CC BY
14Probl. Anal. Issues Anal. Vol. 6(24), No. 2, 2017, pp. 57-80
DOI: 10.15393/j3.art.2017.3850
57
UDC 517.98, 514.745.4
Ya. A. Kopylov
ORLICZ SPACES OF DIFFERENTIAL FORMS ON RIEMANNIAN MANIFOLDS: DUALITY AND COHOMOLOGY
Abstract. We consider Orlicz spaces of differential forms on a Riemannian manifold. A Riesz-type theorem about the func-tionals on Orlicz spaces of forms is proved and other duality theorems are obtained therefrom. We also extend the results on the Holder-Poincare duality for reduced Lq,p-cohomology by Gol'dshtein and Troyanov to -cohomology, where and
are N-functions of class A2 R V2.
Key words: Riemannian manifold,, differential form, exterior differential, Orlicz space, Orlicz cohomology
2010 Mathematical Subject Classification: 58A12, 46E30
Introduction. This article is devoted to the study of the dual spaces of Orlicz spaces of differential forms on an oriented Riemannian manifold X.
Lp-theory of differential forms on Riemannian manifolds has been the subject of many papers and several books since the beginning of the 1980s. In 1976, Atiyah defined L2-cohomology for a Riemannian manifold and initiated various applications of L2-methods to the study of noncom-pact manifolds and quotient spaces of Riemannian manifolds by discrete groups of isometries. The L2-cohomology of such manifolds was studied by Gromov, Cheeger-Gromov and others (see, for example, [2, 3, 12]). In the 1980's, Goldshtein, Kuz'minov, and Shvedov defined the Lp-de Rham complex on a Riemannian manifold M for arbitrary p G [1, to] and began to investigate its cohomology, which they called the Lp-cohomology of M; they obtained many results concerning the density of smooth forms in Lp (see, for example, [5]); the nontriviality and the Hausdorff property of Lp-cohomology on important classes of manifolds (see, for instance, [7, 8, 17]),
©Petrozavodsk State University, 2017
[MglHl
duality for Lp-related spaces of differential forms and the induced duality for Lp-cohomology in [6]; compactly-supported approximation of Lp-forms (see, for example, [16]). In studying the asymptotic invariants of infinite groups and manifolds with pinched negative curvature, Gromov and Pansu also considered Lp-differential forms and lp-simplicial cochains (see [12, 18, 19]). Gol'dstein and Troyanov obtained deep results about the Lqp-cohomology of Riemannian manifolds for q = p in [9, 10, 11].
Like Orlicz function spaces, the Orlicz spaces L^ of differential forms are a natural nonlinear generalization of the spaces Lp. Orlicz spaces of differential forms on domains in Rn were first considered by Iwaniec and Martin in [13] and then by Agarwal, Ding, and Nolder in [1] (see also [4, 14]). In [13], Iwaniec and Martin established a Riesz-type theorem for an Orlicz space of differential forms on a domain in Rn. Orlicz spaces of differential forms on a Riemannian manifold were apparently first examined by Panenko and the author in [15], where de Rham regularization operators were introduced and studied for Orlicz spaces of differential forms.
We prove a Riesz-type theorem for Orlicz spaces of differential forms on a Riemannian manifold and then, using it, describe the dual spaces of Orlicz-Sobolev-type spaces of differential forms, thus generalizing the results of Gol'dshtein, Kuz'minov, and Shvedov obtained in [6] for Lp-related spaces. The so-obtained results are applied for establishing the Holder-Poincare duality for the reduced Orlicz cohomology of X, which extends the Holder-Poincare duality for Lq p-cohomology proved by Gol'dshtein and Troyanov in [11].
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 give the definition of Orlicz spaces of differential forms on a Rie-mannian manifold. The Riesz-type theorem for Orlicz spaces of differential forms (Theorem 3.1) is the contents of Section 3. Then, in Section 4, we examine the structure of the dual spaces to some -related spaces of differential forms. Finally, in Section 5, we establish a theorem on the Poincare duality for the L$I -cohomology of an oriented Riemannian manifold (Theorem 5.8).
1. N-functions and Orlicz function spaces. Definition 1.1.
A function $ : R ^ R is called an N-function if
(i) $ is even and convex;
(ii) $(x) = 0 ^^ x = 0;
(iii) lim *X£l = 0; lim ^Xe! = to.
x^Q x x^-tt x
An N-function $ has left and right derivatives (which can differ only on an at most countable set, see, for instance, [20, Theorem 1, p. 7]). The left derivative p of $ is left continuous, nondecreasing on (0, to), and such that 0 < y>(t) < to for t > 0, ^(0) = 0, lim y>(t) = to. The function
= inf{t > 0 : p(t) > s}, s > 0,
is called the left inverse of
The functions $, ^ given by
|x| |x|
$w = / ,rn, *(x) = /mdt
are called complementary N-functions.
