Annales Mathématiques Blaise Pascal

Teodor Banica

ANNALES MATHÉMATIQUES
BLAISE PASCAL
Teodor Banica
A Note on Free Quantum Groups
Volume 15, no 2 (2008), p. 135-146.
<http://ambp.cedram.org/item?id=AMBP_2008__15_2_135_0>
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Annales mathématiques Blaise Pascal 15, 135-146 (2008)
A Note on Free Quantum Groups
Teodor Banica
Abstract
We study the free complexification operation for compact quantum groups,
G → Gc. We prove that, with suitable definitions, this induces a one-to-one cor-
respondence between free orthogonal quantum groups of infinite level, and free
unitary quantum groups satisfying G = Gc.
Une Note sur les Groupes Quantiques Libres
Résumé
On étudie l’opération de complexification libre pour les groupes quantiques
compacts, G→ Gc. On montre qu’avec des définitions convenables, cette opération
induit une bĳection entre groupes quantiques orthogonaux libres de niveau infini,
et groupes quantiques unitaires libres satisfaisant G = Gc.
Introduction
In this paper we present some advances on the notion of free quantum
group, introduced in [3]. We first discuss in detail a result mentioned
there, namely that the free complexification operation G → Gc studied
in [2] produces free unitary quantum groups out of free orthogonal ones.
Then we work out the injectivity and surjectivity properties of G → Gc,
and this leads to the correspondence announced in the abstract. This
correspondence should be regarded as being a first general ingredient for
the classification of free quantum groups.
We include in our study a number of general facts regarding the oper-
ation G → Gc, by improving some previous work in [2]. The point is that
now we can use general diagrammatic techniques from [4], new examples,
and the notion of free quantum group [3], none of them available at the
time of writing [2].
Work supported by the CNRS and by the Fields Institute.
Keywords: Free quantum group.
Math. classification: 16W30.
135
T. Banica
The paper is organized as follows: 1 contains some basic facts about the
operation G → Gc, and in 2-5 we discuss the applications to free quantum
groups.
1. Free complexification
A fundamental result of Voiculescu [6] states if (s1, . . . , sn) is a semicircular
system, and z is a Haar unitary free from it, then (zs1, . . . , zsn) is a circular
system. This makes appear the notion of free multiplication by a Haar
unitary, a → za, that we call here free complexification. This operation
has been intensively studied since then. See Nica and Speicher [5].
This operation appears as well in the context of Wang’s free quantum
groups [7], [8]. The main result in [1] is that the universal free biuni-
tary matrix is the free complexification of the free orthogonal matrix. In
other words, the passage O+n → U+n is nothing but a free complexification:
U+n = O
+c
n . Moreover, some generalizations of this fact are obtained, in
an abstract setting, in [2].
In this section we discuss the basic properties of A → Ã, the functional
analytic version of G → Gc. We use an adaptation of Woronowicz’s axioms
in [9].
Definition 1.1. A finitely generated Hopf algebra is a pair (A, u), where
A is a C∗-algebra and u ∈ Mn(A) is a unitary whose entries generate A,
such that
∆(uij) =
∑
uik ⊗ ukj
ε(uij) = δij
S(uij) = u∗ji
define morphisms of C∗-algebras (called comultiplication, counit and an-
tipode).
In other words, given (A, u), the morphisms ∆, ε, S can exist or not. If
they exist, they are uniquely determined, and we say that we have a Hopf
algebra.
The basic examples are as follows:
(1) The algebra of functions A = C(G), with the matrix u = (uij)
given by g = (uij(g)), where G ⊂ Un is a compact group.
136
A Note on Free Quantum Groups
(2) The group algebra A = C∗(Γ), with the diagonal matrix u =
diag(g1, . . . , gn), where Γ =< g1, . . . , gn > is a finitely generated
group.
Let T be the unit circle, and let z : T → C be the identity function,
z(x) = x. Observe that (C(T), z) is a finitely generated Hopf algebra,
corresponding to the compact group T ⊂ U1, or, via the Fourier transform,
to the group Z =< 1 >.
Definition 1.2. Associated to (A, u) is the pair (Ã, ũ), where Ã ⊂ C(T)∗
A is the C∗-algebra generated by the entries of the matrix ũ = zu.
It follows from the general results of Wang in [7] that (Ã, ũ) is indeed a
finitely generated Hopf algebra. Moreover, ũ is the free complexification of
u in the free probabilistic sense, i.e. with respect to the Haar functional.
See [2].
A morphism between two finitely generated Hopf algebras f : (A, u) →
(B, v) is by definition a morphism of ∗-algebras As → Bs mapping uij →
vij , where As ⊂ A and Bs ⊂ B are the dense ∗-subalgebras generated
by the elements uij , respectively vij . Observe that in order for a such
a morphism to exist, u, v must have the same size, and that if such a
morphism exists, it is unique. See [2].
Proposition 1.3. The operation A → Ã has the following properties:
(1) We have a morphism (Ã, ũ) → (A, u).
(2) A morphism (A, u) → (B, v) produces a morphism (Ã, ũ) → (B̃, ṽ).
(3) We have an isomorphism ( ˜̃A, ˜̃u) = (Ã, ũ).
Proof. All the assertions are clear from definitions, see [2] for details. 
Theorem 1.4. If Γ =< g1, . . . , gn > is a finitely generated group then
C̃∗(Γ) ' C∗(Z ∗ Λ), where Λ =< g−1i gj | i, j = 1, . . . , n >.
Proof. By using the Fourier transform isomorphism C(T) ' C∗(Z) we
obtain C̃∗(Γ) = C∗(Γ̃), with Γ̃ ⊂ Z ∗ Γ. Then, a careful examination of
generators gives the isomorphism Γ̃ ' Z ∗ Λ. See [2] for details. 
At the dual level, we have the following question: what is the compact
quantum group Gc defined by C(Gc) = C̃(G)? There is no simple answer
to this question, unless in the abelian case, where we have the following
result.
137
T. Banica
Theorem 1.5. If G ⊂ Un is a compact abelian group then C̃(G) = C∗(Z∗
L̂), where L is the image of G in the projective unitary group PUn.
Proof. The embedding G ⊂ Un, viewed as a representation, must come
from a generating system Ĝ =< g1, . . . , gn >. It routine to check that the
subgroup Λ ⊂ Ĝ constructed in Theorem 1.4 is the dual of L, and this
gives the result. 
2. Free quantum groups
Consider the groups Sn ⊂ On ⊂ Un, with the elements of Sn viewed as
permutation matrices. Consider also the following subgroups of Un:
(1) S′n = Z2 × Sn, the permutation matrices multiplied by ±1.
(2) Hn = Z2 o Sn, the permutation matrices with ± coefficients.
(3) Pn = T×Sn, the permutation matrices multiplied by scalars in T.
(4) Kn = T o Sn, the permutation matrices with coefficients in T.
Observe that Hn is the hyperoctahedral group. It is convenient to collect
the above definitions into a single one, in the following way.
Definition 2.1. We use the diagram of compact groups
Un ⊃ Kn ⊃ Pn
∪ ∪ ∪
On ⊃ Hn ⊃ S∗n
where S∗ denotes at the same time S and S′.
In what follows we describe the free analogues of these 7 groups. For
this purpose, we recall that a square matrix u ∈ Mn(A) is called:
(1) Orthogonal, if u = ū and ut = u−1.
(2) Cubic, if it is orthogonal, and ab = 0 on rows and columns.
(3) Magic’, if it is cubic, and the sum on rows and columns is the
same.
138
A Note on Free Quantum Groups
(4) Magic, if it is cubic, formed of projections (a2 = a = a∗).
(5) Biunitary, if both u and ut are unitaries.
(6) Cubik, if it is biunitary, and ab∗ = a∗b = 0 on rows and columns.
(7) Magik, if it is cubik, and the sum on rows and columns is the same.
Here the equalities of type ab = 0 refer to distinct entries on the same
row, or on the same column. The notions (1, 2, 4, 5) are from [7, 3, 8, 7],
and (3, 6, 7) are new. The terminology is of course temporary: we have
only 7 examples of free quantum groups, so we don’t know exactly what
the names name.
Theorem 2.2. C(Gn) with G = OHS∗UKP is the universal commu-
tative C∗-algebra generated by the entries of a n × n orthogonal, cubic,
magic*, biunitary, cubik, magik matrix.
Proof. The case G = OHSU is discussed in [7, 3, 8, 7], and the case G =
S′KP follows from it, by identifying the corresponding subgroups. 
We proceed with liberation: definitions will become theorems and vice
versa.
Definition 2.3. Ag(n) with g = ohs∗ukp is the universal C∗-algebra
generated by the entries of a n × n orthogonal, cubic, magic*, biunitary,
cubik, magik matrix.
The g = ohsu algebras are from [7, 3, 8, 7], and the g = s′kp ones are
new.
Theorem 2.4. We have the diagram of Hopf algebras
Au(n) → Ak(n) → Ap(n)
↓ ↓ ↓
Ao(n) → Ah(n) → As∗(n)
where s∗ denotes at the same time s and s′.
Proof. The morphisms in Definition 1.1 can be constructed by using the
universal property of each of the algebras involved. For the algebras Aohsu
this is known from [7, 3, 8, 7], and for the algebras As′kp the proof is
similar. 
139
T. Banica
3. Diagrams
Let F =< a, b > be the monoid of words on two letters a, b. For a given
