Comptes Rendus Mathématique

Tatsien Li

Bopeng Rao

Uniqueness theorem for a coupled system of wave equations with incomplete internal observation and application to approximate controllability

Sahiba Arora

Jochen Glück

Comptes Rendus
Mathématique
Sahiba Arora and Jochen Glück
Stability of (eventually) positive semigroups on spaces of continuous
functions
Volume 360 (2022), p. 771-775
<https://doi.org/10.5802/crmath.323>
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
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Comptes Rendus
Mathématique
2022, Vol. 360, p. 771-775
https://doi.org/10.5802/crmath.323
Functional analysis, Operator theory / Analyse fonctionnelle, Théorie des opérateurs
Stability of (eventually) positive semigroups
on spaces of continuous functions
Sahiba Aroraa and Jochen Glückb
a Technische Universität Dresden, Institut für Analysis, Fakultät für Mathematik,
01062 Dresden, Germany
b Universität Passau, Fakultät für Informatik und Mathematik, 94032 Passau, Germany
E-mails: sahiba.arora@mailbox.tu-dresden.de, jochen.glueck@uni-passau.de
Abstract. We present a new and very short proof of the fact that, for positive C0-semigroups on spaces of
continuous functions, the spectral and the growth bound coincide. Our argument, inspired by an idea of
Vogt, makes the role of the underlying space completely transparent and also works if the space does not
contain the constant functions – a situation in which all earlier proofs become technically quite involved.
We also show how the argument can be adapted to yield the same result for semigroups that are only
eventually positive rather than positive.
Mathematical subject classification (2010). 47D06, 47B65, 47A10.
Funding. The first named author was supported by Deutscher Akademischer Austauschdienst (Forschungs-
stipendium-Promotion in Deutschland).
Manuscript received 15 December 2021, accepted 4 January 2022.
Positive semigroups
A classical question in the theory of positive C0-semigroups is whether, on a function space E ,
the spectral bound
s(A) := sup{Reλ : λ is a spectral value of A} ∈ [−∞,∞)
of (e t A)t≥0 coincides with its growth bound
ω0(A) := inf
{
ω ∈R : (e t (A−ω))t≥0 is bounded} ∈ [−∞,∞).
Recall that the inequality s(A) ≤ ω0(A) is true for all C0-semigroups, and the question for which
semigroups equality holds is relevant in order to derive the long term behaviour of a semigroup
from spectral properties of their generator.
The answer to the above question is positive on spaces of continuous functions, but the proofs
that are known for this result (for instance, in [10, Theorem B-IV-1.4] or [2, Theorems 5.3.8]) tend
to be technical in case that the space does not contain the constant functions (a situation which
occurs most prominently for the space C0(L) of continuous functions that vanish at infinity, on a
locally compact Hausdorff space L). Specifically, the proofs rely on the structure of the dual space
ISSN (electronic) : 1778-3569 https://comptes-rendus.academie-sciences.fr/mathematique/
772 Sahiba Arora and Jochen Glück
E ′ and the corresponding semigroup (e t A
′
)t≥0 which, in general, is not C0. Because of this reason,
the proofs employ the so-called sun-dual semigroup.
The first purpose of our paper is to provide a short and transparent proof of this classical result.
Theorem 1. Let (e t A)t≥0 be a positive C0-semigroup on an AM-space E. Then s(A) =ω0(A).
We have used the following terminology in the theorem: an AM-space E is a Banach lattice
with the additional property ‖sup{ f , g }‖ = sup{‖ f ‖,‖g‖} for all 0 ≤ f , g ∈ E . Typical spaces of
continuous functions, such as C (K ) for a compact space K or C0(L) for a locally compact space
L, are AM-spaces. For the proof of Theorem 1, we only need that every non-empty relatively
compact subset of an AM-space E has a supremum in E [13, Proposition II.7.6].
Proof of Theorem 1. By a rescaling argument, it suffices to show that (e t A)t≥0 is bounded when-
ever s(A) < 0. Indeed, if this implication is true, then the rescaled semigroup (e t (A−s))t≥0 is
bounded for all s > s(A). Consequently, ω0(A) ≤ s for all s > s(A) which yields ω0(A) ≤ s(A).
So suppose that s(A) < 0 and let 0 ≤ f ∈ E be fixed. Since the positive cone spans E , we only
need to show that the orbit of f is bounded. Boundedness of the semigroup will then follow from
the uniform boundedness theorem.
By the property of AM-spaces mentioned before the proof, the compact set {e t A f : t ∈ [0,1]}
has a supremum g ≥ 0 in E .
Fix a time t ≥ 1 and let I = [t −1, t ]. Then e t A f = e s Ae(t−s)A f ≤ e s A g for all s ∈ I , thus
0 ≤ e t A f =
∫
I
e t A f ds ≤
∫
I
e s A g ds ≤
∫ ∞
0
e s A g ds =: g̃ ∈ E ;
here the last integral exists as an improper Riemann integral due to the positivity of the C0-
semigroup and the assumption s(A) < 0 (see, for instance, [2, Theorem 5.3.1 and Proposi-
tion 5.1.4]). The result now follows using ‖e t A f ‖ ≤ ‖g̃‖ for all t ≥ 1. 
The main idea that we used above is an adaptation of an argument of Vogt, who recently
presented an intriguingly easy proof of s(A) = ω0(A) on Lp -spaces [14] (while earlier proofs
due to Weis were much more involved [15, 16]). For more historical details on the question
whether s(A) =ω0(A) for positive semigroups, we refer for instance to [2, p. 389]. Recently, some
conditions were given in [11, Section 4] which ensure that the above equality holds for positive
semigroups on non-commutative Lp -spaces. For a deeper connection between this question and
geometric properties of Banach spaces, see [12, Section 5].
Remark 2. The proof of Theorem 1 cannot directly be generalized to a larger class of Banach lat-
tices since AM-spaces are the only Banach lattices in which every non-empty relatively compact
set has a supremum (see, for instance, [17, p. 275]).
However, the argument directly generalizes to ordered Banach spaces with closed, normal, and
generating cones, under the assumption that every relatively compact subset is order bounded;
conditions for this latter property can be found in [17, Theorem 1]. For this reason, our argument
also yields a new and simple proof of [4, Theorem 4]. In particular, Theorem 1 is true on every
ordered Banach space which contains an order unit and has normal cone.
Remarks 3. Our proof of Theorem 1 uses two ingredients:
(a) The fact that every relatively compact set in an AM-space has a supremum.
For concrete functions spaces such as C0(L), where L is locally compact, this is not
difficult to show by employing the Arzelà–Ascoli theorem (and for general AM-spaces,
the proof is essentially the same). For C (K ), where K is compact, the situation is even
simpler since every norm bounded set in C (K ) is order bounded, which (as mentioned in
Remark 2) suffices for the proof of Theorem 1.
Sahiba Arora and Jochen Glück 773
(b) The fact that, if s(A) < 0 and g ∈ E , then ∫ ∞0 e s A g ds converges in E as an improper
Riemann integral.
This non-trivial result from the theory of positive semigroups is true on all Banach
lattices (and even on ordered Banach spaces with a normal generating cone); see the
above-quoted [2, Theorem 5.3.1 and Proposition 5.1.4]. Throughout the literature, proofs
that s(A) = ω0(A) holds for all positive semigroups on particular classes of function
spaces, require this result as an important ingredient. The same is true for our proof;
our main contribution is that we significantly simplify the rest of the argument, which is
the only place where the specific structure of the space is used.
Eventually positive semigroups
While the theory of positive semigroups can be considered a classical topic in analysis, many
evolution equations have been recently discovered to only exhibit eventually positive behaviour –
i.e., for positive initial values, the solution first changes sign but then becomes and stays positive.
This gives rise to the following notion:
A C0-semigroup (e t A)t≥0 on a Banach lattice E is said to be individually eventually positive if
for every positive initial datum f ∈ E , there exists t0 ≥ 0 such that e t A f is also positive for all t ≥ t0.
The semigroup is called uniformly eventually positive if t0 can be chosen to be independent of f .
Many examples of eventually positive semigroups occur in the study of concrete PDEs; see
for instance [9, Proposition 2 and Examples 6 and 7], [1, Section 3], [5, Proposition 5.5], and [8,
Section 7] for recently discovered examples. The prevalence of eventually positive semigroups
in concrete differential equations makes it a worthwhile goal to understand their behaviour at a
general and theoretical level, an endeavour which started with the papers [6, 7]; general results
about the spectrum and long-term behaviour of eventually positive semigroups have recently
been obtained in [3].
In the rest of this article, we specifically study whether s(A) =ω0(A) in the eventually positive
case. The property of positive C0-semigroups mentioned in Remark 3(b) surprisingly holds
even if the semigroup is only individually eventually positive; see [7, Proposition 7.1]. As a
consequence, it was shown in [7, Theorem 7.8] that on L1- and L2-spaces, as well as on C (K ),
the property s(A) = ω0(A) is true for individually eventually positive semigroups. On Lp for
other values of p, the aforementioned argument of Vogt carries over to uniformly eventually
positive semigroups [14]; the individually eventually positive case remains currently open on Lp
for p ∈ (1,∞)\{2}. In the following, we settle the question for all AM-spaces (while the proof given
for C (K ) in [7, Theorem 7.8] does not work on spaces not containing an order unit). The following
argument even works for individually eventually positive semigroups.
Theorem 4. Let (e t A)t≥0 be an individually eventually positive C0-semigroup on an AM-space E.
Then s(A) =ω0(A).
We point out that the argument given in [7, Theorem 7.8] for the special case E =C (K ) assumes
that the generator A is real, while we make no such assumption. Furthermore, as mentioned
before Theorem 1, the proofs known for positive semigroups on C0(L), for a locally compact
Hausdorff space L, employ the sun-dual semigroup on the dual space E ′. This works since the
closure of the domain dom A′ of the dual operator can be shown to be an ideal in E ′ when the
semigroup is positive. The proof does not carry over to eventually positive semigroups, and it is
unclear whether the same ideal property is still true for them.
On the other hand, the proof that we presented for Theorem 1 can be modified for the
eventually positive case, which is a major advantage of our approach. We are therefore able to
provide an affirmative answer to a question posed after [7, Theorem 7.8].
774 Sahiba Arora and Jochen Glück
Proof of Theorem 4. We adjust the proof of Theorem 1 to the present situation: Assume that
s(A) < 0 and fix a vector 0 ≤ f ∈ E ; again, it suffices to show that the orbit of f is bounded.
We can find a time t f ≥ 0 such that e t A f ≥ 0 for all t ≥ t f , owing to the individual positivity
assumption on the semigroup. As in the proof of the previous theorem, the supremum g :=
sup{e t A f : t ∈ [t f , t f +1]} exists, since E is an AM-space. In addition, the vector g is positive (being
a supremum of positive vectors), so there exists tg ≥ 0 such that e t A g ≥ 0 for all t ≥ tg .
Note that the set P := {g − e t A f : t ∈ [t f , t f +1]} is a compact subset of the positive cone in E ,
and due to the individual eventual positivity of the semigroup, it is covered by its closed subsets
Pn :=
{
h ∈ P : e t Ah ≥ 0 for all t ≥ n},
where n runs throughN. Hence, by the Baire category theorem, there exists N ∈N such that (PN )◦
is non-void; here, the interior is taken within the space P .
Let us now consider the mapΦ : [t f , t f +1] → P defined byΦ(t ) = g−e t A f for all t ∈ [t f , t f +1]. It
is both continuous and surjective, therefore the pre-image J :=Φ−1((PN )◦) is non-void and open
in [t f , t f +1]. Consequently, there exist t0 ≥ 0 and 0 < ` ≤ 1 such that [t0, t0 +`] ⊆ J ⊆ [t f , t f +1].
Letting t1 = max{N , tg }, we conclude that the vectors e s A
(
g − e t A f ) are positive for all s ≥ t1 and
all t ∈ [t0, t0 +`].
Finally, fix t ≥ t0 + t1 +` and let I := [t − t0 −`, t − t0] ⊆ [t1,∞). With this notation, we have
e t A f = e s Ae(t−s)A f ≤ e s A g for all s ∈ I , and so
0 ≤ e t A f = 1
`
∫
I
e t A f ds ≤ 1
`
∫
I
e s A g ds ≤ 1
`
∫ ∞
t1
e s A g ds =: g̃ ∈ E ;
here we used that the integral on the right exists as an improper Riemann integral since (e t A)t≥0 is
individually eventually positive and s(A) < 0 [7, Proposition 7.1]; moreover, for the first inequality
we used that t ≥ t0 ≥ t f , and for the last inequality we used t1 ≥ tg .
We conclude that ‖e t A f ‖ ≤ ‖g̃‖ for all t ≥ t0 + t1 + `, which proves that the orbit of f is
bounded. 
Acknowledgements
The authors are indebted to Hendrik Vogt for various fruitful discussions and for sharing an early
version of [14] with them. This paper was written during a pleasant visit of the first author to
Universität Passau.
References
[1] D. Addona, F. Gregorio, A. Rhandi, C. Tacelli, “Bi-Kolmogorov type operators and weighted Rellich’s inequalities”,
NoDEA, Nonlinear Differ. Equ. Appl. 29 (2022), article no. 13.
[2] W. Arendt, C. J. K. Batty, M. Hieber, F. Neubrander, Vector-valued Laplace transforms and Cauchy problems, 2 ed.,
Monographs in Mathematics, vol. 96, Birkhäuser, 2011, xii + 539 pages.
[3] S. Arora, J. Glück, “Spectrum and convergence of eventually positive operator semigroups”, Semigroup Forum 103
(2021), no. 3, p. 791-811.
[4] C. J. K. Batty, E. B. Davies, “Positive semigroups and resolvents”, J. Oper. Theory 10 (1983), p. 357-363.
[5] S. Becker, F. Gregorio, D. Mugnolo, “Schrödinger and polyharmonic operators on infinite graphs: Parabolic well-
posedness and p-independence of spectra”, J. Math. Anal. Appl. 495 (2021), no. 2, p. 124748.
[6] D. Daners, J. Glück, J. B. Kennedy, “Eventually and asymptotically positive semigroups on Banach lattices”, J. Differ.
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[7] ——— , “Eventually positive semigroups of linear operators”, J. Math. Anal. Appl. 433 (2016), no. 2, p. 1561-1593.
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Sahiba Arora and Jochen Glück 775
[10] R. Nagel (ed.), One-parameter semigroups of positive operators, Lecture Notes in Mathematics, vol. 1184, Springer,
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Stability of (eventually) positive semigroups on spaces of continuous functions