The N-function ^ complementary to an N-function $ can also be expressed as
^(y) = sup{x|y| - $(x) : x > 0}, y E R.
N-functions are classified in accordance with their growth rates as follows:
Definition 1.2. An N-function $ is said to satisfy the A2-condition for large x (for small x, for all x), which is written as $ E A2(to) ($ E A2 (0), or $ E A2), if there exist constants xQ > 0, K > 2 such that $(2x) < < K$(x) for x > xQ (for 0 < x < xQ, or for all x > 0); and it satisfies the V2-condition for large x (for small x, or for all x), which is denoted symbolically as $ E V2(to) ($ E V2 (0), or $ eV2) if there are constants xQ > 0 and c > 1 such that $(x) < 2c$(cx) for x > xQ (for 0 < x < xQ, or for all x > 0).
Henceforth, let $ be an N-function and let (fi, E,p.) be a measure space.
Definition 1.3. The set L* = L* (fi) = L* (fi, E,^) is defined to be the set of measurable functions f : fi ^ R such that
P*(f) := j $(f № < to. o
Definition 1.4. The linear space
L* = L* (ft) = L* (ft, E,p) =
= {/ : ft ^ R measurable : p*(a/) < to for some a > 0}
is called an Orlicz space on (ft, E,p).
The corresponding Morse-Transue space is the space
M* = M*(ft) = M*(ft, E, p) =
= {/ : ft ^ R measurable : p*(a/) < to for all a > 0}.
For an Orlicz space L* = L* (ft, E,p), the N-function $ is called A2 -regular if $ G A2(to) when p(ft) < to or $ G A2 when p(ft) = to or $ G A2 (0) for ^ the counting measure on countable ft. Let ^ be the complementary N-function to $.
Below we as usual identify two functions equal outside a set of measure zero.
If / G L* then the functional || ■ ||* (called the Orlicz norm) defined
by
$ = llf IIl* (Q) = suP
J
q
: < i
is a seminorm. It becomes a norm if ß satisfies the finite subset property (see [20, p. 59]): if A G E and ß(A) > 0 then there exists B G E, B C A, such that 0 < ß(B) < to.
The equivalent gauge (or Luxemburg) norm of a function / G is defined by the formula
II/Il(*) = II/IIl(*)(q) = inf jk> 0 : k) < •
This is a norm without any constraint on the measure ß (see [20, p. 54, Theorem 3]).
We will need the following familiar assertion (see [20, item (ii), p. 57]): Lemma 1.5. Let
0 < /l < /2 < ■ ■ ■ < /m < • • •
be an increasing sequence of nonnegative measurable functions in the Or-licz space L*(fi) ((fi, E,p) is a measure space) and let fm ^ f a.e. Then ||fm||(*) < llf ||(*) < to. 2. Orlicz spaces of differential forms. Let X be a Riemannian manifold of dimension n. Given x E X, denote by (u(x),6(x)) the scalar product of exterior k-forms u(x) and 6(x) on TxX. This gives a function x ^ (u(x),0(x)) on X.
Let $ : R ^ R and ^ : R ^ R be two complementary N-functions. Denote by L*(X, Ak) the class of all measurable k-forms u such that
p*(u) := J $(|u(x)|)d^x < to. x
Here d^x stands for the volume element of the Riemannian manifold X. We will identify k-forms differing on a set of measure zero.
Given a (not necessarily orientable) Riemannian manifold X, introduce the space L*(X, Ak) as the class of all measurable k-forms u satisfying the condition
p* (au) < to for some a > 0.
The corresponding Morse-Transue space M* (X, Ak) is defined as the class of all measurable k-forms u such that
p*(au) < to for all a > 0.
Obviously, L* (X, Ak) C L* (X, Ak).
As in the case of Orlicz function spaces, the space L*(X, Ak) is endowed with two equivalent norms: the gauge norm
M|(*) =f K> 0 : p^K) ^
and the Orlicz norm
= sup
(u(x), 0(x)) dßx
X
0 g L*(X, Ak), p*(d) < 1
As in the case of function spaces, it can be proved that L*(X, Ak) endowed with one of these norms is a Banach space.
Obviously, the gauge norm of a k-form u is nothing but the gauge norm of its modulus function |u|. The same holds for the Orlicz norm
([15, Lemma 2.1]). Moreover, similarly to the case of Orlicz function spaces ([20, Proposition 10, p. 81]), we have
Lemma 2.1. The Orlicz and gauge norms of a k-form u G L$ (X, Ak) can be calculated by the formulas
u||$ = S^ := sup
eeM *(x,Ak ) ll^lw <1
X
and
|u||($) = XL := sup
eeM *(x,Ak ),
1
X
Proof. For 0 G M*(X, Ak) with ||0||w < 1 we have
(u(x), 0(x))dpX
X
< J |w(x)||0(x)|dpx <
X
< sup
geM *(X ), llsHc®) <1
|u(x)|g(x)dpx
X
= |||u|
.
