Reduced <span class="mathjax-formula">$L_{q,p}$</span>-Cohomology of Some Twisted Products

BLAISE PASCAL

Vladimir Gol⁰dshtein & Yaroslav Kopylov

Reduced L_q,p-Cohomology of Some Twisted Products Volume 23, n^o2 (2016), p. 151-169.

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Reduced L

-Cohomology of Some Twisted Products

Vladimir Gol⁰dshtein Yaroslav Kopylov

Abstract

Vanishing results for reduced Lq,p-cohomology are established in the case of twisted products, which are a generalization of warped products. Only the case q≤pis considered. This is an extension of some results by Gol⁰dshtein, Kuz⁰minov and Shvedov about theLp-cohomology of warped cylinders. One of the main obser- vations is the vanishing of the “middle-dimensional” cohomology for a large class of manifolds.

La cohomologie L_q,p réduite de quelques produits twistés

Résumé

On établit des résultats d’annulation de la cohomologieLq,préduite pour les produits twistés, une généralisation des produits tordus dans le cas de q≤p. Le résultats obtenus sont des généralisations de certains résultats par Gol⁰dshtein, Kuz⁰minov et Shvedov sur la cohomologieLpdes cylindres tordus. Une des obser- vations principales est la trivialité de la cohomolgie en dimension “moyenne” pour une large classe de variétés.

1. Introduction

TheL_q,p-cohomologyH_q,p^k (M) of a Riemannian manifold (M, g) is defined to be the quotient of the space of closed p-integrable differential k-forms by the exterior differentials of q-integrable k-forms. The quotient space of H_q,p^k (M) by the closure of zero is called the reduced Lq,p-cohomology H^k_q,p(M). If p =q then L_q,p-cohomology is usually referred to as simply Lp-cohomology and the indexpis used instead ofp, pin all the notations.

The second author was partially supported by the State Maintenance Program for the Leading Scientific Schools and Junior Scientists of the Russian Federation.

Keywords:differential form, reducedLq,p-cohomology, twisted cylinder.

Math. classification:58A10, 58A12.

A twisted product X×_hY of two Riemannian manifolds (X, g_X) and (Y, g_Y) is the direct product manifoldX×_gY endowed with a Riemannian metric of the form

g:=g_X +h²(x, y)g_Y, (1.1) where h : X ×Y → R is a smooth positive function (see [4]). If X is a half-interval [a, b[ then the twisted product X×_h Y is called a twisted cylinder.

We refer to an m-dimensional Riemannian manifold (M, g_M) as an as- ymptotic twisted product (respectively, as anasymptotic twisted cylinder) if, outside anm-dimensional compact submanifold, it is bi-Lipschitz equiv- alent to a twisted product (respectively, to a twisted cylinder).

In this paper, we prove some vanishing results for the (reduced) L_q,p- cohomology of twisted cylinders [a, b)×_hN for a positive smooth function h : [a, b)×N → Rin the case where the baseN is a closed manifold and p≥q >1.

If in (1.1) the functionhdepends only onx then we obtain the familiar notion of a warped product (see [1]). Twisted products were the object of recent investigations [2, 3, 5, 6, 10, 14]. The reduced Lq,p-cohomology of warped cylinders [a, b)×_hN, i.e., of product manifolds [a, b)×N endowed with a warped product metric

g=dt²+h²(t)gN,

where gN is the Riemannian metric ofN and h: [a, b)→ Ris a positive smooth function, was studied by Gol⁰dshtein, Kuz⁰minov, and Shvedov [7], Kuz⁰minov and Shvedov [12, 13] (for p =q), and Kopylov [11] forp, q ∈ [1,∞), ¹_p −¹_q < _dim¹_N+1.

The main results of this paper are technical. Here we mention a “uni- versal” consequence of the main results on the vanishing of the “middle- dimensional” cohomology:

Let N be a closed smooth n-dimensional Riemannian manifold. If p≥q >1 and ⁿ_p is an integer then H

n

q,pp ([0,∞)×_hN) = 0.

In particular, if1< q≤2andnis even thenH

n 2

q,2([0,∞)×_hN) = 0.

The result was not known even for L₂-cohomology. It does not depend on the type of the warped Riemannian metric. Of course, the result leads to the vanishing of the “middle-dimensional” cohomology for asymptotic twisted cylinders.

