Comptes Rendus Mathématique

Haim Brezis

Alfred Bruckstein

Comptes Rendus
Mathématique
Haim Brezis and Alfred Bruckstein
A sharp relative isoperimetric inequality for the square
Volume 359, issue 9 (2021), p. 1191-1199
<https://doi.org/10.5802/crmath.243>
© Académie des sciences, Paris and the authors, 2021.
Some rights reserved.
This article is licensed under the
Creative Commons Attribution 4.0 International License.
http://creativecommons.org/licenses/by/4.0/
Les Comptes Rendus. Mathématique sont membres du
Centre Mersenne pour l’édition scientifique ouverte
www.centre-mersenne.org
Comptes Rendus
Mathématique
2021, 359, n 9, p. 1191-1199
https://doi.org/10.5802/crmath.243
Partial differential equations / Equations aux dérivées partielles
A sharp relative isoperimetric inequality for
the square
Haim Brezisa, b, c and Alfred Bruckstein∗, d
a Dept. of Mathematics, Rutgers University, Hill Center, Busch Campus, 110
Frelinghuysen Road, Piscataway, NJ 08854, USA
b Depts. of Mathematics and Computer Science,Technion – IIT, Haifa 32000, Israel
c Laboratoire J.-L. Lions, Sorbonne Universités, UPMC, Université Paris-6, 4 place
Jussieu, 75005 Paris, France
d Dept. of Computer Science, Technion – IIT, Haifa 32000, Israel
E-mails: brezis@math.rutgers.edu, freddy@cs.technion.ac.il
Abstract. We compute the exact value of the least “relative perimeter” of a shape S, with a given area,
contained in a unit square; the relative perimeter of S being the length of the boundary of S that does not
touch the border of the square.
Manuscript received and accepted 8th July 2021.
1. Introduction
Let Q denote the unit cube in RN , N ≥ 2. Given a measurable set S ⊂ Q (S stands for shape) we
denote by 1S the characteristic function of S, by |S| = ‖1S‖L1 the volume of S (i.e., the area of S
when N = 2), and by P (S,Q), or simply P (S), the relative perimeter of S, i.e., taking into account
only the part of the boundary of S inside Q; in other words P (S) is the total mass of the measure
∇1S (possibly infinite if S is not rectifiable).
Our goal is to give an explicit formula when N = 2, for the function fN (t ) defined for 0 ≤ t ≤ 1
by
fN (t ) = inf
{
P (S); S is a measurable subset of Q such that |S| = t} . (1)
Clearly
fN (t ) = fN (1− t ) ∀ t ∈ [0,1] ; (2)
just replace S by Q \ S, and thus we will often assume that 0 ≤ t ≤ 12 .
The main result of this note is the following:
∗Corresponding author.
ISSN (electronic) : 1778-3569 https://comptes-rendus.academie-sciences.fr/mathematique/
1192 Haim Brezis and Alfred Bruckstein
Theorem 1. Assume N = 2, then
f2(t ) =
{
(πt )
1
2 if 0 ≤ t ≤ 1π
1 if 1π ≤ t ≤ 12
. (3)
Moreover the infimum in (1) is achieved by explicit shapes whose boundary consists or arcs of circles
or line segments.
Remark 2. We do not know any similar result for fN (t ) when N ≥ 3. In particular, it is not known
whether fN (t ) ≡ 1 in a neighborhood of t = 12 . There is however a simple lower bound for fN (t )
valid for all N ≥ 2. More precisely
fN (t ) ≥ 4t (1− t ) ∀ N ≥ 2, ∀ t ∈
[
0,
1
2
]
, (4)
and the constant 4 in (4) is sharp. This inequality is originally due to H. Hadwiger [7] when the
infimum in (1) is restricted to polyhedral subsets S of the cube Q. Far-reaching variants appeared
subsequently in the literature (see e.g. S. G. Bobkov [5, 6], D. Bakry and M. Ledoux [3], F. Barthe
and B. Maurey [4], and their references). The version stated as (4) (i.e., for measurable sets S) was
proved in its full generality by L. Ambrosio, J. Bourgain, H. Brezis and A. Figalli in [2, Appendix],
where it plays an essential role. Note that when N = 2 inequality (4) is consistent with the explicit
formula (3) since
(πt )
1
2 ≥ 4t (1− t ) ∀ t ∈
(
0,
1
π
)
, (5)
or equivalently
π
1
2
4
≥ s (1− s2) ∀ s ∈ (0, 1
π
1
2
)
. (6)
Indeed the function s(1− s2) is increasing on the interval (0, 1
3
1
2
) and thus (5) reduces to
π
1
2
4
≥ 1
π
1
2
(
1− 1
π
)
,
which is obvious.
Remark 3. Y. Altshuler and A. Bruckstein [1] established earlier a version of Theorem 1 where
the infimum in (1) is restricted to “nice” connected sets S. Their strategy of proof enters as an
ingredient in this note.
Remark 4. The conclusion of Theorem 1 is probably known to the experts even though we could
not find a reference in the literature. E. Milman suggested an alternative approach by considering
the result of H. Howards cited in [8, Section 7], and concerning the isoperimetric problem on a
flat 2D torus.
Acknowledgments
We thank Emanuel Milman for useful discussions concerning the proof of Theorem 1. We also
thank Jean Mawhin and Petru Mironescu for enlightening exchanges related to Lemma 6.
2. Some simple facts
The proof of Theorem 1 relies on three simple facts.
Fact 1 (The classical planar isoperimetric inequality). Given any shape S in the plane we have
P (S) = P (S,R2)≥ 2pπ√|S|.
C. R. Mathématique — 2021, 359, n 9, 1191-1199
Haim Brezis and Alfred Bruckstein 1193
Fact 2 (The half-plane isoperimetric inequality). Given any shape S in the half-plane denoted
1
2R
2 we have
P (S) = P
(
S,
1
2
R2
)
≥p2π
√
|S|.
Proof. If S touches the boundary of the half-plane, we reflect it across the boundary, thereby
generating a (symmetric) shape S′ of area 2|S| and such that
P
(
S′,R2
)= 2P (S, 1
2
R2
)
.
Applying 1 to S′ we obtain
P
(
S′,R2
)≥ 2pπ√|S′|,
which yields
P
(
S,
1
2
R2
)
≥pπ
√
2|S|. 
Fact 3 (The quarter-plane isoperimetric inequality). Given any shape S in a quater-plane
denoted 14R
2 we have
P
(
S,
1
4
R2
)
≥pπ
√
|S|.
Proof. If S touches the two orthogonal boundaries of the quarter-plane we reflect it symmetri-
cally into the three quarters plane, generating a shape S′ of area 4|S| and such that
P
(
S′,R2
)= 4P (S, 1
4
R2
)
.
Applying 1 to S′ we obtain
P
(
S′,R2
)≥ 2pπ√|S′|,
which yields
P
(
S,
1
4
R2
)
≥pπ
√
|S|. 
3. Proof of Theorem 1
Since we consider only the case N = 2, we will write simply f (t ) instead of f2(t ). Set
g (t ) =
{
(πt )
1
2 if 0 ≤ t ≤ 1π
1 if 1π ≤ t ≤ 12
. (7)
The goal is to prove that f (t ) = g (t ) ∀ t ∈ [0, 12 ]. The proof is divided into 7 steps.
Step 1. We have
f (t ) ≤ g (t ) ∀ t ∈
[
0,
1
2
]
. (8)
Assume first that t ≤ 1π and consider the set S as in Figure 1, where R = 2
√
t
π ≤ 1, so that
|S| = πR24 = t and P (S) = 2πR4 = (πt )
1
2 . Therefore (by definition of f (t )), f (t ) ≤ (πt ) 12 = g (t ).
Assume now that 1π ≤ t ≤ 12 and consider the set S as in Figure 2, so that |S| = t . On the other
hand P (S) = 1. Therefore, (by definition of f (t )), f (t ) ≤ 1 = g (t ). 
In what follows we concentrate on the lower bound
f (t ) ≥ g (t ) ∀ t ∈
[
0,
1
2
]
. (9)
Let S be a minimizer in (1). We know from abstract theory (see A. Ros [12, Theorem 1] and
the references therein) that ∂S is smooth and consists of arcs of circle - possibly straight lines;
C. R. Mathématique — 2021, 359, n 9, 1191-1199
1194 Haim Brezis and Alfred Bruckstein
S
R
Figure 1.
S
t
Figure 2.
moreover if ∂Q ∩ ∂S 6= ;, then ∂Q meets ∂S orthogonally. In view of this fundamental result it
suffices to establish the lower bound
P (S) ≥ g (|S|) (10)
when S is restricted to the above class i.e., ∂S is smooth, ∂S consists of arcs of circle (or straight
lines) and ∂S meets ∂Q orthogonally; but S need not be connected. Our next step allows to
assume that S is also connected.
Step 2. Reduction to the case where S is also connected
Assume we have established (10) under the additional assumption that the shape is con-
nected. Consider now some S which is not connected, and write S = ⋃
i
Si where here (Si ) are
the connected components of S. Then
|S| =∑
i
|Si | (11)
and
P (S) =∑
i
P (Si ). (12)
Assume first that |S| ≤ 1π ; then |Si | ≤ 1π ∀ i and by (10) applied to Si we have
P (Si ) ≥
√
π|Si | ∀ i
Thus
P (S) ≥∑
i
√
π|Si | ≥
√
π
∑
i
|Si | =
√
π|S|
i.e., (10) holds for S.
Assume next that 1π ≤ |S| ≤ 12 . We distinguish two cases:
C. R. Mathématique — 2021, 359, n 9, 1191-1199
Haim Brezis and Alfred Bruckstein 1195
Case 1. |Si | ≤ 1π ∀ i .
Then, as above,
P (S) ≥
√
π|S| ≥ 1 = g (|S|),
i.e., (10) holds for S.
Case 2. |Si | > 1π for some i = i0.
By (10) applied to Si0 we have P (Si0 ) ≥ 1, and thus P (S) ≥ P (Si0 ) ≥ 1, i.e., (10) also holds for S.
We are therefore reduced to the situation investigated by Altshuler and Bruckstein [1], even
under the additional assumption that ∂S consists of arcs of circle (or straight lines) meeting ∂Q
orthogonally. We follow the strategy of their argument. 
Step 3. S touches 0 side of Q.
In this case the classical isoperimetric inequality (1 in Section 2) yields
P (S) ≥ 2
√
π|S| ≥ g (|S|) ∀ S, with |S| ≤ 1
2
. 
A typical example is as in Figure 3:
S
Figure 3.
Step 4. S touches 1 side of Q
A typical example is as in Figure 4:
Figure 4.
In this case, the half-plane isoperimetric inequality (see 2 in Section 2) yields
P (S) ≥
√
2π|S| ≥ g (|S|) ∀ S,with |S| ≤ 1
2
. 
C. R. Mathématique — 2021, 359, n 9, 1191-1199
1196 Haim Brezis and Alfred Bruckstein
Figure 5.
Step 5. S touches 2 sides of Q.
In this case we have either S touches two opposite sides of Q or two adjacent sides of Q. The
two opposite sides correspond to Figure 5.
(Here we use the assumption that S is connected.) In this configuration
P (S) ≥ 2 ≥ g (|S|) for all such S.
The two adjacent sides correspond to Figure 6.
Figure 6.
In this configuration the quarter plane isoperimetric inequality (see 3 in Section 2) yields
P (S) ≥
√
π|S| ≥ g (|S|) ∀ S, with |S| ≤ 1
2
. 
Step 6. S touches 3 sides of Q.
A typical example is as in Figure 7, where a portion of the boundary of S has to join two
opposite sides of Q.
Figure 7.
In this case
P (S) ≥ 1 ≥ g (|S|) ∀ S, with |S| ≤ 1
2

C. R. Mathématique — 2021, 359, n 9, 1191-1199
Haim Brezis and Alfred Bruckstein 1197
Step 7. S touches the 4 sides of Q.
A typical example is as in Figure 8.
Figure 8.
Let T =Q \ S and denote by (Ti ) the connected components of T .
Lemma 5. Each Ti touches 0,1, or 2 adjacent sides of Q.
The proof of Lemma 5 relies on the following assertion which appears without proof in a paper
by H. Poincaré [11, p. 67].
Lemma 6. Let A1, B1 be points of ∂Q belonging to opposite sides of Q, and let C1 be a curve in Q
connecting A1 to B1. Let A2, B2 be points of ∂Q belonging to a distinct pair of opposite sides of Q,
and let C2 be a curve in Q connecting A2 to B2. Then
C1 ∩C2 6= ;. (13)
(see Figure 9)
× ×
×
×A1
A2
B1
B2
C1
C2
Figure 9.
Proof of Lemma 6. Let (p1(t ), q1(t )) (resp. (p2(t ), q2(t ))), 0 ≤ t ≤ 1, be a parametrization of
C1 (resp. C2) such that (p1(0), q1(0)) = A1, (p1(1), q1(1)) = B1, (resp. (p2(0), q2(0)) = A2,
(p2(1), q2(1)) = B2). Consider the map F : Q →R2 defined by
F (t , s) = (F1(t , s),F2(t , s)) =
(
p1(t )−p2(s), q1(t )−q2(s)
)
.
C. R. Mathématique — 2021, 359, n 9, 1191-1199
1198 Haim Brezis and Alfred Bruckstein
We have to show that there exists some (t , s) ∈Q such that F (t , s) = 0. Note that
F1(t ,0) = p1(t )−p2(0) = p1(t ) > 0, ∀ t ∈ [0,1], (14)
F1(t ,1) = p1(t )−p2(1) = p1(t )−1 < 0, ∀ t ∈ [0,1], (15)
F2(0, s) = q1(0)−q2(s) = 1−q2(s) > 0, ∀ s ∈ [0,1], (16)
and
F2(1, s) = q1(1)−q2(s) =−q2(s) < 0, ∀ s ∈ [0,1]. (17)
We deduce from the Poincaré–Miranda theorem (see W. Kulpa [9], J. Mawhin [10], and the
references therein) that there exists (t , s) ∈Q (and in fact (t , s) ∈Q) such that F (t , s) = 0. 
Proof of Lemma 5. Assume by contradiction that Ti touches (at least) 2 opposite sides of Q. Fix
a path C1 connecting these 2 opposite sides within Ti (this is possible because Ti is connected).
Consider the remaining 2 opposite sides of Q and fix a path C2 connecting them within S; this is
possible because S touches (by assumption) the 4 sides of Q and S is connected. From Lemma 6
we know that C1 ∩C2 6= ;. But this is impossible since C1 ⊂ Ti , C2 ⊂ S and Ti ∩S =;. 
Proof of Step 7. We now complete the proof of Step 7. By 1, 2 and 3 in Section 2, we have for
every i
P (Ti ) ≥ min
{p
π,
p
2π,2
p
π
}√
|Ti | =
p
π
√
|Ti |. (18)
Thus
P (S) = P (T ) =∑
i
P (Ti ) ≥
p
π
∑
i
√
|Ti |. (19)
From the obvious inequality ∑
i
√
|Ti | ≥
√∑
i
|Ti |, (20)
we deduce that
P (S) ≥pπ
√∑
i
|Ti | =
p
π
√
|T | =pπ
√
1−|S|. (21)
On the other hand
p
1−|S| ≥p|S| since |S| ≤ 12 , and therefore
P (S) ≥pπ
√
|S| ≥ g (|S|) ∀ S, with |S| ≤ 1
2
(22)