The last equality here holds by [20, Proposition 10, p. 81]. Thus,
S^ = sup
eeM * (x,Ak ), lWlc®) <i
(u(x), 0(x))
X
<||M
$.
On the other hand, let (gm)meN be a sequence of functions in M*(X) with ||gm||(^) < 1 such that
|w(x)|gm(x)dpx
X
— || |u| ||$ as m —y to.
Since
|u(x)|gm(x)dpx
< J |u(x)||gm(x)|dpx < || |u| ||$,
X
we also have
|w(x)||gm(x)|d^x ^ || ||$ as m ^ to.
x
Consider the sequence (#m)meN of k-forms 0m defined by
|gm(x)| ^Jf if u(x) = 0,
0m (x) =
Then ||0m||(*) = ||gm|| < 1 and
otherwise.
(u(x),0m(x))dßx
x
|^(x)||gm (x)|d^X
x
as m ^ to. Therefore,
||$ < sup
eeM *(x,Afe),
(u(x), 0(x))dßx
K«) ^
<1
x
= |M|$.
Thus, we get the desired equality for the Orlicz norm.
For the gauge norm, the equality ||u||(*) = |||u||(*) is obvious, and one must only prove that
TL = yM^^
which is done in the same manner as for the Orlicz norm with the use of [20, Proposition 10, p. 81]. □
Below, when this does not lead to confusion, we use the abbreviations L* = (L*, ||-||*), L(*! = (L*, |.|(*));
M* = (M*, || ■ ||*), M(*! = (M*, || ■ ||(*)).
3. The Riesz theorem. Let X be an oriented n-dimensional Rie-mannian manifold.
For a k-form u on X, let *u be the Hodge dual of u (an (n — k)-form).
The bilinear function
(w,0) = J w a 0 (1)
defines a pairing
between L*(X, Ak) and L(^(X, Ak) (and between L(*)(X, Ak) and L*(X, Ak)). The integral on the right-hand side of (1) exists because
w A 0 = (-1)kn-k(w, *0)dpx, |(w, *0)x |<|w|x |* 0|x = |w|x |0|x . Hence, we obtain two versions of the Holder inequality:
|(w,0)|<||w||* ||0N(^) (2)
and
|(W,0)I<|M|(*) ||0|^. (3)
Assign to each form 0 G L(*)(X, An-k) the functional
F0 (w) = ^ w A 0. (4)
X
By (2) and (3), we have
F(w)| < ||wN*N0N(^); F(w)| < ||w||(*)||0|k. (5)
Theorem 3.1. If $ is an N-function then the correspondence 0 ^ yields isometric isomorphisms
L«(X, An-k) 4 (M*(X, Ak))'; L*(X, An-k) 4 (M(*) (X, Ak))'.
Proof. Let us prove the first isomorphism.
By (5), ||F^|| < ||0|(^). Show that an arbitrary continuous functional F G (M*(X, Ak)) is representable uniquely in the form (4). Let h : V ^ Rn, V C X be a local chart of X and let U be an open set with compact closure clX U C V; then U is endowed with two metrics: the metric p of the Riemannian manifold X and the metric p induced by h from the standard metric on Rn. It is not hard to see that the L*-spaces (M*-spaces) of k-forms on U L* (U, Ak ,p) and L(*) (U, Ak ,p)
(M*(U, Ak, p) and M(*)(U, Ak, p)) corresponding to these metrics coincide and have equivalent norms. Making use of the Riesz theorem on the general form of a linear functional on the function space M*, we, involving the coordinate representation of differential forms, conclude that every functional / G (M*(U, Ak, p)) is uniquely representable in the form
/(a) = y a A 0f, 0f G LW(U, An-k,p).
X
By the equivalence of the norms in M*(U, Ak, p) and M*(U, Ak, pi), the same holds for functionals in M*(U, Ak,p). Therefore, for FG (M*(X, Ak))' and an open set U with compact closure, there is a unique form 0U G G LW(U, An-k) such that
F(w) ^ y w A 0u for every w G M* (U, Ak).
U
Given two sets Ui and U2 as above, the forms 0Ux and 0U2 coincide on U1 fl U2 by the uniqueness of 0UinU2. Thus, all forms 0U defined for different U agree with each other and thus define an (n — k)-form 0 on X. The form 0 belongs to L(*) (X, An-k) locally, satisfies the condition
F(w) = y w A 0 for all w G M* (X, Ak) with compact support,
X
and is defined by this condition uniquely.
Consider a compact set Y C X. Let g G M* (X) be a function with compact support contained in Y having ||g|* < 1. Let be the k-form on X defined by the formula
^ (x) ,'(—1)k(n-k) |I(X| (*0(x)) if X G Y and 0(x) = 0; g 0 otherwise.