corepresentation u we let ua = u, ub = ū, then we define the tensor powers
uα with α ∈ F arbitrary, according to the rule uαβ = uα ⊗ uβ.
Definition 3.1. Let (A, u) be a finitely generated Hopf algebra.
(1) CA is the collection of linear spaces {Hom(uα, uβ)|α, β ∈ F}.
(2) In the case u = ū we identify CA with {Hom(uk, ul)|k, l ∈ N}.
A morphism (A, u) → (B, v) produces inclusions
Hom(uα, uβ) ⊂ Hom(vα, vβ)
for any α, β ∈ F , so we have the following diagram:
CAu(n) ⊂ CAk(n) ⊂ CAp(n)
∩ ∩ ∩
CAo(n) ⊂ CAh(n) ⊂ CAs∗(n)
We recall that CAs(n) is the category of Temperley-Lieb diagrams.
That is, Hom(uk, ul) is isomorphic to the abstract vector space spanned
by the diagrams between an upper row of 2k points, and a lower row of
2l points. See [3].
In order to distinguish between various meanings of the same dia-
gram, we attach words to it. For instance
⋂
∩ab,
⋂
∩ba are respectively in
Ds(∅, ab), Ds(∅, ba).
Lemma 3.2. The categories for Ag(n) with g = ohsukp are as follows:
(1) CAo(n) =<
⋂
∩ >.
(2) CAh(n) =<
⋂
∩, | ∪∩ | >.
(3) CAs′(n) =<
⋂
∩, | ∪∩ |, ∪∩ >.
(4) CAs(n) =<
⋂
∩, | ∪∩ |,∩ >.
(5) CAu(n) =<
⋂
∩ab,
⋂
∩ba >.
140
A Note on Free Quantum Groups
(6) CAk(n) =<
⋂
∩ab,
⋂
∩ba, | ∪∩ |abab, | ∪∩ |baba >.
(7) CAp(n) =<
⋂
∩ab,
⋂
∩ba, | ∪∩ |abab, | ∪∩ |baba, ∪∩
a
a,
∪
∩
b
b >.
Proof. The case g = ohs is discussed in [3], and the case g = u is discussed
in [4]. In the case g = s′kp we can use the following formulae:
∪
∩ =
∑
ij
eij
| ∪∩ | =
∑
i
eii ⊗ eii
The commutation conditions | ∪∩ | ∈ End(u⊗ ū) and | ∪∩ | ∈ End(ū⊗ u)
correspond to the cubik condition, and the extra relations ∪∩ ∈ End(u)
and ∪∩ ∈ End(ū) correspond to the magik condition. Together with the
fact that orthogonal plus magik means magic’, this gives all the g = s′kp
assertions. 
We can color the diagrams in several ways: either by putting the se-
quence xyyxxyyx . . . on both rows of points, or by putting α, β on both
rows, then by replacing a → xy, b → yx. We say that the diagram is col-
ored if all the strings match, and half-colored, if there is an even number
of unmatches.
Theorem 3.3. For g = ohs∗ukp we have CAg(n) = span(Dg), where:
(1) Ds(k, l) is the set of all diagrams between 2k points and 2l points.
(2) Ds′(k, l) = Ds(k, l) for k − l even, and Ds′(k, l) = ∅ for k − l odd.
(3) Dh(k, l) consists of diagrams which are colorable xyyxxyyx . . ..
(4) Do(k, l) is the image of Ds(k/2, l/2) by the doubling map.
(5) Dp(α, β) consists of diagrams half-colorable a → xy, b → yx.
(6) Dk(α, β) consists of diagrams colorable a → xy, b → yx.
(7) Du(α, β) consists of double diagrams, colorable a → xy, b → yx.
Proof. This is clear from the above lemma, by composing diagrams. The
case g = ohsu is discussed in [3, 4], and the case g = s′kp is similar. 
141
T. Banica
Theorem 3.4. We have the following isomorphisms:
(1) Au(n) = Ão(n).
(2) Ah(n) = Ãk(n).
(3) Ap(n) = Ãs∗(n).
Proof. It follows from definitions that we have arrows from left to right.
Now since by Theorem 3.3 the spaces End(u⊗ ū⊗ u⊗ . . .) are the same
at right and at left, Theorem 5.1 in [2] applies, and gives the arrows from
right to left. 
Observe that the assertion (1), known since [1], is nothing but the iso-
morphism U+n = O+cn mentioned in the beginning of the first section.
4. Freeness, level, doubling
We use the notion of free Hopf algebra, introduced in [3]. Recall that a mor-
phism (A, u) → (B, v) induces inclusions Hom(uα, uβ) ⊂ Hom(vα, vβ).
Definition 4.1. A finitely generated Hopf algebra (A, u) is called free if:
(1) The canonical map Au(n) → As(n) factorizes through A.
(2) The spaces Hom(uα, uβ) ⊂ span(Ds(α, β)) are spanned by dia-
grams.
It follows from Theorem 3.3 that the algebras Aohs∗ukp are free.
In the orthogonal case u = ū we say that A is free orthogonal, and in
the general case, we also say that A is free unitary.
Theorem 4.2. If A is free orthogonal then Ã is free unitary.
Proof. It is shown in [2] that the tensor category of Ã is generated by the
tensor category of A, embedded via alternating words, and this gives the
result. 
Definition 4.3. The level of a free orthogonal Hopf algebra (A, u) is the
smallest number l ∈ {0, 1, . . . ,∞} such that 1 ∈ u⊗2l+1.
As the level of examples, for As(n) we have l = 0, and for Aohs′(n) we
have l = ∞. This follows indeed from Theorem 3.3.
142
A Note on Free Quantum Groups
Theorem 4.4. If l < ∞ then Ã = C(T) ∗A.
Proof. Let < r > be the algebra generated by the coefficients of r. From
1 ∈< u > we get z ∈< zuij >, hence < zuij >=< z, uij >, and we are
done. 
Corollary 4.5. Ap(n) = C(T) ∗As(n).
Proof. For As(n) we have 1 ∈ u, hence l = 0, and Theorem 4.4 applies. 
We can define a “doubling” operation A → A2 for free orthogonal
algebras, by using Tannakian duality, in the following way: the spaces
Hom(uk, ul) with k − l even remain by definition the same, and those
with k− l odd become by definition empty. The interest in this operation
is that A2 has infinite level.
At the level of examples, the doublings are Aohs∗(n) → Aohs′(n).
Proposition 4.6. For a free orthogonal algebra A, the following are equiv-
alent:
(1) A has infinite level.
(2) The canonical map A2 → A is an isomorphism.
(3) The quotient map A → As(n) factorizes through As′(n).
Proof. The equivalence between (1) and (2) is clear from definitions, and
the equivalence with (3) follows from Tannakian duality. 
5. The main result
We know from Theorem 3.4 that the two rows of the diagram formed by
the algebras Aohs∗ukp are related by the operation A → Ã. Moreover, the
results in the previous section suggest that the correct choice in the lower
row is s∗ = s′. The following general result shows that this is indeed the
case.
Theorem 5.1. The operation A → Ã induces a one-to-one correspon-
dence between the following objects:
(1) Free orthogonal algebras of infinite level.
(2) Free unitary algebras satisfying A = Ã.
143
T. Banica
Proof. We use the notations γk = abab . . . and δk = baba . . . (k terms
each).
We know from Theorem 4.2 that the operation A → Ã is well-defined,
between the algebras in the statement. Moreover, since by Tannakian du-
ality an orthogonal algebra of infinite level is determined by the spaces
Hom(uk, ul) with k− l even, we get that A → Ã is injective, because these
spaces are:
Hom(uk, ul) = Hom((zu)γk , (zu)γl)
It remains to prove surjectivity. So, let A be free unitary satisfying
A = Ã. We have CA = span(D) for certain sets of diagrams D(α, β) ⊂
Ds(α, β), so we can define a collection of sets D2(k, l) ⊂ Ds(k, l) in the
following way:
(1) For k − l even we let D2(k, l) = D(γk, γl).
(2) For k − l odd we let D2(k, l) = ∅.
It follows from definitions that C2 = span(D2) is a category, with du-
ality and involution. We claim that C2 is stable under ⊗. Indeed, for k, l
even we have:
D2(k, l)⊗D2(p, q) = D(γk, γl)⊗D(γp, γq)
⊂ D(γkγp, γlγq)
= D(γk+p, γl+q)
= D2(k + p, l + q)
For k, l odd and p, q even, we can use the canonical antilinear isomor-
phisms Hom(uγK , uγL) ' Hom(uδK , uδL), with K, L odd. At the level of
diagrams we get equalities D(γk, γL) = D(δK , δL), that can be used in the
following way:
D2(k, l)⊗D2(p, q) = D(δk, δl)⊗D(γp, γq)
⊂ D(δkγp, δlγq)
= D(δk+p, δl+q)
= D2(k + p, l + q)
144
A Note on Free Quantum Groups
Finally, for k, l odd and p, q odd, we can proceed as follows:
D2(k, l)⊗D2(p, q) = D(γk, γl)⊗D(δp, δq)
⊂ D(γkδp, γlδq)
= D(γk+p, γl+q)
= D2(k + p, l + q)
Thus we have a Tannakian category, and by Woronowicz’s results in
[10] we get an algebra A2. This algebra is free orthogonal, of infinite level.
Moreover, the spaces End(u⊗ ū⊗ u⊗ . . .) being the same for A and A2,
Theorem 5.1 in [2] applies, and gives Ã2 = Ã. Now since we have A = Ã,
we are done. 
References
[1] T. Banica, Le groupe quantique compact libre U(n), Comm. Math.
Phys. 190 (1997), 143–172.
[2] , Representations of compact quantum groups and subfactors,
J. Reine Angew. Math. 509 (1999), 167–198.
[3] T. Banica, J. Bichon, and B. Collins, The hyperoctahedral quantum
group, J. Ramanujan Math. Soc. 22 (2007), 345–384.
[4] T. Banica and B. Collins, Integration over compact quantum groups,
Publ. Res. Inst. Math. Sci. 43 (2007), 377–302.
[5] A. Nica and R. Speicher, Lectures on the combinatorics of free prob-
ability, Cambridge University Press, Cambridge, 2006.
[6] D.V. Voiculescu, Circular and semicircular systems and free product
factors, Progress in Math. 92 (1990), 45–60.
[7] S. Wang, Free products of compact quantum groups, Comm. Math.
Phys. 167 (1995), 671–692.
[8] , Quantum symmetry groups of finite spaces, Comm. Math.
Phys. 195 (1998), 195–211.
[9] S.L. Woronowicz, Compact matrix pseudogroups, Comm. Math. Phys.
111 (1987), 613–665.
[10] , Tannaka-Krein duality for compact matrix pseudogroups.
Twisted SU(N) groups, Invent. Math. 93 (1988), 35–76.
145
T. Banica
Teodor Banica
Department of Mathematics
Paul Sabatier University
118 route de Narbonne
31062 Toulouse, France
banica@picard.ups-tlse.fr
146


