]

Therefore,

[TABLE]

By Proposition 4.1,

[TABLE]

The theorem is proved. ∎

Corollary 5.4**.**

Suppose that $q\geq p\geq 1$, $D\subset\mathbb{R}^{n}$ is a convex set, $\alpha:[a,b)\to\mathbb{R}$ is an admissible weight, $\beta,\gamma:D\to\mathbb{R}$ are positive smooth functions. If the conditions

[TABLE]

are fulfilled for some $\overline{p}$, $1\leq\overline{p}\leq p$ (for $\overline{p}=p$, we put $\frac{p\overline{p}}{p-\overline{p}}=\infty$), then the inequality

[TABLE]

where

[TABLE]

holds for any $\omega\in C^{\infty}L^{p}(D,\Lambda^{k})$.

Proof.

By Theorem 5.3,

[TABLE]

If $\overline{p}<p$ then, using Hölder’s inequality, we have

[TABLE]

Inequality (4) also holds for $\overline{p}=p$.

The corollary follows. ∎

Corollary 5.5**.**

Suppose that $q\geq p\geq 1$, $\frac{1}{p}-\frac{1}{q}<\frac{q-1}{q(n+1)}$, $U$ is a bounded convex set in $\mathbb{R}^{n}$, $D=[a,b)\times U$, $\alpha:[a,b)\to\mathbb{R}$ is an admissible weight, and $\beta,\gamma:[a,b)\to\mathbb{R}$ are positive smooth functions. If the conditions $\|\beta\|_{L^{q}([a,b))}\!\!<\!\!\infty$, $\|\tau\beta(\tau)\|_{L^{q}([a,b))}\!\!<\!\!\infty$, and $\|\gamma^{-1}\|_{L^{p\overline{p}/(p-\overline{p})}([a,b))}<\infty$ are fulfilled for some $\overline{p}$, $1\leq\overline{p}\leq p$ (for $\overline{p}=p$, we put $\frac{p\overline{p}}{p-\overline{p}}=\infty$), then the inequality

[TABLE]

*with some constant depending $k$,$p$,$q$,$n$,$\alpha$,$\beta$, and $\gamma$ holds for any $\omega\in C^{\infty}L^{p}(D,\Lambda^{k},\gamma)$. *

Proof.

Suppose that a number $\overline{p}\leq p$ satisfies the conditions of the corollary.

If $x\in D$ then $x=(\tau,w)$, where $\tau\in[a,b)$ and $w\in U$. By Corollary 4.4, since $\frac{1}{\overline{p}}-\frac{1}{q}<\frac{q-1}{q(n+1)}$ and $\|\beta\|_{L^{q}([a,b))}<\infty$, we have

[TABLE]

On the other hand, since $\|\tau\beta(\tau)\|_{L^{q}([a,b))}<\infty$, we have by Corollary 4.4:

[TABLE]

The relations $\|\tau\beta(\tau)\|_{L^{q}([a,b))}<\infty$ and $\||x|\beta(\tau)\|_{L^{q}(D)}<\infty$ enable us to apply Corollary 5.4 and obtain the desired assertion. ∎

6. Globalization: the Sobolev–Poincare Inequality on a Cylinder

Here we globalize the Sobolev–Poincaré inequality to cylinders. The main assertion of the section is

Theorem 6.1**.**

Suppose that $M$ is the cylinder $[a,b)\times N$, where $N$ is a closed manifold of dimension $n$, $q\geq p\geq 1$, $\frac{1}{p}-\frac{1}{q}<\frac{q-1}{q(n+1)}$, and $\beta,\gamma:[a,b)\to\mathbb{R}$ be positive smooth functions. Let $\omega$ be an exact $k$-form in $C^{\infty}L^{p}(M,\Lambda^{k},\gamma)$. If the conditions $\|\beta\|_{L^{q}([a,b))}<\infty$, $\|t\beta(t)\|_{L^{q}([a,b))}<\infty$, and $\|\gamma^{-1}\|_{L^{p\overline{p}/(p-\overline{p})}([a,b))}<\infty$ are fulfilled for some $\overline{p}$, $1\leq\overline{p}\leq p$ (for $\overline{p}=p$, we put $\frac{p\overline{p}}{p-\overline{p}}=\infty$), then there exists a $(k-1)$-form $\xi\in C^{\infty}L^{q}(M,\Lambda^{k-1},\beta)$ such that

