Comptes Rendus Mathématique

Lukas Lewark

Comptes Rendus
Mathématique
Lukas Lewark
The second coefficient of the Alexander polynomial as a satellite obstruction
Volume 363 (2025), p. 7-11
Online since: 6 March 2025
https://doi.org/10.5802/crmath.694
This article is licensed under the
Creative Commons Attribution 4.0 International License.
http://creativecommons.org/licenses/by/4.0/
C EN T R E
MER S ENN E
The Comptes Rendus. Mathématique are a member of the
Mersenne Center for open scientific publishing
www.centre-mersenne.org — e-ISSN : 1778-3569
Comptes Rendus. Mathématique
2025, Vol. 363, p. 7-11
https://doi.org/10.5802/crmath.694
Research article / Article de recherche
Geometry and Topology / Géométrie et Topologie
The second coefficient of the Alexander
polynomial as a satellite obstruction
Le deuxième coefficient du polynôme d’Alexander
comme obstruction satellitaire
Lukas Lewark a
a ETH Zürich, Rämistrasse 101, 8092 Zürich, Switzerland
URL: https://people.math.ethz.ch/~llewark/
E-mail: llewark@math.ethz.ch
Abstract. A set P of links is introduced, containing positive braid links as well as arborescent positive Hopf
plumbings. It is shown that for links in P , the leading and the second coefficient of the Alexander polynomial
have opposite sign. It follows that certain satellite links, such as (n,1)-cables, are not in P .
Résumé. Un ensemble P de liens est introduit, contenant les clôtures de tresses positives ainsi que les
plombages arborescents de bandes de Hopf positives. Il est démontré que pour les liens appartenant à P ,
le premier et le deuxième coefficient du polynôme d’Alexander sont de signe opposé. Il s’ensuit que certains
liens satellites, tels que les câbles (n,1), n’appartiennent pas à P .
Keywords. Alexander polynomial, Hopf plumbings, satellite knots, arborescent knots, positive braids.
Mots-clés. Polynôme d’Alexander, plombages de Hopf, nœuds satellites, nœuds arborescents, tresses posi-
tives.
2020 Mathematics Subject Classification. 57K10, 57K14.
Manuscript received 11 March 2024, revised 9 October 2024, accepted 13 October 2024.
In this short note, we introduce a certain set P of fibered links in S3, which in particular
contains non-split positive braid links and arborescent positive Hopf plumbings, and is closed
under connected sum. We then prove the following.
Theorem 1. For all links in P , the second coefficient of the Alexander polynomial is negative.
Let us briefly define the above-mentioned terms. A link L, i.e. a non-empty compact oriented
one-dimensional smooth submanifold of S3, is fibered if there exists a fibration S3 \ L → S1
whose every fiber is the interior of a Seifert surface of L, i.e. a compact connected oriented two-
dimensional smooth submanifold of S3 with boundary L. The fiber surface of a fibered link L is
uniquely determined up to isotopy, and it is the unique Seifert surface of minimal genus for L (see
e.g. [12, Proposition 2.19] and [4, Lemma 5.1]). For example, a disk is the fiber surface of the
unknot; and an unknotted band with a left-handed full-twist, called a positive Hopf band, is the
fiber surface of the positive Hopf link. On the other hand, split links are not fibered. Here, a link L
is called split if it admits a 2-sphere S ⊂ S3 \L such that each component of S3 \S contains at least
one component of L. Further examples of fibered links may be constructed as follows: given a
Seifert surface Σ0 and a properly embedded arc h ⊂ Σ0, let Σ+ be the union of Σ0 with a positive
ISSN (electronic): 1778-3569 https://comptes-rendus.academie-sciences.fr/mathematique/
8 Lukas Lewark
L+ L0 L−
Figure 1. The links L+,L0,L− appearing in the skein relation.
Hopf band H along a square S, such that S ⊂Σ0 is a product neighborhood of h, S ⊂ H is a product
neighborhood of a cocore of H , and H is contained in a small neighborhood of h in S3, as shown
in Figure 2. Then Σ+ is a positive Hopf plumbing of Σ0, and Σ+ is a fiber surface if Σ0 is.