2. Basic Definitions

In this section, we recall the main definitions and notations. In what fol- lows, we tacitly assume all manifolds to be oriented. Let M be a smooth Riemannian manifold. Denote by D^k(M) := C₀^∞(M,Λ^k) the space of all smooth differential k-forms with compact support contained in M\∂M and designate as L¹_loc(M,Λ^k) the space of locally integrable differential k-forms. Denote byL^p(M,Λ^k) the Banach space of locally integrable dif- ferentialk-forms endowed with the normkθk_Lp(M,Λ^k):= (^R_M|θ|^pdx)^1p <∞ (as usual, we identify forms coinciding outside a set of measure zero).

Definition 2.1. We call a differential (k+ 1)-form θ ∈ L¹_loc(M,Λ^k+1) the weak exterior derivative (or differential) of a differential k-form φ ∈ L¹_loc(M,Λ^k) and write dφ=θ if

Z

M

θ∧ω= (−1)^k+1 Z

M

φ∧dω for anyω ∈ D^n−k−1(M).

Remark 2.2. Note that the orientability of M is not substantial 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, the space of q-integrable forms with p-integrable weak exterior derivative:

Ω^k_q,p(M) =ⁿω∈L^q(M,Λ^k)|dω∈L^p(M,Λ^k+1)^o, This is a Banach space for the graph norm

kωk_Ωk

q,p(M)=kωk²_Lq(M,Λ^k)+kdωk²_Lp(M,Λ^k+1)

1/2

.

The space Ω^k_q,p(M) is a reflexive Banach space for any 1< q, p <∞. This can be proved using standard arguments of functional analysis.

Denote by Ω^k_q,p,0(M) the closure ofD^k(M) in the norm of Ω^k_q,p(M). We now define our basic ingredients (for three parameters r, q, p).

Definition 2.3. Put

(a) Z_p,r^k (M) = Ker[d: Ω^k_p,r(M)→L^r(M,Λ^k+1)].

(b) B_q,p^k (M) = Im[d: Ω^k−1_q,p (M)→L^p(M,Λ^k)].

Lemma 2.4. The subspaceZ_p,r^k (M) does not depend on r and is a closed subspace in L^p(M,Λ^k).

Proof. The lemma is in fact [9, Lemma 2.4 (i)]. However, we now repeat the proof for the reader’s convenience. Note thatZ_p,r^k (M) is a closed subspace in Ω^k_p,r(M) because it is the kernel of the bounded operator d. It is also a closed subspace of L^p(M,Λ^k) since kαk_Ωk

p,r(M) = kαk_Lp(M,Λ^k) for any

α ∈Z_p,r^k (M).

This allows us to use the notation Z_p^k(M) for all Z_p,r^k (M). Note that Z_p^k(M) ⊂ L^p(M,Λ^k) is always a closed subspace but this is in general not true for B_q,p^k (M). Denote by B^k_q,p(M) its closure in the L^p-topology.

Observe also that since d◦d= 0, one hasB^k_q,p(M)⊂Z_p^k(M). Thus, B_q,p^k (M)⊂B^k_q,p(M)⊂Z_p^k(M) =Z^k_p(M)⊂L^p(M,Λ^k).

Definition 2.5. Suppose that 1 ≤ p, q ≤ ∞. The Lq,p-cohomology of (M, g) is defined as the quotient

H_q,p^k (M) :=Z_p^k(M)/B_q,p^k (M),

and thereduced L_q,p-cohomology of (M, g) is, by definition, the space H^k_q,p(M) :=Z_p^k(M)/B^k_q,p(M).

Since B_q,p^k is not always closed, the Lq,p-cohomology is in general a (non-Hausdorff) semi-normed space, while the reduced Lq,p-cohomology is a Banach space. Considering only the forms equal to zero on some neighborhood (depending on the form) of a subset A ⊂ M and taking closures in the corresponding spaces, we obtain the definition of the rela- tive spaces L^p(M, A,Λ^k) and Ω_q,p(M, A) and therelative nonreduced and reduced cohomology spaces H_q,p^k (M, A) and H^k_q,p(M, A).

Similarly, one can define the L_q,p-cohomology with compact support (interior cohomology)H_q,p;0^k (M, A) andH^k_q,p;0(M, A). The interior reduced cohomology is dual to the reduced cohomology:

Theorem 2.6([9]). Let(M, g)be an orientedm-dimensional Riemannian manifold. If 1 < p, q < ∞ then H^k_q,p(M) is isomorphic to the dual of H^m−k_p⁰_,q⁰_;0(M), where _p¹0 +¹_p = _q¹0 +¹_q = 1.