Remark 7. The same argument as above applies to the case where Q is replaced by a rectangle
D(X ,Y ) of dimensions X and Y such that X ≤ Y . By analogy with the above we define for
0 ≤ t ≤ X Y ,
f (t ) = inf{P (S); S is a measurable subset of D(X ,Y )such that|S| = t } .
Clearly
f (t ) = f (X Y − t ) ∀ t ∈ [0, X Y ].
The analogue of Theorem 1 is:
Theorem 8. We have
f (t ) =
{
(πt )
1
2 if 0 ≤ t ≤ X 2π
X if X
2
π ≤ t ≤ 12 X Y
.
C. R. Mathématique — 2021, 359, n 9, 1191-1199
Haim Brezis and Alfred Bruckstein 1199
References
[1] Y. Altshuler, A. M. Bruckstein, “On Short Cuts or Fencing in Rectangular Strips”, https://arxiv.org/abs/1011.5920,
2010.
[2] L. Ambrosio, J. Bourgain, H. Brezis, A. Figalli, “BMO-Type Norms Related to the Perimeter of Sets”, Commun. Pure
Appl. Math. 69 (2016), no. 6, p. 1062-1086.
[3] D. Bakry, M. Ledoux, “Lévy–Gromov’s isoperimetric inequality for an infinite-dimensional diffusion generator”,
Invent. Math. 123 (1996), no. 2, p. 259-281.
[4] F. Barthe, B. Maurey, “Some remarks in isoperimetry of Gaussian type”, Ann. Inst. Henri Poincaré, Probab. Stat. 36
(2000), no. 4, p. 419-434.
[5] S. G. Bobkov, “A functional form of the isoperimetric inequality for the Gaussian measure”, J. Funct. Anal. 135 (1996),
no. 1, p. 39-49.
[6] ——— , “An isoperimetric inequality on the discrete cube, and an elementary proof of the isoperimetric inequality
in Gauss space”, Ann. Probab. 25 (1997), no. 1, p. 206-214.
[7] H. Hadwiger, “Gitterperiodische Punktmengen und Isoperimetrie”, Monatsh. Math. 76 (1972), p. 410-418.
[8] H. Howards, M. Hutchings, F. Morgan, “The Isoperimetric Problem on Surfaces”, Amer. Math. Monthly 106 (1999),
no. 5, p. 430-439.
[9] W. Kulpa, “The Poincaré–Miranda theorem”, Amer. Math. Monthly 104 (1997), no. 6, p. 545-550.
[10] J. Mawhin, “Simple proofs of the Hadamard and Poincaré–Miranda theorems using the Brouwer fixed point theo-
rem”, Amer. Math. Monthly 126 (2019), no. 3, p. 260-263.
[11] H. Poincaré, “Sur certaines solutions particulières du problème des trois corps”, Bull. Astronomique 1 (1884), p. 65-
74.
[12] A. Ros, “The isoperimetric problem”, in Global theory of minimal surfaces (D. Hoffman, ed.), Clay Mathematics
Proceedings, vol. 2, American Mathematical Society, 2005, Clay Mathematics Institute 2001 summer school, Berkeley,
CA, USA, June 25–July 27, 2001, p. 175-209.
C. R. Mathématique — 2021, 359, n 9, 1191-1199


A sharp relative isoperimetric inequality for the square

Roya Bahramian

Neda Ahanjideh

Comptes Rendus
Mathématique
Roya Bahramian and Neda Ahanjideh
p-parts of co-degrees of irreducible characters
Volume 359, issue 1 (2021), p. 79-83.
<https://doi.org/10.5802/crmath.158>
© Académie des sciences, Paris and the authors, 2021.
Some rights reserved.
This article is licensed under the
Creative Commons Attribution 4.0 International License.
http://creativecommons.org/licenses/by/4.0/
Les Comptes Rendus. Mathématique sont membres du
Centre Mersenne pour l’édition scientifique ouverte
www.centre-mersenne.org
Comptes Rendus
Mathématique
2021, 359, n 1, p. 79-83
https://doi.org/10.5802/crmath.158
Group theory / Théorie des groupes
p-parts of co-degrees of irreducible characters
Roya Bahramiana and Neda Ahanjideh∗, a
a Department of Pure Mathematics, Faculty of Mathematical Sciences, Shahrekord
University, P. O. Box 115, Shahrekord, Iran
E-mails: roya.bahramian98@gmail.com, ahanjideh.neda@sku.ac.ir
Abstract. For a character χ of a finite group G , the co-degree of χ is χc (1) = [G :kerχ]χ(1) . Let p be a prime and
let e be a positive integer. In this paper, we first show that if G is a p-solvable group such that pe+1 - χc (1),
for every irreducible character χ of G , then the p-length of G is not greater than e. Next, we study the finite
groups satisfying the condition that p2 does not divide the co-degrees of their irreducible characters.
Mathematical subject classification (2010). 20C15, 20D10, 20D05.
Manuscript received 22nd April 2020, revised and accepted 24th November 2020.
1. Introduction and preliminaries
In this paper, G is a finite group, p is a prime number and e is a positive integer. Let Z (G) be the
center of G and let Op (G) and Op ′ (G) be the largest normal p-subgroup and the largest normal
p ′-subgroup of G , respectively. Also, Op
′
(G) is the largest normal subgroup of G whose index in G
is co-prime to p. For a p-solvable group G , the p-length of G , denoted by `p (G), is the minimum
possible number of factors that are p-groups in any normal series of G which every factor is either
a p-group or a p ′-group. Let Irr(G) denote the set of (complex) irreducible characters of G . For a
normal subgroup N of G and a character θ of N , let IG (θ) denote the inertia group of θ in G and
let Irr(G|θ) be the set of the irreducible constituents of the induced character θG . Also, we use ep
to show the p-part of e. For a character χ of G , the number χc (1) = [G :kerχ]χ(1) is called the co-degree
of χ (see [11]). Set Codeg(G) = {χc (1) : χ ∈ Irr(G)}. In [1–3, 11], some properties of the co-degrees
of irreducible characters of finite groups have been studied.
In [1], it has been proved that the p-length of a finite p-solvable group is not greater than the
number of the distinct co-degrees of its irreducible characters which are divisible by p. In this
paper, we prove that:
Theorem 1. If G is a p-solvable group and pe+1 -χc (1), for every χ ∈ Irr(G), then `p (G) ≤ e.
In [8–10], it has been shown that if p2 -χ(1), for every χ ∈ Irr(G), then [G : Op (G)]p ≤ p3. In this
paper, we also prove that:
∗Corresponding author.
ISSN (electronic) : 1778-3569 https://comptes-rendus.academie-sciences.fr/mathematique/
80 Roya Bahramian and Neda Ahanjideh
Theorem 2. Let G be a non-p-solvable group. If χc (1)p ≤ p, for every χ ∈ Irr(G), then |G|p = p.
Corollary 3. If χc (1)p ≤ p, for every χ ∈ Irr(G), then the Sylow p-subgroups of G are elementary
abelian p-groups.
In Examples 9, 10 and 11, we show that in Theorem 2, “non-p-solvability” cannot be substi-
tuted with “non-solvability” and in Corollary 3, there is not necessarily an upper bound for |G|p
or |G/Op (G)|p .
2. Proofs of the main results
We first state a lemma that will be used frequently in this paper without explicit reference.
Lemma 4 (cf. [11, Lemma 2.1]). Let N be a normal subgroup of G. Then, Codeg(G/N ) ⊆
Codeg(G). Also, if ψ ∈ Irr(N ), then ψc (1) |χc (1), for every χ ∈ Irr(G|ψ).
Lemma 5. Let S be a non-abelian simple group.
(i) If p is a prime divisor of the order of the Schur multiplier of S, then |S|p ≥ p2.
(ii) If p is a prime divisor of |Out(S)| such that p divides |S|, then |S|p ≥ p2.
Proof. Since S is a non-abelian simple group, |S|2 ≥ 4. So, the lemma follows when p = 2. Next,
assume that p ≥ 3. If p is a prime divisor of the order of the Schur multiplier of S, then since
p ≥ 3, [7, Section 5.1] shows that S ∼= PSLn(q), p | q −1 and p | n, S ∼= PSUn(q), p | q +1 and p | n,
S ∈ {PSL2(9), Al t7,PSU4(3),G2(3), J3, M22,F i22, Mcl ,Suz,B3(3), 2E6(4),F i ′24,O′N }
(under isomorphism) and p = 3, S ∼= E6(q) and p = 3 | q −1 or S ∼= 2E6(q) and p = 3 | q +1. Thus,
we can check at once that |S|p ≥ p2, as desired in (i). Next, let p be a prime divisor of |Out(S)| and
|S|. Then, [8, Lemma 3.1] shows that |S|p > |Out(S)|p ≥ p. Thus, |S|p ≥ p2, as wanted. 
In order to prove the main results, we need to prove the following propositions:
Proposition 6. Let N be a minimal normal subgroup of G.
(i) If N is abelian and χ ∈ Irr(G) such that N 6≤ kerχ, then |N | divides χc (1).
(ii) If p | |N | and χc (1)p ≤ p, for every χ ∈ Irr(G), then |N |p = p and N is a simple group.
Proof. (i). Since N is a minimal normal subgroup of G and N 6= N ∩ kerχEG , N ∩ kerχ =
{1}. So, N ∼= N kerχ/kerχ is an abelian normal subgroup of G/kerχ. By Ito’s theorem (see [6,
Theorem 6.15]), χ(1) | [ Gkerχ : N kerχkerχ ]= [G : N kerχ]. Thus, |N | |χc (1), as desired in (i).
(ii). First suppose that N ≤ Op (G), θ ∈ Irr(N ) − {1N } and χ ∈ Irr(G|θ). Then, N 6≤ kerχ. So,
|N | | χc (1), by (i). Thus, |N |p ≤ χc (1)p ≤ p. However, N is a p-group. Hence, |N | = p, as desired.
Now, let N be non-abelian. Then, N = S1 × ·· ·×St , where S1, . . . ,St are isomorphic non-abelian
simple groups. For every i ∈ {1, . . . , t }, p | |Si | and there exists θi ∈ Irr(Si )− {1Si } such that p - θi (1),
by [6, Corollary 12.2]. Set θ = θ1 × ·· · × θt and let χ ∈ Irr(G|θ). Then, θ ∈ Irr(N ), kerθ = {1} and
p - θ(1). Thus, |N |p | θc (1), hence |N |p | χc (1). So, |N |p ≤ p. Consequently, t = 1 and N is a non-
abelian simple group. 
Proposition 7. Let N ≤ Z (G) and G/N be a non-abelian simple group. If p divides |N | and |G/N |,
then there exists χ ∈ Irr(G) such that χc (1)p ≥ p2.
Proof. Since N ≤ Z (G), N is abelian. Hence, N has a maximal normal subgroup M such that
|N /M | = p. However, M ≤ N ≤ Z (G). Thus, M EG and N /M ≤ Z (G/M). If we can show that
there exists χ ∈ Irr(G/M) such that χc (1)p ≥ p2, then Lemma 4 completes the proof. So, without
loss of generality, assume that |N | = p. Then, G ′ ∩ N = {1} or N , where G ′ denotes the derived
C. R. Mathématique, 2021, 359, n 1, 79-83
Roya Bahramian and Neda Ahanjideh 81
subgroup of G . Since G/N is a non-abelian simple group, G ′N = G . Thus, either G ′×N = G and
G ′ ∼=G/N or N ≤G ′ =G . In the former case, there exists θ ∈ Irr(G ′)− {1G ′ } such that p - θ(1), by [6,
Corollary 12.2]. Set χ= θ×ϕ, for some ϕ ∈ Irr(N )− {1N }. Then, χ ∈ Irr(G), p - χ(1) and kerχ= {1}.
Thus, |G|p divides χc (1)p . Hence, χc (1)p ≥ p2, as desired. In the latter case, G is a quasi-simple
group with Z (G) = N . Consequently, |N | divides |M(G/N )|, the order of the Schur multiplier
of G/N . Therefore, p | |M(G/N )|. It follows from Lemma 5(i) that |G/N |p ≥ p2. Since G/N is a
non-abelian simple group, there exists ψ ∈ Irr(G/N ) such that p - ψ(1) and kerψ = {1}, by [6,
Corollary 12.2]. So, |G/N |p divides ψc (1). Consequently, ψc (1)p ≥ p2. Hence, the proposition
follows because Codeg(G/N ) ⊆ Codeg(G). 
Proof of Theorem 1. Let G be a minimal counterexample. Then, since the hypothesis is inher-
ited by quotients and normal subgroups and `p (G/Op ′ (G)) = `p (G) = `p (Op ′ (G)), we can as-
sume that Op ′ (G) = {1} and Op ′ (G) = G . Thus, every minimal normal subgroup M of G is a p-
group and `p (G/M) ≤ e. Suppose that M and W are two distinct minimal normal subgroups
of G . Then, since M ∩ W = {1}, `p (G/W ) ≤ e and `p (G/M) ≤ e, `p (G) = `p (G/(M ∩ W )) ≤
max{`p (G/M),`p (G/W )} ≤ e, by [5, VI. 6.4]. This is a contradiction. Now let M be the unique
minimal normal subgroup of G . Let l = `p (G/M) and define a normal series {1} = P0(G/M)E
M0(G/M)EP1(G/M)EM1(G/M)E · · ·EPl (G/M)EMl (G/M) =G/M of G/M such that Mi (G/M)Pi (G/M) =
Op ′
( G/M
Pi (G/M)
)
and Pi (G/M)Mi−1(G/M) = Op
( G/M
Mi−1(G/M)
)
. Set Pi /M = Pi (G/M) and Mi /M = Mi (G/M). We
claim that Op ′ (G/M) 6= {1}. If not, M0 = P0. Thus, P1 =Op (G) and {1}EP1EM1E · · ·EPlEMl =G
is a normal series of G such that MiPi =Op ′
( G
Pi
)
and PiMi−1 =Op
( G
Mi−1
)
, for every 1 ≤ i ≤ l . Therefore,
`p (G) = l = `p (G/M) ≤ e. This is a contradiction. Thus, Op ′ (G/M) 6= {1}. Set N /M =Op ′ (G/M). By
Schur–Zassenhaus theorem, N has a p-complement L. Then, G = N NG (L) = M NG (L). Since M
is abelian, M ∩ NG (L)EG . However, M is a minimal normal subgroup of G and Op ′ (G) = {1}.
Thus, we can check that M ∩ NG (L) = {1}. So, every λ ∈ Irr(M) extends to IG (λ), by [6, Exer-
cise 6.18]. Let 1M = λ1, . . . ,λt be the representatives of the action of G on Irr(M). If Oi is the G-
orbit of λi , then 1 +Σti=2|Oi |λi (1)2 = Σλ∈Irr(M)λ(1)2 = |M | ≡p 0. Hence, there exists i > 1 such
that p - |Oi | = [G : IG (λi )]. So, IG (λi ) contains a Sylow p-subgroup P of G . Since λi extends to
IG (λi ), there exists λ̂i ∈ Irr(IG (λi )) such that λ̂i M =λi . Set χ= λ̂Gi . By Clifford theory (see [6, The-
orem 6.4]), χ ∈ Irr(G) and χ(1) = [G : IG (λi )]. Also, kerχ∩ M is a normal subgroup of G and M
is a minimal normal subgroup of G . Thus, either kerχ∩M = M or kerχ∩M = {1}. In the former
case, M ≤ kerχ, so χM is trivial and λi = λ1, which is a contradiction. Therefore, kerχ∩M = {1}.
Consequently, kerχ = {1}, because M is the unique minimal normal subgroup of G . Hence,
χc (1) = |IG (λi )|, which is divisible by |P |. So, |G|p ≤ pe . Therefore, `p (G) ≤ e, which is a con-
tradiction. Now, the proof is complete. 
Note that in some parts of the proof of Theorem 1, we follow the ideas in the proof of [4,
Theorem 2.3].
Proof of Theorem 2. First, let G be a non-p-solvable group of the minimal order such that
|G|p ≥ p2. Since the hypothesis is inherited by quotients and normal subgroups, we may assume
that Op ′ (G) = {1} and Op ′ (G) =G . We continue the proof in the following cases:
Case a. Assume that Op (G) 6= {1}. Let N ≤ Op (G) be a minimal normal subgroup of G . Then,
|N | = p, by Proposition 6(ii). However, G/N is not p-solvable and Codeg(G/N ) ⊆ Codeg(G). So,
|G/N |p = p, by minimality of G . Hence, |G|p = p2. Since |N | = p and G/CG (N ) is isomorphic to a
subgroup of Aut(N ), we have Op
′
(G) ≤CG (N ). However, Op ′ (G) =G . So, CG (N ) =G . Consequently,
N ≤ Z (G). Set G =G/N and let M/N = M be a minimal normal subgroup of G . If M ≤Op ′ (G), then
|N | and |M/N | are co-prime. By Schur–Zassenhaus theorem, M has a p-complement H . Since
N ≤ Z (G), M = H ×N . Thus, M = H N /N ∼= H/(H ∩N ) = H = Op ′ (M) ≤ Op ′ (G) = {1}, which is a
contradiction. Now let Op ′ (G) = {1}. Then, since G is not p-solvable and |G|p = p, we get that M
C. R. Mathématique, 2021, 359, n 1, 79-83
82 Roya Bahramian and Neda Ahanjideh
is the unique minimal normal subgroup of G and |M |p = p. So, M is a non-abelian simple group.
By Proposition 7, there exists θ ∈ Irr(M) such that θc (1)p ≥ p2. Therefore, χc (1)p ≥ p2, for every
χ ∈ Irr(G|θ). This is a contradiction.
Case b. Let Op (G) = {1}. Since Op ′ (G) = {1}, every minimal normal subgroup of G is a non-abelian
simple group of order divisible by p, by Proposition 6(ii). If G has two distinct minimal normal
subgroups M1 and M2, then p | |M1|, |M2|. However, |G|p = p2 and Op ′ (G) =G . Thus, G = M1×M2.
So, there exists θ = θ1 ×θ2 ∈ Irr(M1)× Irr(M2) = Irr(G) such that p - θ(1) and kerθ = {1}. Therefore,
p2 = |G|p | θc (1), which is a contradiction. Next let G have the unique minimal normal subgroup,
say M . Then, CG (M) = {1}. Consequently, G . Aut(M). By Proposition 6(ii), |M |p = p. Thus,
p | |G/M |, hence p | |Out(M)|. Lemma 5(ii) shows that |M |p ≥ p2. This is a contradiction.
So, |G|p = p, as desired. 
Remark 8. If χc (1)p ≤ p, for every irreducible character χ of G , then by Theorems 1 and 2, either
|G|p = p or G is a p-solvable group of p-length one.
Proof of Corollary 3. If G is non-p-solvable, then |G|p = p, by Theorem 2. Thus, the corollary
follows. Now, let G be p-solvable. By Theorem 1, G has p-length one. Let L = Op ′ (G) and K /L =
Op (G/L). Then, K /L is isomorphic to a Sylow p-subgroup of G . By Lemma 4, Codeg(K /L) = {1, p}.
Thus, [3, Lemma 2.4] forces K /L to be elementary abelian, as desired. 
Example 9. Assume that G = L1 × ·· · ×Lt , where L1, . . . ,Lt are Symmetric groups of degrees 3.
Let χ ∈ Irr(G)− {1G }. Then, there exist θ1 ∈ Irr(L1), . . . ,θt ∈ Irr(Lt ) such that χ = θ1 × ·· · × θt . Set
Ω1 = {1 ≤ i ≤ t : θi (1) = 2} andΩ2 = {1 ≤ i ≤ t : i 6∈Ω1}. Let χ1 =Πi∈Ω1θi and χ2 =Πi∈Ω2θi . If χ=χ1,
then χ(1) = |G|2. Hence, χc (1)2 = 1. Otherwise, fix H = Πi∈Ω2 Li . Then, H/H ′ is an elementary
abelian 2-group of order |H |2 and χ2 ∈ Irr(H/H ′). Therefore, |kerχ2|2 = |H |2/2, by [3, Lemma 2.4].
Since, χ = χ1 ×χ2, (Πi∈Ω1 1Li )×kerχ2 ≤ kerχ. Thus, 2|Ω2|−1 ≤ |kerχ|. Also, χ(1) = 2|Ω1|. Therefore,
χc (1)2 ≤ 2. This example shows that in Corollary 3, |G/Op (G)|p is not necessarily bounded.
Example 10. Let K be an elementary abelian 3-group of order 3n . Then, the cyclic group P = 〈z〉
of order 2 acts on K by xz = x2, for every x ∈ K . Let G be a semi-direct product K o P and
let χ ∈ Irr(G) − {1G }. If K ≤ kerχ, then χc (1) = 2. Otherwise, there exists θ ∈ Irr(K ) − {1K } such
that 〈χK ,θ〉 6= 0. By [3, Lemma 2.4], |kerθ| = 3n−1. It is easy to check that kerθEG . Therefore,
kerθ ≤ kerχ. Thus, χc (1)3 | |G/kerθ|3 = 3. Consequently, χc (1)3 ≤ 3. This example shows that in
Corollary 3, |Op (G)| and |G/Z (G)|p are not necessarily bounded.
Example 11. Let p 6= 2 and S be a non-abelian simple group such that p - |S|. Suppose that P is
an elementary abelian p-group of order pn and G = P ×S. For every χ ∈ Irr(G), we can see that
pn−1 | |kerχ|, so χc (1)p | pn/pn−1 = p. This example shows that in Theorem 2, “non-p-solvability”
cannot be substituted with “non-solvability”.
Acknowledgments
The authors would like to thank the referees for careful reading and many constructive com-
ments, which substantially helped to improve the quality of the paper.
References
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C. R. Mathématique, 2021, 359, n 1, 79-83