A Note on Free Quantum Groups

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Non-commutative entropy computations for continuous fields and cross-products

Fares Gherbi

Tarek Rouabhi

Hyper–(Abelian–by–finite) groups with many subgroups of finite depth

Hammadi Abidi

Taoufik Hmidi

Résultats d’existence dans des espaces critiques pour le système de la MHD inhomogène

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Cyclically valued rings and formal power series

Toussaint Joseph Rabeherimanana

Grandes déviations et loi fonctionnelle du logarithme itéré pour les processus de diffusions avec réflexion

Satoshi  Ishiwata

ANNALES MATHÉMATIQUES
BLAISE PASCAL
Satoshi Ishiwata
Discrete version of Dungey’s proof for the gradient heat
kernel estimate on coverings
Volume 14, no 1 (2007), p. 93-102.
<http://ambp.cedram.org/item?id=AMBP_2007__14_1_93_0>
© Annales mathématiques Blaise Pascal, 2007, tous droits réservés.
L’accès aux articles de la revue « Annales mathématiques Blaise Pas-
cal » (http://ambp.cedram.org/), implique l’accord avec les condi-
tions générales d’utilisation (http://ambp.cedram.org/legal/). Toute
utilisation commerciale ou impression systématique est constitutive
d’une infraction pénale. Toute copie ou impression de ce fichier doit
contenir la présente mention de copyright.
Publication éditée par le laboratoire de mathématiques
de l’université Blaise-Pascal, UMR 6620 du CNRS
Clermont-Ferrand — France
cedram
Article mis en ligne dans le cadre du
Centre de diffusion des revues académiques de mathématiques
http://www.cedram.org/
Annales mathématiques Blaise Pascal 14, 93-102 (2007)
Discrete version of Dungey’s proof for the
gradient heat kernel estimate on coverings
Satoshi Ishiwata
Abstract
We obtain another proof of a Gaussian upper estimate for a gradient of the
heat kernel on cofinite covering graphs whose covering transformation group has
a polynomial volume growth. It is proved by using the temporal regularity of the
discrete heat kernel obtained by Blunck [2] and Christ [3] along with the arguments
of Dungey [7] on covering manifolds.
1. Introduction
Let X = (V,E) be an oriented locally finite connected graph. We consider
the reversible random walk on X defined by functions p : E → R>0 and
m : V → R>0 satisfying
p(e)m(o(e)) = p(e)m(t(e))
and ∑
e∈Ex
p(e) = 1 x ∈ V,
where o(e) is the origin of e, t(e) is the end of e, e is the inverse edge of e
and Ex = {e ∈ E | o(e) = x}. Here, p(e) is the probability that a particle
at o(e) moves to t(e) in a unit time. The function m on V is a measure
on V . Then the transition probability pn(x, y), x, y ∈ V is given by
pn(x, y) =
∑
(e1,e2,...,en)∈Cx,n,t(en)=y
p(e1)p(e2) · · · p(en),
The author is partially supported by the Overseas Advanced Educational Research
Practice Support Program by Japanese Ministry of Education, Culture, Sports, Science
and Technology.
Keywords: Gradient estimates, Random walks, Gaussian estimates for the heat kernel.
Math. classification: 60J10, 58J35, 58J37.
93
S. Ishiwata
where Cx,n is the set of paths from x with length n. The transition oper-
ator P associated with the random walk generates a discrete semigroup
{Pn}n∈N acting on functions on V defined by
Pnf(x) =
∑
y∈V
pn(x, y)f(y).
Then the kernel of Pn on the weighted graph (X, m) is given by
kn(x, y) = pn(x, y)m(y)−1.
The purpose of this paper is to obtain a Gaussian upper estimate for the
gradient of kn on X assuming that the latter admits a cofinite group action
with polynomial volume growth. By the results of Gromov [9], without
loss of generality, we can always assume that the covering transformation
group Γ of X is a nilpotent group of order D. Moreover, we also assume
that the random walk on such X is Γ-invariant, namely, p : E → R>0 and
m : V → R>0 are Γ-invariant.
Under our assumptions, the Gaussian upper estimate for kn
kn(x, y) ≤ Cn−D/2e−C
′d(x,y)2/n ∀x, y ∈ V (1.1)
is known (see [12] and also [10]). Here d(x, y) is the graph distance, the
length of the shortest path from x to y and C, C ′ are some positive
constants.
Moreover, the following theorem for the Gaussian upper estimate for
the gradient of kn has been proved by [12] along the method of [10]. Let
∇1 be the gradient with respect to the first variable given by
∇1kn(x, y) =
 ∑
d(x,ω)≤1
|kn(ω, y)− kn(x, y)|2p1(x, ω)
1/2
=
∑
e∈Ex
|dkn(e, y)|2p(e)
1/2 ,
where d is the exterior derivative defined by
df(e) = f(t(e))− f(o(e)), e ∈ E
94
Discrete gradient heat kernel on coverings
for a function f on V . Similarly we denote by ∇2kn(x, y) the gradient with
respect to the second variable, namely
∇2kn(x, y) =
 ∑
d(y,ω)≤1
|kn(x, ω)− kn(x, y)|2p1(y, ω)
1/2
=
∑
e∈Ey
|dkn(x, e)|2p(e)
1/2 .
Then we have
Theorem 1.1. Let X = (V,E) be a non-bipartite covering graph whose
covering transformation group has polynomial volume growth of order D.
Then there exist C, C ′ > 0 such that
∇1kn(x, y) ≤ Cn−(D+1)/2e−C
′d(x,y)2/n (1.2)
for n ∈ N∗ and x, y ∈ V .
It should be noted that the estimate (1.2) is closely related to the bound-
edness of the Riesz transform (see [1], [13] and [11]). Let ∆ be a discrete
Laplacian on X given by ∆ = I − P . Then we have
Theorem 1.2. Let X be as above. For 1 < p < ∞, there exists Cp > 0
such that
‖∇∆−1/2f‖Lp ≤ Cp‖f‖Lp
for all finitely supported functions f on V . Here ‖ · ‖Lp is the Lp norm
with respect to the measure m on V .
On the other hand, Dungey proved (1.2) for the heat kernel on covering
manifolds with polynomial volume growth in [6]. Recently, in [7], he gave
a new proof of (1.2) using the well-known temporal regularity of the heat
kernel (see for instance [5]). In this paper, we give a shorter proof of
Theorem 1.1 along with the arguments of the recent result by Dungey [7].
Let
∂1kn(x, y) = kn+1(x, y)− kn(x, y).
We use the following discrete version of the temporal regularity proved by
Blunck [2], Christ [3]:
Theorem 1.3. Let X be an oriented non-bipartite graph satisfying
C−1rD ≤ m(B(x, r)) ≤ CrD x ∈ V, r ≥ 1
95
S. Ishiwata
and (1.1), where B(x, r) is the ball centered at x ∈ V with radius r. Then
there exist positive constants C and C ′ such that
∂1kn(x, y) ≤ Cn−(D+2)/2e−C
′d(x,y)2/n (1.3)
for n ∈ N∗ and x, y ∈ V .
We remark that Dungey gave a short proof of (1.3) recently in [8].
2. Proof of Theorem 1.1
Let ν = m+n be a positive integer, where we choose m = n or m = n+1
depending on whether ν is even or odd. By the Cauchy-Schwarz inequality,
it is easy to see that
erd
2(u,z)/ν∇1kν(u, z) ≤ ‖e2rd
2(u,·)/n∇1kn(u, ·)‖L2‖e2rd
2(·,z)/mkm(·, z)‖L2
for r > 0.
By the Gaussian upper bound (1.1) for kn, for small r > 0, there exists
Cr > 0 such that
‖e2rd2(·,z)/mkm(·, z)‖L2 ≤ Crm−D/4. (2.1)
Let F ⊂ V be a fundamental domain for the action of the transformation
group Γ on V . Namely, F is a subset in V such that for all x ∈ V , there
exists a unique pair γx ∈ Γ and x0 ∈ F satisfying x = γxx0. Then we
denote Fx = γxF . The following lemma gives a comparison of the weighted
integral for ∇1kn and ∇2kn. Similar arguments can be found in [12].
Lemma 2.1. There exists a positive constant C > 0 such that
‖e2rd2(u,·)/n∇1kn(u, ·)‖2L2 ≤ C
∑
y0∈Fu
‖e2rd2(y0,·)/n∇2kn(y0, ·)‖2L2m(y0)
for all u ∈ X.
Proof. Since m is Γ-invariant, there exists C > 0 such that
C−1 < min{m(x) |x ∈ V }. Then we have
‖e2rd2(u,·)/n∇1kn(u, ·)‖2L2
≤ C
∑
v0∈Fu
∑
y∈V
∣∣∣e2rd2(v0,y)/n∇1kn(v0, y)∣∣∣2 m(y)m(v0)
= C
∑
v0,y0∈Fu
∑
γ∈Γ
∣∣∣e2rd2(v0,γy0)/n∇1kn(v0, γy0)∣∣∣2 m(γy0)m(v0).
96
Discrete gradient heat kernel on coverings
From the Γ-invariance of the distance function d and kn, the latter is
C
∑
v0,y0∈Fu
∑
γ∈Γ
∣∣∣e2rd2(γ−1v0,y0)/n∇1kn(γ−1v0, y0)∣∣∣2 m(y0)m(γ−1v0).
By replacing γ−1 with γ in the sum of Γ, we get
C
∑
v0,y0∈Fu
∑
γ∈Γ
∣∣∣e2rd2(γv0,y0)/n∇1kn(γv0, y0)∣∣∣2 m(y0)m(γv0).
Since ∇1kn(γv0, y0) = ∇2kn(y0, γv0), this is also
C
∑
v0,y0∈Fu
∑
γ∈Γ
∣∣∣e2rd2(y0,γv0)/n∇2kn(y0, γv0)∣∣∣2 m(γv0)m(y0)
= C
∑
y0∈Fu
∑
v∈V
∣∣∣e2rd2(y0,v)/n∇2kn(y0, v)∣∣∣2 m(v)m(y0)
= C
∑
y0∈Fu
‖e2rd2(y0,·)/n∇2kn(y0, ·)‖2L2m(y0).