[TABLE]

Let $\tilde{\mathcal{U}}=\{\tilde{U}_{x}\}$, $x\in N$, be a coordinate open cover of the base $N$. At each point $x\in N$, consider a geodesic ball $U_{x}$ that is geodesically convex (small balls are geodesically convex, see [7, Proposition 4.2]) and such that its closure (a compact set) is contained in $\tilde{U}_{x}$. Then $\mathcal{U^{0}}=\{\tilde{U}_{x}\}$ is an open cover of $N$. Extract a finite subcover $\mathcal{U}=\{U_{i}\}$, $i=1,\dots,l$, from $U^{0}$. Since $\mathcal{U}$ consists of geodesic balls, it is a good cover, i.e., all finite intersections $U_{I}=U_{i_{0}}\cap\dots\cap U_{i_{s-1}}$, $I=(i_{0},\dots,i_{s-1})$, are bi-Lipschitz diffeomorphic to convex open sets with compact closure in $\mathbb{R}^{n}$. With such a cover $\mathcal{U}$, we associate the corresponding cover $\mathcal{V}=\{V_{i}=[a,b)\times U_{i}\}$, $i=1,\dots,l$, of $M$ and put $V_{I}=V_{i_{0}}\cap\dots\cap V_{i_{s-1}}$ for $I=(i_{0},\dots,i_{s-1})$. Then each intersection $V_{I}$ is bi-Lipschitz diffeomorphic to a cylinder of the form $[a,b)\times U_{\mathbb{R}^{n}}$, where $U_{\mathbb{R}^{n}}$ is a convex set with compact closure in $\mathbb{R}^{n}$. By analogy with [22], we put

[TABLE]

Given $\varkappa\in K^{r,s}$, denote by $\varkappa_{I}$, $I=(i_{0},\dots,i_{s-1})$, $i_{0}<\dots<i_{s}$, the components of $\varkappa$. Define a coboundary operator $\delta:K^{k,s}\to K^{k,s+1}$ as follows:

[TABLE]

Let $L^{q}(K^{k,s})$ be the space of elements $\varkappa\in K^{k,s}$ with the finite norm

[TABLE]

As usual, if $\varkappa\in K^{k,s}$ has components $\varkappa_{I}$, $I=(i_{0},\dots,i_{s-1})$, $i_{0}<\dots<i_{s}$, and $\nu$ is a permutation of the set $\{0,\dots,s-1\}$ then $\alpha_{\nu(I)}=\alpha_{I}\mathrm{sign}\nu$.

The following proposition is a modification for our case of [22, Proposition 3.6], which is in turn an adaptation of [3, Propositions 8.3 and 8.5].

Proposition 6.2**.**

$(K^{k,\bullet},\delta)$* is an exact complex. Moreover, if $\lambda\in L^{q}(K^{k,s+1},\beta)$ satisfies $\delta\lambda=0$ then there exists $\varkappa\in L^{q}(K^{k,s},\beta)$ such that $\lambda=\delta\varkappa$ and*

• •

$\|\varkappa\|_{L^{q}(K^{k,s},\beta)}\leq\mathrm{const\,}\|\lambda\|_{L^{q}(K^{k,s+1},\beta)}$**

• •

$\|d\varkappa\|_{L^{q}(K^{k+1,s},\beta)}\leq\mathrm{const\,}\left(\|\lambda\|_{L^{q}(K^{k,s+1},\beta)}+\|d\lambda\|_{L^{q}(K^{k+1,s+1},\beta)}\right)$.

Proof.

The fact that $(K^{k,\bullet},\delta)$ is an exact complex was established in [3, Propositions 8.3 and 8.5] but we will give the standard argument for completeness. If $\varkappa\in L^{q}(K^{k,s},\beta)$ then

[TABLE]

Suppose that $\lambda\in L^{q}(K^{k,s+1},\beta)$ is such that $\delta\lambda=0$. Let $\tilde{\rho}_{j}$ be a partition of unity subordinate to the cover $\{U_{i}\}$ of $N$. Then the functions $\rho_{j}:M\to\mathbb{R}$, $\rho_{j}(t,x)=\tilde{\rho}_{j}(x)$ for all $(t,x)\in M=[a,b)\times N$, constitute a partition of unity subordinate to the cover $\{V_{i}\}$ of $M$. Put

[TABLE]

Show that $\delta\varkappa=\lambda$.