$p$-parts of co-degrees of irreducible characters

Olivier Rahavandrainy

Familles de polynômes unitairement parfaits sur $\protect \mathbb{F}_2$

Sergio Conti

Adriana Garroni

Comptes Rendus
Mathématique
Sergio Conti and Adriana Garroni
Sharp rigidity estimates for incompatible fields as a consequence of the
Bourgain Brezis div-curl result
Volume 359, issue 2 (2021), p. 155-160.
<https://doi.org/10.5802/crmath.161>
© Académie des sciences, Paris and the authors, 2021.
Some rights reserved.
This article is licensed under the
Creative Commons Attribution 4.0 International License.
http://creativecommons.org/licenses/by/4.0/
Les Comptes Rendus. Mathématique sont membres du
Centre Mersenne pour l’édition scientifique ouverte
www.centre-mersenne.org
Comptes Rendus
Mathématique
2021, 359, n 2, p. 155-160
https://doi.org/10.5802/crmath.161
Analysis / Analyse
Sharp rigidity estimates for incompatible
fields as a consequence of the Bourgain Brezis
div-curl result
Estimées de rigidité pour les champs incompatibles
comme conséquence du résultat div-rot de Bourgain et
Brezis
Sergio Contia and Adriana Garroni∗, b
a Institute for Applied Mathematics, University of Bonn, Endenicher Allee 60, 53115
Bonn, Germany
b University of Rome, Sapienza, P.le A. Moro 2, 00185 Rome, Italy
E-mails: sergio.conti@uni-bonn.de (S. Conti), garroni@mat.uniroma1.it (A. Garroni)
Abstract. In this note we show that a sharp rigidity estimate and a sharp Korn’s inequality for matrix-
valued fields whose incompatibility is a bounded measure can be obtained as a consequence of a Hodge
decomposition with critical integrability due to Bourgain and Brezis.
Résumé. Dans cette note, nous démontrons qu’une estimée de rigidité et une inégalité de Korn pour des
champs avec des valeurs matricielles dont l’incompatibilité est une mesure bornée peuvent être obtenues
comme conséquence d’une décomposition de Hodge avec intégrabilité critique dû à Bourgain et Brezis.
Keywords. rigidity estimates, div-curl systems, Korn inequality, plasticity, incompatible fields, Hodge decom-
position.
Mots-clés. estimée de rigidité, systèmes div-rot, inégalité de Korn, plasticité, champs incompatibles, décom-
position de Hodge.
2020 Mathematics Subject Classification. 49Q20, 74C15, 53C24.
Funding. Partially funded by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation)
via project 211504053 - SFB 1060 and project 390685813 - HCM and by MIUR via PRIN 2017 Variational
methods for stationary and evolution problems with singularities and interfaces, 2017BTM7SN.
Manuscript received 13th June 2020, revised and accepted 30th November 2020.
∗Corresponding author.
ISSN (electronic) : 1778-3569 https://comptes-rendus.academie-sciences.fr/mathematique/
156 Sergio Conti and Adriana Garroni
1. Introduction
The celebrated rigidity estimate of Friesecke, James, and Müller [8, 9] is a fundamental tool
for the analysis of variational, geometrically nonlinear models in elasticity. Indeed it allows to
obtain compactness of the deformation gradient (and therefore, for example, linearization or
dimension-reduction results) when the elastic energy has a degeneracy due to frame indifference.
The linear counterpart of this estimate is the classical Korn inequality, which permits to deal with
problems in linear elasticity where the energy, due to the approximation of the frame indifference,
does not control the antisymmetric part of the displacement gradient, the so called infinitesimal
rotations.
In the presence of plastic deformations, and then beyond the elastic regime mentioned above,
the variational models should account for plastic slips that occur at microscopic scale and for the
presence of topological defects which induce elastic distortion. Therefore, the relevant variable is
not the gradient of a deformation but a more general matrix-valued field, the elastic strain, which
may not be a gradient and may have a non trivial curl, possibly concentrated on low-dimensional
sets.
This framework calls for a version of the rigidity result, and of its linear counterpart, for fields
that may be incompatible, in the sense that are not gradients, but for which one knows that the
curl is a bounded measure. Such results do not seem to be available in the literature in dimension
greater than two, and are stated in Theorems 1 and 2.
In the sequel, given a matrix-valued function β ∈ L1(Ω;Rn×n), Curlβ denotes the tensor-
valued distribution whose rows are the curl, in the sense of distributions, of the rows of β, in the
sense that Curlβi j k := ∂kβi j −∂ jβi k , moreover M (Ω;Rk ) denotes the space of Radon measures
in Ω with values in Rk . In dimension 3, Curlβ can as usual be identified with a matrix-valued
distribution αi l :=
∑
j ,k εl j k Curlβi j k . The natural space is then the one obtained by the Sobolev-
conjugate exponent 1∗ := n/(n −1).
Theorem 1. Given an open, bounded, connected, and Lipschitz setΩ inRn , with n ≥ 2, there exists
a constant C =C (n,Ω) > 0 such that for every β ∈ L1(Ω; Rn×n) with Curlβ ∈M (Ω; Rn×n×n), there
exists a rotation R ∈ SO(n) such that
∥∥β−R∥∥L1∗ (Ω) ≤C (∥∥dist(β,SO(n))∥∥L1∗ (Ω) + ∣∣Curlβ∣∣ (Ω)) . (1)
Theorem 2. Given an open, bounded, connected, and Lipschitz setΩ inRn , with n ≥ 2, there exists
a constant C = C (n,Ω) > 0 such that for every β ∈ L1(Ω;Rn×n) with Curlβ ∈ M (Ω;Rn×n×n) there
exists an antisymmetric matrix A such that
∥∥β− A∥∥L1∗ (Ω) ≤C (∥∥β+βT ∥∥L1∗ (Ω) + ∣∣Curlβ∣∣ (Ω)) . (2)
In the above results |Curlβ|(Ω) denotes the total variation of the measure Curlβ. In the case
when Curlβ is absolutely continuous with respect to the Lebesgue measure it is simply given
by the L1 norm of Curlβ. We remark that the main point in this estimate is that it provides an
estimate with the total variation and therefore allows for concentrated incompatibilities, which
correspond to concentration of crystal defects. Korn’s inequalities with incompatibility involving
the Lp norm of Curlβ, with p > 1, instead of the total variation, can be found in [15] and the
references therein.
One can see Theorem 2 as a a linearization of Theorem 1. Indeed, it could be proven that way,
using a suitable density argument to reduce to fields β ∈ L∞. This procedure is normally not used
in the case that β is a gradient, as the linear result is a key ingredient in the proof of the nonlinear
rigidity estimate.
C. R. Mathématique, 2021, 359, n 2, 155-160
Sergio Conti and Adriana Garroni 157
In 2 dimensions we have 1∗ = 2 and Korn and rigidity estimates for incompatible field
(Theorems 2 and 1 with n = 2) have been obtained in [10, Theorem 11] and [16, Theorem 3.3],
respectively. A key ingredient in the proof is an interpolation inequality due to Bourgain and
Brezis which provides an estimate of the H−1 norm of a field in terms of the L1 norm and the
H−2 norm of its divergence, [2].
In higher dimension a recent result by Lauteri and Luckhaus, [14], shows that the rigidity
estimate of Theorem 1 can be obtained in the spaces Lp with p < 1∗ and the Lorentz space L1∗,∞,
but their results do not cover the case L1
∗
which has the natural scaling.
Here, we show that the result in dimension three and above, Theorems 1 and 2, can be
obtained very easily via Hodge decomposition using a sharp regularity result due to Bourgain
and Brezis for the div-curl system [1, 2] (see also [13] and [18]) in the case that Ω is a cube, and
then generalized to Lipschitz domains using a Whitney-type argument.
A similar Hodge decomposition argument had been used in the subcritical case in [4]. We had
used the Bourgain–Brezis regularity result for the div-curl system to show existence of plastic
strain in the presence of dislocations in [6]. After this paper was submitted, a related argument
has been used in [11] to obtain similar rigidity estimates with zero boundary data or on cubes for
other differential operators, including the deviatoric strain.
2. Proof of the results for n ≥ 3
Lemma 3. Let n ≥ 3 and Y ∈ L1(Q; Rn) be such that divY = 0 distributionally and Y n = 0 on ∂Q.
Then
‖Y ‖L1∗ (Q) ≤ c(n)‖curlY ‖L1(Q) . (3)
Here and below Q := (0,1)n ; the assumption on Y means that extending Y by zero to Rn we
obtain a divergence-free field.
We recall that this assertion does not hold for n = 2, as one can see with curlY = δx0 for some
x0 ∈ Q, see for example [1, Remark 2]. Indeed, the proof of Theorem 1 and Theorem 2 for n = 2
given in [10, Theorem 11] and [16, Theorem 3.3] is different, although it also uses in a crucial way
the construction by Bourgain and Brezis.
We remark that for n = 3 this implies that for every f ∈ L1(Ω; R3), with div f = 0, there exists a
unique solution Z ∈ L3/2(Q; R3) of the system
div Z = 0 in Q,
curl Z = f in Q,
Z ·n = 0 on ∂Q
(4)
and it satisfies
‖Z‖L3/2(Q) ≤C‖ f ‖L1(Q) (5)
for some constant independent on f . By an approximation argument it follows that the same
result holds for measures, i.e., f ∈ M (Q; R3), with div f = 0 in the sense of distributions. By [3,
Theorem 4.1] this result can be extended to smooth bounded sets Ω ⊂ Rn . This argument was
used for the study of geometrically linear dislocations in [6].
Proof. By density, it suffices to prove the bound for regular fields Y (one can reflect Y across all
boundaries and then mollify). Let X ∈ Ln(Q; Rn). By [2, Corollary 18’] there are ϕ ∈ W 1,n(Q) and
γ ∈W 1,n0 ∩L∞(Q;Rn×n) such that γ+γT = 0, X = Dϕ+divγ, and∥∥ϕ∥∥W 1,n (Q) +∥∥γ∥∥L∞(Q) +∥∥γ∥∥W 1,n (Q) ≤ c(n)‖X ‖Ln (Q). (6)
C. R. Mathématique, 2021, 359, n 2, 155-160
158 Sergio Conti and Adriana Garroni
We estimate∫
Q
Y ·X dx =
∫
Q
Y ·Dϕ dx +
∫
Q
Y ·divγ dx =
∫
∂Q
Y ·γn dH n−1 −
∫
Q
DY ·γ dx
=−1
2
∫
Q
(
DY −DY T ) ·γd x =−1
2
∫
Q
curlY ·γ dx
≤ 1
2
‖curlY ‖L1(Q)
∥∥γ∥∥L∞(Q) ≤ c(n)‖curlY ‖L1(Q) ‖X ‖Ln (Q) .
(7)