Remark 2.2. In this proof, we use only the Γ-invariance of d, ∇1kn and
∇1kn(x, y) = ∇2kn(y, x). Therefore, there are other definitions of ∇ so
that the previous lemma holds. For example, we can obtain the same
results for
∇p1kn(x, y) :=
( ∑
d(ω,x)≤1
|kn(ω, y)− kn(x, y)|pp1(x, ω)
)1/p
, 1 < p < ∞,
∇∞1 kn(x, y) := sup
d(ω,x)≤1
|kn(ω, y)− kn(x, y)|,
which are comparable with each other.
By the same arguments as the continuous case ([4]), we obtain the
following by “discrete integration by parts”. Similar arguments can be
found in [10] and [12].
Lemma 2.3. For e ∈ E, let m(e) be a weight of e defined by m(e) =
p(e)m(o(e)). Then we have
‖e2rd2(y0,·)/n∇2kn(y0, ·)‖2L2 =−
∑
e∈E
de4rd
2(y0,e)/nkn(y0, o(e))dkn(y0, e)m(e)
− 2
∑
v∈V
e4rd
2(y0,v)/nkn(y0, v)∂1kn(y0, v)m(v).
97
S. Ishiwata
Proof.
∑
v∈V
∣∣∣e2rd2(y0,v)/n∇2kn(y0, v)∣∣∣2 m(v)
=
∑
v∈X
e4rd
2(y0,v)/n
∑
e∈Ev
|dkn(y0, e)|2p(e)m(v)
=
∑
e∈E
e4rd
2(y0,o(e))/n|kn(y0, t(e))− kn(y0, o(e))|2p(e)m(o(e))
=
∑
e∈E
e4rd
2(y0,o(e))/n
(
kn(y0, t(e))2 − 2kn(y0, t(e))kn(y0, o(e))
+ kn(y0, o(e))2
)
m(e)
=
∑
e∈E
e4rd
2(y0,o(e))/nkn(y0, t(e)) (kn(y0, t(e))− kn(y0, o(e)))m(e)
+
∑
e∈E
e4rd
2(y0,o(e))/nkn(y0, o(e)) (kn(y0, o(e))− kn(y0, t(e)))m(e).
Since m(e) = m(e), by replacing e with e in the sum of E in the first term,
=
∑
e∈E
e4rd
2(y0,t(e))/nkn(y0, o(e)) (kn(y0, o(e))− kn(y0, t(e)))m(e)
+
∑
e∈E
e4rd
2(y0,o(e))/nkn(y0, o(e)) (kn(y0, o(e))− kn(y0, t(e)))m(e)
=
∑
e∈E
(
e4rd
2(y0,t(e))/n − e4rd2(y0,o(e))/n
)
kn(y0, o(e))
· (kn(y0, o(e))− kn(y0, t(e)))m(e)
+ 2
∑
e∈E
e4rd
2(y0,o(e))/nkn(y0, o(e)) (kn(y0, o(e))− kn(y0, t(e)))m(e)
=−
∑
e∈E
de4rd
2(y0,e)/nkn(y0, o(e))dkn(y0, e)m(e)
− 2
∑
v∈V
e4rd
2(y0,v)/nkn(y0, v)
∑
e∈Ev
(kn(y0, t(e))− kn(y0, o(e)))p(e)m(v).
98
Discrete gradient heat kernel on coverings
Since
∑
e∈Ev
(kn(y0, t(e))− kn(y0, o(e)))p(e)
=
∑
e∈Ev
kn(y0, t(e))p(e)− kn(y0, v)
= kn+1(y0, v)− kn(y0, v)
= ∂1kn(y0, v),
the lemma is proved. 
Finally, we apply the temporal regularity for kn along with the argument
of Lemma 2.3 in [4]. Let
I(n, y0) =
∑
v∈V
∣∣∣e2rd2(y0,v)/n∇2kn(y0, v)∣∣∣2 m(v),
I1(n, y0) =− 2
∑
v∈V
e4rd
2(y0,v)/nkn(y0, v)∂1kn(y0, v)m(v),
I2(n, y0) =−
∑
e∈E
de4rd
2(y0,e)/nkn(y0, o(e))dkn(y0, e)m(e).
Lemma 2.3 says that I(n, y0) = I1(n, y0)+I2(n, y0). Using (1.1) and (1.3),
for sufficiently small r > 0, there exists Cr > 0 such that
|I1(n, y0)| ≤ Crn−D/2−1.
By the Cauchy-Schwarz inequality, we have that |I2(n, y0)| is less than
∑
v∈V
∑
e∈Ev
|de4rd2(y0,e)/n|2p(e)
1/2∑
e∈Ev
|dkn(y0, e)|2p(e)
1/2kn(y0, v)m(v)
=
∑
v∈V
∑
e∈Ev
|de4rd2(y0,e)/n|2p(e)
1/2∇2kn(y0, v)kn(y0, v)m(v).
99
S. Ishiwata
Using the mean value theorem for f(x) = e4rx2/n, there exists
d(y0, o(e)) < s < d(y0, t(e)) such that
e4rd
2(y0,t(e))/n − e4rd2(y0,o(e))/n =8rs
n
e4rs
2/n(d(y0, t(e))− d(y0, o(e)))
≤8r(d(y0, o(e)) + 1)
n
e4r(d(y0,o(e))+1)
2/n
≤ Cr√
n
e16rd
2(y0,o(e))/n
for some Cr > 0. Then we have that |I2(n, y0)| is less than
Cr√
n
∑
v∈V
e16rd
2(y0,v)/n∇2kn(y0, v)kn(y0, v)m(v)
≤ Cr√
n
(∑
v∈V
e28rd
2(y0,v)/n|kn(y0, v)|2m(v)
)1/2
·
(∑
v∈V
|e2rd2(y0,v)/n∇2kn(y0, v)|2m(v)
)1/2
.
For sufficiently small r > 0, we obtain
|I2(n, y0)| ≤ C ′rn−D/4−1/2I(n, y0)1/2.
Then we conclude that
I(n, y0) ≤ Crn−D/2−1 + C ′rn−D/4−1/2I(n, y0)1/2,
namely,
I(n, y0) ≤ Cn−D/4−1/2
for some C > 0. Together with (2.1), this completes the proof of Theorem
1.1.
Acknowledgment. This work was done while the author was visiting
University of Cergy-Pontoise. He would like to thank Professor Thierry
Coulhon for his encouragements. He also thanks Professor Nick Dungey
for reading the manuscript carefully and useful comments.
100
Discrete gradient heat kernel on coverings
References
[1] P. Auscher, T. Coulhon, X. T. Duong & S. Hofmann – Riesz
transform on manifolds and heat kernel regurality, Ann. Scient. Éc.
Norm. Sup. 37 (2004), p. 911–957.
[2] S. Blunck – Perturbation of analytic operators and temporal regu-
larity, Colloq. Math. 86 (2000), p. 189–201.
[3] M. Christ – Temporal regularity for random walk on discrete nilpo-
tent groups, Proceedings of the Conference in Honor of Jean-Pierre
Kahane (Orsay, 1993). J. Fourier Anal. Appl. Special Issue (1995),
p. 141–151.
[4] T. Coulhon & X. T. Duong – Riesz transforms for 1 ≤ p ≤ 2,
Trans. Amer. Math. Soc. 351 (1999), p. 1151–1169.
[5] E. B. Davies – Non-gaussian aspects of heat kernel behaviour, J.
London Math. Soc. 55 (1997), p. 105–125.
[6] N. Dungey – Heat kernel estimates and Riesz transforms on some
riemannian covering manifolds, Math. Z. 247 (2004), p. 765–794.
[7] , Some gradient estimates on covering manifolds, Bull. Pol.
Acad. Sci. Math. 52 (2004), p. 437–443.
[8] , A note on time regularity for discrete time heat kernel, Semi-
group Forum 72 (2006), p. 404–410.
[9] M. Gromov – Groups of polynomial growth and expanding maps,
Inst. Hautes Études Sci. Publ. Math. 53 (1981), p. 53–73.
[10] W. Hebisch & L. Saloff-Coste – Gaussian estimates for Markov
chains and random walks on groups, Ann. Probab. 21 (1993), p. 673–
709.
[11] S. Ishiwata – Asymptotic behavior of a transition probability for
a random walk on a nilpotent covering graph, Contemp. Math. 347
(2004), p. 57–68.
[12] , A Berry-Esseen type theorem on nilpotent covering graphs,
Canad. J. Math. 56 (2004), p. 963–982.
[13] E. Russ – Riesz transform on graphs for p > 2, unpublished manu-
script.
101
S. Ishiwata
Satoshi Ishiwata
Institute of Mathematics
University of Tsukuba
1-1-1 Tennoudai, 305-8571
Ibaraki
JAPAN
ishiwata@math.tsukuba.ac.jp
102


Discrete version of Dungey’s proof for the gradient heat kernel estimate on coverings