We have
F (^g ) = y ^g A0 = (— 1)k(n-k) y ^ (*0(x))A0(x) = J g(x)|0(x)|d^X .
Y Y Y
Since ||g||$ < 1, this gives
g(x)|9(x)|d^x
Y
= IF (ßg )|<||F |
Hence, using Lemma 2.1, we obtain
H^Iy ||(*) = |||% |||(*) = sup
geM*(Y); ||gH®<1
g(x)I6(x)Idßx
Y
< IIFII.
Let Yi C Y2 C ■ ■ ■ C Ym C ■ ■ ■ C X be an exhaustion of X by compact sets and let 9m be the restriction of 9 to Ym. Put fm = |9m|. Then the sequence {fm}meN satisfies the conditions of Lemma 1.5. Since 11fmH(^) < ||FH, the function lim fm = |9| lies in L(*)(X), and so 9 G
G L(*)(X, An-k) and
m—^^o
||9|(^) = lim ||9m||w < HF|
m—^
(6)
The functionals F and Fq coincide on the set of forms in M* (X, Ak) having compact support, which is, as in the case of Orlicz function spaces, dense in M*(X, Ak). Thus,
F(u) = u A 0
for all u E M*(X, Ak). Combining (2) and (6), we infer that ||FQ|| = = ||0||W.
Let us now establish the second isomorphism
L* (X, An-k) 4 (M(*! (X, Ak))'.
Let F E (M(*! (X, Ak)). Then, as above, we see that there exists a unique (n — k)-form 0 belonging to L* locally that satisfies the condition
F(u) = J u A 0 for all u E M(*!(X, Ak) with compact support. x
Using Lemma 2.1, we verify in the same manner as for || ■ ||* that, given any compact set Y C X,
IY H* <
F.
Because of the inequalities
|| ■ < || ■ ^ < 2| ■ ||(^),
we have
||0|y||(*) < ||F||.
Taking an exhaustion Y1 C Y2 C ■ ■ ■ C Ym C ■ ■ ■ C X of X by compact sets, we as above conclude that 0 G L*.
Now, the functionals F and F# coincide on the dense set of forms with compact support in M(*)(X, Ak) and hence on M(*)(X, Ak). By Lemma 2.1,
|F || = ||Ffl || = sup ll^lm <i
wA0
X
*.
The theorem is completely proved. □
4. The dual spaces to L*-related spaces of differential forms.
Throughout this section, X is an oriented smooth Riemannian manifold of dimension n and ($1, and ($2, ) are pairs of conjugate N-functions. Introduce some spaces of differential forms.
For A G {L, M} and ($*) G {$*, ($*)}, denote by Ak*lM*2)(X) the space A*1 (X, Ak) © A*2 (X, Ak+1) with the norm
||(a,^)||(*1 ),(*2) = ||a||<*i) + ||^|(*2) . Given (a, ft) G Mf*^^)(X) and (w,0) G L^-^(X), where
T^) = if ($i) = $i,
( ') if ($i) = ($i),
we put
((a,ft), (w,0)) = (—1)k (a, 0) + (ft,w). (7)
Theorem 3.1 implies that the pairing (7) defines an isometric isomorphism
(Mf*1 ),<*2 )(X ))' = (X ).
Moreover,
|((a,ft), (w,0))| < ||(a,ft)11<*i),<*2) ■ |(w,0)|.
A differential (k + 1)-form 0 E L11oc (X, Ak+1) on X is called the weak exterior differential (or derivative) of a k-form u E L1oc(X, Ak) (which is written as du = 0) if,
J 0 A u = (— 1)k+iy u A du xx
for any u E Dn-k-1 (X), where D1 (X) is the set of smooth l-forms on X with compact support included in Int X.
Let $1 and $2 be N-functions. For 0 < k < n, put
fik*lM*2)(X) = { u E L<*^(X, Ak) : du E L<*2)(X, Ak+1)} .
This is a Banach space with the norm
||u||<*1 ),<*2 ) = ||u||<*1) + |du|{*2) .
From now on we assume that $1, $2 E A2 n V2, and hence also , ^2 E A2 n V2.
If $ E A2 n V2 then, as is well known, the spaces L* and M* coincide and hence, by Theorem 3.1, the space L* is reflexive. Thus, there is no need in the spaces M*,+. We will often assume that the space fik*) <*2) (X)
is embedded in Lk*) <*2) (X) by identifying a form a E fik*) <*2) (X) with
the pair (a,da) E Lk*1),<*2)(X).
Given a subspace H C Lk* *), denote by H^ the annihilator of H
in (X) with respect to the pairing (7). Since this pairing satisfies
((a,P), (u,0)> = (—1)k(n"k-1)((u,0), (a,P)>,
there is no difference between the pairings between Lk*) <*) (X) and
L^-k"-^(X) and between L^-k-^(X) and Lk** N ,* N(X1).