We have

[TABLE]

Since $\lambda$ is a cocycle,

[TABLE]

Hence,

[TABLE]

Thus, $(K^{k,\bullet},\delta)$ is indeed an exact complex.

The element $\varkappa$ defined by (6) admits the estimates of the norms mentioned in the proposition.

Indeed, we infer

[TABLE]

which gives the first estimate of the proposition.

Let us prove the second estimate. We have

[TABLE]

Therefore,

[TABLE]

∎

Now, applying the general scheme of [22], we first construct some elements $\xi^{s}\in L^{q}(K^{k-s-1,s+1},\beta)$ and then elements $x^{s}\in L^{q}(K^{k-s-1,s},\beta)$ such that $\xi=x^{0}\in C^{\infty}L^{q}(M,\Lambda^{k-1},\beta)$ is an element satisfying the claim of Theorem 6.1.

Construction of the elements $\xi^{s}\in L^{q}(K^{k-s-1,s+1},\beta)$.

Put $\xi^{-1}=\omega$ and define (by induction) $\xi^{s}$ by setting its component $(\xi^{s})_{I}$ to be a solution to the equation

[TABLE]

in $V_{I}$, $I=(i_{0},\dots,i_{s})$ such that

[TABLE]

for $0\leq s\leq k-1$.

Note that such a solution always exists due to the local Sobolev–Poincaré inequality (Corollary 5.5) since $V_{I}$ is bi-Lipschitz diffeomorphic to a cylinder over a convex subset in $\mathbb{R}^{n}$ of finite volume.

We have the following estimate of the weighted $q$-norm of $\xi^{s}$:

Proposition 6.3**.**

If $I=(i_{0},\dots,i_{s})$ then

[TABLE]

Proof.

Use induction on $s$. For $s=0$, the assertion follows from the local Sobolev–Poincaré inequality. Let now $s>0$. We infer

[TABLE]

∎

Note that $\xi^{k-1}$ is a collection of [math]-forms satisfying the condition $d\delta\xi^{k-1}=0$. Thus, the functions $(\delta\xi^{k-1})_{I}$ are constants on each set $V_{I}$, $I=(i_{0},\dots,i_{k})$. The global constant functions $(\delta\xi^{k-1})_{I}$ on $M$ belong to $L^{q}(M,\beta)$ due to the hypotheses on $\beta$.

The following assertion is Theorem 3.10 in [22]:

Lemma 6.4**.**

There exists $c\in K^{0,k}$ with constant components $c_{I}$, $I=(i_{0},\dots,i_{k-1})$, such that

[TABLE]

In addition, there exist numbers $b_{I,L}\in\mathbb{R}$, $I=(i_{0},\dots,i_{k-1})$, $L=(i_{0},\dots,i_{k})$, such that

[TABLE]

where $b_{i,L}$ depend on the chosen cover $\mathcal{U}$ of $N$.

We have

Proposition 6.5**.**

The constants $c_{I}$ of Lemma 6.4 satisfy the estimate

[TABLE]

Proof.

By Lemma 6.4, each $c_{I}$ is representable as $c_{I}=\sum_{L}b_{I,L}(\delta\xi^{k-1})_{L}$. Hence,

[TABLE]

Since $(\delta\xi^{k-1})_{L}$ is a globally defined constant function on $M$ as in the proof of Proposition 6.3, we have

[TABLE]

This gives the estimate of the proposition. ∎

Construction of the elements $x^{s}\in L^{q}(K^{k-s-1,s},\beta)$.

Let us now glue all the forms $\xi^{s}$, $s-0,\dots,k-1$, into a global form $\xi$ satisfying (5). Construct by induction elements $x^{s}\in L^{q}(K^{k-s-1,s},\beta)$, $s=k-1,\dots,1,0$, such that $\xi=x^{0}$ is a desired form on $M$.