Giuseppe Geymonat

ANNALES MATHÉMATIQUES
BLAISE PASCAL
Giuseppe Geymonat
Trace Theorems for Sobolev Spaces on Lipschitz Domains.
Necessary Conditions
Volume 14, no 2 (2007), p. 187-197.
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Annales mathématiques Blaise Pascal 14, 187-197 (2007)
Trace Theorems for Sobolev Spaces on Lipschitz
Domains. Necessary Conditions
Giuseppe Geymonat
Abstract
A famous theorem of E. Gagliardo gives the characterization of traces for
Sobolev spaces W 1, p (Ω) for 1 ≤ p < ∞ when Ω ⊂ RN is a Lipschitz domain.
The extension of this result to W m, p (Ω) for m ≥ 2 and 1 < p < ∞ is now
well-known when Ω is a smooth domain. The situation is more complicated for
polygonal and polyhedral domains since the characterization is given only in terms
of local compatibility conditions at the vertices, edges, .... Some recent papers give
the characterization for general Lipschitz domains for m=2 in terms of global
compatibility conditions. Here we give the necessary compatibility conditions for
m ≥ 3 and we prove how the local compatibility conditions can be derived.
1. Introduction
Let Ω be a Lipschitz bounded and connected subset of RN whose bounded
and orientable boundary is denoted by Γ. For 1 ≤ p < ∞ and m integer
Wm, p (Ω) denotes the Sobolev space of functions of Lp (Ω) whose distribu-
tional derivatives up to the order m also belong to Lp (Ω). For m ≥ 1 the
restriction γ0(u) = u|Γ to Γ of a function u ∈Wm, p (Ω) is well-defined and
belongs to Lp (Γ). A famous result of E. Gagliardo [6] gives, for m = 1, the
caracterization of the range of γ0. More precisely, Gagliardo proves that
the operator γ0 is linear and continuous from W 1, p (Ω) onto W 1−1/p, p (Γ)
for 1 ≤ p <∞ and has a continuous right inverse for p > 1.
When u ∈ W 2, p (Ω) then ∂u∂xj ∈ W
1, p (Ω) for j = 1, . . . , N ; therefore
γ1(u) = ∂u∂n =
∑N
j=1 γ0
(
∂u
∂xj
)
nj ∈ Lp (Γ) since n = (n1, . . . , nN ) is defined
almost everywhere and belongs to (L∞ (Γ))N . J. Nečas [9] proves that
γ0(u) ∈ W 1, p (Γ) and that the map u −→ (γ0(u), γ1(u)) is linear and
continuous from W 2, p (Ω) into W 1, p (Γ) × Lp (Γ). A natural question is
to characterize the range of the map (γ0, γ1). A first answer has been
obtained for polygonal-type domains Ω ⊂ R2 by Kondratev and Grisvard
(see e.g. [8] for full references) in terms of compatibility conditions at
187
G. Geymonat
the corners and then the results have been extended to polyhedral-type
domains (N = 3). These characterizations have been extensively used
in order to give regularity results for different types of boundary-value
problems.
For general Lipschitz domains a first characterization of the range of
(γ0, γ1) has been obtained for N = 2 and p = 2 in [7] as a byproduct of
the study of the Airy function; this result has been extended to general
p > 1 by [5]. During a visit at the Istituto di Analisi Numerica del CNR in
Pavia the following equivalent statement came out after some discussions
the with F. Brezzi and A. Buffa.
Theorem 1.1. The range of (γ0, γ1) is the set of (f0, f1) ∈ W 1, p (Γ) ×
Lp (Γ) such that:
∂f0
∂t
t + f1n ∈ (W 1−1/p, p (Γ))2. (1.1)
A first consequence of (1.1) has been a general characterization of the
range of (γ0, γ1) for N = 3 (see [2]) that also works for all N ≥ 3.
Let us remark that the compatibility conditions at a corner follow from
the characterization of W 1−1/p, p (Γ) and the exchange of n and t at the
crossing of the corner.
The statement of the theorem 1.1 allows an easy interpretation of the
necessity of the condition (1.1). Indeed, when u ∈W 2, p (Ω) then grad u =(
∂u
∂x1
, ∂u∂x2
)
∈
(
W 1, p (Ω)
)2 and γ0(grad u) ∈ (W 1−1/p, p (Γ))2. Hence the
necessity of (3.1) follows from
γ0(grad u) =
∂γ0(u)
∂t
t + γ1(u)n. (1.2)
In this paper we give general necessary conditions for the traces of the
elements of Wm, p (Ω) for all integer m ≥ 2, all p > 1 and all N ≥ 2. The
proof of [7] can be adapted in order to prove that these conditions are also
sufficient when N = 2 and p = 2.
The author is indebted to many people. At first to F. Krasucki whose
questions about the mechanical meaning of Grisvard results on the traces
of Airy functions were the starting point for the extension of the Gagliardo
theorem to Wm, p (Ω) in the case m=2 and p=2 for 2-dimensional domains
[7]. During a visit to the IAN of the CNR the discussions with F. Brezzi
and particularly with A. Buffa allowed the extension to the case m=2 and
1 < p <∞ for general N-dimensional domains [2]. At last the discussions
188
Trace Theorems. Necessary Conditions
with F. Murat during the Colloque were stimulating to obtain the actual
formulation of the conditions for m=3.
2. Preliminaries
Let Ω be a Lipschitz bounded and connected subset of RN whose bounded
and orientable boundary is denoted by Γ. This means (see [9], [8] and
[1]) that for every x ∈ Γ there exists a neighborhood V (x) of x in RN
and a new orthonormal coordinate system {y1, ..., yN} such that V (x) =∏N
i=1]−αi, αi[ = V ′(x)×]−αN , αN [ and there exists a Lipschitz continuous