<*2),<*1 y ' <*2 <*1 )'<*2)V '
The definition of fik*) <* )(X) implies that
fik*1),<*2)(X) = (Dn_k_1(X))±.
put fik*1),<*2),o(X) = (finík;(*0 (X))^. Since Dn-k-1 X C we have fik*1 ) , <*2 ),0 (X) C fik*1),<*2) (X).
Observe that if Ok* \ ,* N 0(X) = 0k \ /* \ (X) then (X) =
<*1),<*2) ,0V > <*1) , <*21 <*2 ) ,<*1 ) ,0V '
= (X)-
Lemma 4.1. The following hold for G A2 f V2:
(1) Smooth forms constitute a dense set in ) <*2 )(X)•
(2) Smooth forms with compact support constitute a dense set in 0k*1) , <*2 ) ,0 (X).
Proof. Item (1) stems from the only theorem of [15] about the properties of the de Rham regularization operators in Orlicz spaces of differential forms. Prove (2). Denote the closure of Dk(X) in L<*1),<*2) (X) by Dk(X). Then, by [21, Theorem 4.7],
•>k\_ f ok f V\\ _ ok
Dk (X) = ((Dk= (x )) = ).
□
Lemma 4.2. If , $2 G A2 nV2 and a form u G ) (^) (X) has compact support then u G ) ($2) o(X).
Proof. Suppose that u G ) ($2)(X) has compact support. Assume first that 9 is a smooth (n — k — 1)-form. By Lemma 4.1, there exists a sequence {uj } of smooth forms with compact support such that Uj ^ u in norm as j ^ to. Then
((u, du), (9, du)) = lim ((u,, duj), (9, d9)) =
= lim I [(—1)kuj A d9 + duj A 9] = lim d(u A 9) = 0. (8)
j^^ J j^^
X
The last equality in (8) is due to the Stokes theorem. Now, let 9 be an arbitrary form in O^^L.(X). By Lemma 4.1, there is a sequence
{9j} of smooth forms converging to 9 in norm as j ^ to. Then
((w,dw), (0, dw)) = lim ((w,dw), (0j,d0j)) =0.
Thus, 0 G 0k*1),<*2),0(X). □
Each pair of forms (w,0) G L^-(X) defines by (7) a continuous linear functional on Lk*) <* )(X) and hence on ) <*2) (X) and
fik*i) <*2) o(X). On the last two spaces, this functional is defined by the formula
F(a) = J[(—1)ka A 0 + da A u]. (9)
x
Theorem 4.3. If $1, $2 E A2 n V2 and are the correspon-
ding complementary functions then any continuous linear functional on fik*i) <*) (X) (on fik*) <*2) 0(X)) can be represented in the form (9). A pair of forms (u,0) defines the zero functional on fik*) <*)(X) (on fik*1),<*2),o(X)) if and only if u E fi^-y--*),0(X) and 0 = du (u E
E fi^—(X) and 0 = du). The norm of the functional (9) on
<*2), <*1) V ' J V 7
fik*1),<*2) (x ) (on fik*1),<*2),o (x )) has the form
||F|| = inf {||0 + dP||^ + ||u + PH^ : P E fin-y^.D^o
( HFH =inf {|0 + ^fe + ||u + Pfe : P E fin-k-*o(X)}) .
Proof. In accordance with [21, Theorem 4.9], if H is a closed subspace in a Banach space Y then Y'/H^ = H', where the isomorphism is induced by the canonical pairing between Y and Y'. Therefore,
(fik*1),<*2) (x ))'=Ln-k-c*!) (X) / (fik*1 ),<*2 )(X ))x =
=(X ,o(x);
vfik*1),<*2),o(X)) = Ln-7,<*7)(X) / (fik*1)'<*2),o(X0 =
=L
--4-1 d (x Vfi™(X)
<*2), <*1) / <*2),<*1) □
Theorem 4.4. If $1, $2 E A2 n V2 and are their complementary
N-functions then the dual of the space fik*) <*2 )(X) is isomorphic to the
completion of Dn-k (X) with respect to the norm
||u|| =inf {||u + d0||<*-y + ||0N(*2y : 0 e Dn-k-1(X)} . (10)
This isomorphism is given by the action
(a,w) = (-l)fcy a Л w. (11)
X
Proof. Consider the embedding j : L<*1 )(X, An-k) ^ L^-4-^.(X) defined by j (w) = (0,w). Let
n: Ln-k-4o(X) ^ LT-y-fc(X V°T-T-fc,o(X)
be the canonical projection. It is not hard to see that n o j is a monomor-phism. Since the set S = {(w,0) : w G Dn-k-1(X),0 G Dn-k(X)} is
dense in Ln-k-1 , (X), n(S) is dense in L^-^-(X) /o^-^- (X). Let w G Dn-k-1(X), 0 G Dn-k(X). Since (w,dw) G O^-^- Q(X), we have n(w, 0) = n(0, 0 — dw) = n o j(0 — dw). Hence, the set n o j(Dn-k) is dense in L^-^.(XWo^zk-^ (X). Moreover,
<*2 ),<*1 P V <*2),<*1 ),0V ! '
||n o j(w)|| . , / . =
II A ^n^-fe-1 (XWnmkri_ (X)
\ v > t\\ /\T/. \ nv ^
<Ф2 <Ф2 )><ф1 )'0
inf {||w + +ne||w : в e ,0(X^.