Proof of Theorem 1. To simplify notation, we write s := 1∗ = n/(n−1) and denote by c a constant
that depends only on the spatial dimension n, and that may change from line to line. It suffices
to prove the assertion if the right-hand side is finite.
First we prove the result in the case thatΩ=Q = (0,1)n .
By density it suffices to prove the assertion for β smooth. Indeed, it is sufficient to extend
β by reflection and then to mollify. We define u ∈ W 2,1(Q;Rn) as the solution of ∆u = divβ in
Q, ∂nu = βn on ∂Q, and set Y := β−Du. Since divY = 0, Y n = 0 on ∂Q, and curlY = curlβ,
application of Lemma 3 to each row of Y yields
‖Y ‖Ls (Q) ≤ c
∥∥curlβ∥∥L1(Q) . (8)
Therefore, since Du =β−Y ,
‖dist(Du,SO(n))‖Ls (Q) ≤
∥∥dist(β,SO(n))∥∥Ls (Q) +‖Y ‖Ls (Q)
≤∥∥dist(β, SO(n))∥∥Ls (Q) + c ∥∥curlβ∥∥L1(Q) . (9)
We then use the geometric rigidity estimate by Friesecke, James, and Müller [8, 9] (see also [5]
for the version in Lp ) which states that for any bounded Lipschitz connected domain Ω and any
p ∈ (1,∞) there is a constant c = c(Ω,n, p) such that for any u ∈ W 1, p (Ω; Rn) there is R ∈ SO(n)
such that
‖Du −R‖Lp (Ω) ≤ c ‖dist(Du,SO(n))‖Lp (Ω) (10)
and obtain the assertion. By scaling invariance, we have shown that for any cube Qr := x∗
+ (−r,r )n and any β : Qr →Rn×n there is R ∈ SO(n) such that∥∥β−R∥∥Ls (Qr ) ≤ c (‖dist(Du,SO(n))‖Ls (Qr ) +∥∥curlβ∥∥L1(Qr )) . (11)
We now deal with a generic bounded Lipschitz connected set Ω. We consider a Whitney
covering ofΩ by countably many cubes Q j := x j + (−r j ,r j )n with finite overlap, so that
χΩ ≤
∑
j
χQ̂ j ≤
∑
j
χQ j ≤ cχΩ, (12)
where
Q̂ j := x j +
(
−1
2
r j ,
1
2
r j
)n
,
and such that r j ≤ dist(Q j ,∂Ω) ≤ cr j , see for example [17, Section VI.1] or [7, Section 6.5].
Condition (12) states that the smaller cubes cover the domain Ω and the bigger ones still have
finite overlap. This permits to construct a partition of unity ϕ j ∈ C∞c (Q j ) subordinated to the
cubes Q j , with
∑
j ϕ j = 1 inΩ and such that |Dϕ j | ≤ c/r j .
We apply the estimate (11) on each cube Q j and obtain rotations R j ∈ SO(n). If Q j ∩Qk 6= ;,
then by a triangular inequality we obtain∫
Q j ∩Qk
∣∣R j −Rk ∣∣s dx ≤ c ∫
Q j ∩Qk
∣∣β−R j ∣∣s dx + c ∫
Q j ∩Qk
∣∣β−Rk ∣∣s dx. (13)
We define R :Ω→Rn×n by R :=∑ j ϕ j R j , and observe that∥∥β−R∥∥sLs (Ω) ≤ c ∑
j
∥∥β−R j ∥∥sLs (Q j ) . (14)
C. R. Mathématique, 2021, 359, n 2, 155-160
Sergio Conti and Adriana Garroni 159
From
∑
j ϕ j = 1 we obtain
∑
j Dϕ j = 0 onΩ, so that
DR =∑
j
Dϕ j R j =
∑
j
Dϕ j
(
R j −β
)
. (15)
Recalling that dist(Q j ,∂Ω) ≤ cr j and (12),∫
Ω
dists (x,∂Ω) |DR|s dx ≤ c
∑
j
∫
Q j
r sj
∣∣Dϕ j ∣∣s ∣∣β−R j ∣∣s dx ≤ c ∑
j
∫
Q j
∣∣β−R j ∣∣s dx. (16)
Since (11) holds in each cube Q j ,∫
Ω
dists (x,∂Ω) |DR|s dx ≤ c
∑
j
[∥∥dist(β,SO(n))∥∥sLs (Q j ) + (∣∣curlβ∣∣ (Q j ))s]
≤ c ∥∥dist(β,SO(n))∥∥sLs (Ω) + c (∣∣curlβ∣∣ (Ω))s−1 ∑
j
∣∣curlβ∣∣ (Q j )
≤ c ∥∥dist(β,SO(n))∥∥sLs (Ω) + c (∣∣curlβ∣∣ (Ω))s .
(17)
By the weighted Poincaré inequality [12, Theorem 8.8] there is a matrix R∗ ∈Rn×n such that
‖R −R∗‖sLs (Ω) ≤ c
∫
Ω
dists (x,∂Ω) |DR|s dx. (18)
Let now R̂ ∈ SO(n) be a matrix such that |R̂ −R∗| = dist(R∗,SO(n)). Then, recalling (14),
|Ω|1/s ∣∣R̂ −R∗∣∣≤‖R∗−R‖Ls +∥∥R −β∥∥Ls +∥∥dist(β,SO(n))∥∥Ls
≤c ∥∥dist(β,SO(n))∥∥Ls (Ω) + c ∣∣curlβ∣∣ (Ω). (19)

Proof of Theorem 2. This follows by the same proof, replacing geometric rigidity in Ls , Equa-
tion (10), by Korn’s inequality in Ls , which states that for every bounded Lipschitz connected do-
main Ω and any p ∈ (1,∞) there is a constant c = c(Ω,n, p) such that for any u ∈ W 1, p (Ω; Rn)
there is R ∈ SO(n) such that
‖Du −R‖Lp (Ω) ≤ c
∥∥Du +DuT ∥∥Lp (Ω) (20)
(see, for example, [5] and references therein). 
References
[1] J. Bourgain, H. Brezis, “New estimates for the Laplacian, the div-curl, and related Hodge systems”, C. R. Math. Acad.
Sci. Paris 338 (2004), no. 7, p. 539-543.
[2] ——— , “New estimates for elliptic equations and Hodge type systems”, J. Eur. Math. Soc. 9 (2007), no. 2, p. 277-315.
[3] H. Brezis, J. Van Schaftingen, “Boundary estimates for elliptic systems with L1 data”, Calc. Var. Partial Differ. Equ. 30
(2007), no. 3, p. 369-388.
[4] A. Chambolle, A. Giacomini, M. Ponsiglione, “Piecewise rigidity”, J. Funct. Anal. 244 (2007), no. 1, p. 134-153.
[5] S. Conti, G. Dolzmann, S. Müller, “Korn’s second inequality and geometric rigidity with mixed growth conditions”,
Calc. Var. Partial Differ. Equ. 50 (2014), no. 1-2, p. 437-454.
[6] S. Conti, A. Garroni, M. Ortiz, “The line-tension approximation as the dilute limit of linear-elastic dislocations”, Arch.
Ration. Mech. Anal. 218 (2015), no. 2, p. 699-755.
[7] L. C. Evans, R. F. Gariepy, Measure theory and fine properties of functions, Studies in Advanced Mathematics, CRC
Press, 1992.
[8] G. Friesecke, R. D. James, S. Müller, “A theorem on geometric rigidity and the derivation of nonlinear plate theory
from three-dimensional elasticity”, Commun. Pure Appl. Math. 55 (2002), no. 11, p. 1461-1506.
[9] G. Friesecke, S. Müller, R. D. James, “Rigorous derivation of nonlinear plate theory and geometric rigidity”, C. R.
Math. Acad. Sci. Paris 334 (2002), no. 2, p. 173-178.
[10] A. Garroni, G. Leoni, M. Ponsiglione, “Gradient theory for plasticity via homogenization of discrete dislocations”, J.
Eur. Math. Soc. 12 (2010), no. 5, p. 1231-1266.
[11] F. Gmeineder, D. Spector, “On Korn–Maxwell–Sobolev Inequalities”, https://arxiv.org/abs/2010.03353, 2020.
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[12] A. Kufner, Weighted Sobolev Spaces, Teubner-Texte zur Mathematik [Teubner Texts in Mathematics], vol. 31, BSB B.
G. Teubner Verlagsgesellschaft, 1980.
[13] L. Lanzani, E. Stein, “A note on div curl inequalities”, Math. Res. Lett. 12 (2005), no. 1, p. 57-61.
[14] G. Lauteri, S. Luckhaus, “Geometric rigidity estimates for incompatible fields in dimension ≥ 3”, https://arxiv.org/
abs/1703.03288, 2017.
[15] P. Lewintan, P. Neff, “Nečas-Lions lemma reloaded: An Lp -version of the generalized Korn inequality for incompati-
ble tensor fields”, https://arxiv.org/abs/1912.08447, 2019.
[16] S. Müller, L. Scardia, C. I. Zeppieri, “Geometric rigidity for incompatible fields and an application to strain-gradient
plasticity”, Indiana Univ. Math. J. 63 (2014), no. 5, p. 1365-1396.
[17] E. M. Stein, Singular Integrals and differentiability properties of functions, Princeton University Press, 1970.
[18] J. Van Schaftingen, “Estimates for L1-vector fields”, C. R. Math. Acad. Sci. Paris 339 (2004), no. 3, p. 181-186.
C. R. Mathématique, 2021, 359, n 2, 155-160


Sharp rigidity estimates for incompatible fields as a consequence of the Bourgain Brezis div-curl result