function ϕ : V ′(x) −→]−αN , αN [ such that |ϕ (y′)| ≤ αN/2 for every y′ =
(y1, ..., yN−1) ∈ V ′(x) and such that Ω∩V (x) = {y = (y′, yN ) ∈ V (x) | yN
> ϕ (y′)} and Γ ∩ V (x) = {y = (y′, yN ) ∈ V (x) | yN = ϕ (y′)}.
Since Γ is compact there exists M open and connected subsets Γi such
that Γ =
⋃M
i=1 Γi and there exists M points ai ∈ Γi and M Lipschitz con-
tinuous function ϕi such that Γi = Γ∩V (ai). The induced parametrization
(ϕi,Γi) of Γ is defined for i = 1, . . . ,M by
y′ = (y1, ..., yN−1) 7−→
(
y′, ϕi
(
y′
))
.
This parametrization induces N − 1 linearly independent tangent vectors
defined a.e. on Γi:
tk =
(
eN−1k , ∂ϕi/∂yk
)
, k = 1, . . . , N − 1
where eN−1k = (δh,k)h=1,...,N−1. A corresponding set of orthonormalized
vectors Tk =
(
T kh
)
h=1,...,N
∈ (L∞ (Γ))N can defined as follows for k = 1:
T1 =
−1√
1 + (∂ϕi/∂y1)
2
t1 (2.1)
and for k = 2, . . . , N − 1,
T kh =
−∂ϕi/∂yh√
1 +
∑k−1
l=1 (∂ϕi/∂yl)
2
∂ϕi/∂yk√
1 +
∑k
l=1 (∂ϕi/∂yl)
2
, h = 1, . . . , k − 1,
(2.2)
T kk =
√
1 +
∑k−1
l=1 (∂ϕi/∂yl)
2√
1 +
∑k
l=1 (∂ϕi/∂yl)
2
(2.3)
189
G. Geymonat
T kh = 0 , h = k, . . . , N − 1 (2.4)
T kN =
1√
1 +
∑k−1
l=1 (∂ϕi/∂yl)
2
∂ϕi/∂yk√
1 +
∑k
l=1 (∂ϕi/∂yl)
2
. (2.5)
At a.e. point x = (y′, ϕi (y′)) ∈ Γi the vectors (T1, . . . ,TN−1) span the
tangent space TxΓ who is an hyperplane of RN . Any other orthonormal
basis of TxΓ is obtained applying a rotation to the previous one. The unit
outward normal vector n = (n1, n2, . . . , nN ) ∈ (L∞ (Γ))N is defined a.e.
on Γi by
y′ 7−→ (∂ϕi/∂y1, . . . , ∂ϕi/∂yN−1, −1)√
1 +
∑N−1
k=1 (∂ϕi/∂yk)
2
.
For a.e. x ∈ Γ the vectors (T1, . . . ,TN−1,n) are a positively oriented basis
of TxΩ ≈ RN . As in the case of regular domains the definitions are intrinsic
since not depending from the choice of the parametrization (ϕi,Γi).
Since Γ is a Lipschitz-continuous manifold without boundary the Sobolev
spaces W s, p (Γ) are well defined (independently from the parametrization
(ϕi,Γi)) for −1 ≤ s ≤ 1 and 1 < p <∞. More precisely, ψ ∈ Lp (Γ) means
that for i = 1, . . . ,M one has ψ (y′, ϕi (y′)) ∈ Lp (V ′(xi)) and ψ ∈W 1, p (Γ)
when ψ (y′, ϕi (y′)) ∈W 1, p (V ′(xi)). This means that for k = 1, . . . , N − 1
one has ∂ (ψ (y′, ϕi (y′))) /∂yk ∈ Lp (V ′(xi)) where
∂ψ (y′, ϕi (y′))
∂yk
=
(
∂ψ
∂yk
) (
y′, ϕi
(
y′
))
+
(
∂ψ
∂yN
) (
y′, ϕi
(
y′
)) ∂ϕi (y′)
∂yk
. (2.6)
Using the tangent fields Tk we define the tangential derivatives
∂Tkψ =
N∑
h=1
(
∂ψ
∂yh
) (
y′, ϕi
(
y′
))
T kh (2.7)
and for ψ ∈ W 1, p (Γ) the tangential vector field ∇Γψ =
∑N−1
k=1 (∂Tkψ)T
k.
It follows that∇Γψ belongs to (Lp (Γ))N and its definition does not depend
on the parametrization (ϕi,Γi).
For u ∈ Wm, p (Ω), m ≥ 1, the restriction to Γ is defined on every Γi
by u|Γi = u (y
′, ϕi (y′)). This restriction, denoted γ0(u) = u|Γ , belongs to
Lp (Γ).
When m = 2 then Du =
(
∂u
∂x1
, . . . , ∂u∂xN
)
∈
(
W 1, p (Ω)
)N . Hence it
follows from (2.6) that γ0(u) ∈ W 1, p (Γ) since for i = 1, . . . ,M on Γi one
190
Trace Theorems. Necessary Conditions
has for k = 1, . . . , N − 1:
∂γ0(u) (y′, ϕi (y′))
∂yk
= γ0
(
∂u
∂yk
)
+ γ0
(
∂u
∂yN
)
∂ϕi
∂yk
∈ Lp (Γi) . (2.8)
The exterior normal derivative of u on Γ, denoted γ1(u), is defined a.e. on
Γi by: (
∂u
∂n
) (
y′, ϕi
(
y′
))
=
N∑
j=1
(
∂u
∂yj
)(
y′, ϕi
(
y′
))
nj
(
y′
)
. (2.9)
The normal derivative γ1(u) so defined belongs to Lp (Γ) since the unit
normal vector n = (n1, . . . , nN ) belongs to (L∞ (Γ))N .
In a similar way, when m = 3, Du ∈
(
W 2, p (Ω)
)N and the hessian
Hu = (Dαu)|α|=2 belongs to W
1, p
(
Ω;MNsym
)
where MNsym denotes the
vector space of symmetric N × N matrices. The second order exterior
normal derivative, denoted γ2(u), is defined on Γi by:(
∂2u
∂n2
)(
y′, ϕi
(
y′
))
=
N∑
l,j=1
(
∂2u
∂yl∂yj
)(
y′, ϕi
(
y′
))
nl
(
y′
)
nj
(
y′
)
. (2.10)
Hence γ2(u) ∈ Lp (Γ).
More generally, when u ∈ Wm, p (Ω), m ≥ 4, all γ0 (Dαu) ∈ Lp (Γ) for
|α| ≤ m−1 and then the exterior normal derivatives γk(u) up to the order
k = m− 1, are defined on Γi by:(
∂ku
∂nk
)(
y′, ϕi
(
y′
))
=
∑
|α|=k
k!
α!
(Dαu)
(
y′, ϕi
(
y′
))
nα
(
y′
)
(2.11)
for all k ≥ 1, γk(u) ∈ Lp (Γ).
3. Necessary conditions
From the previous definitions it follows that u 7−→ (γ0(u), . . . , γm−1(u)) is
a linear and continuous map from Wm, p (Ω) into W 1, p (Γ)× (Lp (Γ))m−1.
A natural question is the characterization of the range of such map.
When the boundary Γ of Ω is more regular (for instance of class C∞),
the extension of the Gagliardo’s theorem states that, for p > 1, u 7−→
(γ0(u), . . . , γm−1(u)) is a linear and continuous map from Wm, p (Ω) onto∏m−1
k=0 W
m−k−1/p, p (Γ) and it has a continuous right inverse.
191
G. Geymonat
For polygonal-type domains (when N = 2) and for polyhedral-type
domains (when N = 3) the caracterization of the range of the map
(γ0, . . . , γm−1) has been obtained in terms of compatibility conditions.
A first step toward the characterization of the range for general Lips-
chitz domains is the obtention of the necessary conditions.
Proposition 3.1. Let be (f0, f1) ∈W 1, p (Γ)×Lp (Γ). A necessary condi-
tion in order that (f0, f1) ∈ range(γ0, γ1) is:
∇Γf0 + f1n ∈ (W 1−1/p, p (Γ))N . (3.1)