Put $\tilde{\xi}^{k-1}_{I}=\xi^{k-1}_{I}-c_{I}$, where $c_{I}$ is as in Lemma 6.4, $I=(i_{0},\dots,i_{k-1})$. We have $d\tilde{\xi}^{k-1}_{I}=d\xi^{k-1}_{I}$ and $\delta\tilde{\xi}^{k-1}_{I}=0$. By Proposition 6.2, there exists $x^{k-1}\in L^{q}(K^{0,k-1},\beta)$ such that $\delta x^{k-1}=\tilde{\xi}^{k-1}$ and

[TABLE]

Propositions 6.3 and 6.5 yield

[TABLE]

and

[TABLE]

Suppose that $x^{k-(r-1)}$ is already constructed. By Proposition 6.2, there exists $x^{k-r}$ such that

[TABLE]

where

[TABLE]

and

[TABLE]

Here the last inequality stems from the fact that

[TABLE]

The above considerations imply the following

Proposition 6.6**.**

The forms $x^{s}$ admit the estimates:

(1)* $\|x^{k-r}\|_{L^{q}(K^{r-1,k-r},\beta)}\leq\mathrm{const\,}\|\omega\|_{L^{p}(M,\Lambda^{k},\gamma)}$;*

(2)* $\|dx^{k-r}\|_{L^{q}(K^{r,k-r},\beta)}\leq\mathrm{const\,}\|\omega\|_{L^{p}(M,\Lambda^{k},\gamma)}$.*

Proof.

Use induction on $r$. For $r=1$, (1) and (2) are just estimates (9) and (10). Assume that $r>1$. For proving estimate (2), observe that, by the induction hypothesis and (12),

[TABLE]

Now, Proposition 6.3 and estimates (11) and (2) yield

[TABLE]

∎

Finally, put $\xi=x^{0}$. Then $d\xi=\omega$. Indeed, we have

[TABLE]

Since $\delta(\omega-dx^{0})_{i}=(\omega-dx^{0})|_{V_{i}}$, we infer that $\omega=dx_{0}$ on $M$. By Proposition 6.6,

[TABLE]

Theorem 6.1 is completely proved.

7. $L_{q,p}$-Cohomology of a Twisted Cylinder

Theorem 7.1**.**

Suppose that $N$ is a closed manifold of dimension $n$, $H^{k}_{\mathrm{DR}}(N)=0$, $q\geq p\geq 1$, and $\frac{1}{p}-\frac{1}{q}<\frac{q-1}{q(n+1)}$. If

[TABLE]

and

[TABLE]

for some $\overline{p}$, $1\leq\overline{p}\leq p$ (for $\overline{p}=p$, we put $\frac{p\overline{p}}{p-\overline{p}}=\infty$), then $H_{q,p}^{k}(C_{a,b}^{h}N)=0$.

Proof.

Let $\overline{M}$ be the cylinder $[a,b)\times N$ with the usual product metric. By the Künneth formula for the de Rham cohomology, we have

[TABLE]

Using expression (3) for the norm and the definition of $f_{l,p}$, we infer

[TABLE]

Thus, $\omega\in C^{\infty}L^{p}(\overline{M},\Lambda^{k},\min(f_{k-1,p},f_{k,p}))$. Since the de Rham cohomology $H^{k}_{\mathrm{DR}}(\overline{M})$ is trivial, $\omega$ is exact, and we can apply Theorem 6.1, by which there exists $\xi\in C^{\infty}L^{q}(\overline{M},\Lambda^{k},\max(F_{k-2,q}(t),F_{k-1,q}(t))$ with

[TABLE]

For this form $\xi$, we have

[TABLE]

Combining (13),(14), and (15), we obtain

[TABLE]

Thus, $C^{\infty}H_{q,p}^{k}(C_{a,b}^{h}N)=0$, and hence, by Theorem 2.5, also $H_{q,p}^{k}(C_{a,b}^{h}N)=0$. ∎

8. $L_{q,p}$-Cohomology of an Asymptotic Twisted Cylinder

Recall the following definition, given in [10]:

Definition 8.1**.**

We refer to a pair $(M,X)$ consisting of an $m$-dimensional manifold $M$ and an $m$-dimensional compact submanifold $X$ with boundary as an asymptotic twisted cylinder $AC_{a,b}^{h}\partial X$ if $M\setminus X$ is bi-Lipschitz diffeomorphically equivalent to the twisted cylinder $C_{a,b}^{h}\partial X$.

For asymptotic twisted cylinders, Theorem 7.1 gives:

Theorem 8.2**.**

Let $(M,X)=AC_{a,b}^{h}\partial X$ be an asymptotic twisted cylinder with $\dim M=\dim X=m=n+1$. Assume that $q\geq p\geq 1$, $\frac{1}{p}-\frac{1}{q}<\frac{q-1}{qm}$, and $H_{\mathrm{DR}}^{k}(X)=0$. If

[TABLE]

and

[TABLE]

for some $\overline{p}$, $1\leq\overline{p}\leq p$ (for $\overline{p}=p$, we put $\frac{p\overline{p}}{p-\overline{p}}=\infty$), then $H_{q,p}^{k}(M)=0$.