Let us now define the above-mentioned two classes of fibered links. Firstly, an arborescent pos-
itive Hopf plumbing is associated with a plane tree T in the following way (see [3, Chapter 12],
[2,5]): place a positive Hopf band at each vertex of T , and plumb any two bands whose corre-
sponding vertices are connected by an edge; the embedding of T into R2 dictates the order of
the (disjoint) plumbing squares around each band. Secondly, consider Artin’s braid group on n
strands with its standard generators σ1, . . . ,σn−1 [1]. By closing off its strands, a braid β gives rise
to a link called the closure of β. A braid is called positive if it can be written as a word in the gen-
erators σi (without their inverses). Closures of positive braids are called positive braid links; they
are fibered if they are non-split [13].
The Alexander polynomial ∆L(t ) of a link L is an integer Laurent polynomial in t 1/2. We use
the Conway normalization; that is to say, ∆U (t ) = 1 for the unknot U , and the skein relation
∆L+ (t ) =∆L− (t )+
(
t 1/2 − t−1/2)∆L0
holds, whenever L±,L0 are three links as shown in Figure 1. These equations determine ∆L(t )
uniquely for all links L. Non-zero Alexander polynomials are of the form
∆L(t ) = ad t d +ad−1t d−1 +·· ·+a−d+1t−d+1 +a−d t−d
with d(L) = d ≥ 0 an integer and ai = a−i if L has an odd number of components, and d(L) ≥ 1/2
a half-integer and ai = −a−i if L has an even number of components. For ∆L ̸= 0, let us call
α(L) = ad ̸= 0 the leading coefficient and β(L) = ad−1 the second coefficient. It is well-known that
for fibered links L (such as the links in P ), the Alexander polynomial ∆L is non-zero, d(L) equals
b1(Σ)/2 for Σ⊂ S3 a fiber surface of L, and α(L) =±1. We shall see in the proof of Theorem 1 that
α(L) = 1 for all L ∈ P . Thus, if one prefers the convention that the Alexander polynomial is only
well-defined up to multiplication with ±t k , one may state Theorem 1 as follows: for all links L
in P , the leading and second coefficient of ∆L have opposite sign (i.e. α(L) ·β(L) < 0).
Theorem 1 and its proof are inspired by Ito’s theorem [6, Corollary 2] that −β(L) equals the
number of prime connected summands of L if L is a positive braid knot. Our main motivation
for Theorem 1 is the satellite obstruction stated below in Corollary 2. Given a so-called pattern P ,
which is a link in the solid torus, and a companion, which is a knot in S3, the satellite link P (K ) is
the link obtained from P by tying the knot K into the solid torus containing P , preserving standard
longitudes. The winding number w(P ) of the pattern is the algebraic intersection number of P
with a meridional disk of the solid torus, or equivalently, the class of P in the first homology group
of the solid torus identified with Z.
Corollary 2. Let P be a pattern with winding number not equal to ±1 such that the product of the
leading and the second coefficient of ∆P (U ) is non-negative. Then P (K ) ̸∈P for all knots K .
Proof. The Alexander polynomial of P (K ) equals (see e.g. [9, Theorem 6.15 and its proof])
∆P (K )(t ) =∆K
(
t w(P )
) ·∆P (U )(t ).
Lukas Lewark 9
L+ L0
h
L−
Figure 2. The links L+,L0,L−, the fiber surfaces Σ+,Σ0, the arc h, and the Seifert surface Σ−
featuring in Definition 3.
It follows that
α
(
P (K )
)=α(K ) ·α(P (U )),
β
(
P (K )
)=α(K ) ·β(P (U )).
Multiplying the two equations, one sees that the hypothesis α(P (U )) · β(P (U )) ≥ 0 implies
that α(P (K )) ·β(P (K )) ≥ 0. By Theorem 1, it follows that P (K ) ̸∈P . □
The hypotheses for P in Corollary 2 are e.g. satisfied by the (n,1)-cable patterns for n ≥ 2, since
P (U ) is the unknot (which satisfies α(U ) = 1,β(U ) = 0). So in particular, Corollary 2 recovers
Krishna’s recent theorem that (n,1)-cables of non-trivial knots are never positive braid knots [8,
Theorem 1.5].