Giancarlo Lucchini Arteche

Comptes Rendus
Mathématique
Giancarlo Lucchini Arteche
On homogeneous spaces with finite anti-solvable stabilizers
Volume 360 (2022), p. 777-780
<https://doi.org/10.5802/crmath.339>
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
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Comptes Rendus
Mathématique
2022, Vol. 360, p. 777-780
https://doi.org/10.5802/crmath.339
Algebraic geometry / Géométrie algébrique
On homogeneous spaces with finite
anti-solvable stabilizers
Giancarlo Lucchini Artechea
a Departamento de Matemáticas, Facultad de Ciencias, Universidad de Chile, Las
Palmeras 3425, Ñuñoa, Santiago, Chile
E-mail: luco@uchile.cl
Abstract. We say that a group is anti-solvable if all of its composition factors are non-abelian. We consider
a particular family of anti-solvable finite groups containing the simple alternating groups for n 6= 6 and all
26 sporadic simple groups. We prove that, if K is a perfect field and X is a homogeneous space of a smooth
algebraic K -group G with finite geometric stabilizers lying in this family, then X is dominated by a G-torsor.
In particular, if G = SLn , all such homogeneous spaces have rational points.
Keywords. homogeneous spaces, rational points, non-abelian cohomology, finite simple groups.
2020 Mathematics Subject Classification. 14M17, 12G05, 14G05.
Funding. The author was partially supported by Fondecyt Grant 1210010.
Manuscript received 23 June 2021, revised 26 November 2021, accepted 26 January 2022.
We say that a finite group F is aut-split if the exact sequence
1 → Inn(F ) → Aut(F ) → Out(F ) → 1,
splits, where Inn(F ) denotes the group of inner automorphisms of F and Out(F ) is simply defined
as the quotient Aut(F )/Inn(F ). A full characterization of aut-split finite simple groups was given
in [7]. These include the alternating group An for n = 5 or n ≥ 7 and all 26 sporadic groups. As
for simple groups of Lie type, some notation is needed. If F is a finite simple group of Lie type
over Fq with q = pm and p prime, denote by d the order of the group of diagonal automorphisms
(i.e. those induced by conjugation by diagonal matrices) modulo inner automorphisms. Then F
is aut-split if and only if one of the following holds:
• F is a Chevalley group1, not of type D`(q), and
( q−1
d ,d ,m
)= 1;
• F = D`(q) and
( q`−1
d ,d ,m
)= 1;
• F is a twisted group, not of type 2D`(q), and
( q+1
d ,d ,m
)= 1;
• F = 2D`(q) and either ` is odd or p = 2.
1In the sense of finite simple groups, not of algebraic groups.
ISSN (electronic) : 1778-3569 https://comptes-rendus.academie-sciences.fr/mathematique/
778 Giancarlo Lucchini Arteche
Recall that, by the Jordan–Hölder Theorem (cf. [4, 3.4, Thm. 22]), every finite group admits
a composition series and its composition factors (which are simple groups) are unique up to
permutation. We say that F is anti-solvable if all of its composition factors are non-abelian. In this
note we are interested in anti-solvable groups whose composition factors are aut-split (i.e. they
belong to the list above). This includes of course the groups in the list themselves. The main result
of this note is the following.
Theorem 1. Let K be a perfect field and let G be a smooth algebraic K -group. Let X be a
homogeneous space of G with geometric stabilizer F̄ . Assume that F̄ is finite, anti-solvable, and
that all of its composition factors are aut-split. Then X is dominated by a G-torsor. In particular, if
G = SLn , then X has a rational point.
Of course, the particular case of G = SLn is a natural consequence of the triviality of torsors
under SLn , which is valid over any field K . Now, recall that the Springer class, which is class in
a non-abelian 2-cohomology set associated to every homogeneous space X , corresponds to the
obstruction to being dominated by a G-torsor (cf. [3, §2] or [5, §1], or [9] for the original article by
Springer). Using this tool we immediately see that Theorem 1 is implied by the following result
on non-abelian Galois cohomology.
Theorem 2. Let K be a perfect field and let L be a finite K -lien (or K -kernel, or K -band) whose
underlying group is anti-solvable and all of its composition factors are aut-split. Then H 2(K ,L)
has exactly one class and it is neutral.
As an anonymous referee pointed out, the proof of this statement uses nothing in particular
about Galois groups and works actually for profinite groups in general. In this context, it is better
to consider the point of view of extensions for nonabelian 2-cohomology (cf. for instance [10],
or [5, 1.18], or [9, 1.13–14], see also [3, §2.2] for a full comparison between classes of extensions,
of gerbes and of nonabelian 2-cocyles). Theorem 2 follows immediately then from the following
statement.
Theorem 3. Let Γ be a profinite group and let L be a finite Γ-lien (or Γ-kernel, or Γ-band) whose
underlying group F is anti-solvable and all of its composition factors are aut-split. Then H 2(Γ,L)
has exactly one class and it is neutral. In particular, every extension of profinite groups
1 → F → E → Γ→ 1,
(i.e. F is a closed, discrete normal subgroup of E and the quotient arrow is open) is split.
The rest of this note is devoted to proving Theorem 3. We start with the particular case where
the group is itself aut-split (for instance, if there is only one composition factor).
Proposition 4. Let Γ be a profinite group and let L be a finite Γ-lien whose underlying group is
aut-split. Then H 2(Γ,L) has a neutral class (in particular, the Γ-lien L is representable). If moreover
L has a trivial center, then this is the only class in H 2(Γ,L).
The idea of this proof comes from [2, Prop. 3.1], which actually follows ideas from Douai’s PhD
thesis.
Proof. Let F be the underlying finite group of L. By the definition of aΓ-lien we have a continuous
homomorphism κ : Γ→ Out(F ). Since F is aut-split, we can compose this with a section Out(F ) →
Aut(F ) and get a continuous group action ofΓon F . This defines a an extension ofΓby F by means
of the corresponding semi-direct product and, since this action lifts κ, it corresponds to a neutral
class η ∈ H 2(Γ,L). The second assertion follows immediately from [9, Prop. 1.17] (see also [8, IV,
Thm. 8.8]). 
Giancarlo Lucchini Arteche 779
Proof of Theorem 3. Let L be aΓ-lien and let F be its underlying group as in Theorem 3. Since F is
anti-solvable, it has trivial center and hence the Γ-lien L is representable (cf. [8, IV, Thm. 8.7]). We
fix then a class η ∈ H 2(Γ,L), which is unique again by the triviality of the center and [9, Prop. 1.17].
This class corresponds to an extension of Γ by F . We will prove that this extension is split, i.e. that
the class is neutral.
Let E be an extension of Γ by F . If F has no proper nontrivial characteristic subgroups (i.e. it is
characteristically simple), then F =Gn with G a finite simple group (cf. [6, Ch. 2, Thm. 1.4]), which
has to be aut-split then by hypothesis. Since every automorphism of such a product permutes its
direct factors, we see that Aut(F ) is isomorphic to Aut(G)noSn (with the obvious action of Sn) and
hence Out(F ) is isomorphic to Out(G)noSn . It is evident then that the section s : Out(G) → Aut(G)
and the obvious section t : Sn → Aut(F ) define a section (sn o t ) : Out(F ) → Aut(F ), proving that F
is aut-split as well. We are done then by Proposition 4.
Assume now that F has a proper nontrivial characteristic subgroup N . It is clear that both
N and F /N also satisfy the hypotheses of Theorem 3, so we argue by induction. Since N is
characteristic, the arrow Aut(F ) → Aut(F /N ) is well-defined and thus the homomorphism κ : Γ→
Out(F ) gives a homomorphism Γ→ Out(F /N ), which gives us a Γ-lien L′ whose underlying group
is F /N . By induction hypothesis, we know that H 2(Γ,L′) has exactly one class η′ and that it is
neutral. The class η′ corresponds to a split extension which, since the arrow H 2(K ,L) → H 2(K ,L′)
clearly sends η to η′, fits into a commutative diagram
1 // F //

E //
π

ΓK // 1
1 // F /N // E ′ // ΓK //
s
{{
1.
Consider the preimage π−1(s(ΓK )) ⊂ E , which fits into the following commutative diagram
1 // N //