Therefore, if M is complete then H^k_p(M) = H^k_p;0(M) and H^k_p(M) is isomorphic to dual of H^m−k_p0 (M), where ¹_p+ _p¹0 = 1 [9].

Below |X|stands for the volume of a Riemannian manifold (X, g); the notation |ω|_X means the modulus of a differential form on (X, g).

3. L_q,p-Cohomology and Smooth Forms

It follows from the results of [8] that, under suitable assumptions on p, q, theL_q,p-cohomology of a Riemannian manifold can be expressed in terms of smooth forms.

Introduce the notations:

C^∞L^p(M,Λ^k) :=C^∞(M,Λ^k)∩L^p(M,Λ^k);

C^∞Ω^k_q,p(M) :=C^∞(M,Λ^k)∩Ω^k_q,p(M).

Theorem 3.1([8, Theorem 12.8 and Corollary 12.9]). Let(M, g)be anm- dimensional Riemannian manifold and suppose that p, q ∈ (1,∞) satisfy

1

p − ¹_q ≤ _m¹. Then the cohomology H_q,p^∗ (M) can be represented by smooth forms.

More precisely, any closed form in Z_p^k(M)is cohomologous to a smooth form inL^p(M). Furthermore, if two smooth closed formsα, β ∈C^∞(M)∩ Z_p^k(M) are cohomologous modulo dΩ^k−1_q,p (M) then they are cohomologous modulo dC^∞Ω^k−1_q,p (M).

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

In what follows, unless otherwise specified, we always assume that p, q∈(1,∞) and ¹_p−¹_q ≤ _dim¹_M.

4. The Homotopy Operator

From now on,C_a,b^h N is the twisted cylinderI×_hN, that is, the product of a half-interval I := [a, b) and a closed smooth n-dimensional Riemannian manifold (N, g_N) equipped with the Riemannian metric dt²+h²(t, x)g_N, where h:I×N →Ris a smooth positive function.

Every differential form on I×N admits a unique representation of the form ω = ωA+dt∧ωB, where the forms ωA and ωB do not contain dt

(cf. [7]). It means thatω_Aandω_Bcan be viewed as one-parameter families ω_A(t) andω_B(t),t∈I, of differential forms onN. Given a formω defined on I×N and numbers c, t ∈[a, b), consider the form ^R_c^tω =^R_c^tω_B(τ)dτ on N and the form S_cω on [a, b)×N, (S_cω)(t) = ^R_c^tω for all t ∈ [a, b).

The domains of of these operators will be specified below.

The modulus of a form ω of degree k on C_a,b^h N is expressed via the moduli of ω_A(t) andω_B(t) onN as follows:

|ω(t, x)|_M =h^−2k(t, x)|ω_A(t, x)|²_N +h^−2(k+1)(t, x)|ω_B(t, x)|²_X^1/2 (4.1) Consequently,

kωk_Lp(C_a,b^h N,Λ^k)

=

"

Z b a

Z

N

h^2(np−k)(t, x)|ω_A(t, x)|²_N+h^2(np−k+1)(t, x)|ω_B(t, x)|²_N

p 2dxdt

#¹

p

. (4.2) In the particular case of ω = ω_A, call the form ω horizontal. If ω is a horizontal form then

kωk_Lp(C_a,b^h N,Λ^k)=

"

Z b a

Z

N

|ω(t)|^ph^n−kp(t, x)dxdt

#1/p

. (4.3) Put

fk,p(t) = min

x∈N

h^np−k(t, x) and

Fk,p(t) = max

x∈N

h^np−k(t, x) . Remarks 4.1.

(1) Suppose thatk = ⁿ_p is an integer. Then F_n/p,p(t) =f_n/p,p(t) ≡1.

For example, if nis even andp= 2 thenF_n/2,2(t) =f_n/2,2(t)≡1.

(2) For warped products (h depends only on x), f_k,p(t) = F_k,p(t) = h^np−k(t).

Let

π:I×N →N, π(t, x) =x

be the natural projection. For c∈[a, b), putN_c:={c} ×N. Leti_t:N → N_t⊂[a, b)×N and i_c:N →N_c⊂[a, b)×N be the natural immersions.