Mickaël Matusinski

Simon Müller

Comptes Rendus
Mathématique
Mickaël Matusinski and Simon Müller
Quantifier elimination for quasi-real closed fields
Volume 359, issue 3 (2021), p. 291-295.
<https://doi.org/10.5802/crmath.169>
© Académie des sciences, Paris and the authors, 2021.
Some rights reserved.
This article is licensed under the
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http://creativecommons.org/licenses/by/4.0/
Les Comptes Rendus. Mathématique sont membres du
Centre Mersenne pour l’édition scientifique ouverte
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Comptes Rendus
Mathématique
2021, 359, n 3, p. 291-295
https://doi.org/10.5802/crmath.169
Mathematical logic / Logique mathématique
Quantifier elimination for quasi-real closed
fields
Mickaël Matusinskia and SimonMüller∗, b
a Univ. Bordeaux, IMB, UMR 5251, F-33400 Talence, France
b Universität Konstanz, 78467 Konstanz, Germany
E-mails: mickael.matusinski@math.u-bordeaux.fr (M. Matusinski),
simon.2.mueller@uni-konstanz.de (S. Müller)
Abstract. We prove quantifier elimination for the theory of quasi-real closed fields with a compatible valua-
tion. This unifies the same known results for algebraically closed valued fields and real closed valued fields.
Résumé. Nous prouvons l’élimination des quantificateurs pour la théorie des corps quasi-réels clos munis
d’une valuation compatible. Cela reprend et unifie les mêmes résultats connus pour les corps algébriquement
clos et les corps réels clos.
2020 Mathematics Subject Classification. 03C10, 03C64, 12J10, 12J15, 12L12.
Manuscript received 26th May 2020, accepted 13th December 2020.
1. Introduction
Ordered fields and valued fields share several similar features, most notably via the notion of
(order-)compatible valuations (see e.g. [9, 10, 13] and below). In the present note we illustrate
this by subsuming the following major model theoretic results: the (first order) theories of
algebraically closed fields with a non-trivial valuation, respectively real closed fields with a
compatible non-trivial valuation, both admit quantifier elimination (cf. [11], respectively [3]).
We prove that the same applies if we consider the theory of the union of these two classes in a
common language, the one of quasi-ordered fields. Quasi-ordered fields were introduced in [7]
and provide a uniform axiomatisation of ordered and valued fields. We derive a notion of quasi-
real closed field, which subsumes the classes of algebraically closed fields and real closed fields.
Note that real closed fields were developed by Artin and Schreier in [1] as maximal ordered fields,
in analogy to the notion of algebraic closure of a field.
∗Corresponding author.
ISSN (electronic) : 1778-3569 https://comptes-rendus.academie-sciences.fr/mathematique/
292 Mickaël Matusinski and Simon Müller
2. Preliminaries on quasi-ordered fields
Throughout this note, all orderings and quasi-orderings are always assumed to be total. An
ordered field is a field K equipped with an ordering ≤ (i.e. a binary, reflexive, transitive and anti-
symmetric relation) such that the following axioms are satisfied for all x, y, z ∈ K :
(O1) x ≤ y ∧ 0 ≤ z ⇒ xz ≤ y z,
(O2) x ≤ y ⇒ x + z ≤ y + z.
Orderings are in one-to-one correspondence with so-called positive cones, i.e. subsets P ⊆ K
such that PP ⊆ P, P +P ⊆ P, P ∪ (−P ) = K and −1 ∉ P : 0 ≤ x ⇔ x ∈ P .
A valuation v on a field K (in the sense of Krull [8]) is known to endow K with a quasi-ordering,
i.e. a binary, reflexive, transitive relation ¹, as follows:
a ¹ b ⇐⇒ v(b) ≤ v(a).
If ¹ is a quasi-ordering, we write x ³ y as shorthand for x ¹ y ¹ x. Note that if ¹ is an ordering,
then ³ is just equality, whereas if ¹ is induced by a valuation v as above, then x ³ y if and only if
v(x) = v(y). Moreover, we write x ≺ y as shorthand for x ¹ y and x 6³ y.
Pushing the analogy of ordered and valued fields further, Fakhruddin introduced the notion
of quasi-ordered field, that is a field K equipped with a quasi-ordering ¹ such that the following
axioms are satisfied for all x, y, z ∈ K :
(Q0) x ³ 0 ⇒ x = 0,
(Q1) x ¹ y ∧ 0 ¹ z ⇒ xz ¹ y z,
(Q2) x ¹ y ∧ z 6³ y ⇒ x + z ¹ y + z.
Evidently, quasi-ordered fields arise from ordered fields by replacing anti-symmetry with the
weaker axiom (Q0), and by weakening the compatibility with addition. The condition z 6³ y in
(Q2) relates to the fact that v(y) 6= v(z) implies v(y+z) = min{v(y), v(z)} for any valuation v on K .
Theorem 1 (Fakhruddin’s dichotomy [7, Theorem 2.1]). Let K be a field and ¹ a binary relation
on K . Then (K ,¹) is a quasi-ordered field if and only if it is either an ordered field or there is a
valuation v on K such that x ¹ y ⇔ v(y) ≤ v(x) for all x, y ∈ K .
Note that ¹ is induced by a valuation v on K if and only if 0 ≺ −1. In that case v is unique up
to equivalence of valuations, and we denote the quasi-ordering ¹ also by ¹v .
Remark 2. In [5], a locality on a field is, by definition, either an ordering or a valuation. Hence,
by Fakhruddin’s dichotomy, localities and quasi-orderings are, in fact, equivalent notions.
Baer and Krull exhibited another strong relation between orderings and valuations on fields [2,
8]: any ordering ≤ on K induces a valuation on K , the so-called natural valuation of (K ,≤). It is
the valuation on K whose valuation ring is the convex hull of Z in K , i.e. the smallest convex
subring of K w.r.t. ≤ . An ordering ≤ and a valuation v on K are called compatible, if Kv is convex.
We also say that v, respectively Kv , is convex. In [9], the authors generalised this notion to quasi-
orderings: given a quasi-ordered field (K ,¹), a valuation v on K – or equivalently a quasi-ordering
¹v – is said to be compatible with ¹, if the valuation ring Kv is ¹-convex, or equivalently, if
0 ¹ x ¹ y ⇒ x ¹v y for all x, y ∈ K . Note that if¹ is induced by a valuation w, then¹v is compatible
with ¹w if and only if v is a coarsening of w. For a further discussion on this subject we refer the
interested reader to [9] and [10].
We conclude this section by recalling some basic facts about real closed fields. An ordered field
(K ,≤) is said to be real closed if one of the following equivalent conditions is satisfied:
(RC1) ≤ does not extend to any proper algebraic extension of K .
(RC2) (K ,≤) satisfies the following conditions:
(i) any x ∈ K with 0 ≤ x has a square root in K ,
C. R. Mathématique — 2021, 359, n 3, 291-295
Mickaël Matusinski and Simon Müller 293
(ii) any polynomial f ∈ K [X ] of odd degree has a root in K .
Any ordered field (K ,≤) admits a real closure, i.e. an algebraic field extension which is real
closed. It is unique up to isomorphism of ordered fields (cf. [1, Satz 8]).
3. Quasi-real closed fields
In the present section we introduce quasi-real closed fields, that way unifying algebraically closed
fields and real closed fields. To this end, we impose (RC1) on quasi-ordered fields.
Definition 3. A quasi-ordered field (K ,¹) is called quasi-real closed, if ¹ does not extend to any
proper algebraic extension of K .
Lemma 4. Let K be a field. The following are equivalent:
(1) K is algebraically closed,
(2) (K ,¹v ) is quasi-real closed for some valuation v on K ,
(3) (K ,¹v ) is quasi-real closed for any valuation v on K .
Proof. If K is algebraically closed, then K admits no proper algebraic extension, whence (K , v)
is quasi-real closed for any valuation v on K . Consequently, (1) implies (3). Moreover, since any
field admits a valuation, clearly (3) implies (2). Finally, suppose that (2) holds. Note that v extends
to any field extension L of K by an immediate consequence of Chevalley’s Extension Theorem [6,
Theorems 3.1.1 and 3.1.2]. Since (K , v) is quasi-real closed, this means that K does not admit any
proper algebraic extension, i.e. that K is algebraically closed. Thus, (2) implies (1). 
Theorem 5. The following are equivalent for any quasi-ordered field (K ,¹) :
(a) (K ,¹) is quasi-real closed.
(b) (QRC1) either ¹ is an ordering and (K ,¹) is a real closed field, or ¹ is induced by a
valuation and K is an algebraically closed field.
(c) (QRC2) (K ,¹) satisfies the following conditions:
(i) any x ∈ K with 0 ¹ x has a square root in K .
(ii) any polynomial f ∈ K [X ] of odd degree has a root in K .
Proof. The equivalence of (a) and (b) is an immediate consequence of Fakhruddin’s dichotomy
(Theorem 1) and (RC1) (if ¹ is an ordering), respectively Lemma 4 (if ¹ is induced by a valuation).
It remains to show that (b) and (c) are equivalent. If ¹ is an ordering, then this follows from
(RC2). So let ¹ be induced by some valuation. If (K ,¹) is quasi-real closed, then K is algebraically
closed according to Lemma 4, whence (c) is fulfilled. Conversely, suppose that (c) holds. Since
K = {0 ¹ x}, we obtain by (i) that all elements in K have a square root in K . Along with condition (ii)
this implies that K is algebraically closed according to [12, Theorem 2]. Hence, (K ,¹) is quasi-real
closed again by Lemma 4. 
Definition 6. Let (K ,¹) be a quasi-ordered field. A quasi-ordered field (L,¹′) is called a quasi-real
closure of (K ,¹), if
(1) (L,¹′) is a quasi-real closed field,
(2) L|K is an algebraic field extension,
(3) ¹′ is an extension of ¹, i.e. K ∩¹′=¹ .
Proposition 7. Any quasi-ordered field (K ,¹) admits a quasi-real closure. It is unique up to
isomorphism of quasi-ordered fields.
Proof. Let us consider an algebraic closure K̃ of K and the set{
(L,¹′) : (L,¹′) is a quasi-ordered field, K ⊆ L ⊆ K̃ , K∩¹′=¹} ,
C. R. Mathématique — 2021, 359, n 3, 291-295
294 Mickaël Matusinski and Simon Müller
partially ordered by
(L1,¹′1) ≤ (L2,¹′2) ⇐⇒ L1 ⊆ L2 and L1∩¹′2 =¹′1 .
By Zorn’s lemma, there is a maximal quasi-ordered algebraic field extension (L,¹′) of (K ,¹).
Hence, (L,¹′) is quasi-real closed and a quasi-real closure of (K ,¹).
The proof of the uniqueness relies on Fakhruddin’s dichotomy and (QRC1). So let (K1,¹1) and
(K2,¹2) be quasi-real closures of (K ,¹). If ¹ is an ordering, then (K1,¹1) and (K2,¹2) are real
closures of (K ,¹), whence they are order-isomorphic according to [1, Satz 8].
Likewise, if ¹ is induced by some valuation on K , then K1 and K2 are algebraic closures of K .
Hence, they are isomorphic as fields. Let ¹1=¹v1 and ¹2=¹v2 . Then the corresponding valuation
rings Kv1 and Kv2 are conjugated by an element of Gal(K̃ |K ) (cf. [6, Conjugation Theorem 3.2.15]).
By composition, we obtain an isomorphism between (K1,¹1) and (K2,¹2). 
4. The theory of quasi-real closed fields and quantifier elimination
We conclude this note by proving that the theory of quasi-real closed fields with a non-trivial
compatible valuation admits quantifier elimination. Moreover, we deduce that this theory is the
model companion of the theory of quasi-ordered fields with a non-trivial compatible valuation.
To this end, we exploit Fakhruddin’s dichotomy and the fact that the theories of real closed valued
fields and algebraically closed valued fields both admit quantifier elimination.
Theorem 8 (cf. [4, Theorems 4.4.2 and 4.5.1]).
(1) The theory of algebraically closed fields with a non-trivial valuation v admits quantifier
elimination in the language of fields adjoined with ¹v .
(2) The theory of real closed fields with a non-trivial compatible valuation v admits quantifier
elimination in the language of ordered fields adjoined with ¹v .
The characterization (QRC2) from Theorem 5 allows us to formalise quasi-real closed fields in
a first order language. As language of quasi-ordered fields with a compatible valuation v , we fix
L = {+, ·,−,¹,¹v ,0,1}. Moreover, we denote by ΣQRC the following set of L -sentences:
(1) the axioms stating that (K ,+, ·,−,0,1) is a field.
(2) the axioms stating that (K ,¹) is a quasi-real closed field.
(3) the axioms stating that (K ,¹v ) is a quasi-ordered field with v non trivial.
(4) the following L -sentences that determine the relationship of ¹ and ¹v :
(i) φ1 ≡ 0 ≺−1 → (∀ x, y : 0 ¹ x ¹ y ↔ x ¹v y)
(ii) φ2 ≡−1 ≺ 0 → (∀ x, y : 0 ¹ x ¹ y → x ¹v y)
The L -sentences φ1 and φ2, respectively, state that ¹=¹v if ¹ is induced by a valuation, and
that v is ¹-compatible if ¹ is an ordering. That way, we have unified the languages of real closed
valued fields and algebraically closed valued fields that are used in Theorem 8. We refer to the
theory ofΣQRC as the theory of quasi-real closed fields with a non-trivial compatible valuation. We
also consider the theory of quasi-ordered fields with a non-trivial compatible valuation, denoted
by ΣQO, which consists of (1), (3), (4), and the axioms (Q0), (Q1) and (Q2) (see Section 2).
Theorem 9. The theory of ΣQRC admits quantifier elimination.
Proof. Any model of ΣQRC is either an algebraically closed field with a non-trivial valuation or a
real closed field with a non-trivial compatible valuation (Theorem 5). Therefore, the result follows
from the fact that each of these two classes admits quantifier elimination (Theorem 8). 
Definition 10. Let Σ ⊆ Sent(L ) for a given language L . A set Σ∗ ⊆ Sent(L ) is called a model
companion of Σ, if
(1) every model of Σ∗ is a model of Σ,
C. R. Mathématique — 2021, 359, n 3, 291-295
Mickaël Matusinski and Simon Müller 295
(2) every model of Σ can be extended to a model of Σ∗,
(3) Σ∗ is model complete.
Corollary 11 (cf. [4, Corollaries 4.4.3 and 4.5.4]). The theory of ΣQRC is model complete. It is the
model companion of the theory of ΣQO.
Proof. Property 10(1) is obviously satisfied. Model completeness follows immediately from the
quantifier elimination that we obtained in Theorem 9. It remains to show that condition (2) of
Definition 10 holds in our setting.
So let (K ,¹,¹v ) be some quasi-ordered field with a non-trivial compatible valuation, and let
(L,¹′) be a quasi-real closure of (K ,¹) (Proposition 7). Then (L,¹′) is a quasi-real closed field.
Moreover, L|K is algebraic, whence [5, Theorem 20.1.1(a)] yields a unique extension ¹′ν of ¹v
from K to L such that ν is compatible with ¹′ . If ¹ is a valuation (i.e. ¹v=¹), then the uniqueness
tells us that also ¹′ν=¹′ . Furthermore, ν is non-trivial since v is non-trivial. Hence, (L,¹′,¹′ν) is
an extension of (K ,¹,¹v ) and a model of the theory of quasi-real closed fields with a non-trivial
compatible valuation. 
Remark 12. The theory of ΣQRC is not complete. For example the L -sentence ∃ x : x2 +1 = 0 is
false for any real closed field, but true for any algebraically closed field. Alternatively, this follows
from the fact that for algebraically closed fields the characteristic is not fixed.
Open Question. Can we replace the L -sentencesφ1 andφ2 inΣQRC with the single L -sentence
φ≡∀ x, y : x ¹ y → x ¹v y? This is equivalent to the question, whether algebraically closed fields
admit quantifier elimination in the language of fields adjoined by ¹v and ¹w , where v and w are
valuations on K such that v is coarser than w .
Acknowledgment
The authors wish to thank Salma Kuhlmann for initiating and supporting their collaboration, and
also for helpful discussion on the subject. Without her, this paper would not have come to exist.
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C. R. Mathématique — 2021, 359, n 3, 291-295