π−1(s(ΓK )) //

ΓK // 1
1 // F // E // ΓK // 1.
The top row corresponds to a class in H 2(K ,L′′) for some K -lien L′′ with underlying group N . By
induction hypothesis again we know that this class is neutral and hence the extension is split,
giving a section ΓK → π−1(s(ΓK )) and hence a section ΓK → E by composition, proving that the
class η is neutral. 
Remark 5. In [1], Bercov characterizes anti-solvable finite groups whose composition factors are
aut-split as iterated twisted wreath products of simple groups. Note however that we did not need
this result in the proof above.
Remark 6. Theorems 1 and 2 are false if we take away the aut-split hypothesis on the compo-
sition factors, already for simple groups. Indeed, consider the group A6 and the order 2 element
φ ∈ Out(A6) coming from the nontrivial class of outer automorphisms of S6. Taking K =R, we can
define a K -lien L via the homomorphism κ : ΓR → Out(A6) that sends the nontrivial element in
ΓR to φ. By the same arguments used above we see that H 2(K ,L) has only one class η. But this
class is not neutral, since this would amount to the existence of a lift of φ to an automorphism
ϕ ∈ Aut(A6) of order 2, which does not exist. Using [3, Cor. 3.3] we may construct a homogeneous
space X of SLn over Rwhose Springer class is η. This implies that X is not dominated by a torsor.
Acknowledgements
The author would like to thank an anonymous referee for their useful comments.
780 Giancarlo Lucchini Arteche
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stabilisateurs finis”, Compos. Math. 155 (1900), no. 8, p. 1568-1593.
[4] D. S. Dummit, R. M. Foote, Abstract Algebra, 3rd ed., John Wiley & Sons, 2004.
[5] Y. Z. Flicker, C. Scheiderer, R. Sujatha, “Grothendieck’s theorem on non-abelian H2 and local-global principles”, J.
Am. Math. Soc. 11 (1998), no. 3, p. 731-750.
[6] D. Gorenstein, Finite Groups, 2nd ed., Chelsea Publishing, 1980.
[7] A. Lucchini, F. Menegazzo, M. Morigi, “On the existence of a complement for a finite simple group in its automor-
phism group”, Ill. J. Math. 47 (2003), no. 1-2, p. 395-418.
[8] S. Mac Lane, Homology, Classics in Mathematics, Springer, 1995, Reprint of the 1975 edition.
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On homogeneous spaces with finite anti-solvable stabilizers

Prasanna Kumar

Comptes Rendus
Mathématique
Prasanna Kumar
On the Erdős–Lax Inequality
Volume 360 (2022), p. 1081-1085
https://doi.org/10.5802/crmath.141
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
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e-ISSN : 1778-3569
Comptes Rendus
Mathématique
2022, Vol. 360, p. 1081-1085
https://doi.org/10.5802/crmath.141
Complex analysis and geometry / Analyse et géométrie complexes
On the Erdős–Lax Inequality
Prasanna Kumara
a Department of Mathematics, Birla Institute of Technology and Science Pilani, K K
Birla Goa Campus, Goa, India 403726
E-mail: prasannak@goa.bits-pilani.ac.in
Abstract. The Erdős–Lax Theorem states that if P (z) =∑nν=1 aνzν is a polynomial of degree n having no zeros
in |z| < 1, then
max|z|=1 |P
′(z)| ≤ n
2
max|z|=1 |P (z)|. (1)
In this paper, we prove a sharpening of the above inequality (1). In order to prove our result we prove a sharp-
ened form of the well-known Theorem of Laguerre on polynomials, which itself could be of independent
interest.
Mathematical subject classification (2010). 30A10.
Manuscript received 8 September 2020, revised 1 October 2020, accepted 28 October 2020.
1. Introduction
If P (z) is a polynomial of degree n, then the well-known Bernstein’s inequalities [2] on polynomi-
als are given by
max
|z|=1
|P ′(z)| ≤ n max
|z|=1
|P (z)| (2)
and
max
|z|=R
|P (z)| ≤ Rn max
|z|=1
|P (z)|, (3)
whenever R ≥ 1.
The inequality (2) is a direct consequence of Bernstein’s Theorem on the derivative of a
trigonometric polynomial [9] and the inequality (3) follows from the maximum modulus theorem
(see [8, Problem 269]). For the class of polynomials having no zeros inside the unit circle, it was
Erdős [3] who conjectured, and later proved by Lax [6] that, if P (z) is a polynomial of degree n
having no zeros in |z| < 1, then
max
|z|=1
|P ′(z)| ≤ n
2
max
|z|=1
|P (z)|. (4)
Equality holds in (4) if all zeros of P (z) lie on the circle |z| = 1.
Although, the above inequality (4) is best possible with equality holding for polynomials
P (z) = ∑nν=1 aνzν having no zeros in |z| < 1, satisfying |a0| = |an |, in particular for polynomials
having all its zeros on |z| = 1, it should be possible to improve upon the bound for polynomials
P (z) = ∑nν=1 aνzν having no zeros in |z| < 1, satisfying |a0| 6= |an |. Here we consider this problem
and in this regard prove the following.
ISSN (electronic) : 1778-3569 https://comptes-rendus.academie-sciences.fr/mathematique/
1082 Prasanna Kumar
Theorem 1. If P (z) =∑nν=1 aνzν is a polynomial of degree n, having no zeros in |z| < 1, then
max
|z|=1
|P ′(z)| ≤ n
2
[
1− |a0|− |an |
n(|a0|+ |an |)
]
max
|z|=1
|P (z)| (5)
whenever max|z|=1 |P (z)| ≥ 2|a0|, and (4) otherwise.
Since P (z) has all its zeros in |z| ≥ 1, hence for any polynomial P (z) = ∑nν=1 aνzν, we
have |a0|−|an |n(|a0|+|an |) ≥ 0. Therefore, for all polynomials satisfying the hypotheses of Theorem 1 except-
ing those satisfying |a0| = |an |, and so in particular for polynomials satisfying P (z) ≡ znP (1/z)
which includes polynomials having all their zeros on |z| = 1, our above inequality (5) sharpens
Erdős–Lax Inequality (4).
Ankeny and Rivlin [1] improved the inequality (3) for the class of polynomials having no zeros
in the unit disk by proving that, if P (z) is a polynomial of degree n having no zeros in |z| < 1, then
max
|z|=R
|P (z)| ≤ 1+R
n
2
max
|z|=1
|P (z)|, (6)
for any R ≥ 1. Ankeny and Rivlin used Erdős–Lax Inequality (4) in order to prove the above
Inequality (6), and so if instead of (4) we use its sharpened form (5) we easily get the following
which is a refinement of (6).
Corollary 2. If P (z) =∑nν=1 aνzν is polynomial of degree n having no zeros in |z| < 1 then
max
|z|=R
|P (z)| ≤ (1+R
n)−λ(Rn −1)
2
max
|z|=1
|P (z)|, (7)
for any R ≥ 1, and λ= |a0|−|an |n(|a0|+|an |) whenever max|z|=1 |P (z)| ≥ 2|a0|, and (6) otherwise.
2. Lemmas
The first result in this section is a simple exercise, and can be easily verified by principle of
induction on n and hence the proof is omitted.
Lemma 3. Let xi ≥ 1,1 ≤ i ≤ n. Then
n∑
i=1
1
1+xi
≤ n −1
2
+ 1
1+x1x2 . . . xn
.
The following few results seek attention to some well-known facts associated with the polyno-
mial P (z) having no zeros in the unit disc. In that aspect, the next lemma describes one of the
fundamental properties of the class of polynomials having no zeros in the unit disc.
Lemma 4. If P (z) =∑nν=1 aνzν is a polynomial of degree n ≥ 1 having no zeros in |z| < 1, then for
all z on |z| = 1 for which P (z) 6= 0
Re
(
zP ′(z)
P (z)
)
≤ n
2
− |a0|− |an |
2(|a0|+ |an |)
. (8)
Proof. Suppose P (z) = an ∏nk=1(z − zk ). By hypothesis each zk satisfies |zk | ≥ 1,1 ≤ k ≤ n. Then
for all z on |z| = 1 for which P (z) 6= 0, we will have
Re
(
zP ′(z)
P (z)
)
=
n∑
k=1
Re
z
z − zk
≤
n∑
k=1
1
1+|zk |
.
Now it follows from the Lemma 3 that
Re
(
zP ′(z)
P (z)
)
≤ n −1
2
+ 1
1+ |a0||an |
(9)
for all z on |z| = 1 for which P (z) 6= 0. Note that (9) and (8) are equivalent and hence the proof is
complete. 
Prasanna Kumar 1083
The well-known Theorem of Laguerre [5, 10] states that, if P (z) is a polynomial of degree n
having no zeros in the disc |z| < 1, then the polynomial nP (z)+(α−z)P ′(z) has no zeros in |z| < 1
for every complex number α with |α| < 1. What happens if we replace n by n −δ where δ is a
non-negative real number less than n? This question is very intriguing and is answered in the
following result for some specific values of δ that depends on the coefficients of the underlying
polynomial.
Lemma 5. If P (z) =∑nν=1 aνzν is a polynomial of degree n having no zeros in the disc |z| < 1, then
the polynomial
n
(
1− |a0|− |an |
n(|a0|+ |an |)
)
P (z)+ (α− z)P ′(z)
has no zeros in |z| < 1 for every α with |α| < 1.
Proof. From Lemma 4, it follows that, for all z on |z| = 1, for which P (z) 6= 0, we have
Re
 zP ′(z)
n
(
1− |a0|−|an |n(|a0|+|an |)
)
P (z)
≤ 1
2
,
and hence ∣∣∣∣∣∣1−
 zP ′(z)
n
(
1− |a0|−|an |n(|a0|+|an |)
)
P (z)
∣∣∣∣∣∣≥
∣∣∣∣∣∣
 zP ′(z)
n
(
1− |a0|−|an |n(|a0|+|an |)
)
P (z)
∣∣∣∣∣∣
for all z on |z| = 1, for which P (z) 6= 0. Therefore∣∣∣∣n (1− |a0|− |an |n(|a0|+ |an |)
)
P (z)− zP ′(z)
∣∣∣∣≥ |P ′(z)|
for all z on |z| = 1. But then for any α with |α| < 1 and |z| < 1, we have∣∣∣∣n (1− |a0|− |an |n(|a0|+ |an |)
)
P (z)− zP ′(z)
∣∣∣∣> |αP ′(z)|,
or in other words, for any α with |α| < 1 and |z| < 1, we have
n
(
1− |a0|− |an |
n(|a0|+ |an |)
)
P (z)− zP ′(z)+αP ′(z) 6= 0. 
Lemma 5 is crucial in proving the next result as well as the main theorem in this paper. Note
that Lemma 5 holds also true if we replace |a0| by any positive real number x such that |a0| ≥ x.
Lemma 6. If D is the open unit disc and S is an arbitrary point set of complex numbers, and
P (z) = a0 +a1z +·· ·+an zn is a polynomial of degree n satisfying max|z|=1 |P (z)| ≥ |s|,∀s ∈ S then
we have for any z ∈ D and α ∈ D that
(α− z)P ′(z)
n
(
1− |a0|−|an |n(|a0|+|an |)
) +P (z) ∈ S,
whenever max|z|=1 |P (z)| ≥ 2|a0|.
Proof. Suppose δ is outside S. Then P (z) 6= δ for any z ∈ D. Now applying Lemma 5 to the
polynomial P (z)−δ, with the fact that |δ| > max|z|=1 |P (z)| ≥ 2|a0|, it follows that
(α− z)P ′(z)+n
(
1− |a0|− |an |
n(|a0|+ |an |)
)
P (z) 6= n
(
1− |a0|− |an |
n(|a0|+ |an |)
)
δ,
which is equivalent to
(α− z)P ′(z)
n
(
1− |a0|−|an |n(|a0|+|an |)
) +P (z) 6= δ
for all z ∈ D, α ∈ D and any δ 6∈ S. This completes the proof. 
1084 Prasanna Kumar
3. Proof of Theorem 1
Since P (z) 6= 0 in |z| < 1, we have from Lemma 5
ζP ′(z) 6= zP ′(z)−n
(
1− |a0|− |an |
n(|a0|+ |an |)
)
P (z) (10)
for |ζ| < 1, and |z| < 1. With an appropriate choice of the argument of ζ in (10) we can get
|ζ||P ′(z)| 6= |zP ′(z)−n
(
1− |a0|− |an |
n(|a0|+ |an |)
)
P (z)|. (11)
Thus we have
|ζ||P ′(z)| < |zP ′(z)−n
(
1− |a0|− |an |
n(|a0|+ |an |)
)
P (z)| (12)
for |ζ| < 1, and |z| < 1, because otherwise inequality is violated for sufficiently small values of |ζ|.
Taking |ζ|→ 1, in (12) we get for |z| ≤ 1
|P ′(z)| ≤ |zP ′(z)−n
(
1− |a0|− |an |
n(|a0|+ |an |)
)
P (z)|. (13)
On the other hand, from Lemma 6, for |α| < 1 and |z| < 1,
(α− z)P ′(z)
n
(
1− |a0|−|an |n(|a0|+|an |)
) +P (z) ∈ S (14)
where S is as defined in Lemma 6. Hence for |z| = 1 and taking |α|→ 1 with appropriate choice of
argument of α we obtain
|zP ′(z)|+ |zP ′(z)−n
(
1− |a0|− |an |
n(|a0|+ |an |)
)
P (z)| ≤ n
(
1− |a0|− |an |
n(|a0|+ |an |)
)
max
|z|=1
|P (z)|. (15)
From (13) and (15), the result follows.
4. Polynomials having no zeros in |z| < K , K ≥ 1
It is quite natural to seek the extension of Theorem 1 to the class of polynomials having no zeros
in any open disc of radius K ≥ 1. Let P (z) = a0+a1z+·· ·+an zn be a polynomial of degree n having
no zeros in |z| < K , K ≥ 1. Then Malik [7] (see also [4]) proved that
max
|z|=1
|P ′(z)| ≤ n
1+K max|z|=1 |P (z)|. (16)
Let us sharpen (16) as well. Observe that P (K z) is a polynomial of degree n having no zeros in
|z| < 1. Now using P (K z) in place of P (z) in the proof of Theorem 1 and proceeding similarly by
noting that the inequality (13) becoming for |z| ≤ 1,
K |P ′(z)| ≤ |zP ′(z)−n
(
1− |a0|− |an |K
n
n(|a0|+ |anK n |)
)
P (z)|,
through some simple steps, we will obtain the following generalization of Theorem 1.
Theorem 7. If P (z) = a0 + a1z + ·· · + an zn is a polynomial of degree n, having no zeros in
|z| < K ,K ≥ 1, then
max
|z|=1
|P ′(z)| ≤ n
1+K
[
1− |a0|− |an |K
n
n(|a0|+ |an |K n)
]
max
|z|=1
|P (z)| (17)
whenever max|z|=K |P (z)| ≥ 2|a0|, and (16) otherwise.
Theorem 7 is the sharpened form of result due to Malik [7] (see also [4]) on the generalization
of Erdős–Lax inequality (4) to the class of polynomials having no zeros in |z| < K , K ≥ 1.
Prasanna Kumar 1085
Acknowledgments
The author is extremely grateful to the referee for their careful reading of the paper, and for their
comments and detailed suggestions which enhanced the quality of the paper considerably. The
work in the paper was supported partially by research grants from the National Board for Higher
Mathematics, and Council of Scientific and Industrial Research, Government of India.
References
[1] N. C. Ankeny, T. J. Rivlin, “On a theorem of S. Bernstein”, Pac. J. Math. 5 (1955), p. 849-852.
[2] S. N. Bernshteı̆n, Leçons sur les propriétés extrémales et la meilleure approximation des fonctions analytiques d’une
variable réelle, Collection de monographies sur la théorie des fonctions, Gauthier-Villars, 1926.
[3] P. Erdős, “On extremal properties of derivatives of polynomials”, Ann. Math. 41 (1940), p. 310-313.
[4] N. K. Govil, Q. I. Rahman, “Functions of exponential type not vanishing in a halfplane and related polynomials”,
Trans. Am. Math. Soc. 137 (1969), p. 501-517.
[5] E. Laguerre, “Oeuvres Vol. 1”, Nouv. Ann. Math. 17 (1878), no. 2, p. 20-25.
[6] P. D. Lax, “Proof of a conjecture of P. Erdős on the derivative of a polynomial”, Bull. Am. Math. Soc. 50 (1944), p. 509-
513.
[7] M. A. Malik, “On the derivative of a polynomial”, J. Lond. Math. Soc. 1 (1969), p. 57-60.
[8] G. Pólya, G. Szegő, Aufgaben und Lehrsätze aus der Analysis, Grundlehren der Mathematischen Wissenschaften,
vol. 19, Springer, 1925.
[9] A. C. Schaeffer, “Inequalities of A. Markoff and S. Bernstein for polynomials and related functions”, Bull. Am. Math.
Soc. 47 (1941), p. 565-579.
[10] G. Szegő, “Bemerkungen zu einem Satz von J. H. Grace über die Wurzeln algebraischer Gleichungen”, Math. Z. 13
(1922), p. 28-55.