Every smooth k-form ω onC_a,b^h N satisfies the homotopy relations d_N

Z t c

ω

+ Z t

c

dω=i^∗_tω−i^∗_cω, (4.4) dScω+Scdω=ω−π^∗i^∗_cω. (4.5) Here dN stands for the exterior derivative on N and d designates the exterior derivative on [a, b)×N.

The homotopy relations cannot be used automatically for ω ∈ Ω^k_q,p(C_a,b^h N) because of the problem of the existence of traces on subman- ifolds. However, by Theorem 3.1, we can take only smooth forms in all considerations concerning both reduced and nonreducedL_q,p-cohomology.

For the reader’s convenience, we repeat the classical proofs of (4.4) and (4.5).

Using the representation ω=ω_A+dt∧ω_B, we have dω=d(ωA(t) +dt∧ωB(t)) =dt∧∂ωA

∂t (t) +dNωA(t)−dt∧dNωB(t), dN

Z _t

c

ω

=dN

Z _t

c

ωB(τ)dτ

= Z t

c

dNωB(τ)dτ, Z t

c

dω= Z t

c

(dω)_B(τ)dτ = Z t

c

∂ω_A

∂t (τ)

dτ− Z t

c

d_Nω_B(τ)dτ.

Hence, d_N

Z t c

ω

+ Z t

c

dω= Z t

c

∂ω_A

∂t (τ)

dτ =ω_A(t)−ω_A(c) =i^∗_tω−i^∗_cω.

Similarly,

S_cω= Z t

c

ω_B(τ)dτ dS_cω=d

Z t c

ω_B(τ)dτ

=dt∧ω_B(t) + Z t

c

d_Nω_B(τ)dτ, Scdω=

Z t c

dω= Z t

c

∂ωA

∂t (τ)

dτ− Z t

c

dNωB(τ)dτ.

Therefore,

dS_cω+S_cdω=dt∧ω_B(t) + Z t

c

∂ω_A

∂t (τ)

dτ

=dt∧ω_B(t) +ω_A(t)−ωc(t)

=ω−i^∗_cω.

Lemma 4.2. Suppose that g is a function locally integrable on [a, b), Rb

a|g(t)|dt = ∞, and ω ∈ C^∞L^k_p(C_a,b^h N). Then every set of full mea- sure on [a, b) contains a sequence {t_j} that converges toband is such that g(t_j)6= 0 and ki^∗_tjωk_Lp(N,Λ^k)=o([f_k,p(t_j)]⁻¹|g(t_j)|^1/p) as j→ ∞.

Proof. Equality (4.3) implies:

Z b a

ki^∗_tωk^p

L^p(N,Λ^k)f_k,p^p (t)dt≤ Z b

a

Z

N

|ω_A(t)|^ph^n−kp(t, x)dxdt

≤ kωk^p

L^p(C_a,b^h N,Λ^k)

<∞,

which yields the lemma.

Lemma 4.3. Suppose that ω ∈ L^k_p(C_a,b^h N), c ∈ [a, b), p ≥ q > 1. Then the form ^R_c^tω is defined for each t ∈ [a, b) and belongs to L^q(N,Λ^k−1).

Moreover, the norm of the linear operator ^R_c^t:L^k_p(C_a,b^h N)→L^k−1_q (C_a,b^h N) satisfies the inequality

Z t c

≤ |N|^1q−1p

Z t c

f_k−1,p^−p0 (τ)dτ

1/p⁰

.

The same holds for c=bif Z b

a

f_k−1,p^−p0 (τ)dτ <∞.

Proof. By Hölder’s inequality

Z t c

ω

_Lq(N,Λk−1)

≤ |N|^1q−1p

Z t c

ω

_Lp(N,Λk−1)

=|N|^1q−1p Z

N

Z t c

ω_B(τ)dτ

p N

dx ^1/p

≤ |N|^1q−1p Z

N

Z t c

|ω_B(τ)|_Ndτ

p

dx 1/p

≤ |N|^1q−1p

"

Z

N

Z t c

h^{−(np−k+1)p0}(x, τ)dτ Z t

c

|ω_B(τ)|^p_Nh^n−kp+p(τ, x)dτ dx

#^1/p

≤ |N|^1q−1p

"

Z

N

Z t c

f_k−1,p^−p0 (τ, x)dτ Z t

c

|ω_B(τ)|^p_Nh^n−kp+p(τ, x)dτ

dx

#1/p

≤ |N|^1q−1p

Z t c

f_k−1,p^−p0 (τ)dτ

kωk_Lp(C_a,b^h N,Λ^k).