By Lemma 4.1(2), the set Dn-k-1(X) is dense in O^-^q(X). Hence,
||n o j(w)|| . , / . =
11 JV JULn-kr1_(XWnmkri_ (X)
<«2 >,<«1>V 7 <«2 >,<«1 >,0V 7
= inf {||w + d0||<^ + ||0N<*2y : 0 G Dn"k"1
Thus, the space L^-^-^(X) /(X) is isomorphic to the completion of Dn-k (X) with respect to the norm (10). Now, in view of [21, Theorem 4.9], if H is a closed subspace in a Banach space Y then (Y/H)' = = Hwhere the isomorphism is induced by the canonical pairing between Y and Y'. Thus, (l^-i (XWo^-* (X))' =
' V <*2 ),<*1)V V <*2 ),<*1 ),0V >)
= (O^-k-^. (X))± = Ok* v ,* v(X), and the first claim of the theo-
V <*2 ),<*1),0V ') <*1), <*2) ^
rem is established.
Further, since
((a, da), (0, u)> = (-1)^ a A u,
x
the form a E fik*) <*2)(X) acts at the forms n o j(u), u E Dn-k(X), by the formula
(a, n o j(u)> = (-1)k J a A u.
x
The theorem is proved. □
5. Holder—Poincare duality for L* */7-cohomology. Let X be
an oriented Riemannian manifold of dimension n.
Given N-functions $/ and $//, consider the spaces
Zk*„) (X) = {u E L<*») (X, Ak) : du = 0};
Bk*i),<*„)(X) = {u E L<*») (X, Ak) :
u = dp for some P E L<*) (X, Ak-1)}.
_k 7
Denote by B<*7 ),<*„ )(X) the closure of Bk* ),<*//)(X) in L<*") (X, Ak). The quotient spaces
Hf*i),<*„)(X) := z{W)(X)/bk*i),<*„)(X)
and
Hk*i),<*„)(X) := Z{k*/7)(X)/Bk*i),<*n)(X)
are called the kth L<*7),<*/7) -cohomology and the kth reduced L<*7),<*/7) -cohomology of the Riemannian manifold X, the latter cohomology being a Banach space.
If $/ = $// = $ then we use the notations fik*)(X), H{{*)(X), and
h k*) (x ) instead of fik*),<*)(x), Hf*),<*) (x ), and h {*),<*)(x ) respectively. Thus, the L<*)-cohomology Hk*)(X) (respectively, the reduced _k
L<*)-cohomology H<*)(X)) is the kth cohomology (respectively, the kth reduced cohomology) of the cochain complex {fi**)(X),d}.
The kth interior reduced L<*),<$/7)-cohomology of a Riemannian manifold X is the Banach space
),<*„),o(X) = ),o(X) jdDk-1 (X) ,
where dDk-1 (X) is the closure of dDk (X) in ) (X, Ak) and
),<*„),o(X) = Ke^d : ^^) ^ ftf+1 ^ JnZ*(X>■<*" >.
Thus, a k-form 6 belongs to ) ) 0 (X) if and only if 6 G
G )(X, Ak), d6 = 0, and there is a sequence is a weakly closed forms 6^ G (X) such that
l|6j — 6||<$7) ^ 0 and ||d6j||{*7) ^ 0 as j ^ to.
The quotient (semi)norm on each of the above-introduced cohomolo-gy spaces depends on the choice of the norm on and but the resulting topology does not.
From now on, we assume all N-functions under consideration to belong to A2 n V2.
In [11], Gol'dshtein and Troyanov realized the kth Lq,p-cohomology as the kth cohomology of some Banach complex. Here we apply this approach to L<*),<$/7)-cohomology.
Fix an (n + 1)-tuple of N-functions F = |$0, ..., and put
0F(X) = ,$fe+x(X); ^kF)(X) = ^(W),($fe+i)(X).
Use the unified notation fi(F)(X) for OF (X) and fi|F-)(X). Since the weak exterior differential is a bounded operator d : ^(f) (X) ^ ^(i)1 (X), we obtain a Banach complex
0 ^ ^0f) (X) ^ ^1F)(X) ^ ■ ■ ■ ^ 0(F) (X) ^ ■ ■ ■ ^ ^(!F) (X) ^ 0.
The L<f) -cohomology H(F) (X) (respectively, the reduced L<f) -cohomology _k
H<f) (X)) of X is the kth cohomology (respectively, the kth reduced cohomology) of the Banach complex (fi*F),d).