On the Erdős–Lax Inequality

Erik Walsberg

Comptes Rendus
Mathématique
Erik Walsberg
Topological proofs of results on large fields
Volume 360 (2022), p. 1187-1192
Published online: 8 December 2022
https://doi.org/10.5802/crmath.305
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
e-ISSN : 1778-3569
Comptes Rendus
Mathématique
2022, Vol. 360, p. 1187-1192
https://doi.org/10.5802/crmath.305
Algebra, Algebraic geometry / Algèbre, Géométrie algébrique
Topological proofs of results on large fields
Erik Walsberg∗, a
a Department of Mathematics, University of California, Irvine, USA
URL: https://www.math.uci.edu/~ewalsber
E-mail: ewalsber@uci.edu
Abstract. We use the recently introduced étale open topology to prove several known facts on large fields. We
show that these facts lift to a quite general topological setting.
Manuscript received 3 July 2021, revised 20 October 2021, accepted 26 November 2021.
Throughout K ,L are fields, L is infinite, and AmK ,A
m
L is m-dimensional affine space over K ,L,
respectively. A K -variety is a separated K -scheme of finite type, not assumed to be reduced. If
K is a subfield of L and V is a K -variety then VL = V ×SpecK SpecL is the base change of V , and
if f : V → W is a morphism of K -varieties then fL : VL → WL is the base change of f . Given a
K -variety V we let V (K ) be the set of K -points of V , K [V ] be the coordinate ring of V when V is
affine, and K (V ) be the function field of V when V is integral.
The field L is large if every smooth integral L-curve with an L-point has infinitely many
L-points. Most fields of interest fall, or conjecturally fall, into one of the following mutually
exclusive categories:
(1) K is large.
(2) K is finitely generated over its prime subfield (Equivalently: K is a number field, a
function field1 over a number field, or a function field over a finite field).
(3) K is a function field over a large field. (This is nontrivially equivalent to: K is finitely
generated over a large field and not large.)
Local fields, real closed fields, separably closed fields, fields which admit Henselian valuations,
quotient fields of Henselian domains, pseudofinite fields, infinite algebraic extensions of finite
fields, PAC fields, p-closed fields, and fields that satisfy a local-global principle are all large.
Function fields are not large. It is an open question whether the maximal abelian or maximal
solvable extension of Q is large. See [9] and [2] for more background on large fields. Large
fields where introduced for Galois-theoretic purposes, in particular Pop solved the inverse Galois
problem over K (x) for K large [8]. Large fields are now central in Galois theory and have been a
subject of increasing interest from other directions.
We will give topological proofs of Facts A, B, and C below. Fact A is [9, Proposition 2.6].
∗Corresponding author.
1Finitely generated purely transendental proper extension.
ISSN (electronic) : 1778-3569 https://comptes-rendus.academie-sciences.fr/mathematique/
1188 Erik Walsberg
Fact A. Suppose that L is large and V is an irreducible L-variety with a smooth L-point. Then V (L)
is Zariski dense in V .
Facts B and C are due to Fehm. Fact B is proven in [3]. Note that Fehm uses “ample” for “large”
(this is one of a surprisingly large number of names used in the literature.)
Fact B. Suppose that L is large, K is a proper subfield of L, and V is a positive-dimensional
irreducible K -variety with a smooth K -point. Then |V (L) \V (K )| = |L|.
Fact B is a strengthening of the fact that if L is large and V is a positive-dimensional irreducible
L-variety with a smooth L-point then |V (L)| = |L|. This was previously proven by Pop, see [5,
Proposition 3.3]. Fact B is used to show that if L is large and C is a smooth projective integral
L-curve then the absolute Galois group of L(C ) is quasi-free, see [5, Proposition 3.4] and [6,
Corollary 4.4]. We give a separate proof of this fact in Section 3. Secondly, Fact B and the fact
that an algebraic extension of a large field is large yields the following: if K is large, V is a
positive dimensional irreducible K -variety with a smooth K -point, and L/K is algebraic then
|V (L) \ V (K )| = |L|. (We also give a topological proof of the fact that large fields are closed under
algebraic extensions in Section 1.)
Fact C is from [4]. We let td(E/F ) be the transendence degree of a field extension E/F .
Fact C. Suppose K is a subfield of L, L is large, and V is a smooth geometrically integral K -variety.
Then the following are equivalent:
(1) td(L/K ) ≥ dimV and V (L) ̸= ;,
(2) there is a K -algebra embedding K (V ) → L.
Another proof of Fact C is given in [1, Proposition 1.1], they reduce to the one-dimensional case
which then follows directly by Fact B. The implication (2) ⇒ (1) is routine and does not require
largeness. We describe a geometric statement equivalent to (1) ⇒ (2). Suppose that p ∈V (L) and
p ∉ W (L) for any proper closed subvariety W of V . Let U be an affine open subvariety of V , so
p ∈U (L). Note that K (U ) = K (V ) and K (V ) is the fraction field of K [U ]. Now p gives a morphism
SpecL →U , which is dual to an K -algebra morphism K [U ] → L. Note that K [U ] → L is injective as
p ∉W (L) for any proper closed subvariety W of V . So K [U ] → L extends to a K -algebra morphism
K (V ) = K (U ) → L. We prove the following.
Alternate form of Fact C. Suppose that L is large, K is a subfield of L with td(L/K ) ≥ m, and V
is a smooth geometrically integral m-dimensional K -variety with V (L) ̸= ;. Then there is p ∈V (L)
such that p ∉W (L) for any proper closed subvariety W of V .
We now discuss our proof technique. Each fact says that V (L) is large in some sense. Fix
a smooth p ∈ V (L). There is an open subvariety U of V containing p and an étale morphism
f : U → AmL , and f (U (L)) is a nonempty étale open subset of Lm . This allows us to reduce each
of the facts above to a statement saying non-empty étale open subsets of Lm are large in some
sense. In each case the statement holds in a very broad setting which we now describe.
A system of topologies. T over L is a choice of topology on V (L) for each L-variety V such that
the following holds for any morphism f : V →W of L-varieties:
(1) the induced map V (L) →W (L) is continuous,
(2) if f is an open immersion then V (L) →W (L) is a topological open embedding, and
(3) if f is a closed immersion then V (L) →W (L) is topological closed embedding.
If τ is a Hausdorff field topology on L then we produce a system of topologies by equipping
each V (L) with the usual τ-topology, the other familiar example of a system is the Zariski topology.
The étale open topology is a system of topologies, which may or may not be induced by a
Hausdorff field topology on L. It is easy to see that the T-topology on L = A1L(L) is discrete if
Erik Walsberg 1189
and only if the T-topology on V (L) is discrete for every L-variety V , and we say that T is discrete
if these conditions hold. We show in [7, Theorem C.1] that L is large if and only if the étale open
topology over L is not discrete.
Facts A, B, C follows from Propositions A, B, C, respectively.
Proposition A. Suppose that T is a non-discrete system of topologies over L and O is a nonempty
T-open subset of Lm . Then O is Zariski dense in AmL .
Proposition B. Suppose that T is a non-discrete system of topologies over L, K is a proper subfield
of L, and O is a nonempty T-open subset of Lm . Then |O \ K m | = |L|.
Note that if a = (a1, . . . , am) ∈ Lm then td(K (a1, . . . , am)/K ) is the minimum dimension of a
closed subvariety W ofAmK such that a ∈W (L).
Proposition C. Suppose that T is a non-discrete system of topologies over L, m ≥ 1, K is a subfield
of L with td(L/K ) ≥ m, and O is a nonempty T-open subset of Lm . Then there is (a1, . . . , am) ∈ O
such that td(K (a1, . . . , am)/K ) = m. Equivalently there is a ∈ O such that a ∉ W (L) for any proper
closed subvariety W ofAmK .
Acknowledgements
The basic facts about the étale open topology that we use were developed jointly with Will
Johnson, Chieu-Minh Tran, and Vincent Ye. The ideas in the proof of Fact B come from work
of Arno Fehm. Arno Fehm also read an earlier version of this note, made helpful suggestions, and
pointed out mistakes.
0.1. Background
It is worth noting that any system of topologies refines the Zariski topology, i.e. if T is a system of
topologies over L and V is an L-variety then the T-topology on V (L) refines the Zariski topology.
Fact 1 is proven in [7, Lemma 7.24]. The T-topology and the product of the T-topologies on
(V ×W )(L) =V (L)×W (L) may not agree.
Fact 1. Suppose that T is a system of topologies over L and V ,W are L-varieties. Then the
projection V (L)×W (L) →V (L) is a T-open map.
We will also make frequent use the obvious fact that the T-topology on L is affine invariant,
i.e. the map L → L, x 7→ ax +b is a homeomorphism for all a ∈ L×,b ∈ L. In particular this implies
that T is discrete if and only if there is a non-empty finite T-open subset of L.
Let V be a L-variety. An étale image in V (L) is a set of the form h(W (L)) for an étale mor-
phism h : W → V of L-varieties. We emphasize that Fact 2 follows from standard facts on étale
morphisms. Fact 2 is also proven in [7, Theorem A, Lemma 5.3].