This proves the lemma.

5. The Main Results

5.1. Absolute reduced L_q,p-cohomology Using the results of the previous section, we prove

Theorem 5.1. LetN be a closed smoothn-dimensional Riemannian man- ifold and let p≥q >1. If

I_a,b:=

Z b a

F_k,p^p (t)dt=∞; J_δ0,b:=

Z b δ0

f_k,p^p (τ) Z _τ

a

F_k,p^p (t)dt −1

dτ =∞ for some δ0∈[a, b) thenH^k_q,p(C_a,b^h N) = 0.

Remark 5.2. The condition “Jδ0,b = ∞ for some δ0 ∈ [a, b)” is in fact equivalent to “J_δ0,b =∞ for every δ₀ ∈ [a, b)”. In this connection, below we sometimes write Jb instead of Jδ0,b.

Proof. Suppose that ω ∈L^k_p(C_a,b^h N) is weakly differentiable and dω = 0.

Therefore, ω ∈ Ω^k_p,p(C_a,b^h N). By Theorem 3.1, we may assume that ω is a smooth form. If ^R_δ^b

0f_k,p^p (τ)^R_a^τF_k,p^p (t)dt⁻¹dτ =∞ for δ0 ∈(a, b) then the function g(τ) =f_k,p^p (τ)^R_a^τF_k,p^p (t)dt⁻¹ is not integrable on intervals of the form (c, b), δ₀ ≤ c < b. By Lemma 4.2, there exists a sequence {τ_j} ⊂(δ0, b)⊂(a, b) such thatτj →b and

ki^∗_τ

jωk_Lp(N,Λ^k)=o Z _τj

a

F_k,p^p (t)dt −1/p!

asj→ ∞.

Consider the form

ωj =

(ω on C_a,τ^h _jN, π^∗i^∗_τjω on C_τ^h_j,bN .

It is easy to verify thatωj is weakly differentiable, belongs to Ω^k_q,p(C_a,b^h N), and satisfies dωj = 0. Indeed, if u ∈ D^n−k−1(C_a,b^h N) then, denoting the restrictions of the corresponding forms to C_a,τ^h _j and C_τ^h

j,bN by the same symbols for simplicity and using (4.5), we get

Z

C^h_a,bN

ω_j∧du= Z

C_a,τj^h

ω∧du+ Z

C_{τj ,b}^h

(ω−dS_τjω)∧du

= (−1)^k+1 Z

C_a,τj^h

dω∧u+ (−1)^k+1 Z

C^h

τj ,b

d(ω−dSτjω)∧u= 0, which implies that dωj = 0.

We infer kω_jk^p

L^p(C_a,b^h N,Λ^k)≤ Z τj

a

F_k,p^p (t)dt ki^∗_τ

jωk^p

L^p(N,Λ^k)+kωk^p

L^p(C^h

τj ,bN,Λ^k). Therefore, kω_jk_Lp(C_a,b^h N,Λ^k) →0 asj → ∞; moreover, the formω−ωj

is equal to 0 on [τj, b) × N. Hence, Sτj(ω −ωj) ∈ Ω_q,p(C_a,b^h N) and dS_τj(ω−ω_j) = ω−ω_j. Thus, the cocycle ω is zero in the reduced co-

homology H^k_q,p(C_a,b^h N).

From Remark 4.1 and Theorem 5.1 we obtain

Theorem 5.3. LetN be a closed smoothn-dimensional Riemannian man- ifold. If p≥q >1, b=∞ and ⁿ_p is an integer then H

n

q,pp (C_a,∞^h N) = 0.

In particular, if 1< q≤2 and n is even thenH

n 2

q,2(C_a,∞^h N) = 0.

Proof. In this case, Ia,∞=Ja,∞=∞, and by Theorem 5.1 H

n

q,pp (C_a,∞^h N) = 0.