Let us now give the definition of P and prove our results.
Definition 3 (see Figure 2). Let P be the smallest set of fibered links in S3 that contains the positive
Hopf link and satisfies the following. Let L0 ∈ P with fiber surface Σ0, and h ⊂ Σ0 a properly
embedded arc. Let Σ− be the surface obtained from Σ0 by cutting along h, and let Σ+ be the surface
obtained from Σ0 by plumbing a positive Hopf band to Σ0 along h. Let us write L± = ∂Σ±. If the
following holds:
“The coefficient of t b1(Σ−)/2 in ∆L− (t ) is less than or equal to 1.” (∗)
then L+ ∈P .
Let us make some remarks regarding this definition; note that (∗) is in particular satisfied if L−
is fibered (because then the coefficient is ±1 or 0), or if L− is split (because then ∆L− is zero, and
so is the coefficient). Also remark that by definition, for all links L ∈ P with the exception of the
positive Hopf link, there exists links L− and L0 such that L plays the role of L+. Finally, note that
as plumbings of positive Hopf bands, all links in P are strongly quasipositive [11].
Proposition 4.
(1) P is closed under connected sum.
(2) P contains all non-trivial closures of non-split positive braids.
(3) P contains all non-trivial arborescent positive Hopf plumbings.
Proof.
(1). Let L,L′ ∈ P . Let us show that L#L′ ∈ P (where the connected sum # is taken along any
choice of components), by induction over the first Betti number b of the fiber surface of L#L′.
By definition of P , the Hopf link is the only link with Betti number 1 in P . So if b = 2, then L
and L′ are both the positive Hopf link, and so L#L′ ∈ P . Otherwise, if b > 2, then L or L′ is not
the positive Hopf link; without loss of generality, suppose L is not. Then, by definition of P ,
10 Lukas Lewark
one may pick links L− and L0 for L+ = L as in Definition 3. Hence L+#L′ is obtained from L0#L′
by plumbing a positive Hopf band along an arc h, such that cutting along h yields L−#L′. The
link L0#L′ is in P by induction. Since α(L′) = 1 (see proof of Theorem 1) and L− satisfies (∗), so
does L−#L′. It follows that L+#L′ = L#L′ ∈P .
(2). Let L be a non-trivial non-split positive braid link. We show by induction over the first
Betti number b of the fiber surface of L that L ∈ P . If b = 1, then L is the positive Hopf link,
and so L ∈ P . Let us now assume b ≥ 2. An elegant argument by Rudolph [10, proof of last
proposition] shows that a positive braid link that is not an unlink can be written as the closure of
a positive braid that starts with the square of a generator, i.e. is of the formσ2kβwith positiveβ. So
let us pick such a braid σ2kβ with closure L. Let L0 and L− be the links obtained as the closures of
σkβ and β, respectively. One may choose an arc h in the fiber surface of L0 such that plumbing a
positive Hopf band along h yields L+ = L, and cutting along h yields L−. Since b ≥ 2, it follows that
L0 is non-trivial. Moreover, L0 is the closure of the non-split positive braid σkβ. Hence L0 ∈ P
follows by the induction hypothesis. Finally, L− is the closure of a positive braid. Thus L− is either
split or fibered. In either case, L− satisfies condition (∗). Thus L+ = L ∈P by definition of P .
(3). Let T be a plane tree, and L the associated link. We proceed by induction over the number
of vertices of T . If T consists of a single vertex, then L is a positive Hopf link, so L ∈P . Otherwise,
pick a leaf v of T with parent w . Let L0 be the link associated with the plane tree T \ {v}. There
is an arc h in the fiber surface of L0 such that plumbing a positive Hopf band along h yields the
fiber surface of L+ = L, and cutting along h yields a link L− that is a connected sum of the links
associated with the plane tree components of the forest T \{v, w}. By induction, L0 ∈P . Moreover,
L− satisfies (∗) since it is fibered (though it may not be in P , namely if T \ {v, w} is empty and L−
is thus the unknot). By definition of P , it follows that L+ = L ∈P . □
Proof of Theorem 1. Let L ∈P . Let us write b for the first Betti number of the fiber surface of L,
so that d(L) = b/2. We show α(L) = ab/2(L) = 1 and β(L) = ab/2−1(L) ≤ −1 by induction over b.