_k
The above-defined cohomology spaces H{F)(X) and H f (X) in fact depend only on $k-1 and :
H<F)(X) = Hf$fe_i),{$fe)(X) = )(X) /Bi$fe_i),{$fe);
H<F) (X) = H<$fe-i),{$fc) (X) = )(X) /),{$fc) .
Denote by ftkF) 0 (X) the closure of Dk (X) in ftkF)(X). The interior reduced L<f)-cohomology of X is the reduced cohomology of the Banach complex
0 ^ ^0F),O(X) ^ ^1F),0(X) ^ ■ ■ ■ ^ ft<F),0(X) ^ ■ ■ ■ ^ ^F),O(X) ^ 0;
_k _k . /-(X Afc)
H<F),0(X)= H<*fe),{*fe+i),0(X) = Zf*fe),{*fe+i),o(X)/ dD<-1 (X). ^ j
The dual of an (n + 1)-tuple of N-functions F = |$0, ..., is the (n + 1)-tuple F' = |^0, ,..., where and $n-k are complementary N-functions for all k. Henceforth, we assume all N-functions to belong to the class A2 HV2.
Fix an (n + 1)-tuple of N-functions F = |$0, $1,..., and let F' = |^0, ,..., } be its dual (n + 1)-tuple. For -1 < k < n, introduce the vector spaces
P(F) (X) = ),{*„+i) (X) = ) (X, Ak) © L^+^X, Ak+1)
(here ) (X, Ak) = 0 for k = -1,n + 1). If (a, ft) G P{F)(X) with a G ) (X, Ak) and ft G L^+0 (X, Ak+1) then P{F)(X) is endowed with the norm
ll(a,ft)l|p(f> (X) = ^«^{^fe) + llftll{$fe + i). Let dp : P<kF)(X) ^ P<k+)1 (X) be defined as
dp (a, ft) = (ft, 0).
The so-obtained Banach complex ^P*f)(X),dpj has trivial cohomology.
Lemma 5.1. Let F = |$0, ,..., } be an (n+1)-tuple of N-functions and let F' = , ..., } be its dual (n + 1)-tuple. Then the spaces
PkF) (X) and ) (here, as above, the bar changes the type of the
norm) are dual with respect to the pairing
((a,P), (u,0)) = J ((-1)ka A u + P A 0) . (12)
Lemma 5.1 easily follows from Theorem 4.3. Lemma 5.2. The operators
d : PiF)(X) ^ PL(X) and d : P^-1^) ^ P^ (X) are adjoint.
Proof. If (a,P) G Pk-1 (X) and (u,0) G Pn-k-1 (X) then (d(a, P), (u, 0)) = ((P, 0), (u, 0)) = j(-1)kP A
X
((a, P), d(u, 0)) = ((a, P), (0,0)) = / P A 0. □
X
Put
(X) = {(u,dU) G P(F") (X) : U G )} ;
£<f),o(X) = {(u,du) G P(F)(X) : u G fifo^X)} .
Clearly, these spaces form Banach complexes S{f) (X) and £{f),0(X) which are isomorphic to ^{f)(X) and fi{F),0(X) respectively. Introduce the following quotient complex of P^F)(X):
AFy (X ) = P{F) (X ) ,o(x ).
What was said above implies: Proposition 5.3. The graded vector space A^^-(X) possesses the following properties:
(1) A^p-(X) is a Banach space with respect to the norm
y(u, 0)||a = inf {||u + p||<^y + ||0 + } .
(2) AL(X) is dual to E^-*"1^) with respect to the pairing (12).
(3) The differential dp : pL(X) ^ pM1 (X) induces a differential
dA : A^(X) ^ Ak+1 (X) and (Af—"(X),d^) is a Banach com-
{F ) {F ) {F )
plex.
(4) The operators dA : A<-1 (X) ^ A^-(X) and dE : Sn"k-1 (X) ^
{F/) {F) {F)
^ )< (X) are adjoint up to sign with respect to the pairing (12).
Examine the cohomology of the Banach complex (A^^-(X)(X),dA).
{F )
If we put
Zk (A^(X)) = Ker dA : A^(X) ^ A^(X)
and
Bk (a^(X)) = Im dA (A^Fry(X))
and denote by B< (a^-(X)) the closure of Bk (a^(X)) then the cohomology and the reduced cohomology of A^^-(X) are the spaces
{FT)
H< (aF)(X)) = Z< (aFT(X)) /B< (aFT(X)) ; H (af)(X)) = Z< (ATFt)(X)) /B (aFt)(X)) .
We will need the following assertion [11, Lemma 3.1]:
Lemma 5.4. Let I : Y0 x Y1 ^ R be a duality between two reflexive Banach spaces. Let B0, B1, A0, A1 be linear subspaces such that
Bo C Ao = B^ c Yo ; Bi c Ai = BO1 C Yi.