Quantifier elimination for quasi-real closed fields

Mètolidji Moquilas Raymond Affossogbe

Guy Martial Nkiet

Carlos Ogouyandjou

Comptes Rendus
Mathématique
Mètolidji Moquilas Raymond Affossogbe, Guy Martial Nkiet
and Carlos Ogouyandjou
Dimension reduction in spatial regression with kernel SAVE method
Volume 359, issue 4 (2021), p. 475-479
<https://doi.org/10.5802/crmath.187>
© Académie des sciences, Paris and the authors, 2021.
Some rights reserved.
This article is licensed under the
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Comptes Rendus
Mathématique
2021, 359, n 4, p. 475-479
https://doi.org/10.5802/crmath.187
Statistics / Statistiques
Dimension reduction in spatial regression
with kernel SAVE method
Réduction de la dimension en régression spatiale avec la
méthode SAVE à noyau
Mètolidji Moquilas Raymond Affossogbea, GuyMartial Nkiet∗, b
and Carlos Ogouyandjoua
a Institut de Mathématiques et de Sciences Physiques,Porto Novo, Bénin
b Université des Sciences et Techniques de Masuku, Franceville, Gabon
E-mails: metolodji.affossogbe@imsp-uac.org, guymartial.nkiet@univ-masuku.org,
ogouyandjou@imsp-uac.org
Abstract. We consider the smoothed version of sliced average variance estimation (SAVE) dimension reduc-
tion method for dealing with spatially dependent data that are observations of a strongly mixing random field.
We propose kernel estimators for the interest matrix and the effective dimension reduction (EDR) space, and
show their consistency.
Résumé. Nous considérons la version lisse de la méthode SAVE pour prendre en compte des observations
spatialement dépendantes émanant d’un champ aléatoire fortement mélangeant. Nous proposons des esti-
mateurs à noyau pour la matrice d’intérêt et l’espace de rédution de la dimension, et montrons leur conver-
gence.
Manuscript received 22nd September 2019, accepted 7th February 2021.
Version française abrégée
Soit le modèle semi-paramétrique de régression (1) ; on cherche à estimer l’espace de réduction
de la dimension, obtenu par la méthode SAVE, sur la base d’observations sur le domaine In = {i =
(i1, i2, . . . , iL) ∈ ZL ,1 ≤ ik ≤ nk ,k = 1,2, . . . ,L} d’un champ aléatoire {Wi, i ∈ (N∗)L}, où Wi = (Zi,Yi)
a la même loi que (Z ,Y ), avec Z = Σ−1/2(X −E(X )), la matrice Σ étant la matrice de covariance
de X supposée inversible. La méthode SAVE étant basée sur l’analyse spectrale de la matrice Γ
définie en (2), l’estimation recherchée est obtenue à partir de l’analyse spectrale de l’estimateur
à noyau Γ̂n de Γ défini en (3). Posant n̂ = n1 × n2 · · · × nL et n = (n1, . . . ,nL) ∈ (N∗)L , on écrit
n → +∞ lorsque min{ni , i = 1,2, . . . ,L} → +∞. Soit une suite (bn) de réels strictement positifs
∗Corresponding author.
ISSN (electronic) : 1778-3569 https://comptes-rendus.academie-sciences.fr/mathematique/
476 Mètolidji Moquilas Raymond Affossogbe, label2 and Carlos Ogouyandjou
tendant vers 0 lorsque n → +∞, constituant la fenêtre de l’estimateur à noyau précédent, on
pose φn = bkn + 1bn
√
log n̂
n̂ , et on obtient :
Théorème 8. Sous les hypothèses 2–6 avec χ(t ) = O(t−θ), t > 0, θ > 2L et n̂b3n(log n̂)−1 → 0,
n̂bθ1n (log n̂)
−1 →+∞ où θ1 = 4L+θθ−2L , on a :
Γ̂n −Γ=Op
(
1p
n̂
)
+Op
(
φn
en
)
+Op
(
φ2n
e2n
)
+Op
(
φ3n
e3n
)
+Op
(
φ4n
e4n
)
+Op
(
bkn +
φ2n
e2n
)
.
Corollaire 9. Sous les hypothèses 2–7 avec χ(t ) = O(t−θ), t > 0, θ > 2L et n̂b3n(log n̂)−1 → 0,
n̂bθ1n (log n̂)
−1 →+∞ où θ1 = 4L+θθ−2L , on a Γ̂n −Γ=Op
(
n̂−1/2
)
.
Supposons que τ1,τ2, . . . ,τN sont des vecteurs propres orthonormaux de Γ associés aux va-
leurs propres λ1, . . . ,λN respectivement, telles que λ1 >λ2 > ·· · >λN > 0. Soient τ̂1, τ̂2, . . . , τ̂N des
vecteurs propres orthonormaux de Γ̂n associés aux valeurs propres λ̂1, . . . , λ̂N respectivement,
telles que λ̂1 > λ̂2 > ·· · > λ̂N > 0. Pour tout j ∈ {1, . . . , N }, on aβ j =Σ−1/2τ j et on pose β̂ j = Σ̂−1/2n τ̂ j ,
où Σ̂n = 1n̂
∑
i∈In
(
Xi −X
)(
Xi −X
)T et X = 1n̂ ∑i∈In Xi.
Corollaire 10. Sous les hypothèses du Théorème 8, on a pour tout j ∈ {1, . . . , N }, ‖β̂ j −β j ‖ = op (1).
1. Introduction
Let us consider the semiparametric regression model introduced by Li [8] and defined as
Y = g (βT1 X ,βT2 X , . . . ,βTN X ,ε), (1)
where Y (resp. X ) is a random variable with values in R (resp. Rd , d ≥ 2), N is an integer
such that N < d , the parameters β1,β2, . . . ,βN are d-dimensional linearly independent vectors,
ε is a random variable that is independent of X , and g is an arbitrary unkown function. The
estimation of the space spanned by the βk ’s, called the effective dimension reduction (EDR)
space, is a crucial issue for achieving reduction dimension. For this problem, Li [8] introduced
the Sliced Inverse Regression (SIR) method whereas an alternative method, called sliced average
variance estimation (SAVE), that is more comprehensive since it uses first and second moments
was proposed in [4]. Smoothed versions of these methods, based on kernel estimators, have
been proposed later in [13] and [14]. Recently, nonparametric statistical methods have evolved
with the existence of spatially dependent data. So, kernel nonparametric estimation of the
spatial regression function have been studied (cf. [1, 2, 6, 7, 10, 11]). For dimension reduction
in spatial context, Loubes and Yao [9] investigated the kernel SIR method under strong mixing
conditions. In this note, we study the case of kernel SAVE, which had never been done before.
In Section 2, we introduce a kernel estimate of SAVE based on spatially dependent observations.
Then, assumptions and consistency results are given in Section 3. An outline of the proofs of
theorems is postponed in Section 4.
2. Kernel estimation of SAVE based on spatial data
In all of the paper, we assume that E
(‖X ‖2) < +∞, where ‖·‖ is the usual Euclidean norm of Rd ,
and that the covariance matrix Σ of X is invertible. Putting Z = Σ−1/2(X −E(X )) and denoting by
Cov(Z |Y ) the conditional covariance matrix of Z conditionally to Y , it is shown in [4] that the
EDR space is fully obtained from the spectral analysis of the matrix
Γ := E[(Id −Cov(Z |Y ))2] = Id −2E(C (Y ))+E
[
C (Y )2
]
, (2)
C. R. Mathématique — 2021, 359, n 4, 475-479
Mètolidji Moquilas Raymond Affossogbe, label2 and Carlos Ogouyandjou 477
where Id is the d ×d identity matrix and C (Y ) := Cov(Z |Y ) = R(Y )− r (Y )r (Y )T , where R(Y ) =
E(Z Z T |Y ) and r (Y ) = E(Z |Y ). From the variance decomposition theorem, we have that Id =
E(C (Y ))+Ψ, where Ψ = Cov(E(r (Y )) = E[r (Y )r (Y )T ]. Therefore, Γ = −Id + 2Ψ+Λ, where Λ =
E[C (Y )2], and the estimation of Γ boils down to that of the matrices Ψ and Λ. From now on,
we assume that Y admits a density such that f (y) > 0 for all y ∈ R. Let us consider a stationary
random field {Wi, i ∈ (N∗)L} where Wi = (Zi,Yi) has the same distribution than (Z ,Y ). We suppose
that this process is observed on a region In = {i = (i1, i2, . . . , iL) ∈ ZL ,1 ≤ ik ≤ nk ,k = 1,2, . . . ,L},
where n = (n1, . . . ,nL) ∈ (N∗)L . We put n̂ = n1 × n2 · · · × nL and write n → +∞ if min{ni , i =
1,2, . . . ,L} →+∞. For defining our estimators, we consider a sequence (bn) of strictly positive real
numbers converging to zero as n →+∞, and a kernel function K defined on R. An estimator of f
is then given by f̂en (y) = max{en, f̂n(y)}, where (en) is a sequence of strictly positive real numbers
such that limn→+∞ en = 0, and
f̂n(y) = 1n̂bn
∑
i∈In
K
(
y −Yi
bn
)
.
Then, we consider
m̂n(y) = 1n̂bn
∑
i∈In
K
(
y −Yi
bn
)
Zi, M̂n(y) = 1n̂bn
∑
i∈In
K
(
y −Yi
bn
)
Zi Z
T
i ,
r̂n(y) = m̂n(y)
f̂en (y)
, R̂n(y) = M̂n(y)
f̂en (y)
,
and we take as estimator of Γ the random matrix
Γ̂n =−Id +2Ψ̂n + Λ̂n, (3)
where
Ψ̂n = 1n̂
∑
i∈In
r̂n(Yi)r̂n(Yi)
T −Z Z T , Λ̂n = 1n̂
∑
i∈In
Ĉn(Yi)
2
with Z = 1n̂
∑
i∈In Zi and Ĉn(Yi) = R̂n(Yi)− r̂n(Yi)r̂n(Yi)T .
3. Assumptions and asymptotic results
In order to establish the asymptotic results, the following assumptions will be considered.
Assumption 1. Γ is a positive-definite matrix.
Assumption 2. The kernel K is a density function with compact support, is of order k (where
k ≥ 3) and satisfies ∫ |u|k K (u)du = 1 and |K (x)−K (y)| ≤C |x − y | for some C > 0.
Assumption 3. The functions f , r and R belong to C k (R) and supy∈R | f (k)(y)|, supy∈R ‖m(k)(y)‖
and supy∈R ‖M (k)(y)‖ are bounded, where m(y) = f (y)r (y) and M(y) = f (y)R(y).
Assumption 4.
p
n̂E[‖R(Y )‖21{ f (Y )≤en}] = o(1),
p
n̂E[‖r (Y )‖41{ f (Y )≤en}] = o(1) and
p
n̂E[‖R(Y )‖×
‖r (Y )‖21{ f (Y )≤en}] = o(1).
Assumption 5. ‖Z‖ ≤ D , where D is a strictly positive constant.
Assumption 6. The process {Wi, i ∈ (Z∗)L} is strongly mixing, i.e. there exists a function χ from
R+ to itself satisfying χ(t ) ↓ 0 as t →+∞, such that for all subsets S and S′ of (Z∗)L ,
α(B(S),B(S′)) := sup
A∈B(S),B∈B(S′)
|P(A∩B)−P(A)P(B)| ≤χ(δ(S,S′))
where B(S) (resp. B(S′)) denotes the Borel σ-fields generated by {Wi, i ∈ S} (resp. {Wi, i ∈ S′}) and
δ(S,S′) denotes the Euclidean distance between S and S′.
C. R. Mathématique — 2021, 359, n 4, 475-479
478 Mètolidji Moquilas Raymond Affossogbe, label2 and Carlos Ogouyandjou
Assumption 7. bn ∼ n̂−c1 and en ∼ n̂−c2 , where c1 and c2 are real numbers satisfying c1 > 0,
0 < c2 < 2k−14(2k+1) and c2k + 14k < c1 < 12 −2c2.
Putting φn = bkn + 1bn
√
log n̂
n̂ , we have:
Theorem 8. Under Assumptions 2–6, if χ(t ) = O(t−θ), t > 0, θ > 2L and n̂b3n(log n̂)−1 → 0,
n̂bθ1n (log n̂)
−1 →+∞ with θ1 = 4L+θθ−2L , then we have:
Γ̂n −Γ=Op
(
1p
n̂
)
+Op
(
φn
en
)
+Op
(
φ2n
e2n
)
+Op
(
φ3n
e3n
)
+Op
(
φ4n
e4n
)
+Op
(
bkn +
φ2n
e2n
)
.
Corollary 9. Under Assumptions 2–7, if χ(t ) = O(t−θ), t > 0, θ > 2L and n̂b3n(log n̂)−1 → 0,
n̂bθ1n (log n̂)
−1 →+∞ with θ1 = 4L+θθ−2L , then we have Γ̂n −Γ=Op
(
n̂−1/2
)
.
For dealing with the β̂ j ’s we assume that τ1,τ2, . . . ,τN are orthonormal eigenvectors of Γ asso-
ciated with eigenvalues λ1, . . . ,λN respectively, such that λ1 > λ2 > ·· · > λN > 0. Let τ̂1, τ̂2, . . . , τ̂N
be orthonormal eigenvectors of Γ̂n associated with the eigenvalues λ̂1, . . . , λ̂N respectively, such
that λ̂1 > λ̂2 > ·· · > λ̂N > 0. For j ∈ {1, . . . , N }, we have β j = Σ−1/2τ j and we put β̂ j = Σ̂−1/2n τ̂ j ,
where Σ̂n = 1n̂
∑
i∈In
(
Xi −X
)(
Xi −X
)T and X = 1n̂ ∑i∈In Xi. Then, we have:
Corollary 10. Under Assumptions 1–6, if χ(t ) = O(t−θ), t > 0, θ > 2L and n̂b3n(log n̂)−1 → 0,
n̂bθ1n (log n̂)
−1 →+∞ with θ1 = 4L+θθ−2L , then we have for any j ∈ {1, . . . , N }, ‖β̂ j −β j ‖ = op (1).
4. Outline of proofs
We first have to prove that, under the assumptions of Theorem 8, we have supy∈R ‖M̂n(y) −
M(y)‖ = Op
(
φn
)
. For doing that, we consider the inequality supy∈R ‖M̂n(y) − M(y)‖ ≤
supy∈R |M̂n(y) − E[M̂n(y)]‖ + supy∈R ‖E[M̂n(y)] − M(y)‖. From the equality E[M̂n(y)] − M(y) =∫
K (v)
[
M(y − vbn)−M(y)
]
dv we obtain supy∈R ‖E[M̂n(y)]− M(y)‖ = Op (bkn) by using a Taylor
expansion of M and the properties of K given in Assumption 2. For dealing with the other term,
we consider a real ε > 0 and a sequence (an) of non-negative real numbers converging to +∞,
and we obtain
P
(
sup
y∈R
∥∥M̂n(y)−E[M̂n(y)]∥∥> ε;‖Z Z T ‖ ≤ an
)
≤P(Sn > ε),
where Sn = ∑i∈InΘi,n with Θi,n = C2n̂bn {‖Zi Z Ti ‖ + E[‖Zi Z Ti ‖]}1{‖Zi Z Ti ‖≤an}, C2 being a constant
that bound K . Then, using the spatial block decomposition of [12] together with Lemma 3.6
in [3], Markov inequality and Bernstein inequality, and taking ε = εn = 1bn
√
log n̂
n̂ and an =
(log n̂)1/4 we show that P(Sn > ε) → 0 and P(‖Z Z T ‖ > an) → 0 as n → +∞. Hence
P
(
supy∈R
∥∥M̂n(y)−E[M̂n(y)]∥∥> εn)→ 0 as n →+∞, that is supy∈R∥∥M̂n(y)−E[M̂n(y)]∥∥=Op (εn).
We have Γ̂n − Γ = 2(Σ̂n −Σ) + (Λ̂n −Λ); it is shown in [9] that Σ̂n −Σ = Op
(
bkn + φ
2
n
e2n
)
, then for
proving Theorem 8 it remains to treat the second term. This is done from the decomposition
Λ̂n −Λ = A1n + A2n + A3n, where A1n = 1n̂
∑
i∈In C (Yi)
2 −E[Cov(Z |Y )2], A2n = 1n̂ ∑i∈In Ce,n(Yi)2 −
1
n̂
∑
i∈In C (Yi)
2 and A3n = 1n̂
∑
i∈In Ĉn(Yi)
2 − 1n̂
∑
i∈In Ce,n(Yi)
2. Using Lemma 6.3 of [9], we show
that A1n = Op (1/n̂) = Op
(
1p
n̂
)
. The second term is decomposed as A2n = A21n − A22n with
A21n = A211n + A212n, where
E
(p
n̂‖A211n‖
)
≤
p
n̂E
(‖R(Y )‖21{ f (Y )<en})+pn̂E(‖R(Y )‖ ‖r (Y )‖21{ f (Y )<en})
and
E
(p
n̂‖A22n‖
)
≤
p
n̂E
(‖R(Y )‖21{ f (Y )<en})
+2
p
n̂E
(‖R(Y )‖ ‖r (Y )‖21{ f (Y )<en})+pn̂E(‖r (Y )‖41{ f (Y )<en}) .
C. R. Mathématique — 2021, 359, n 4, 475-479
Mètolidji Moquilas Raymond Affossogbe, label2 and Carlos Ogouyandjou 479
From Assumption 4 and Markov inequality we deduce thatA21n = Op
(
1/
p
n̂
)
, A22n = Op
(
1/
p
n̂
)
,
and consequently that A2n =Op
(
1/
p
n̂
)
. The third term has a decomposition A3n = A31n−A32n+
A33n with A31n = A311n + A312n + A313n + A314n, where ‖A311n‖ ≤ 2D4en ‖ f − f̂n‖∞+
2D2
en
‖M − M̂n‖∞,
and
‖A312n‖ ≤ 2D4
‖ f − f̂n‖2∞
e2n
+2D2 ‖m −m̂n‖
2∞
e2n
+2D3 ‖m −m̂n‖∞ ‖ f − f̂n‖∞
e2n
.
Then from Lemma 6.5 in [9] and the above result ‖M̂n − M‖∞ = Op
(
φn
)
, we obtain A311n =
Op
(
φn
en
)
and A312n = Op
(
φ2n
e2n
)
. Similar arguments lead to A313n = Op
(
φn
en
)
and A314n = Op
(
φn
en
)
,
and we conclude that A31n = Op (φnen )+Op (
φ2n
e2n
). It can be noticed that ‖A33n‖ = ‖A31n‖. Thus,
A33n = Op (φnen )+Op (
φ2n
e2n
). Similarly, we also obtain A32n = Op (φnen )+Op (
φ2n
e2n
)+Op (φ
3
n
e3n
)+Op (φ
4
n
e4n
).
We can then conclude that A3n = Op (φnen )+Op (
φ2n
e2n
)+Op (φ
3
n
e3n
)+Op (φ
4
n
e4n
). Corollary 3.1 is obtained
from the fact that n̂
1/2
en
φn ∼ n̂−1/2+c2+c1
√
log n̂. Since −1/2+ c2 + c1 < 0, we obtain that n̂1/2en φn → 0
as n →+∞. Then, φnen = Op (n̂−1/2) = op (1) and, consequently,
φ`n
e`n
= Op
(
φn
en
)
= Op (n̂−1/2) for ` =
2,3,4. For proving Corollary 3.2, we write Σ̂n −Σ= 1n̂
∑
i∈In Vi −
(
X −E(X )
)
X T −E(X )
(
X −E(X )
)T
,
where Vi := Xi X Ti −E(X )E(X )T . Then using 6.3 of [9], we obtain 1n̂
∑
i∈In Vi = Op (1/n̂) = op (1) and
X − E(X ) = Op (1/n̂) = op (1). Therefore, Σ̂n −Σ = Op (1/n̂) = op (1) and ‖Σ̂−1/2n −Σ−1/2‖ = op (1).
Applying Lemma 1 of [5] and Theorem 8 permits to obtain t ‖τ̂ j −τ j ‖ = op (1) for j = 1,2, . . . , N .
Finally, from ‖β̂ j −β j ‖ ≤ ‖Σ̂−1/2n −Σ−1/2‖ ‖τ̂ j ‖+‖Σ−1/2‖ ‖τ̂ j −τ j ‖ we conclude that ‖β̂ j −β j ‖ =
op (1).
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C. R. Mathématique — 2021, 359, n 4, 475-479