Proof. When u ∈ W 2, p (Ω) then Du =
(
∂u
∂x1
, . . . , ∂u∂xN
)
∈
(
W 1, p (Ω)
)N .
It follows from (2.8), the definition of ∇Γ and of Tk that in every point
(y′, ϕi (y′)) ∈ Γi one has for h = 1, . . . , N − 1:
(∇Γγ0(u))h =
∂u
∂yh
−
(
N−1∑
l=1
∂u
∂yl
∂ϕi
∂yl
− ∂u
∂yN
)
∂ϕi/∂yh
1 +
∑N−1
k=1 (∂ϕi/∂yk)
2
and
(∇Γγ0(u))N =
∂u
∂yN
+
(
N−1∑
l=1
∂u
∂yl
∂ϕi
∂yl
− ∂u
∂yN
)
1
1 +
∑N−1
k=1 (∂ϕi/∂yk)
2 .
Hence from (2.9) and the definition of n it follows that
∇Γγ0(u) + γ1(u)n = γ0(Du). (3.2)
One obtains (3.1) since γ0(Du) ∈ (W 1−1/p, p (Γ))N . 
Remark 3.2. For Lipschitz domains the two terms of the sum in (3.1)
belong separately only to (Lp (Γ))N .
Whenm = 3, thenDu ∈
(
W 2, p (Ω)
)N and hence γ0(Du) ∈ (W 1, p (Γ))N .
From the previous proposition it then follows the following result.
Lemma 3.3. Let be (f0, f1, f2) ∈W 1, p (Γ)×Lp (Γ)×Lp (Γ). A necessary
condition in order that (f0, f1, f2) ∈ range(γ0, γ1, γ2) is:
∇Γf0 + f1n ∈ (W 1, p (Γ))N . (3.3)
In order to state the second necessary condition, let be
g0 = ∇Γf0 + f1n =
(
g0h
)
h=1,...,N
∈ (W 1, p (Γ))N
192
Trace Theorems. Necessary Conditions
and let define for k = 1, . . . , N − 1 the vector ∂Tkg0 ∈ (Lp (Γ))N whose
components are (
∂T kg
0
)
h
= ∂T kg
0
h, h = 1, . . . , N (3.4)
where the tangential derivatives are defined in (2.7).
Theorem 3.4. Let be (f0, f1, f2) ∈ W 1, p (Γ)× Lp (Γ)× Lp (Γ). The nec-
essary conditions in order that (f0, f1, f2) ∈ range(γ0, γ1, γ2) are (3.3)
and:
N−1∑
k=1
(
∂T kg
0 ·T k
) (
T k ⊗T k
)
+
+
∑
1≤k<p≤N−1
(
∂T kg
0 ·T p
) (
T k ⊗T p + T p ⊗T k
)
+
+
N−1∑
k=1
(
∂T kg
0 · n
) (
T k ⊗ n + n⊗T k
)
+
+f2n⊗ n ∈W 1−1/p, p
(
Γ;MNsym
)
. (3.5)
Proof. Since g0 = γ0(Du) ∈ (W 1, p (Γ))N it follows from (3.4) and the
definition of Tk that on V ′(xi) one has for h = 1, . . . , N :
∂T kg
0
h = ∂T k
∂u
∂yh
=
N∑
s=1
(
∂2u
∂yh∂ys
)(
y′, ϕi
(
y′
))
T ks
A simple computation then gives:
∂T kg
0 ·T p = 1
2
γ0(Hu) :
(
T k ⊗T p + T p ⊗T k
)
(3.6)
and
∂T kg
0 · n = 1
2
γ0(Hu) :
(
T k ⊗ n + n⊗T k
)
(3.7)
where A : B =
∑N
i,j=1 aijbij denotes the scalar product of the symmetric
matrices A = (aij) and B = (bij). Since the vectors (T1, . . . ,TN−1,n) are
an orthonormal basis of TxΩ, an orthonormal basis of the symmetrized
tensor product TxΩ⊗S TxΩ is given by:{
T k ⊗T k
}
k=1,...,N−1
, n⊗ n
193
G. Geymonat
{
1√
2
(
T k ⊗T p + T p ⊗T k
)
, 1√
2
(
T k ⊗ n + n⊗T k
)}
1≤k<p≤N−1
A development of γ0(Hu) with respect to this basis and the use of (3.6)
and (3.7) gives immediately (3.5). 
Remark 3.5. Let us once more remark that each term of the sum in (3.5)
only belongs to L p
(
Γ;MNsym
)
.
Remark 3.6. With the same procedure one can write the necessary com-
patibility conditions for the traces of u ∈Wm, p (Ω), m ≥ 4.
4. Examples
In order to avoid inessential technicalities, we consider for N = 2 the case
where Ω = {(x, y) ;x > 0, y > 0} is the first quadrant and hence the cor-
ner has opening π/2. On the vertical (resp. horizontal) side of the corner
Γ1 = {(0, y) ; y ≥ 0}, resp. Γ2 = {(x, 0) ;x ≥ 0}, one has T 1 = (0,−1) and
n = (−1, 0), resp.T 1 = (−1, 0) and n = (0,−1).
Example 4.1. We prove that the usual compatibility conditions in a corner
( see e.g. [8]) for u ∈ W 2, p (Ω) follow from (3.1). Indeed this condition
becomes:
g 0 = (∂T1f0) T
1 + f1n ∈ (W 1−1/p, p (Γ))2 .
Since the definition of W 1−1/p, p (Γ) is invariant under the Lipschitz trans-
form Γ −→ R, the previous condition means that:
W 1−1/p, p (R) 3 g01 =
{
−f1 for x < 0
−df0/dx for x > 0
and
W 1−1/p, p (R) 3 g02 =
{
−df0/dx for x < 0
−f1 for x > 0 .
Since obviously, df0/dx, f1 ∈ W 1−1/p, p (R±), one can verify these condi-
tions with the help of the following proposition, whose proof can be found
e.g. in [8].
Proposition 4.2. Let be H± ∈W 1−1/p, p (R±); and let define:
H (x) =
{
H− (x) for x < 0
H+ (x) for x > 0 . (4.1)
194
Trace Theorems. Necessary Conditions
Then H ∈W 1−1/p, p (R)
(i) for 1 < p < 2 without any other condition;
(ii) for p = 2 if and only if∫ +∞
0
∣∣H+ (t)−H− (−t)∣∣2 dt/t < +∞;
(ii) for p > 2 if and only if
H+ (0) = H− (0) .
Example 4.3. In order to prove that the usual compatibility conditions in
a corner for u ∈ W 3, p (Ω) , follow from (3.3)and (3.5) one has at first to
express the conditions g0h ∈ W 1, p (R) for h = 1, 2 with an analogous of
proposition 4.2. Since the orthonormal basis of TxΩ⊗S TxΩ is now given
by:
T 1 ⊗T 1, n⊗ n, 1√
2
(
T 1 ⊗ n + n⊗T 1
)
,
it is a simple exercice to express the different terms of these symmetric
matrices on Γ1,Γ2 and so find the compatibility conditions, always thank
to the proposition 4.2.
When N = 3 with an analogous method one can express the compati-
bility conditions for a vertex or an edge of a polyhedral-type domain. Once
more since the definition of W 1−1/p, p (Γ) and W 1, p (Γ) are invariant under
a Lipschitz transform one can reduce the study of the vertex behaviour to
the following problems. Let be R2 divided in Λ non overlapping sectors Sλ,
λ = 1, . . . ,Λ, with vertex at the origin. Let be given Hλ ∈ W 1−1/p, p (Sλ)
(resp.Hλ ∈W 1, p (Sλ)) for λ = 1, . . . ,Λ; find the conditions such that:
H(x, y) =