Proof.

Since bi-Lipschitz diffeomorphisms preserve $L_{p_{1}}$ and $L_{p_{2}}$ and extension by zero gives a topological isomorphism between the spaces $W_{p_{1},p_{2}}(C_{a,b}^{h}\partial X)$ and $W_{p_{1},p_{2}}(M)$ for all $p_{1},p_{2}$, we have a topological isomorphism

[TABLE]

for all $p_{1}$, $p_{2}$. The theorem now follows from Theorem 7.1. ∎

9. Examples

Let us analyze the conditions of the last theorems for comparatively simple cases. Suppose that $N$ is the $n$-dimensional sphere $S^{n}$. Then $H^{k}_{\mathrm{DR}}(N)=0$ for any $k\neq n$. By the hypothesess of the theorems, $q\geq p\geq 1$ and $\frac{1}{p}-\frac{1}{q}<\frac{q-1}{q(n+1)}$. Put

[TABLE]

Then, by definition,

[TABLE]

[TABLE]

and

[TABLE]

By the hypotheses of the theorems, we must check the integrability of these three functions in the corresponding degrees under the above-mentioned restrictions on $p$ and $q$.

Suppose for simplicity that $s(t)$ and $g(t)$ are smooth increasing functions tending to $\infty$ as $t\to b-0$. Denote the maximal integrability intervals for $s^{u}$ and $g^{v}$ by $(-\infty,\alpha)$ and $(-\infty,\beta)$, i.e $s^{u}$ is integrable on $[a,b)$ for every $u<\alpha$ and is not integrable for every $u>\alpha$ and similarly for $g^{v}$. Let also $\alpha_{1}$ be the supremum of $\mu$ such that $ts^{\mu}(t)$ is integrable on $[a,b)$.

For this case $I_{1,q,k}=s^{\frac{n}{q}-k+2}$, $I_{2,q,k}(t)=ts^{\frac{n}{q}-k+2}(t)$, and $I_{3,p,k}=g^{k-\frac{n}{p}}$.

The conditions of the theorems are fulfilled if

[TABLE]

Note that these inequalities cannot hold simultaneously if $b=\infty$. In this case, $\alpha$, $\alpha_{1}$, and $\beta$ are all negative, whence $\frac{n}{p}-k>\frac{n}{q}-k+2$. We thus have $\frac{1}{p}-\frac{1}{q}>\frac{2}{n}$, which contradicts the hypotheses.

Examine more closely the case of $0\leq a<b<\infty$. The function $t$ is bounded, and hence $\alpha=\alpha_{1}$. Therefore, the inequalities for $I_{1,q,k}$ and $I_{3,p,k}$ can be combined into one inequality

[TABLE]

It means that the additional condition $k-2+\alpha\leq k-\beta$, i.e., $\alpha+\beta\leq 2$, must be fulfilled.

The last condition is $\frac{1}{p}-\frac{1}{q}<\frac{q-1}{q(n+1)}$, i.e., $p\leq q<\frac{np}{n+1-p}$.

Summarizing, we conclude that for known integrability limits $\alpha$ and $\beta$, we need to check two simple conditions for $p$ and $q$:

[TABLE]

and the inequality

[TABLE]

for the degree $k$.

Under these conditions, the cohomology of the warped product $C_{[a,b)}^{f}S^{n}$ vanishes.

For example, if $f(t)=g(t)=(b-t)^{-2}$ then $\alpha=\beta=1/2$. For $p=2$ we have $2\leq q<2\frac{n}{n-1}$.

The last inequality yields

[TABLE]

Let $q$ be an arbitrary number in $(2,\frac{8}{3})$. Then the second inequality $q<\frac{2n}{n-1}$ gives us the constraint $n<q/(q-2)$. Since $q/(q-2)<4$, we can take $n=4$. We have $1/2<(2k-1)/8$, i.e., $k>2$. For $k=3$, the leftmost inequality gives us the fulfilled condition $3/8<1/q$. Note that if $q=2$ then always $q<\frac{2n}{n-1}$. If $n$ is even and $k=\frac{n}{2}+1$ then all inequalities in (16) are fulfilled. Thus, we have

[TABLE]

and

[TABLE]