Dimension reduction in spatial regression with kernel SAVE method

Mourad Choulli

Some stability inequalities for hybrid inverse problems

Şebnem Yıldız

Comptes Rendus
Mathématique
Şebnem Yıldız
A new extension on the theorem of Bor
Volume 359, issue 5 (2021), p. 555-562
<https://doi.org/10.5802/crmath.195>
© Académie des sciences, Paris and the authors, 2021.
Some rights reserved.
This article is licensed under the
Creative Commons Attribution 4.0 International License.
http://creativecommons.org/licenses/by/4.0/
Les Comptes Rendus. Mathématique sont membres du
Centre Mersenne pour l’édition scientifique ouverte
www.centre-mersenne.org
Comptes Rendus
Mathématique
2021, 359, n 5, p. 555-562
https://doi.org/10.5802/crmath.195
Numerical analysis / Analyse numérique
A new extension on the theorem of Bor
Şebnem Yıldıza
a Department of Mathematics, Ahi Evran University, Kırşehir, Turkey
E-mail: sebnemyildiz@ahievran.edu.tr
Abstract. In [8], Bor has obtained a main theorem dealing with Riesz summability factors of infinite series and
Fourier series. In this paper, we generalized that theorem to |A,θn |k summability method for taking power
increasing sequence. Also some new and known results are obtained dealing with some basic summability
methods.
2020 Mathematics Subject Classification. 26D15, 42A24, 40F05, 40G99.
Manuscript received 10th October 2020, revised 26th November 2020, accepted 1st March 2021.
1. Introduction
Let
∑
an be a given infinite series with partial sums (sn). By uαn and t
α
n we denote the nth Cesàro
means of order α, with α>−1, of the sequence (sn) and (nan), respectively, that is (see [9])
uαn =
1
Aαn
n∑
v=0
Aα−1n−v sv and t
α
n =
1
Aαn
n∑
v=1
Aα−1n−v vav , (1)
where
Aαn =
(α+1)(α+2) . . . (α+n)
n!
=O(nα), Aα−n = 0 for n > 0. (2)
The series
∑
an is said to be summable |C ,α|k , k ≥ 1, if (see [11, 13])
∞∑
n=1
nk−1|uαn −uαn−1|k =
∞∑
n=1
1
n
|tαn |k <∞. (3)
If we take α= 1, then |C ,α|k summability reduces to |C ,1|k summability.
Let (pn) be a sequence of positive real numbers such that
Pn =
n∑
v=0
pv →∞ as n →∞, (P−i = p−i = 0, i ≥ 1). (4)
The sequence-to-sequence transformation
wn = 1
Pn
n∑
v=0
pv sv , Pn 6= 0. (5)
defines the sequence (wn) of the Riesz mean or simply the (N , pn) mean of the sequence (sn)
generated by the sequence of coefficients (pn) (see [12]).
ISSN (electronic) : 1778-3569 https://comptes-rendus.academie-sciences.fr/mathematique/
556 Şebnem Yıldız
Let (θn) be any sequence of positive constants. The series
∑
an is said to be summable
|N , pn ;θn |k , k ≥ 1, if (see [18])
∞∑
n=1
θk−1n |wn −wn−1|k <∞. (6)
In the special case if we take θn = Pn/pn , then |N , pn ;θn |k summability reduces to |N , pn |k
summability (see [2]). When θn = n and pn = 1 for all values of n, then we get |C ,1|k summability.
Furthermore, if we take θn = n, then |N , pn ;θn |k summability reduces to |R, pn |k summability
(see [3]).
For any sequence (λn) we write that
∆2λn =∆λn −∆λn+1 and ∆λn =λn −λn+1.
A sequence (λn) is said to be of bounded variation, denoted by (λn) ∈BV , if
∞∑
n=1
|∆λn | <∞.
Let A = (anv ) be a normal matrix, i.e., a lower triangular matrix of nonzero diagonal entries.
Then A defines the sequence-to-sequence transformation, mapping the sequence s = (sn) to
As = (An(s)), where
An(s) =
n∑
v=0
anv sv , n = 0,1, . . . (7)
Let (θn) be any sequence of positive real numbers. The series
∑
an is said to be summable |A,θn |k ,
k ≥ 1, if (see [16, 17])
∞∑
n=1
θk−1n |∆An(s)|k <∞, (8)
where
∆An(s) = An(s)− An−1(s). (9)
If we take anv = pvPn , then |A,θn |k summability reduces to |N , pn ;θn |k summability. If we take
θn = Pnpn , then |A,θn |k summability reduces to |A, pn |k summability (see [19]). And also if we take
θn = Pnpn and anv =
pv
Pn
, then |A,θn |k summability reduces to |N , pn |k summability. Furthermore,
if we take θn = n, anv = pvPn and pn = 1 for all values of n, then |A,θn |k summability reduces to
|C ,1|k summability (see [11]). Finally, if we take θn = n and anv = pvPn , then |A,θn |k summability
reduces to |R, pn |k summability (see [3]).
Definition 1 (cf. [1]). A positive sequence (bn) is said to be an almost increasing sequence if
there exists a positive increasing sequence (cn) and two positive constants M and N such that
Mcn ≤ bn ≤ N cn .
Definition 2 (cf. [20]). A positive sequence X = (Xn) is said to be quasi- f -power increasing
sequence if there exists a constant K = K (X , f ) ≥ 1 such that K fn Xn ≥ fm Xm for all n ≥ m ≥ 1,
where f = { fn(σ,β)} = {nσ(l og n)β,β≥ 0,0 <σ< 1}.
If we take β = 0, then we have a quasi-σ-power increasing sequence (see [15]). Every almost
increasing sequence is a quasi-σ-power increasing sequence for any non-negative σ, but the
converse is not true for σ> 0.
C. R. Mathématique — 2021, 359, n 5, 555-562
Şebnem Yıldız 557
2. The Known Results
Recently, many papers have been done for absolute matrix summability factors of infinite series
and Fourier series (see [5–7, 14, 22, 23]). From these, in [14], Lee explained history of summability
of infinite series and Hüseyin Bor briefly. Now we also used Bor’s new theorem dealing with the
Fourier series and we will extend following theorem.
Theorem 3 (cf. [8]). Let (θn pn/Pn) be a non-increasing sequence. Let (pn) be a sequence of positive
numbers such that
Pn =O(npn) as n →∞. (10)
Let (Xn) be a positive increasing sequence. If the conditions
λn = o(1) as n →∞, (11)
m∑
n=1
nXn |∆2λn | =O(1) as m →∞, (12)
m∑
n=1
θk−1n
(
pn
Pn
)k |tn |k
X k−1n
=O(Xm) as m →∞, (13)
are satisfied, then the series
∑
anλn is summable |N , pn ,θn |k , k ≥ 1.
If we take θn = Pn/pn , then we get a theorem dealing with |N , pn |k summability (see [6]).
3. The Main Result
Given a normal matrix A = (anv ), we associate two lower semimatrices A = (anv ) and Â = (ânv )
as follows:
anv =
n∑
i=v
ani , n, v = 0,1, . . . (14)
and
â00 = a00 = a00, ânv = anv −an−1,v , n = 1,2, . . . (15)
It may be noted that A and Â are the well-known matrices of series-to-sequence and series-to-
series transformations, respectively. Then, we have
An(s) =
n∑
v=0
anv sv =
n∑
v=0
anv av (16)
and
∆An(s) =
n∑
v=0
ânv av . (17)
By using above notations, we generalize Theorem 3 for |A,θn |k summability method by taking
(Xn) as a quasi- f - power increasing sequence.
Theorem 4. Let k ≥ 1 and A = (anv ) be a positive normal matrix such that
an0 = 1, n = 0,1, . . . , (18)
an−1,v ≥ anv , for n ≥ v +1, (19)
1 =O(nann), (20)
n−1∑
v=1
av v |ân,v+1| =O(ann). (21)
C. R. Mathématique — 2021, 359, n 5, 555-562
558 Şebnem Yıldız
Let (θn ann) be a non-increasing sequence and (Xn) be a quasi- f -power increasing sequence for
some σ (0 < σ < 1). If the conditions (11)–(12) of Theorem 3 and (θn) holds for the following
condition,
m∑
n=1
θk−1n a
k
nn
|tn |k
X k−1n
=O(Xm) as m →∞, (22)
are satisfied, then the series
∑
anλn is summable |A,θn |k , k ≥ 1.
We need the following lemmas for the proof of Theorem 4.
Lemma 5 (cf. [21]). By using conditions (14), (15), (18) and (19), we have
n−1∑
v=1
|∆v (ânv )| ≤ ann , (23)
m+1∑
n=v+1
|∆v (ânv )| ≤ av v , (24)
m+1∑
n=v+1
|ân,v+1| =O(1). (25)
Lemma 6 (cf. [4]). Under the conditions of Theorem 3 we have the following
nXn |∆λn | =O(1) as n →∞, (26)
∞∑
n=1
Xn |∆λn | <∞, (27)
Xn |λn | =O(1) as n →∞. (28)
Proof of Theorem 4. Let (In) denotes the A-transform of the series
∑∞
n=1 anλn . Then, by (16) and
(17), we have
∆In =
n∑
v=1
ânv avλv .
Applying Abel’s transformation to this sum, we have that
∆In =
n∑
v=1
ânv avλv
v
v
=
n−1∑
v=1
∆
(
ânvλv
v
) v∑
r=1
r ar + ânnλn
n
n∑
v=1
vav
=
n−1∑
v=1
∆
(
ânvλv
v
)
(v +1)tv + ânnλn n +1
n
tn
=
n−1∑
v=1
∆v (ânv )λv tv
v +1
v
+
n−1∑
v=1
ân,v+1∆λv tv
v +1
v
+
n−1∑
v=1
ân,v+1λv+1
tv
v
+annλn tn n +1
n
= In,1 + In,2 + In,3 + In,4.
To complete the proof of Theorem 4, by Minkowski’s inequality, it is sufficient to show that
∞∑
n=1
θk−1n |In,r |k <∞, for r = 1,2,3,4. (29)
C. R. Mathématique — 2021, 359, n 5, 555-562
Şebnem Yıldız 559
First, by applying Hölder’s inequality with indices k and k ′, where k > 1 and 1k + 1k ′ = 1, we have
that
m+1∑
n=2
θk−1n |In,1|k ≤
m+1∑
n=2
θk−1n
{
n−1∑
v=1
∣∣∣∣ v +1v
∣∣∣∣ |∆v (ânv )||λv ||tv |
}k
=O(1)
m+1∑
n=2
θk−1n
{
n−1∑
v=1
|∆v (ânv )||λv |k |tv |k
}
×
{
n−1∑
v=1
|∆v (ânv )|
}k−1
=O(1)
m+1∑
n=2
(θn ann)
k−1
{
n−1∑
v=1
|∆v (ânv )||λv |k |tv |k
}
=O(1)
m∑
v=1
|λv |k−1|λv ||tv |k
m+1∑
n=v+1
(θn ann)
k−1|∆v (ânv )|
=O(1)
m∑
v=1
(θv av v )
k−1 1
X k−1v
|λv ||tv |k
m+1∑
n=v+1
|∆v (ânv )|
=O(1)
m∑
v=1
(θv av v )
k−1 1
X k−1v
|λv ||tv |k av v
=O(1)
m−1∑
v=1
∆|λv |
v∑
r=1
θk−1r a
k
r r
|tr |k
X k−1r
+O(1)|λm |
m∑
v=1
θk−1v a
k
v v
|tv |k
X k−1v
=O(1)
m−1∑
v=1
|∆λv |Xv +O(1)|λm |Xm
=O(1) as m →∞,
by virtue of the hypotheses of Theorem 4, Lemma 5, and Lemma 6. Now using Hölder’s inequality,
we have that
m+1∑
n=2
θk−1n |In,2|k ≤
m+1∑
n=2
θk−1n
{
n−1∑
v=1
|v +1
v
||ân,v+1||∆λv ||tv |
}k
=O(1)
m+1∑
n=2
θk−1n
{
n−1∑
v=1
|ân,v+1||∆λv ||tv |
}k
=O(1)
m+1∑
n=2
θk−1n
{
n−1∑
v=1
a1−kv v |ân,v+1||∆λv |k |tv |k
}
×
{
n−1∑
v=1
av v |ân,v+1|
}k−1
=O(1)
m+1∑
n=2
(θn ann)
k−1 n−1∑
v=1
|∆λv |k a1−kv v |ân,v+1||tv |k
=O(1)
m∑
v=1
|tv |k a1−kv v |∆λv |k
m+1∑
n=v+1
(θn ann)
k−1|ân,v+1|
=O(1)
m∑
v=1
(θv av v )
k−1|tv |k a1−kv v |∆λv |k
m+1∑
n=v+1
|ân,v+1|
=O(1)
m∑
v=1
(θv av v )
k−1|tv |k a1−kv v |∆λv |k
=O(1)
m∑
v=1
θk−1v a
k
v v |tv |k (v |∆λv |)k−1(v |∆λv |)
=O(1)
m∑
v=1
θk−1v a
k
v v
1
X k−1v
|tv |k (v |∆λv |)
C. R. Mathématique — 2021, 359, n 5, 555-562
560 Şebnem Yıldız
=O(1)
m−1∑
v=1
∆(v |∆λv |)
v∑
r=1
θk−1r a
k
r r
1
X k−1r
|tr |k +O(1)m|∆λm |
m∑
v=1
θk−1v a
k
v v
1
X k−1v
|tv |k
=O(1)
m−1∑
v=1
|∆(v |∆λv |)|Xv +O(1)m|∆λm |Xm
=O(1)
m−1∑
v=1
v Xv |∆2λv |+O(1)
m−1∑
v=1
Xv |∆λv |+O(1)m|∆λm |Xm
=O(1) as m →∞,
by virtue of the hypotheses of Theorem 4, Lemma 5, and Lemma 6. Again, as in In,1, we have that
m+1∑
n=2
θk−1n |In,3|k =
m+1∑
n=2
θk−1n
∣∣∣∣∣n−1∑v=1 ân,v+1λv+1 tvv
∣∣∣∣∣
k
≤
m+1∑
n=2
θk−1n
{
n−1∑
v=1
|ân,v+1||λv+1| |tv |
v
}k
=O(1)
m+1∑
n=2
θk−1n
{
n−1∑
v=1
a1−kv v
1
vk
|ân,v+1||λv+1|k |tv |k
}
×
{
n−1∑
v=1
av v |ân,v+1|
}k−1
=O(1)
m+1∑
n=2
(θn ann)
k−1 n−1∑
v=1
a1−kv v
1
vk
|ân,v+1||λv+1|k |tv |k
=O(1)
m∑
v=1
av v |λv+1|k |tv |k
m+1∑
n=v+1
(θn ann)
k−1|ân,v+1|
=O(1)
m∑
v=1
(θv av v )
k−1av v |λv+1|k |tv |k
m+1∑
n=v+1
|ân,v+1|
=O(1)
m∑
v=1
θk−1v a
k
v v |λv+1|k |tv |k
=O(1)
m∑
v=1
θk−1v a
k
v v |tv |k |λv+1|k−1|λv+1|
=O(1)
m∑
v=1
θk−1v a
k
v v
1
X k−1v
|λv+1||tv |k
=O(1) as m →∞,
by virtue of the hypotheses of Theorem 4, Lemma 5, and Lemma 6. Finally, as in In,1, we have that
m∑
n=1
θk−1n |In,4|k =O(1)
m∑
n=1
θk−1n a
k
nn |λn |k |tn |k =O(1)
m∑
n=1
θk−1n a
k
nn |λn |k−1|λn ||tn |k
=O(1)
m∑
n=1
θk−1n a
k
nn
1
X k−1n
|λn ||tn |k =O(1) as m →∞,
by virtue of hypotheses of Theorem 4, Lemma 5, and Lemma 6. This completes the proof of
Theorem 4. 
4. An application of absolute matrix summability to Fourier series
Let f be a periodic function with period 2π and integrable (L) over (−π,π). The trigonometric
Fourier series of f is defined as
f (x) ∼ 1
2
a0 +
∞∑
n=1
(an cosnx +bn sinnx) =
∞∑
n=0
Cn(x). (30)
C. R. Mathématique — 2021, 359, n 5, 555-562
Şebnem Yıldız 561
Set
φ(t ) = 1
2
{ f (x + t )+ f (x − t )}, (31)
φα(t ) = α
tα
∫ t
0
(t −u)α−1φ(u)du, (α> 0). (32)
It is well known that if φ1(t ) ∈ BV (0,π), then tn(x) = O(1), where tn(x) is the (C ,1) mean of the
sequence (nCn(x)) (see [10]).
The following theorem is known dealing with |N , pn ,θn |k summability factors of Fourier series.
Theorem 7 (cf. [8]). Let
( θn pn
Pn
)
be a non-increasing sequence. Ifφ1(t ) ∈BV (0,π) and the sequences
(pn), (λn), and (Xn) satisfy the conditions of Theorem 3,then the series
∑
Cn(x)λn is summable
|N , pn ,θn |k , k ≥ 1.
Now, we generalize Theorem 7 for |A,θn |k summability method in the following form.
Theorem 8. Let (θn ann) be a non-increasing sequence, and A be a positive normal matrix as in
Theorem 4, and (Xn) be a quasi- f -power increasing sequence for some σ (0 < σ < 1). If φ1(t ) ∈
BV (0,π), and the sequences (pn), (λn), and (Xn) satisfy the conditions of Theorem 4, then the series∑
Cn(x)λn is summable |A,θn |k , k ≥ 1.
It should be noted that if we take (Xn) as a positive increasing sequence and anv = pvPn in
Theorem 8, then we have Theorem 7.
Applications.
(1) If we write
∑n
v=0 pv /Pv , then (Xn) is a positive increasing sequence tending to infinity as
n → ∞. In this case, if we take (Xn) is a positive increasing sequence and anv = pvPn in
Theorem 4, then we have Theorem 3.
(2) If we take θn = n, anv = pvPn and pn = 1 for all values of n in Theorem 4, then we have a
new result concerning |C ,1|k summability method.
(3) If we take θn = n and anv = pvPn in Theorem 4, then we get a new result dealing with |R, pn |k
summability method.
(4) If we take β = 0 and anv = pvPn in Theorem 4, then we have new theorem dealing with
quasi-σ-power increasing sequence.
(5) If we take β = 0 and anv = pvPn in Theorem 8, then we have new theorem dealing with
quasi-σ-power increasing sequence and Fourier series.
Acknowledgments
The author wishes her sincerest thanks to the referee for invaluable suggestions for the improve-
ment of this paper.
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C. R. Mathématique — 2021, 359, n 5, 555-562