Remark 5.4. For a warped cylinder C_a,b^h N, we have f_k,p^p (τ) = F_k,p^p (τ) = h^n−kp(τ). Therefore, if

Z b a

h^n−kp(t)dt=∞ then

J_δ0,b= Z _b

δ0

h^n−kp(τ) Z _τ

a

h^n−kp(t)dt −1

dτ

= Z b

δ0

d dτ log

Z _τ

a

h^n−kp(t)dt

dτ

= lim

τ%b

( log

Z _τ

a

h^n−kp(t)dt

−log Z δ0

a

h^n−kp(t)dt

!)

=∞, and the result of Theorem 5.1 coincides with the corresponding result in [7].

5.2. Relative reduced L_q,p-cohomology

Here we prove a sufficient vanishing condition forH^k_q,p(C_a,b^h N, N_a), where Na={a} ×N.

Theorem 5.5. LetN be a closed smoothn-dimensional Riemannian man- ifold. Assume that p≥q >1,

I˜a,b:=

Z b a

f_k−1,p^−p0 (t)dt=∞, and the integral

Aδ0,b:=

Z b δ0

F_k−1,p^p (τ) f_k−1,p^pp0 (τ)

Z _τ

a

f_k−1,p^−p0 (t)dt ⁻¹

log Z _τ

a

f_k−1,p^−p0 (t)dt

−p

dτ is finite for some δ₀∈[a, b). ThenH^k_q,p(C_a,b^h N, N_a) = 0.

Proof. Suppose that ω ∈C^∞Ω^k_p,p(C_a,b^h N, Na) and dω= 0. The form ω is the limit in Ω^k_p,p(C_a,b^h N) of a sequenceωj of smooth forms each of which is equal to zero on some neighborhood of Na.

As above, fora≤c < d≤bdenote byC_c,d^h N the product [c, d)×N with the metric induced from C_a,b^h N. For every e ∈ (a, b), Lemma 4.3 implies that the operator Sa : Ω^k_p,p(C_a,b^h N) → Ω^k−1_q,p (C_a,e^h N) is bounded. Hence, S_aω_j → S_aω in Ω^k−1_q,p (C_a,e^h N). Each of the forms S_aω_j vanishes on some neighborhood of Na. Therefore,Saω ∈Ω^k−1_q,p (C_a,e^h N, Na) for all e∈(a, b).

Then the fact that i^∗_aω= 0 and relation (4.5) give the equality dSaω=ω.

Consider the functions

I(τ) = Z τ

a

f_k−1,p^−p0 (t)dt, ϕ(τ) = log|logI(τ)|, ϕ_δ(τ) =

(1 ifτ ≤δ, min(1,max(1 +ϕ(δ)−ϕ(τ),0)) ifτ ≥δ.

The function ϕ(t) is defined for τ sufficiently close to b and ϕδ(t) also exists only for δ that are sufficiently close to b.

Since d(ϕ_δS_aω) =dϕ_δ∧S_aω+ϕ_δω, it follows that kd(ϕ_δS_aω)−ωk_Lp(C_a,b^h N,Λ^k)

=kd(ϕ_δSaω)−ωk_Lp(C_δ,b^h N,Λ^k)

≤ kdϕ_δ∧S_aωk_Lp(C_δ,b^h N,Λ^k)+k(ϕ_δ−1)ωk_Lp(C^h_δ,bN,Λ^k). Since |ϕ_δ−1| ≤1, we have

k(ϕ_δ−1)ωk_Lp(C^h_δ,bN,Λ^k) ≤ kωk_Lp(C_δ,b^h N,Λ^k).

By (4.2) and Lemma 4.3 for p=q, forδ sufficiently close tob we infer kdϕ_δ∧S_aωk^p

L^p(C^h_δ,bN,Λ^k) = Z b

δ

h^n−kp+p(τ, x)

dϕ_δ dτ

p

Z τ a

ω

p N

dx dτ

≤ Z b

δ

F_k−1,p^p (τ)

dϕ_δ dτ

pZ τ a

f_k−1,p^−p0 (t)dt

p p0

dτkωk^p

L^p(N,Λ^k)

≤ Z b

δ

F_k−1,p^p (τ) f_k−1,p^pp0 (τ)

Z _τ

a

f_k−1,p^−p0 (t)dt ⁻¹

log Z _τ

a

f_k−1,p^−p0 (t)dt

−p

dτkωk^p_Lp(N,Λk).