If b = 1, then L is the positive Hopf link, with Alexander polynomial t 1/2 − t−1/2 satisfying α = 1
and β = −1. If b ≥ 2, by definition there exists a fibered link L0 ∈ P , whose fiber surface Σ0
contains an arc h ⊂ Σ0 such that plumbing a positive Hopf band along h yields the fiber surface
Σ+ of L+ = L, and cutting along h yields a Seifert surface Σ− with L− = ∂Σ− satisfying (∗). Since
L0 is fibered and b1(Σ0) = b−1, we have d(L0) = b/2−1/2. Since L− is the boundary of the Seifert
surface Σ− (which may or may not be a fiber surface) with b1(Σ−) = b−2, we have d(L−) ≤ b/2−1.
By (∗), the coefficient c of t b/2−1 in ∆L− is less than or equal to 1. Now, the skein relation holds
for L±,L0 (compare Figures 1 and 2). Writing o for any linear combinations of powers of t with
exponent less than b/2−1, we get
α(L+)t b/2 +β(L+)t b/2−1 +o = c · t b/2−1 +
(
t 1/2 − t−1/2)(α(L0)t b/2−1/2 +β(L0)t b/2−3/2)+o.
Equating the coefficients of t b/2 and t b/2−1 gives
α(L+) =α(L0) and β(L+) = c −α(L0)+β(L0).
We have α(L0) = 1 and β(L0) ≤ −1 by induction hypothesis, and c ≤ 1 by (∗). It follows that
α(L+) = 1 and β(L+) ≤−1 as desired. □
While I was writing this text, Ito independently obtained an obstruction similar to Corollary 2.
Namely, Theorem A.2 in [7], whose proof likewise relies on the Alexander polynomial, states that
satellites with patterns satisfying a certain condition (different from the condition in Corollary 2)
are never positive braid knots.
Lukas Lewark 11
Acknowledgments
My gratitude goes to Siddhi Krishna for discussions about which cables are positive braid knots,
to Filip Misev and Peter Feller for inspiring conversations, and to Peter Feller and Louis-Hadrien
Robert for comments on a first version of the text.
Declaration of interests
The author does not work for, advise, own shares in, or receive funds from any organization
that could benefit from this article, and has declared no affiliations other than their research
organizations.
References
[1] E. Artin, “Theorie der Zöpfe”, Abh. Math. Semin. Univ. Hamb. 4 (1925), no. 1, pp. 47–72.
[2] S. Baader, “Hopf plumbing and minimal diagrams”, Comment. Math. Helv. 80 (2005), no. 3,
pp. 631–642.
[3] F. Bonahon and L. C. Siebenmann, “New Geometric Splittings of Classical Knots and the
Classification and Symmetries of Arborescent Knots”, 2016. Online at https://dornsife.usc.
edu/francis-bonahon/preprints-available/.
[4] A. L. Edmonds and C. Livingston, “Group actions on fibered three-manifolds”, Comment.
Math. Helv. 58 (1983), no. 4, pp. 529–542.
[5] D. Gabai, “Genera of the arborescent links”, Mem. Am. Math. Soc. 339 (1986), pp. 1–98.
[6] T. Ito, “A note on HOMFLY polynomial of positive braid links”, Int. J. Math. 33 (2022), no. 4,
article no. 2250031 (18 pages).
[7] T. Ito, “Satellite fully positive braid links are braided satellite of fully positive braid links”,
2024. Online at https://arxiv.org/abs/2402.01129.
[8] S. Krishna, “Taut foliations, braid positivity, and unknot detection”, 2023. Online at https:
//arxiv.org/abs/2312.00196.
[9] W. B. R. Lickorish, An introduction to knot theory, Graduate Texts in Mathematics, Springer,
1997, pp. x+201.
[10] L. Rudolph, “Braided surfaces and Seifert ribbons for closed braids”, Comment. Math. Helv.
58 (1983), pp. 1–37.
[11] L. Rudolph, “Quasipositive plumbing (constructions of quasipositive knots and links. V)”,
Proc. Am. Math. Soc. 126 (1998), no. 1, pp. 257–267.