Fact 2. Given an L-variety V , the collection of étale images in V (L) is a basis for a topology. The
collection of such topologies forms a system of topologies over L. If f : V →W is an étale morphism
of L-varieties and O is an étale open subset of V (L) then f (O) is an étale open subset of W (L).
We refer to this system of topologies as the étale open topology (over L). We are not aware of
any direct connection to the well-known étale topology. We will sometimes refer to it as the EL-
topology when there are multiple fields in play.
1190 Erik Walsberg
1. Algebraic extensions
We give a topological proof of the fact that large fields are closed under algebraic extensions.
Fact 3 is [9, Proposition 2.7].
Fact 3. If K is a subfield of L, K is large, and L/K is algebraic, then L is large.
There is a field K and a finite extension L/K such that L is large and K is not large [10]. Fact 4
is [7, Theorem 5.8]. The proof does not make use of largeness. (The proof of Fact 4, like all other
proofs of Fact 3, uses a form of Weil restriction.)
Fact 4. Suppose that K is a subfield of L, L/K is algebraic, and V is a K -variety. The EK -topology
on V (K ) refines the topology induced by the EL-topology on VL(L) =V (L).
We view Fact 4 as a topological refinement of Fact 3. We prove Fact 3.
Proof. Suppose that L/K is algebraic and L is not large. Then the EL-topology on L is discrete, so
{0} is an EL-open subset of L. By Fact 4 {0} = {0}∩K is an EK -open subset of K . So the EK -topology
on K is discrete, so K is not large. □
2. Fact A
We first prove Proposition A.
Proof. Suppose that O is not Zariski dense in AmL and let W be the Zariski closure of U in A
m
L .
So dimW < m. Fix p ∈ O. A typical line in AmL passing through p will intersect W in only finitely
many points. So there is a closed immersion g : A1L → AmL such that g (0) = p and g (A1L)∩W is
finite. Let O′ be the preimage of O under the induced map L → Lm . So O′ is a nonempty finite
T-open subset of L, hence T is discrete, contradiction. □
We now prove the following stronger version of Fact A. (It is stronger in that it applies to
more subsets of V (K ). Fact A shows that Zariski open subsets of V (K ) are Zariski dense and
Proposition 5 extends this to étale open subsets. This extra strength has been)
Proposition 5. Suppose that L is large, V is an irreducible L-variety with dimV ≥ 1, and O is an
étale open subset of V (L) which contains a smooth L-point. Then O is Zariski dense in V .
Proof. Fix a smooth p ∈ O and let m = dimV . The case m = 0 is trivial so we suppose m ≥ 1. Fix
an open subvariety U of V containing p and an étale morphism f : U →AmL . Let P = f (U (L)∩O),
so P is a non-empty étale open subset of Lm . Suppose that O is not Zariski dense in V and let
W be the Zariski closure of O in V . Then dimW < m hence dimU ∩W < m, and the Zariski
closure of f (U ∩W ) has dimension < m. Therefore P ⊆ f (U ∩W ) is not Zariski dense in AmL ,
contradiction. □
3. Many L-points
Before proving Fact B we prove the following related result.
Proposition 6. Suppose that L is large, V is an irreducible L-variety, and O is a nonempty étale
open subset of V (L) which contains a smooth L-point. Then |O| = |L|.
Proposition 6 follows from a more general fact.
Proposition 7. Suppose that T is a non-discrete system of topologies over L and O is a nonempty
T-open subset of Lm for m ≥ 2. Then |O| = |L|.
Erik Walsberg 1191
Proof. Letπ : Lm → L be a coordinate projection. Fact 1 shows thatπ(O) isT-open. As |O| ≥ |π(O)|
we may suppose that m = 1.
Claim. If O contains 0 then OO−1 = L, hence |O| = |L|.
Proof of the Claim. Suppose that O contains 0 and OO−1 ̸= L. Fix a ∈ L× \ OO−1. Then O ∩aO =
{0}. However, O ∩aO is T-open, so T is discrete, contradiction. □
Note that if b ∈O then |O| = |O −b| = |L|. □
We now prove Proposition 6.
Proof. Suppose that p ∈ O is smooth. Let m = dimV , U be an open subvariety of V , and
f : U → AmL be an étale morphism. Then f (U (K )∩O) is a nonempty étale open subset of Lm .
Apply Proposition 7. □
4. Fact B
We will need a couple lemmas. Fact 8 is a special case of [3, Lemma 3].
Fact 8. Suppose that F is a field, S is an F -vector space of dimension ≥ 2, I is an index set, and Si
is a one-dimensional vector subspace of S for all i ∈ I . If S =⋃i ∈ I Si then |I | ≥ |F |.
Proof. Fix a one-dimensional subspace S′ of S and a ∈ S \ S′. Then |S′| = |F | and it is easy to see
that |Si ∩ (a +S′)| ≤ 1 for all i ∈ I . □
Lemma 9 is essentially in the proof of [3, Lemma 4]. Recall our standing assumption that L is
infinite.
Lemma 9. Suppose that K is a proper subfield of L and X ⊆ L satisfies X X −1 = L. Then |X \K | = |L|.
Proof. Note that |X | = |L|. Suppose that |K | < |L|. Then |X | = |L| > |K | so |X \ K | = |L|. So we may
suppose that |K | = |L|. It now suffices to show that |X \ K | ≥ |K |. We let Y × = Y \ {0} for any Y ⊆ L.
Let A = X ∩K and B = X \ K . So
L = {a/b : (a,b) ∈ X ×X ×}
= {a/b : (a,b) ∈ (A× A×)∪ (A×B×)∪ (B × A×)∪ (B ×B×)}
⊆ K ∪
( ⋃
b∈B×
(1/b)K
)
∪
( ⋃
a∈B
aK
)
∪
( ⋃
a∈B ,b∈B×
(a/b)K
)
.
Consider L to be a K -vector space. Hence L is a union of ≤ 1 + 2|B | + |B |2 one-dimensional
subspaces. By Fact 8 we have 1+2|B |+ |B |2 ≥ |K |. As K is infinite |B | ≥ |K |. □
We now prove Proposition B.
Proof. Let π : Lm → L be the projection onto the first coordinate. By Fact 1 π(O) is T-open. We
have π(O) \ K ⊆ π(O \ K m), so it suffices to show that |π(O) \ K | = |L|. So we may suppose that
O is an T-open subset of L. By the claim in the proof of Proposition 7 (O − b)(O − b)−1 = L for
any b ∈ O. So |O| = |L|. If O ∩K = ; we are done. Suppose otherwise and fix b ∈ O ∩K . By the
claim (O −b)(O −b)−1 = L so by Lemma 9 |(O −b) \ K | = |L|. Note that x 7→ x +b gives a bijection
(O −b) \ K →O \ K . □
1192 Erik Walsberg
We now prove Fact B.
Proof. Let p be a smooth K -point of V and m = dimV . As V is irreducible there is an open
subvariety U of V containing p and an étale morphism f : U → AmK . Let O = fL(UL(L)), note
that fL is étale as étale morphisms are closed under base change. Then O is an étale image in
AmL (L) = Lm and is hence étale open. By Proposition B |U \ K m | = |L|. Note that if p ∈U (K ) then
fL(p) = f (p) ∈ K m . □
5. Fact C
We now prove Proposition C. Given a = (a1, . . . , am) ∈ Lm we let K (a) = K (a1, . . . , am).
Proof. We apply induction on m. Suppose m = 1. Let K ′ be the algebraic closure of K in L. So K ′
is a proper subfield of L. By Proposition B there is a ∈ O \ K ′. So td(K (a)/K ) = 1. Suppose m ≥ 2.
Let π : K m → K m−1 be the projection away from the first coordinate. By Fact 1 π(O) is T-open. By
induction there is b ∈π(O) such that td(K (b)/K ) = m −1. Let Ob = {c ∈ L : (b,c) ∈O}. Note that Ob
is the pre-image of O under the map K → K m given by x 7→ (b, x). So Ob is T-open. As td(L/K ) ≥ m
we have td(L/K (b)) ≥ 1. So there is c ∈Ob such that td(K (b,c)/K (b)) = 1. Let a = (b,c). □
We now prove a stronger version of the second form of Fact C.
Proposition 10. Suppose that L is large, K is a subfield of L with td(L/K ) ≥ m, and V is a smooth
geometrically irreducible m-dimension K -variety. Then the set of p ∈ V (L) such that p ∉ W (L) for
any proper closed subvariety W of V is dense in the étale open topology on V (L) =VL(L).
Proof. Suppose that O is a nonempty étale open subset of V (L). We show that O contains a p as
above. As V is smooth and irreducible there is an open subvariety U of V and an étale morphism
f : U →AmK . By Proposition 5 O intersects UL(L). Let P = fL(UL(L)∩O), so P is a nonempty étale
open subset of Lm . By Proposition C there is a ∈ P such that td(K (a)/K ) = m. Fix p ∈ O ∩UL(L)
such that fL(p) = a. Suppose W is a proper closed subvariety of V , we show that p ∉ W (L).
Let W ′ be the Zariski closure of fL(WL). Then dimW < m, so dimWL < m, so dimW ′ < m. As
td(K (a)/K ) = m we have a ∉W ′. Hence p ∉WL(L) =W (L). □
References
[1] A. Bachmayr, D. Harbater, J. Hartmann, F. Pop, “Large fields in differential Galois theory”, J. Inst. Math. Jussieu 20
(2021), no. 6, p. 1931-1946.
[2] L. Bary-Soroker, A. Fehm, “Open problems in the theory of ample fields”, in Geometric and differential Galois theory,
Séminaires et Congrès, vol. 27, Société Mathématique de France, 2013, p. 1-11.
[3] A. Fehm, “Subfields of ample fields. Rational maps and definability”, J. Algebra 323 (2010), no. 6, p. 1738-1744.
[4] ——— , “Embeddings of function fields into ample fields”, Manuscr. Math. 134 (2011), no. 3-4, p. 533-544.
[5] D. Harbater, “On function fields with free absolute Galois groups”, J. Reine Angew. Math. 632 (2009), p. 85-103.
[6] D. Harbater, K. F. Stevenson, “Local Galois theory in dimension two”, Adv. Math. 198 (2005), no. 2, p. 623-653.
[7] W. Johnson, M. Tran, E. Walsberg, J. Ye, “Étale open topology and the stable field conjecture”, https://arxiv.org/abs/
2009.02319, 2021.
[8] F. Pop, “Embedding problems over large fields”, Ann. Math. 144 (1996), no. 1, p. 1-34.
[9] ——— , “Little survey on large fields - old & new”, in Valuation Theory in Interaction. Proceedings of the 2nd inter-
national conference and workshop on valuation theory, EMS Series of Congress Reports, European Mathematical
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[10] P. Srinivasan, “A virtually ample field that is not ample”, Isr. J. Math. 234 (2019), no. 2, p. 769-776.