Then the pairing I: (A0/B0) x (A1/B1) ^ R (with the bars standing for closures) is well-defined and induces duality between A0/B0 and A1/B1.
Lemma 5.5. The pairing (12) induces a pairing between the reduced cohomologies of A^^-(X) and S*F)(X).
Proof. We have
Bk"1(AfFTy(X)) C Zk-1(AyF7y(X)) = (Bn-k(EJF)(X)))X C A^X)),
and, similarly,
Im d£-k-1 C Ker d£-k = (Im d^-2)X C E^-) (X),
where the equalities are due to the fact that d^ and d^ are adjoint operators. It remains to apply Lemma 5.4 with X0 = A-^-1 and X1 =
= EnF)k (X). □
Lemma 5.6. The reduced cohomology of the Banach complex
,0
a shift:
(A^^- (X),dA) is isomorphic to the interior cohomology of X up to
H ^r (X) = H k-1 (A^- (X)
{F ')(X )= H
ism is in
j (P) = (0,P).
The isomorphism is induced by the mapping j : Z^— (X) ^ P-k-1 (X),
{F ),0 {F )
Proof. Every element in A^^-(X) is represented by an element (a, P) G
G P-k-^!(X) modulo Ek-1 (X); thus, (a, P) and (a1,p1) represent one
{F ) {F ) ,0
element in A^(X) if and only if a — a1 = u and P — P1 = du, where
- e Ek-r,0»
Further, (a,P) G P-^1(X) represents an element of Zk-1 (a^-(X)) whenever dP(a,P) = (P, 0) G E^ q(X), that is, P G Z-^ q(X). Thus,
Z(-1 (af)(X)) = {(a,P) Gp{-)(X) : P G Z{Fy,0(X)} /E(-),0(X).
Similarly, (a, P) represents an element in Bk-1 ^A^^-(Xif there is (7,^) G P-^X) with (a, P) = dA(7,^) = (5, 0) modulo E^(X), which means that P = du G B^^ Q(X). Thus,
(Af)(X0^(a,P) GP|—-(X) : P G BF^^/E(F1),0(X)
and
Bk-1 (A^(X)) = {(a,P) GP^X) : P G £^,0(X)} /e^X)
Therefore,
k i ( \ {("'в) GP^(X): в G Z*о
H k-1 (A^- (X)) = 4-^-^-}
v <F> ; {(а,в) GP^(X) : в g В^0(Х)}
'в) G Р<-У(X) : в G Z<Fy,o(X)} • R Rk
{(0,$ GP<Fy(X) : в G B
<F '>,0
Thus, the embedding j : Z-^- (X) — P-k-!(X), j(ft) = (0,ft), induces
{F ),0 {F )
an algebraic isomorphism j : H-^^ Q(X) -— Hk-1 ^A^^-(XWe also have the relation
'в) GP-k-l(X) : в G Zk
Hk-1 (a^-(X)) = ^-.
V <F> 7 |(0'в) GP^X) : в G £kF>,o(X)}
The quotient on the right-hand side is endowed with the natural quotient norm and j induces an isometric isomorphism j* : Ну^ту,0(X) -=
4 H"-1 Итт (X )\-D ,
Thus, we have
Theorem 5.7. Let X be a smooth n-dimensional oriented Riemannian manifold and let F = (Ф0, Ф1,... Фп) and F' = (Ф0, Ф1,..., Фп) be dual sequences of N-functions with Ф^ G Д2 П V2. Then the Banach spaces H(X) and Hn-Ty0(X) are dual with respect to the pairing (ш,в) = = / ш Л в for ш G ZjL (X) and в G Z^ (X).
This gives the following duality theorem for £ф/,ф//-cohomology: Theorem 5.8. Let X be an oriented n-dimensional Riemannian manifold. If Ф/, Ф/j are N-functions belonging to Д2 П V2 and Ф/ and Фц
_k
are their respective complementary N-functions then Hф/,ф// (X) is isomorphic to the dual of H^-),(Ф7),0(X) and Hкф/),(ф//) (X) is isomorphic
_у
to the dual of Hф//,ф/,0(X). The dualities are given by the pairing
([ш], [в]) = |ш Л в.
X
Proof. The theorem results from Theorem 5.7 by considering any sequence of N-functions ($0,..., with $(-1 = and = $// and its dual sequence. □
Acknowledgment. The author is indebted to the anonymous referees whose helpful comments and suggestions have improved the exposition of the paper.
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Received May 12, 2017.
In revised form, October 6, 2017.
Accepted August 30, 2017.
Published online October 6, 2017.
Sobolev Institute of Mathematics
4, Akad. Koptyuga st., Novosibirsk 630090, Russia;
Novosibirsk State University
2, Pirogova st., Novosibirsk 630090, Russia
E-mail: [email protected]