[12] L. Rudolph, “Knot theory of complex plane curves”, in Handbook of knot theory, Elsevier,
2005, pp. 349–427.
[13] J. R. Stallings, “Constructions of fibred knots and links”, in Algebraic and geometric topology
(Proc. Sympos. Pure Math., Stanford Univ., Stanford, Calif., 1976), Part 2, Proceedings of
Symposia in Pure Mathematics, American Mathematical Society, 1978, pp. 55–60.


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Comptes Rendus
Mathématique
Emu Kondo, Shinya Moritoh and Yumi Tanaka
A direct proof of the weighted Pólya–Knopp inequality following Carleson’s method
Volume 363 (2025), p. 599-602
Online since: 5 June 2025
https://doi.org/10.5802/crmath.745
This article is licensed under the
Creative Commons Attribution 4.0 International License.
http://creativecommons.org/licenses/by/4.0/
C EN T R E
MER S ENN E
The Comptes Rendus. Mathématique are a member of the
Mersenne Center for open scientific publishing
www.centre-mersenne.org — e-ISSN : 1778-3569
Comptes Rendus. Mathématique
2025, Vol. 363, p. 599-602
https://doi.org/10.5802/crmath.745
Research article / Article de recherche
Harmonic analysis / Analyse harmonique
A direct proof of the weighted Pólya–Knopp
inequality following Carleson’s method
Une preuve directe de l’inégalité de Pólya–Knopp à poids
par la méthode de Carleson
Emu Kondo a, ShinyaMoritoh b and Yumi Tanaka c
a Graduate School of Humanities and Sciences, Nara Women’s University, Nara, Japan 630-8506
b Department of Mathematics, Nara Women’s University, Nara, Japan 630-8506
c Center for Diversity and Inclusion, Nara Women’s University, Nara, Japan 630-8506
E-mails: wae_kondo@cc.nara-wu.ac.jp, moritoh@cc.nara-wu.ac.jp, yumi_mrth_logx@hera.eonet.ne.jp
Abstract. The aim of the paper is to provide a direct proof of the weighted Pólya–Knopp inequality. This in-
equality (which is a limiting case of the Ariño–Muckenhoupt inequalities), involving non-increasing func-
tions, was initially established by Sbordone–Wik, who proved its validity under the necessary and sufficient
condition that the weight satisfies an appropriate doubling condition. Our main contribution is to use Car-
leson’s approach to Carleman’s inequality in conjunction with Hardy’s lemma and Sbordone–Wik’s doubling
condition, in order to obtain the weighted Pólya–Knopp inequality.
Résumé. L’objectif de cet article est de fournir une preuve directe de l’inégalité de Pólya–Knopp à poids.
Cette inégalité (qui est un cas limite des inégalités de Ariño–Muckenhoupt), portant sur des fonctions
décroissantes, a été montrée par Sbordone–Wik, qui ont montré qu’elle est valide si elle seulement si le poids
satisfait une condition adaptée de doublement. Notre contribution principale est d’utiliser la méthode de
Carleson pour la preuve de l’inégalité de Carleman, en conjonction avec le lemme de Hardy et la condition
de doublement de Sbordone–Wik, afin de prouver directement l’inégalité de Pólya–Knopp à poids.
Keywords. Hardy’s inequality, Pólya–Knopp’s inequality, weight, doubling condition, nonincreasing.
Mots-clés. Inégalité de Hardy, inégalité de Pólya–Knopp, poids, doublement, convexité.
2020 Mathematics Subject Classification. 26D15, 26A51.
Manuscript received 31 December 2024, revised 20 February 2025 and 29 March 2025, accepted 27 March 2025.
1. Introduction
Let the Ariño–Muckenhoupt class Bp (1 ≤ p < ∞) be defined as the set of all nonnegative
functions W for which a constant B exists such that∫ ∞
r
(
r
x
)p
W (x)dx ≤ B
∫ r
0
W (x)dx for all r > 0,
and let the class B∞ be the union of all such Bp .
ISSN (electronic): 1778-3569 https://comptes-rendus.academie-sciences.fr/mathematique/
600 Emu Kondo, Shinya Moritoh and Yumi Tanaka
Let 1 ≤ p <∞. Then, the Ariño–Muckenhoupt theorem [1, Theorem 1.7] asserts the following.