Topological proofs of results on large fields

Pierre Germain

Simon Myerson

Comptes Rendus
Mathématique
Pierre Germain and Simon Myerson
Bounds for spectral projectors on the Euclidean cylinder
Volume 360 (2022), p. 1257-1262
Published online: 8 December 2022
https://doi.org/10.5802/crmath.378
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
e-ISSN : 1778-3569
Comptes Rendus
Mathématique
2022, Vol. 360, p. 1257-1262
https://doi.org/10.5802/crmath.378
Harmonic analysis / Analyse harmonique
Bounds for spectral projectors on the
Euclidean cylinder
Pierre Germain ∗, a and SimonMyerson b
a Courant Institute of Mathematical Sciences, New York University, 251 Mercer Street,
New York, N.Y. 10012-1185, USA
b Mathematics Institute, University of Warwick, Zeeman Building, Coventry, CV4 7AL,
United Kingdom
E-mails: pmg7@nyu.edu (P. Germain), Simon.Rydin-Myerson@warwick.ac.uk
(P. Myerson)
Abstract. We prove essentially optimal bounds for norms of spectral projectors on thin spherical shells for
the Laplacian on the cylinder (R/Z)×R. In contrast to previous investigations into spectral projectors on tori,
having one unbounded dimension available permits a compact self-contained proof.
Funding. PG was supported by the Simons collaborative grant on weak turbulence. SLRM was supported by
a Leverhulme Early Career Fellowship.
Manuscript received 29 March 2022, accepted 14 May 2022.
1. Introduction
1.1. Spectral projectors on general manifolds and tori
Given a Riemannian manifold with Laplace–Beltrami operator ∆, consider the spectral projector
Pλ,δ on (perhaps generalized) eigenfunctions with eigenvalues within O(δ) of λ. It is defined
through functional calculus by the formula
Pλ,δ = Pχλ,δ =χ
(p−∆−λ
δ
)
,
where χ is a cutoff function, which is irrelevant for our purposes.
An interesting question is to determine the operator norm from L2 to Lp , with p > 2, of this
operator. A theorem of Sogge [5] gives an optimal answer for any Riemannian manifold if δ= 1∥∥Pλ,1∥∥L2 →Lp ≲λ d−12 − dp +λ d−12 ( 12 − 1p ).
∗Corresponding author.
ISSN (electronic) : 1778-3569 https://comptes-rendus.academie-sciences.fr/mathematique/
1258 Pierre Germain and Simon Myerson
While this completely answers the question if δ > 1, the case δ < 1 is still widely open. Under-
standing the case δ < 1 requires a global analysis on the Riemannian manifold, which makes it
very delicate.
In the case of the rational torus Rd /Zd , Lp bounds on eigenfunctions attracted a lot of
attention; this corresponds to the choice δ = 1/λ, since the distance between two consecutive
eigenvalues is ∼ 1λ . The best result in this direction is due to Bourgain and Demeter [3]. More
recently, the authors of the present paper [4] considered the problem for general values of λ and
δ, conjectured the bound for general tori∥∥Pλ,δ∥∥L2→Lp ≲λ d−12 − dp δ1/2 + (λδ) (d−1)2 ( 12 − 1p ) for δ> 1/λ,
and were able to establish this bound for a range of the parameters δ,λ, p.
A full proof of this conjecture seems very challenging in every dimension d . Restricting to the
case d = 2, consider the case (R/Z)×R = T×R instead of T2. The conjecture remains identical,
but a short proof, relying on ℓ2 decoupling, can be provided; this is the main observation of the
present paper. Generalizations to higher dimensions are certainly possible.
1.2. The Euclidean cylinder
OnT×R= (R/Z)×R, we choose coordinates (x, y), with x ∈ [0,1] and y ∈R. The Laplacian operator
is given by
∆= ∂2x +∂2y .
A function f on T×R can be expanded through Fourier series in x and Fourier transform in y :
f (x) = ∑
k ∈Z
∫
R
f̂ (k,η)e2πi(kx+ηy) dη.
The spectral projector can then be expressed as
Pλ,δ f (x) =
∑
k ∈Z
∫
R
χ
(√
k2 +η2 −λ
δ
)
f̂ (k,η)e2πi(kx+ηy) dη.
Theorem 1 (Main theorem). If λ> 1 and δ< 1,∥∥Pλ,δ∥∥L2 →Lp ≲ϵ λϵδ−ϵ [λ 12 − 2p δ 12 + (λδ) 14 − 12p ]
Furthermore, this estimate is optimal, up to the subpolynomial factor λϵδ−ϵ and the multiplicative
constant.
1.3. Strichartz estimates
It is interesting to draw a parallel with Strichartz estimates in dimension 2 for the Schrödinger
equation, in which case the critical exponent equals 4. It was proved in the foundational paper of
Bourgain [2] that ∥∥∥e i t∆ f ∥∥∥
L4([0,1]×T2)
≲s
∥∥ f ∥∥H s (T2) for s > 0.
Takaoka and Tzvetkov [6] proved that the above inequality fails for s = 0, but that, on T×R,∥∥∥e i t∆ f ∥∥∥
L4([0,1]×T×R) ≲
∥∥ f ∥∥L2(T×R) .
Finally, Barron, Christ and Pausader [1] determined the correct global (in time) estimate, for
which a further summation index is needed. These examples suggest that optimal estimates
might differ by subpolynomial factors between T2 and T×R.
Pierre Germain and Simon Myerson 1259
2. Proof of the Theorem 1 (main theorem)
Proof. By Plancherel’s theorem, it suffices to prove∥∥ f ∥∥Lp ≲ϵ λϵδ−ϵ [λ 12 − 2p δ 12 + (λδ) 14 − 12p ]∥∥ f ∥∥L2 (1)
for f a function whose Fourier transform is localized in the corona Cλ,δ of radius λ and with
thickness δ/λ:
Cλ,δ =
{
(k,η) such thatλ−δ<
√
k2 +η2 <λ+δ
}
.
By symmetry, one can furthermore assume that f̂ (k,η) is localized in the first quadrant k,η≥ 0.
The function f can be split into two pieces, which will correspond to the two terms on the
right-hand side of (1).
f (x) =
 ∑
|k−λ|≤ 1
δ
+ ∑
|k−λ|> 1
δ
∫
R
f̂
(
k,η
)
e2πi(kx+ηy) dη= f1(x)+ f2(x).
The Case |k −λ| ≤ 1δ . The Fourier support of f1 is made up of a collection of segments. We will
see in Lemma 2 below that the added length of these segments can be bounded by∣∣Supp f̂1∣∣≲pλδ.
Therefore, by the Cauchy–Schwartz inequality,∥∥ f1∥∥L∞ ≤ ∣∣Supp f̂1∣∣1/2 ·∥∥ f̂1∥∥1/2L2 ≲ (λδ)1/4 ∥∥ f1∥∥L2 .
Interpolating with L2, this gives ∥∥ f1∥∥Lp ≲ (λδ) 14 − 12p ∥∥ f ∥∥L2 .
The Case |k −λ| > 1δ . We start by choosing a function φ ∈ S which is > 1/2 on [−1,1], and has
Fourier support in [−1,1]. We use periodicity in the x variable to expand the range of x from
x ∈ [0,1] to x < δ−1, so that
∥∥ f2∥∥Lp (T×R) ≲ δ1/p
∥∥∥∥∥∥φ (δx) ∑|k−λ|> 1
δ
∫
R
f̂ (k,η)e2πi(kx+ηy) dη
∥∥∥∥∥∥
Lp (R2)
.
We now change variables as follows: X =λx, Y =λy , K = k/λ, H = η/λ,
f2(x, y) =λF (X ,Y ), F (X ,Y ) =φ
(
δX
λ
) ∑
K ∈Z/λ
|K−1|> 1
δλ
∫
R
f̂ (λK ,λH)e2πi (K X+HY ) d H
to obtain ∥∥ f2∥∥Lp (T×R) ≲ δ1/pλ1− 2p ∥F∥Lp (R2) .
The effect of this change of variables is that the function of (X ,Y ) whose Lp norm we want to
evaluate has Fourier transform supported in the corona C1,3δ/λ of radius 1 and width δ/λ, and
also in the first quadrant X ,Y ≥ 0. This enables us to apply the ℓ2 decoupling theorem of Bourgain
and Demeter [3]: for a smooth partition of unity (χθ) corresponding to a suitable almost disjoint
covering of C1,3δ/λ by caps (θ) of size ∼ δλ ×
√
δ
λ ,
∥∥ f2∥∥Lp (T×R) ≲ δ 1p λ1− 2p (δ/λ)− 14 + 32p −ϵ
(∑
θ
∥∥χθ(D)F∥∥2Lp (R2)
)1/2
,
where χθ(D) is the Fourier multiplier with symbol χθ .
1260 Pierre Germain and Simon Myerson
We now apply the inequality ∥g∥Lp (Rd ) ≲ ∥g∥L2 |Supp ĝ |
1
2 − 1p (if p ≥ 2), which follows by apply-
ing in turn the Hausdorff–Young and Hölder inequalities, and then the Plancherel theorem. Since
by Lemma 2 below ∣∣∣Supp áχθ(D)F ∣∣∣= ∣∣SuppχθF̂ ∣∣≲ δ5/2λ−3/2,
it follows that
∥∥ f2∥∥Lp (T×R) ≲ δ 1p λ1− 2p (δ/λ)− 14 + 32p −ϵ (δ5/2λ−3/2) 12 − 1p
(∑
θ
∥∥χθ(D)F∥∥2L2(R2)
)1/2
.
By almost orthogonality, this becomes∥∥ f2∥∥Lp (T×R) ≲ δ 1p λ1− 2p (δ/λ)− 14 + 32p −ϵ (δ5/2λ−3/2) 12 − 1p ∥F∥L2(R2) .
Finally, undoing the change of variables and using once again periodicity in the x variable gives∥∥ f2∥∥Lp (T×R) ≲ δ 1p λ1− 2p (δ/λ)− 14 + 32p −ϵ (δ5/2λ−3/2) 12 − 1p δ−1/2 ∥∥ f ∥∥L2
≲λ
1
2 − 2p δ1/2
Optimality. The optimality of the statement of the theorem is proved through two examples.
The first one is an analog of the Knapp example: assume λ ∈N, and consider the function g given
by its Fourier transform
ĝ (k,η) = 1λ(k)χ
(
ηp
λδ
)
.
Here, 1λ is the indicator function of {λ} and χ is a cutoff function with a sufficiently small support,
so that Supp ĝ ⊂Cλ,δ. In physical space,
g (x, y) =
p
λδe2πiλx χ̂
(p
λδy
)
.
It has Lp norm ∼ (λδ) 12 − 12p , so that
∥g∥Lp
∥g∥L2
∼ (λδ) 14 − 12p .
We now consider the function h given by its Fourier transform
ĥ(k,η) = 1Cλ,δ (k,η)1[0,λ/2](k);
here, 1Cλ,δ is the indicator function of the annulus, and 1[0,λ/2] the indicator function of the
interval. It is easy to check that |Supp ĥ| ∼λδ, so that ∥h∥L∞ ∼λδ and ∥h∥L2 ∼
p
λδ, and finally
∥h∥L∞
∥h∥L2
∼
p
λδ.
By the Bernstein inequality,
∥h∥Lp
∥h∥L2
≳λ−2/p
∥h∥L∞
∥h∥L2
∼λ 12 − 2p δ1/2.
The examples g and h show that the statement of the Theorem 1 is optimal, up to subpolynomial
losses. □
Pierre Germain and Simon Myerson 1261
3. Bounds for the Fourier support
Lemma 2 (Bound on the size of Fourier support). With the notations of the proof of Theorem 1,
(i) The function f1 is a function on T×R. As such, its Fourier transform is supported on a
union of lines, and has one-dimensional measure∣∣Supp f̂1∣∣≲pλδ.
(ii) The function χθ(D)F is a function on R
2. Its Fourier transform is defined on R2, and has
two-dimensional measure ∣∣SuppχθF̂ ∣∣≲ δ5/2λ−3/2.
Proof.
(i) Consider f as in the proof of Theorem 1, namely with Fourier support in Cλ,δ. Since (k,η) range
in Z×Rwith k,η≥ 0, the Fourier support of f is contained in ∪k ∈Z{k}×Eλk , where
Eλk =