A new extension on the theorem of Bor

Trong Nguyen Ngoc

Truong Le Xuan

Do Tan Duc

Boundedness of second-order Riesz transforms on weighted Hardy and $BMO$ spaces associated with Schrödinger operators

Jeffrey D. Carlson

Comptes Rendus
Mathématique
Jeffrey D. Carlson
The K-theory of the conjugation action
Volume 359, issue 7 (2021), p. 795-796
<https://doi.org/10.5802/crmath.235>
© Académie des sciences, Paris and the authors, 2021.
Some rights reserved.
This article is licensed under the
Creative Commons Attribution 4.0 International License.
http://creativecommons.org/licenses/by/4.0/
Les Comptes Rendus. Mathématique sont membres du
Centre Mersenne pour l’édition scientifique ouverte
www.centre-mersenne.org
Comptes Rendus
Mathématique
2021, 359, n 7, p. 795-796
https://doi.org/10.5802/crmath.235
Geometry and Topology / Géométrie et Topologie
The K-theory of the conjugation action
Jeffrey D. Carlsona
a Department of Mathematics, Imperial College London, 180 Queen’s Gate, London
SW7 2AZ, UK
E-mail: j.carlson@imperial.ac.uk
Abstract. In 1999, Brylinski and Zhang computed the complex equivariant K-theory of the conjugation self-
action of a compact, connected Lie group with torsion-free fundamental group. In this note we show it is
possible to do so in under a page.
Manuscript received 7th April 2021, revised and accepted 9th June 2021.
Brylinski and Zhang [2] showed that if G is a compact, connected Lie group with torsion-free
fundamental group, then the equivariant K-theory of its conjugation action GAd is isomorphic to
the ring Ω∗RG/Z of Grothendieck differentials on the complex representation ring RG of G . Their
proof uses results on holomorphic differentials on complex manifolds, a reduction to the case G
is a torus, and some algebraic geometry. We show a more concrete and arguably more natural
expression for the ring K ∗G (G
Ad) can be obtained rapidly using only Hodgkin’s Künneth spectral
sequence [5], in the same manner they already use it, and elementary algebraic considerations.
We then show this purely algebraic isomorphism admits a satisfying geometric interpretation,
and remark finally that this geometric version gives back Brylinski and Zhang’s description in
terms of Grothendieck differentials at no added cost.
Theorem (Brylinski–Zhang [2, Thm. 3.2]). Let G be a compact, connected Lie group with torsion-
free fundamental group. Then K ∗G (G
Ad) is isomorphic to RG ⊗K ∗G as an RG-algebra. Under
this identification, the forgetful map f : K ∗G (G
Ad) −→ K ∗G becomes reduction with respect to the
augmentation ideal IG of RG.
Proof. Write Gbi for G under the (G ×G)-action (h,k) · g = hg k−1. The orbit space of Gbi ×Gbi
under the restricted, free diagonal action of 1×G is (G × 1)-equivariantly diffeomorphic to GAd
via (g ′, g ) 7−→ g ′g−1, so when X = Y = Gbi, Hodgkin’s (Z×Z/2)-graded Künneth spectral se-
quence TorR(G×G)(K ∗G×G X ,K
∗
G×G Y ) =⇒ K ∗G×G (X ×Y ) reduces to TorRG ⊗RG (RG ,RG) =⇒ K ∗G (GAd).
Here the two structure maps RG ⊗RG −→ RG are the both the multiplication of RG . Recall [5,
Prop. 11.1] that under our hypotheses, RG is the tensor product of a polynomial ring on gener-
ators yi ∈ IG and a Laurent polynomial ring on generators t j ∈ 1+ IG . Let P be the free abelian
group on generators qi and w j and let γ : P −→ IG be the linear map taking qi to yi and w j to
t j −1. Then an (RG)⊗2-module resolution of RG is given by RG ⊗ΛP ⊗RG , with differential the
derivation vanishing on RG ⊗Z⊗RG and sending 1⊗z⊗1, for z ∈ P , to 1⊗1⊗γ(z)−γ(z)⊗1⊗1.
To compute the Tor, apply −⊗RG ⊗RG RG to this resolution to obtain the cdga RG ⊗ΛP with 0
differential, RG in bidegree (0,0), and P in bidegree (−1,0).
ISSN (electronic) : 1778-3569 https://comptes-rendus.academie-sciences.fr/mathematique/
796 Jeffrey D. Carlson
The spectral sequence collapses because the differentials dr for r ≥ 2 send all generators into
the right half-plane. Since π1G is torsion-free and X = G is locally contractible of finite covering
dimension, the spectral sequence strongly converges to the intended target. Hence RG ⊗ΛP is the
graded algebra associated to a filtration (Fp )p≤0 of K ∗G (G
Ad) with F0 ∼= RG and F−1/F0 ∼= RG ⊗P .
Since RG and ΛP are both free abelian, there is no additive extension problem, so K ∗G (G
Ad)
is also free abelian as a group. Let z̃k be elements in F−1 lifting 1⊗qi and 1⊗w j under the
isomorphism F−1/F0 ∼= RG ⊗P . Then the z̃k anticommute with each other because they lie in
K 1G (G
Ad) and square to 0 since K ∗G (G
Ad) contains no 2-torsion, and by induction, they generate
K ∗G (G
Ad) as an RG-algebra, so K ∗G (G
Ad) = RG ⊗Λ[z̃k ].
To see the forgetful map f is as claimed, note that forgetting the (G × 1)-action on Gbi
induces a map to the spectral sequence TorRG (Z,Z) =⇒ K ∗G , which again collapses by lacunary
considerations. Computing TorRG (Z,Z) ∼= ΛP with the resolution ΛP ⊗RG of Z shows the map
E2( f ) : RG ⊗ΛP −→ΛP is reduction modulo IG andΛP ∼= K ∗G . 
Remark. We can be completely explicit about the exterior generators. As observed by
Hodgkin [4, Thm. A], the injection U(n) U := lim−−→U(n) induces an additive map β : RG −→ K
1G
descending to a group isomorphism between the module IG/(IG)2 of indecomposables of RG
and the module PK ∗G of primitives of the exterior Hopf algebra K ∗G ∼=ΛPK ∗G . In particular, a
set of generators is given by β(λi ) = β(λi −dimλi ) for λi lifts in G of the fundamental represen-
tations of the commutator subgroup G ′ and β(t j ) = β(t j − 1) for t j : G → G/G ′ U(1) circular
coordinate functions of the torus Gab =G/G ′ ∼= (S1)rkG−rkG ′ . Let Q = {λi , t j }i , j .
The map β in fact factors as f ◦βAd for a map βAd : RG −→ K ∗G (GAd), already giving surjectivity
of f since PK ∗G generates K ∗G = ΛPK ∗G as a ring. Atiyah [1, Lem. 2, pf.] described βAd and
hence β geometrically: given a representation ρ : G −→ U(n), we can build a representative E of
β(ρ) via the clutching construction, taking with two trivial bundles CG ×Cn over the cone CG
on G and gluing them along G ×Cn via the relation (g , v) ∼ (g ,ρ(g )v) to obtain a bundle over
the suspension CG ∪G CG . The action h · (g , v) =
(
hg h−1,ρ(h)v
)
of G on GAd ×Cn preserves this
relation and so induces a G-action on E making it a G-equivariant bundle over the suspension
of GAd.
The RG-module structure on K ∗G (X ) is always given by σ · [E ] =
[
α(σ)⊗E], where if σ : G −→
AutV is a representation, then α(σ) is the trivial bundle X × V equipped with the diagonal
G-action. Thus σ⊗∏k β(ρk ) 7−→ α(σ) · ∏k βAd(ρk ) for ρk ∈ Q gives an explicit isomorphism
RG ⊗K ∗G ∼−→ K ∗G (GAd).
Remark. The first paragraph of our proof is a variant of Brylinski–Zhang’s §4 [2]. Once the ring
structure is determined as in our proof’s second paragraph, replacing §§5–6, one knows their
mapφ : Ω∗RG/Z −→ K ∗G (GAd) from the ring of Grothendieck differentials is an isomorphism as soon
as one knows it is a well-defined RG-algebra map [2, Prop 3.1], for the class βAd(ρ) ∈ K 1G (GAd)
from the previous remark is in fact the same as1 Brylinski–Zhang’s φ(dρ), so that f ◦φ takes a
basis {dρ : ρ ∈Q} of the free RG-moduleΩ1RG/Z to a Z-basis
{
β(ρ) : ρ ∈Q} of PK ∗G .
References
[1] M. F. Atiyah, “On the K -theory of compact Lie groups”, Topology 4 (1965), no. 1, p. 95-99.
[2] J.-L. Brylinski, B. Zhang, “Equivariant K -theory of compact connected Lie groups”, K -Theory 20 (2000), no. 1, p. 23-36.
[3] C.-K. Fok, “The Real K -theory of compact Lie groups”, SIGMA, Symmetry Integrability Geom. Methods Appl. 10 (2014),
article no. 022 (26 pages).
[4] L. Hodgkin, “On the K -theory of Lie groups”, Topology 6 (1967), no. 1, p. 1-36.
[5] ——— , “The equivariant Künneth theorem in K -theory”, in Topics in K -theory, Lecture Notes in Mathematics, vol.
496, Springer, 1975, p. 1-101.
1 the corrected version of—see Fok [3, Rmk. 2.8.1]
C. R. Mathématique — 2021, 359, n 7, 795-796


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