Ariño–Muckenhoupt theorem ([1, Theorem 1.7]). The weighted Hardy inequality∫ ∞
0
(
1
x
∫ x
0
f (t )1/p dt
)p
W (x)dx ≤C
∫ ∞
0
f (x)W (x)dx
holds for all nonnegative, nonincreasing functions f on [0,∞) if and only if W belongs to the
class Bp .
By Riesz [7],
(∫ x
0 f (t )
1/p dt/x
)p decreases and tends to exp(∫ x0 log f (t )dt/x) as p increases
to ∞. Therefore, as a limiting case of this theorem, it is natural to try to characterize the class
B∞ such that the following holds, where we use the notation G f (x) = exp
(∫ x
0 log f (t )dt/x
)
: the
weighted Pólya–Knopp inequality∫ ∞
0
G f (x)W (x)dx ≤C
∫ ∞
0
f (x)W (x)dx (1)
holds for all positive, nonincreasing functions f on [0,∞) if and only if W ∈ B∞.
In Sbordone–Wik [8], the class B∞ is defined using the doubling condition (Da) below. More-
over, it is proved in [8, Theorem 5] that the class B∞ is the union of all Bp (1 ≤ p <∞), and the
inequality (1) is proved in [8, Theorem 6].
The aim of this paper is to provide a direct proof of the inequality (1) by making full use of
Carleson’s approach to Carleman’s inequality.
Following Carleson [4], we express a nonincreasing function f (x) on (0,∞) by using a nonin-
creasing function h(x) as f (x) = eh(x), where h : (0,∞) → R∪ {±∞}. Then G f (x) = eH(x) and the
inequality (1) can be written as∫ ∞
0
eH(x)W (x)dx ≤C
∫ ∞
0
eh(x)W (x)dx, (2)
where
H(x) := 1
x
∫ x
0
h(t )dt , x ∈ (0,∞).
The inequality where W (x) = x p (−1 < p <∞) and C = ep+1 is treated in Carleson [4]. The case
where p = 0 is known as the Pólya–Knopp inequality [5]. See also Carleman [3].
Our paper is organized as follows. In Section 2, we restate the theorem to be considered and
then give a direct proof of the inequality (2) following Carleson’s method. A few comments are
mentioned in Section 3.
2. Theorem and a direct proof of the weighted Pólya–Knopp inequality
For the convenience of the reader, we restate the result to be proved.
Theorem 1. Let the weight function W belong to the class B∞. Then the weighted Pólya–
Knopp inequality (2) holds for all nonincreasing function h : (0,∞) → R∪ {±∞}, where H(x) =
(1/x)
∫ x
0 h(t )dt .
Proof. For a nonnegative function W (x), suppose that it satisfies the B∞ condition, or equiva-
lently, the following doubling condition (Da) for some a > 1:∫ ar
0
W (x)dx ≤Ca
∫ r
0
W (x)dx for all r > 0. (Da)
For this equivalence, see Sbordone–Wik [8, Theorem 5]. The inequality (2) is now to be proved.
Because h is nonincreasing, we have axH(ax) ≤ xH(x)+ (ax − x)h(x) (even in the case where
Emu Kondo, Shinya Moritoh and Yumi Tanaka 601
h(x) = −∞ and thus this inequality becomes −∞ ≤ −∞). We use this inequality and Hölder’s
inequality to obtain, for any A > 0,∫ A
0
eH(x)W (x)dx =
∫ A/a
0
eH(ax)W (ax)a dx
≤
∫ A
0
eH(x)/a e(a−1)h(x)/aW (ax)a dx
≤
(∫ A
0
eH(x)W (ax)a dx
)1/a(∫ A
0
eh(x)W (ax)a dx
)(a−1)/a
.
(3)
We next note that the functions eH(x) and eh(x) are nonincreasing, and that the condition (Da) can
be written as follows: ∫ r
0
W (ax)a dx ≤Ca
∫ r
0
W (x)dx for all r > 0.
We now recall Hardy’s lemma according to [2, Proposition 3.6 in Chapter 2].