; (k ≥ k+) ,(
0,
√
k2+−k2
)
(|k −λ| < δ) ,(√
(λ−δ)2 −k2,
√
k2+−k2
)
(0 ≤ k ≤λ−δ) .
Recalling that f̂ 1 is just f̂ restricted to |k −λ| ≤ 1δ , one can then add up these pieces to get the
bound ∣∣Supp f̂1∣∣≲ ∑
max
{
0,λ− 1
δ
}≤k <k+ |Ek |≲
p
λδ+ ∑
k ≥0
1
δ
≥λ−k>δ
δλp
λ(λ−x)
and as y 7→ 1/py is decreasing this is
≤ 2
p
λδ+
∫ min{λ, 1
δ
}
δ
δλ√
λy
d y ≤ 4
p
λδ.
(ii) Turning to F , it has Fourier support in
⋃
k ∈Z
|k−λ|> 1
δ
[
k
λ
− 2δ
λ
,
k
λ
+ 2δ
λ
]
×Dλk , Dλk =
H , 1− δλ <
√
k2
λ2
+H 2 < 1+ δ
λ
 . (2)
Consider χθ(D)F , for a cap θ with dimensions
∼ δ
λ
×
√
δ
λ
adapted to the corona C1,3δ/λ. Given such a cap, there is j ∈N with 2 j > 1δ such that every point
in the intersection of θ with the set (2) satisfies |λ−k| ∼ 2 j .
There are around
p
δ2 j /2 such values of k for which the vertical strip [ kλ − 2δλ , kλ + 2δλ ] intersects
the cap θ. For each such k, the size of Dλk is ∼ δλ−1/22− j /2. Hence, adding up the contributions
in (2), ∣∣Suppχθ(D)F ∣∣≲pδ2 j /2 ·δλ−1 ·δλ−1/22− j /2 = δ5/2λ−3/2. □
1262 Pierre Germain and Simon Myerson
References
[1] A. Barron, M. Christ, B. Pausader, “Global endpoint strichartz estimates for Schrödinger equations on the cylinder
R×T”, Nonlinear Anal., Theory Methods Appl. 206 (2021), article no. 112172.
[2] J. Bourgain, “Fourier transform restriction phenomena for certain lattice subsets and applications to nonlinear
evolution equations. II: The KdV-equation”, Geom. Funct. Anal. 3 (1993), no. 3, p. 209-262.
[3] J. Bourgain, C. Demeter, “The proof of the l 2 decoupling conjecture”, Ann. Math. 182 (2015), no. 1, p. 351-389.
[4] P. Germain, S. L. R. Myerson, “Bounds for spectral projectors on tori”, https://arxiv.org/abs/2104.13274v1, 2021.
[5] C. D. Sogge, Fourier integrals in classical analysis, Cambridge Tracts in Mathematic, vol. 105, Cambridge University
Press, 1993.
[6] H. Takaoka, N. Tzvetkov, “On 2D nonlinear schrödinger equations with data on R×T”, J. Funct. Anal. 182 (2001), no. 2,
p. 427-442.


Bounds for spectral projectors on the Euclidean cylinder

Eric G. Samperton

Free actions on surfaces that do not extend to arbitrary actions on 3-manifolds

Ilias Ftouhi

Comptes Rendus
Mathématique
Ilias Ftouhi
On a Pólya’s inequality for planar convex sets
Volume 360 (2022), p. 241-246
<https://doi.org/10.5802/crmath.292>
© Académie des sciences, Paris and the authors, 2022.
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
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Comptes Rendus
Mathématique
2022, 360, p. 241-246
https://doi.org/10.5802/crmath.292
Spectral theory / Théorie spectrale
On a Pólya’s inequality for planar convex sets
Ilias Ftouhia
a Friedrich-Alexander-Universität Erlangen-Nürnberg, Department of Mathematics,
Chair of Applied Analysis (Alexander von Humboldt Professorship), Cauerstr. 11,
91058 Erlangen, Germany
E-mail: ilias.ftouhi@fau.de
Abstract. In this short note, we prove that for every bounded, planar and convex setΩ, one has
λ1(Ω)T (Ω)
|Ω| ≤
π2
12
·
(
1+pπ r (Ω)p|Ω|
)2
,
where λ1, T , r and | · | are the first Dirichlet eigenvalue, the torsion, the inradius and the volume. The
inequality is sharp as the equality asymptotically holds for any family of thin collapsing rectangles.
As a byproduct, we obtain the following bound for planar convex sets
λ1(Ω)T (Ω)
|Ω| ≤
π2
12
(
1+ 2
√
2(6+π2)−π2
4+π2
)2
≈ 0.996613. . .
which improves Polyá’s inequality λ1(Ω)T (Ω)|Ω| < 1 and is slightly better than the one provided in [3].
The novel ingredient of the proof is the sharp inequality
λ1(Ω) ≤
π2
4
·
(
1
r (Ω)
+
√
π
|Ω|
)2
,
recently proved in [8].
Funding. This work was partially supported by the project ANR-18-CE40-0013 SHAPO financed by the French
Agence Nationale de la Recherche (ANR).
Manuscript received 28th August 2021, revised and accepted 26th October 2021.
1. Introduction
LetΩ be an open set in Rn with finite Lebesgue measure |Ω|. We denote
T (Ω) := sup
w∈H 10 (Ω)\{0}
(∫
Ωwdx
)2∫
Ω |∇w |2dx
the torsion of the setΩ, and
λ1(Ω) := inf
u∈H 10 (Ω)\{0}
∫
Ω |∇u|2dx∫
Ωu
2dx
the fundamental frequency ofΩwhich corresponds to the first eigenvalue of the Laplace operator
with Dirichlet boundary condition of the setΩ.
ISSN (electronic) : 1778-3569 https://comptes-rendus.academie-sciences.fr/mathematique/
242 Ilias Ftouhi
In 1951, G. Pólya [13] proved the following inequality:
λ1(Ω)T (Ω)
|Ω| ≤ 1,
which was recently improved by M. van den Berg et al. in [3], where the authors prove the
inequality
λ1(Ω)T (Ω)
|Ω| ≤ 1−
2nω2/nn
n +2
T (Ω)
|Ω|1+ 2n
, (1)
whereωn is the measure of the ball with radius 1 in Rn . They also show, by using homogenization
arguments, that the upper bound 1 is optimal in the class of open sets with finite measure, see [3,
Theorem 1.2], but is not optimal for the class of bounded, planar and convex sets. Indeed, in this
case they obtain the following improved estimate
λ1(Ω)T (Ω)
|Ω| ≤ 1−
1
11560
≈ 0.999913. . . (2)
see [3, Theorem 1.5]. The optimal upper bound is conjectured to be given by π
2
12 , which is
asymptotically attained by any sequence of thin collapsing rectangles.
Let us now state the main results of the present note.
Theorem 1. We have for every bounded planar convex setΩ:
λ1(Ω)T (Ω)
|Ω| ≤
π2
12
·
(
1+pπ r (Ω)p|Ω|
)2
. (3)
The inequality is sharp as it is asymptotically attained by any family of thin collapsing rectangles.
Moreover, for every sequence (Ωk ) of planar convex sets such that |Ωk | = 1 and limk→+∞ d(Ωk ) =
+∞, we have
limsup
k→+∞
λ1(Ωk )T (Ωk )
|Ωk |
≤ π
2
12
.
As a byproduct, we obtain the following slight improvement of inequality (2):
Corollary 2. We have for every bounded planar convex setΩ:
λ1(Ω)T (Ω)
|Ω| ≤
π2
12
(
1+ 2
√
2(6+π2)−π2
4+π2
)2
≈ 0.996613. . . (4)
We provide the complete proofs of Theorem 1 and Corollary 2 in Section 2 and state some
comments in Section 3.
2. Proofs of The main results
2.1. Proof of Theorem 1
LetΩ be a bounded, planar and convex set. We combine the following inequality proved in [8]:
λ1(Ω) < π
2
4
·
(
1
r (Ω)
+
√
π
|Ω|
)2
, (5)
with the following Makai’s inequality [11]:
T (Ω) < 1
3
|Ω|r (Ω)2.
We have
λ1(Ω)T (Ω)
|Ω| <
1
|Ω| ×
π2
4
·
(
1
r (Ω)
+
√
π
|Ω|
)2
× 1
3
|Ω|r (Ω)2 = π
2
12
·
(
1+pπ r (Ω)p|Ω|
)2
.
C. R. Mathématique — 2022, 360, 241-246
Ilias Ftouhi 243
The inequality is sharp as it is an asymptotically an equality for any family of thin vanishing
rectangles, see [2, Proposition 3.2].
Let us now consider (Ωk ) a sequence of planar convex sets such that |Ωk | = 1 and
limk→+∞ d(Ωk ) =+∞. We have for every k ∈N:
0 < r (Ωk )√|Ωk | ≤ 1√|Ωk | 2|Ωk |P (Ωk ) ≤ 2
√|Ωk |
2d(Ωk )
= 1
d(Ωk )
−→
k→+∞
0,
where we used the classical inequalities r (Ωk ) ≤ 2|Ωk |P (Ωk ) (see [5]) and 2d(Ωk ) ≤ P (Ωk ).
We then conclude by using inequality (3) that
limsup
k→+∞
λ1(Ωk )T (Ωk )
|Ωk |
≤ π
2
12
.
2.2. Proof of Corollary 2
Let us prove the inequality
λ1(Ω)T (Ω)
|Ω| < 1−
π
π2
4
(p|Ω|
r (Ω) +
p
π
)2 +π . (6)
We have by inequality (1) (see [3, Theorem 1.1]):
λ1(Ω)T (Ω)
|Ω| ≤ 1−π
T (Ω)
|Ω|2 ,
which is equivalent to the inequality:
λ1(Ω)T (Ω)
|Ω| ≤ 1−
π
|Ω|λ1(Ω)+π
.
Indeed:
λ1(Ω)T (Ω)
|Ω| ≤ 1−π
T (Ω)
|Ω|2 ⇐⇒
T (Ω)
|Ω|
(
λ1(Ω)+ π|Ω|
)
< 1
⇐⇒ λ1(Ω)T (Ω)|Ω| <
λ1(Ω)
λ1(Ω)+ π|Ω|
= 1− π|Ω|λ1(Ω)+π
.
Thus, by using the inequality (5), we obtain the inequality (6).
We then combine the inequalities (3) and (6) to conclude:
λ1(Ω)T (Ω)
|Ω| ≤ min
π2
12
·
(
1+pπ r (Ω)p|Ω|
)2
,1− π
π2
4
(p|Ω|
r (Ω) +
p
π
)2 +π