By hypothesis, the last quantity vanishes as δ → b. Thus, d(ϕ_δS_aω) → ω as δ → b. The function ϕ_δ is equal to zero in some neighborhood of b. Hence, ϕ_δS_aω ∈ Ω^k_q,p(C_a,b^h N, N_a). This shows that

H^k_q,p(C_a,b^h N, N_a) = 0.

Remark 5.6. The paper [7] contains the following assertion for a warped cylinder C_a,b^h N [7, Theorem 2]:

If Z b

a

h^{−(np−k+1)p0}(t)dt <∞ then H^k_p,p(C_a,b^h N, Na) = 0.

Suppose that

b

R

a

h^{−(np−k+1)p0}(t)dt <∞. We infer Z b

δ

h^{−(np−k+1)p0}(τ) Rτ

a h^{−(np−k+1)p0}(t)dtlog ^R_a^τh^{−(np−k+1)p0}(t)dt

pdτ

= 1

p−1 Z b

δ

d dτ

log

Z _τ

a

h^{−(np−k+1)p0}(t)dt 1−p

dτ

= 1

p−1

"

log Z δ

a

h^{−(np−k+1)p0}(t)dt

!#1−p

→0 as δ→b.

Thus, Theorem 5.5 generalizes this assertion to twisted cylindersC_a,b^f N with N closed.

Remark 4.1 and Theorem 5.5 imply

Theorem 5.7. LetN be a closed smoothn-dimensional Riemannian man- ifold. Ifp≥q >1,b=∞, and ⁿ_p is an integer thenH

n p+1

q,p (C_a,b^h N, N_a) = 0.

In particular, if 1< q≤2 and n is even thenH

n 2+1

q,2 (C_a,bN, Na) = 0.

Proof. In this case,A_δ0,b=^R_δ^b

0(τ−a)⁻¹(log(τ −a))^−pdτ <∞for anya, b and, by Theorem 5.5, H^k_q,p(C_a,b^h N, Na) = 0.

6. Asymptotic Twisted Cylinders

Definition 6.1. We refer to a pair (M, X) consisting of anm-dimensional manifold M and an m-dimensional compact submanifold X with bound- ary as an asymptotic twisted cylinder AC_a,b^h ∂X if M \X is bi-Lipschitz diffeomorphically equivalent to the twisted cylinder C_a,b^h ∂X.

Theorem 5.5 readily implies

Theorem 6.2. Let (M, X) =AC_a,b^h ∂X be an asymptotic twisted cylinder with dimM = dimX=m=n+ 1. Assume thatp≥q >1,

Z b a

f_k−1,p^−p0 (t)dt=∞, and the integral

Aδ0,b:=

Z b δ0

F_k−1,p^p (τ) f_k−1,p^pp0 (τ)

Z _τ

a

f_k−1,p^−p0 (t)dt ⁻¹

log Z _τ

a

f_k−1,p^−p0 (t)dt

−p

dτ is finite for some δ0∈(a, b).

Then H^k_q,p(M, X) = 0.

Proof. Note that bi-Lipschitz diffeomorphisms preserveLp and Lq. More- over, extension by zero gives a topological isomorphism between the rel- ative spaces Wr,s(C_a,b^h ∂X,(∂X)_a) and Wr,s(M, X) for all r, s. This gives topological isomorphisms

H_r,s^∗ (M, X)∼=H_r,s^∗ (C_a,b^h ∂X,(∂X)a); H^∗_r,s(M, X)∼=H^∗_r,s(C_a,b^h ∂X,(∂X)a) for all r,s. The theorem now follows from Theorem 5.5.

To obtain a version of this theorem forH^k_q,p(M), we will need the exact sequences of a pair for L_q,p-cohomology.

Recall that an exact sequence

0→A→^ϕ B →^ψ C→0 (6.1)

of cochain complexes of vector spaces is called anexact sequence of Banach complexes if A,B,C are Banach complexes and all mappings ϕ^k,ψ^k are bounded linear operators.

A short exact sequence (6.1) of Banach complexes induces an exact sequence in cohomology

. . .−→H^k−1(C)^∂

k−1

−→ H^k(A) ^ϕ

∗k

−→H^k(B) ^ψ

∗k

−→H^k(C)−→. . . with all operators ∂, ϕ^∗, ψ^∗ bounded and a sequence in reduced cohomology

. . .−→H^k−1(C)^∂

k−1

−→ H^k(A) ^ϕ

∗k

−→H^k(B) ^ψ

∗k

−→H^k(C)−→. . . , (6.2)