Hardy’s lemma ([2, Proposition 3.6 in Chapter 2]). Let f1 and f2 be nonnegative functions on
[0,∞), and suppose ∫ r
0
f1(x)dx ≤
∫ r
0
f2(x)dx for all r > 0.
Let g be any nonnegative nonincreasing function on [0,∞). Then,∫ A
0
f1(x)g (x)dx ≤
∫ A
0
f2(x)g (x)dx for all A > 0.
In our case, Hardy’s lemma yields∫ A
0
eH(x)W (ax)a dx ≤Ca
∫ A
0
eH(x)W (x)dx, (4)
and ∫ A
0
eh(x)W (ax)a dx ≤Ca
∫ A
0
eh(x)W (x)dx. (5)
Therefore, from (3), (4) and (5), we obtain∫ A
0
eH(x)W (x)dx ≤Ca
(∫ A
0
eH(x)W (x)dx
)1/a(∫ A
0
eh(x)W (x)dx
)(a−1)/a
. (6)
As it is assumed in Ariño–Muckenhoupt [1, p. 730], we can also assume that h is constant on [0,d ]
for some d > 0. We can add this restriction on h without loss of generality by use of the monotone
convergence theorem. Then H is also constant on [0,d ] and we have∫ A
0
eH(x)W (x)dx <∞.
By rearranging the inequality (6) using this finite-valued term, we obtain∫ A
0
eH(x)W (x)dx ≤C a/(a−1)a
∫ A
0
eh(x)W (x)dx.
Letting A tend to ∞, we obtain the desired inequality (2) with the constant C =C a/(a−1)a . □
Remark 2. In Carleson [4], the power function W (x) = xp (−1 < p <∞) is used as a weight. Then,
the doubling constant is Ca = ap+1, and we have that infa>1 C a/(a−1)a = lima→1 C a/(a−1)a = ep+1,
which is known to be the best constant for the power-weighted Pólya–Knopp inequality.
602 Emu Kondo, Shinya Moritoh and Yumi Tanaka
3. Comments
Comment 3. For the use of Carleson’s approach in the related direction of the weighted Hardy
inequality, see Kwon [6].
Comment 4. Let 1 ≤ p < ∞. Then, the Hardy–Littlewood maximal operator is bounded on
the classical Lorentz space Λp (W ) if and only if W satisfies the Bp condition. See Ariño–
Muckenhoupt [1]. The question that remains is what is the limiting case of this equivalence when
p →∞.
Acknowledgments
The authors would like to express their sincere gratitude to an anonymous referee for their very
valuable comments on the representation of nonincreasing functions and on the treatment of
infinities that may appear in the inequalities. They would also like to express their sincere
gratitude to Professors Hitoshi Ishii (Waseda University) and Eiichi Nakai (Ibaraki University) for
valuable suggestions and encouragement.
Declaration of interests
The authors do not work for, advise, own shares in, or receive funds from any organization
that could benefit from this article, and have declared no affiliations other than their research
organizations.
References
[1] M. A. Ariño and B. Muckenhoupt, “Maximal functions on classical Lorentz spaces and
Hardy’s inequality with weights for nonincreasing functions”, Trans. Am. Math. Soc. 320
(1990), no. 2, pp. 727–735.
[2] C. Bennett and R. Sharpley, Interpolation of operators, Pure and Applied Mathematics,
Academic Press Inc., 1988, pp. xiv+469.
[3] T. T. G. Carleman, “Sur les fonctions quasi-analytiques”, in Conférences faites au cinquième
congrès des mathématiciens scandinaves (Helsingfors, 1922), Akateeminen Kirjakauppa,
1923, pp. 181–196.
[4] L. Carleson, “A proof of an inequality of Carleman”, Proc. Am. Math. Soc. 5 (1954), pp. 932–
933.
[5] K. Knopp, “Uber Reihen mit Positiven Gliedern”, J. Lond. Math. Soc. 3 (1928), no. 3, pp. 205–
211.
[6] E. G. Kwon, “On Carleson’s inequality”, J. Inequal. Appl. 2018 (2018), article no. 91 (6 pages).