H1(x, y) for (x, y) ∈ S1
· · ·
HΛ(x, y) for (x, y) ∈ SΛ
belongs to W 1−1/p, p
(
R2
)
, resp.to W 1, p
(
R2
)
. Some partial answers can be
found in [3] and in [5].
195
G. Geymonat
5. Concluding remarks
The examples suggest that the necessary conditions (3.3) and (3.5) are
also necessary. Indeed in the case N = 2 and p = 2 it is possible to adapt
the reasoning used in [7] to prove the sufficiency of these conditions and
the arguments used in [5] will also give the proof of the sufficiency for all
p > 1.
The proof of the sufficiency in the general case N ≥ 3 seems more
delicate. The extension of the previous results to fractional Sobolev Spaces
W s, p (Ω) seems open (for the case p = 2 and 12 < s <
3
2 see however [4]).
References
[1] R. A. Adams & J. J. F. Fournier – Sobolev spaces. Second edition,
Academic Press, New York, 2003.
[2] A. Buffa & G. Geymonat – On the traces of functions in W 1, p(Ω)
for Lipschitz domains in R3, C. R. Acad. Sci. Paris, Série I 332 (2001),
p. 699–704.
[3] A. Buffa & J. P. Ciarlet – On traces for functional spaces related
to Maxwell’s equations. Part I: an integration by parts formula in
Lipschitz polyedra, Math. Meth. Appl. Sci. 24 (2001), p. 9–30.
[4] Z. Ding – A proof of the trace theorem of Sobolev spaces on Lipschitz
domains, Proc. A. M. S. 124 (1996), p. 591–600.
[5] R. G. Durán & M. A. Muschietti – On the traces of W 2, p (Ω) for
a Lipschitz domain, Rev. Mat. Complutense XIV (2001), p. 371–377.
[6] E. Gagliardo – Caratterizzazioni delle tracce sulla frontiera rela-
tive ad alcune classi di funzioni in n-variabili, Rend. Sem. Mat. Univ.
Padova 27 (1957), p. 284–305.
[7] G. Geymonat & F. Krasucki – On the existence of the airy function
in Lipschitz domains. Application to the traces of H2, C. R. Acad. Sci.
Paris, Série I 330 (2000), p. 355–360.
[8] P. Grisvard – Elliptic boundary value problems in nonsmooth do-
mains, Pitman, London, 1985.
[9] J. Nečas – Les méthodes directes en théorie des équations elliptiques,
Masson, Paris, 1967.
196
Trace Theorems. Necessary Conditions
Giuseppe Geymonat
Laboratoire de Mécanique et de Génie
Civil, UMR 5508
CNRS, Université Montpellier II
Place Eugène Bataillon
34695 Montpellier Cedex 5
France
geymonat@lmgc.univ-montp2.fr
197


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