≤ max
x∈
(
0, 1p
π
]min
(
π2
12
· (1+pπx)2 ,1− π
π2
4
( 1
x +
p
π
)2 +π
)
= π
2
12
(
1+ 2
√
2(6+π2)−π2
4+π2
)2
.
The second inequality is a consequence of the estimate 0 < r (Ω)p|Ω| ≤
r (B)p|B | =
1p
π
, where B is any ball
of R2. Let us now detail the computations leading to the last equality. We introduce the functions
f : x 7−→ π212 ·
(
1+pπx)2 and g : x 7−→ 1− π
π2
4
( 1
x +
p
π
)2+π . The function f is increasing on (0, 1pπ ] and g
is decreasing on (0, 1p
π
], moreover, limx→0+ f (x) = π
2
12 < 1 = limx→0+ g (x) and f ( 1pπ ) = π
2
3 > 11+π2 =
g ( 1p
π
). Thus the function x 7−→ min( f (x), g (x)) attains its maximum on (0, 1p
π
] at the point x0
such that f (x0) = g (x0) (see Figure 1). It remains then to solve the equation f (x) = g (x) on (0, 1pπ ].
C. R. Mathématique — 2022, 360, 241-246
244 Ilias Ftouhi
The equation f (x) = g (x) can be written in the algebraic form ax3+bx2+cx+d = 0, where a,b,c
and d are explicit constants. We then notice that− 1p
π
is a solution, which means that the equation
is equivalent to a(x+ 1p
π
)(x2+b′x+c ′) = 0, where b′ and c ′ are explicit constants. It remains then
to solve the second order equation x2 +b′x + c ′ = 0 that admits two solutions
x1 =− π
3/2
4+π2 −
2
√
2(6+π2)
π
4+π2 < 0 and x0 =
2
√
2(6+π2)
π
4+π2 −
π3/2
4+π2 ∈ (0,
1p
π
].
Thus
max
x∈(0, 1p
π
]
min( f (x), g (x)) = f (x0) = g (x0) = π
2
12
(
1+ 2
√
2(6+π2)−π2
4+π2
)2
.
This ends the proof.
0 0.1 0.2 0.3 0.4 0.5
0.85
0.9
0.95
1
1.05
Figure 1. Graphs of the functions f and g .
3. Some comments
We close the paper by the following comments:
• The result of the paper is limited to the planar case because it mainly relies on the
inequality
λ1(Ω) < π
2
4
(
1
r (Ω)
+
√
π
|Ω|
)2
,
which is obtained by combining the inequalities λ1(Ω) < π24 h(Ω)2 (proved in [12]), where
h is the Cheeger constant, and h(Ω) ≤ 1r (Ω) +
√
π
|Ω| (proved in [8]). Up to our knowledge,
the last inequality is only known for the planar case and a generalization to higher
dimensions seems to be difficult as the proof relies on the explicit characterization of
the Cheeger constant of planar convex sets (see [9]).
C. R. Mathématique — 2022, 360, 241-246
Ilias Ftouhi 245
• Let us denote Br (Ω) a ball of radius r (Ω) inscribed inΩ. The inclusion Br (Ω) ⊂Ω combined
with the monotonicity of λ1 for the inclusion yield the inequality
λ1(Ω) ≤
j 201
r (Ω)2
, (7)
where j01 denotes the first zero of the first Bessel function. We note that even if inequal-
ity (7) is better than (5) for sets that are close to the ball (we refer to [8, Section 5.1.1] for
more details), it is not sufficient to improve the upper bound given in Theorem 2.
• We recall that the infimum of the functional λ1(Ω)T (Ω)|Ω| on the class of open sets is zero
(see [2, Remark 2.4]. As for the class of bounded convex subsets of Rn (with n ≥ 2), it
is conjectured that the infimum is given by the constant π
2
12
6
(n+1)(n+2) , see [1, Conjec-
ture 4.2]. In [3], the authors prove the following estimates
λ1(Ω)T (Ω)
|Ω| ≥
(π
2
)2 1
nn+2(n +2) , for n ≥ 3,
and
λ1(Ω)T (Ω)
|Ω| ≥
π2
48
, for n = 2,
that have been improved in [6, Remark 4.1], where the authors provide the lower bound
λ1(Ω)T (Ω)
|Ω| ≥
(π
2
)2 1
n(n +2) .
We also refer to [4, Theorems 1.4 & 1.5] for finer lower bounds in restricted classes of
convex planar sets.
• An interesting tool to visualize the inequalities relating three functionals is the Blaschke–
Santaló diagram. In our case, we are interested in the functionals: torsion T , fundamental
frequency λ1 and measure | · | in the case of planar convex sets, which leads to consider
set of points
D = {(λ1(Ω),T −1(Ω)) |Ω⊂R2 is convex and |Ω| = 1}.
We note that this diagram has been theoretically studied in [10] and we refer to [1,
7] for results in the case of open sets. In Figure 2, we plot an approximation of the
diagram D obtained by randomly generating 105 convex polygons (we used the algorithm
presented in [14]) for which we compute the involved functionals, we also plot the curves
corresponding to the best known inequalities relating the 3 functionals, namely:
– the Kohler-Jobin inequality
T (Ω)λ1(Ω)
2 ≥ T (B)λ1(B)2,
– the Faber–Krahn inequality
|Ω|λ1(Ω) ≥ |B |λ1(B),
– the inequality (4)
λ1(Ω)T (Ω)
|Ω| ≤
π2
12
(
1+ 2
√
2(6+π2)−π2
4+π2
)2
≈ 0.996613. . .
We also plot in dashed line the curves corresponding to the conjectures
π2
24
< λ1(Ω)T (Ω)|Ω| <
π2
12
.
C. R. Mathématique — 2022, 360, 241-246
246 Ilias Ftouhi
20 25 30 35 40 45 50 55
30
40
50
60
70
80
90
100
Figure 2. Blaschke–Santaló diagram D and inequalities.
Acknowledgements
We want to thank warmly the anonymous referee who allows us to improve the writing and clarity
of this paper.
References
[1] M. van den Berg, G. Buttazzo, A. Pratelli, “On relations between principal eigenvalue and torsional rigidity”, Com-
mun. Contemp. Math. 23 (2021), no. 08, article no. 2050093.
[2] M. van den Berg, G. Buttazzo, B. Velichkov, “Optimization problems involving the first Dirichlet eigenvalue and the
torsional rigidity”, in New trends in shape optimization, ISNM. International Series of Numerical Mathematics, vol.
166, Birkhäuser/Springer, 2015, p. 19-41.
[3] M. van den Berg, V. Ferone, C. Nitsch, C. Trombetti, “On Pólya’s inequality for torsional rigidity and first Dirichlet
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[4] ——— , “On a Pólya functional for rhombi, isosceles triangles, and thinning convex sets”, Rev. Mat. Iberoam. 36
(2020), no. 7, p. 2091-2105.
[5] T. Bonnesen, W. Fenchel, Theorie der konvexen Körper, Springer, 1974, Berichtigter Reprint, vii+164+3 pages.
[6] L. Brasco, D. Mazzoleni, “On principal frequencies, volume and inradius in convex sets”, NoDEA, Nonlinear Differ.
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[7] G. Buttazzo, A. Pratelli, “An application of the continuous Steiner symmetrization to Blaschke–Santaló diagrams”,
ESAIM, Control Optim. Calc. Var. 27 (2021), article no. 36 (13 pages).
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[9] B. Kawohl, T. Lachand-Robert, “Characterization of Cheeger sets for convex subsets of the plane”, Pac. J. Math. 225
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[10] I. Lucardesi, D. Zucco, “On Blaschke–Santaló diagrams for the torsional rigidity and the first Dirichlet eigenvalue”,
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[11] E. Makai, “On the principal frequency of a membrane and the torsional rigidity of a beam”, in Studies in mathemati-
cal analysis and related topics, Stanford University Press, 1962, p. 227-231.
[12] E. Parini, “Reverse Cheeger inequality for planar convex sets”, J. Convex Anal. 24 (2017), no. 1, p. 107-122.
[13] G. Pólya, G. Szegö, Isoperimetric Inequalities in Mathematical Physics, Annals of Mathematics Studies, vol. 27,
Princeton University Press, 1951, xvi+279 pages.
[14] V. Sander, “Generating Random Convex Polygons”, http://cglab.ca/~sander/misc/ConvexGeneration/convex.html.
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