[7] F. Riesz, “Sur les valeurs moyennes des fonctions”, J. Lond. Math. Soc. 5 (1930), no. 2, pp. 120–
121.
[8] C. Sbordone and I. Wik, “Maximal functions and related weight classes”, Publ. Mat., Barc. 38
(1994), no. 1, pp. 127–155.


A direct proof of the weighted Pólya–Knopp inequality following Carleson’s method

Dario Corona

Alessandro Della Corte

Comptes Rendus
Mathématique
Dario Corona and Alessandro Della Corte
Corrigendum to “The critical exponent functions”
Volume 363 (2025), p. 791-792
Online since: 4 July 2025
https://doi.org/10.5802/crmath.667
This article is licensed under the
Creative Commons Attribution 4.0 International License.
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C EN T R E
MER S ENN E
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www.centre-mersenne.org — e-ISSN : 1778-3569
Comptes Rendus. Mathématique
2025, Vol. 363, p. 791-792
https://doi.org/10.5802/crmath.667
Erratum / Erratum
Dynamical systems / Systèmes dynamiques
Corrigendum to “The critical exponent
functions”
Corrigendum de « Les fonctions de l’exposant critique »
Dario Corona a and Alessandro Della Corte a
a School of Science and Technology, University of Camerino, Italy
E-mails: dario.corona@unicam.it, alessandro.dellacorte@unicam.it
Abstract. We give a corrigendum to our paper [1] entitled “The critical exponent functions”.
Résumé. Nous donnons un corrigendum à notre article [1] intitulé « Les fonctions de l’exposant critique ».
Manuscript received 2 July 2024, accepted 9 July 2024.
In [1], a family of interval maps based on the word-combinatorial concept of critical exponent
was introduced, hence referred to as “critical exponent functions”. A purely combinatorial result,
namely [1, Theorem 7], was presented and employed in that paper to prove some analytical and
dynamical properties of such interval maps. However, [1, Theorem 7] does not hold. Denoting
by {0,1}∗ and {0,1}ω the sets of finite and infinite binary words, respectively, and using E(w)
to represent the critical exponent of a word, [1, Theorem 7] was formulated as follows: “Let
w ∈ {0,1}∗. For every α> max{2,E(w)}, there exists y ∈ {0,1}ω such that E(w y) =α”. This claim is
false, as shown for instance by the counterexample w = 00100100. Indeed, in this case we have
E(w) = 8/3, while both w0 and w1 are 3-powers, so that E(w y) ≥ 3 > 8/3 for every y ∈ {0,1}ω.
However, most of the properties obtained through that combinatorial result remain valid with
minor adjustments. Indeed, after reviewing all the properties of κ2, we find that all of them hold,
except for the following ones:
• The property of being left- or right-Darboux at every point, as stated in [1, Proposi-
tion 19]; it remains unclear if this property holds or not.
• The characterization of the almost fixed points of κ2, specifically in [1, Corollary 24], and
the subsequent uncountability of them (cf. [1, Corollary 25]).
• The existence of the fixed point can be granted only on any left-neighbourhood of xτ, not
in a neighbourhood of any almost fixed point (cf. [1, Corollary 26]); however, this suffices
to prove the existence of countably many fixed points, so that [1, Corollary 27] holds.
Analogously, most of the properties of κn hold. The main results that fail are:
(1) being left- or right-Darboux at every point (cf. [1, Corollary 42]);
(2) the characterization of the almost-fixed points.
All the properties presented in [1] for κ hold.
The interested reader can find more details in [2].
ISSN (electronic): 1778-3569 https://comptes-rendus.academie-sciences.fr/mathematique/
792 Dario Corona and Alessandro Della Corte
Acknowledgments
We are grateful to Saúl Rodríguez Martín for pointing out the error.
Declaration of interests
The authors do not work for, advise, own shares in, or receive funds from any organization
that could benefit from this article, and have declared no affiliations other than their research
organizations.
References
[1] D. Corona and A. Della Corte, “The critical exponent functions”, C. R. Math. 360 (2022),
pp. 315–332.
[2] D. Corona, A. Della Corte and M. Farotti, “Dynamical properties of critical exponent func-
tions”, 2024. Online at https://arxiv.org/abs/2406.14487.


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