Comptes Rendus Mathématique

J. Marshall Ash

Stefan Catoiu

Hajrudin Fejzić

Comptes Rendus
Mathématique
J. Marshall Ash, Stefan Catoiu and Hajrudin Fejzić
A new proof of the GGR conjecture
Volume 361 (2023), p. 349-353
https://doi.org/10.5802/crmath.413
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
2023, Vol. 361, p. 349-353
https://doi.org/10.5802/crmath.413
Functional analysis / Analyse fonctionnelle
A new proof of the GGR conjecture
J. Marshall Asha, Stefan Catoiu∗, a and Hajrudin Fejzićb
a Department of Mathematics, DePaul University, Chicago, IL 60614
b Department of Mathematics, California State University, San Bernardino, CA 92407
E-mails: mash@depaul.edu (J. M. Ash), scatoiu@depaul.edu (S. Catoiu),
hfejzic@csusb.edu (H. Fejzić)
Abstract. For each positive integer n, function f , and point x, the 1998 conjecture by Ginchev, Guerragio,
and Rocca states that the existence of the nth Peano derivative f(n)(x) is equivalent to the existence of all
n(n +1)/2 generalized Riemann derivatives,
Dk,− j f (x) = lim
h→0
1
hn
k∑
i=0
(−1)i
(
k
i
)
f (x + (k − i − j )h),
for j ,k with 0 ≤ j < k ≤ n. A version of it for n ≥ 2 replaces all − j with j and eliminates all j = k − 1. Both
the GGR conjecture and its version were recently proved by the authors using non-inductive proofs based on
highly non-trivial combinatorial algorithms. This article provides a second, inductive, algebraic proof to each
of these theorems, based on a reduction to (Laurent) polynomials.
Mathematical subject classification (2010). 26A24, 13F20, 15A03, 26A27.
Funding. This paper is in final form and no version of it will be submitted for publication elsewhere.
Manuscript received 10 June 2021, revised 30 April 2022, accepted 24 August 2022.
Arguably, the most important application of higher order derivatives is Taylor’s theorem,
asserting that an n times differentiable function f at a point c is approximated near c by a
polynomial p with error f (c +h)− p(c +h) = o(hn) as h → 0. It is also well known that Taylor’s
theorem provides only a sufficient condition for this approximation to happen, and all functions
f with this property are said to be n times Peano differentiable at c.
In 1998, Ginchev, Guerragio, and Rocca (GGR) conjectured the following result:
Conjecture (GGR Conjecture). When n ≥ 2, the following two conditions,
(i) f is n −1 times Peano differentiable at c and
(ii) limh→0 1hn
∑n
j=0(−1)n− j
(n
j
)
f (c + ( j −k)h) exists for all k with 0 ≤ k ≤ n −1,
are sufficient to make f an n times differentiable function at c.
They proved the theorem by hand for n = 2,3,4 in [7], and with the use of a computer they
proved it for n = 5,6,7,8 in [8], leaving the rest as a conjecture. The GGR conjecture was recently
proved in [2] and is now a theorem. The result proved in [2] is slightly stronger for n odd, by
eliminating the second condition for k = 0. A variant of the conjecture, where the bounds for k
∗Corresponding author.
ISSN (electronic) : 1778-3569 https://comptes-rendus.academie-sciences.fr/mathematique/
350 J. Marshall Ash, Stefan Catoiu and Hajrudin Fejzić
are replaced by −(n −2) ≤ k ≤ 0, is proved in [4]. Article [6] sheds some light towards extending
the GGR conjecture for n = 1.
The original statement of the GGR conjecture is actually an equivalent version of the above, by
the principle of mathematical induction: If Dn,k denote the above limits, Ginchev, Guerragio, and
Rocca conjectured that the
(n+1
2
)
limits Dm,k for 0 ≤ k < m ≤ n will be enough for f to be n times
Peano differentiable at c. The limit Dn,0 = Dn,0 f (c) is called the nth Riemann derivative of f at c,
and Dn,n/2 f (c) is the nth symmetric Riemann derivative of f at c. Both of these derivatives were
invented by Riemann in the mid 1800s, see [11]. The Peano derivatives were invented by Peano
in [10] in 1892, and then developed greatly by de la Vallée Poussin in [12]. For this reason, they are
often referred to as de la Vallée Poussin derivatives.
The purpose of this note is to provide a new, simple proof to both the GGR conjecture and its
variant.
∗ ∗ ∗
Let Rn(h) be the difference defined recursively by R1(h) = f (c + h) − f (c), and Rn(h) =
Rn−1(2h)−2n−1Rn−1(h) for n ≥ 2. Closed form formulas for generalizations of these differences
are deduced in [3]; they involve the Gaussian or q-binomial coefficients, so they are q-analogues
of the Riemann differences. Other q-analogues of Riemann differences are found in [1] and [5].
We will use the following 1936 result of Marcinkiewicz and Zygmund in [9].
Theorem 1. Suppose f is n −1 times Peano differentiable at c. If limh→0 Rn(h)/hn exists, then f
is n times Peano differentiable at c.
If we denote ∆k (h) as the difference
∑n
j=0(−1)n− j
(n
j
)
f (c + ( j −k)h) in the GGR limit condition,
then this condition can be concisely written as limh→0∆k (h)/hn exists. It is also obvious that if all
GGR limit conditions are met then the following linear combination
lim
h→0
∑
k ck∆k (sk h)
hn
exists, where ck , sk are arbitrary real constants.
Hence, if we can show that R(h) is a linear combination
∑
k ck∆k (sk h), then the GGR conjecture
follows from Theorem 1.
That this is indeed the case will follow from the analogous result for polynomials, via the linear
isomorphism, ∆(h) := ∑c j f (c + b j h) 7→ d(t ) := ∑c j t b j , from the R-space of all differences of
f at c and h with integer nodes (the b j ), to the R-space R[t , t−1] of all Laurent polynomials in
indeterminate t with real coefficients. In this way, (1) if
∆k (h) =
n∑
j=0
(−1)n− j
(
n
j
)
f (c + ( j −k)h),
then
dk (t ) =
n∑
j=0
(−1)n− j
(
n
j
)
t j−k = t−k (t −1)n ;
(2) the polynomial corresponding to ∆k (sh) is dk (t
s ); and (3) if rn(t ) is the polynomial that
corresponds to Rn(h) under this linear isomorphism, then its recursive definition is r1(t ) = t −1,
and rn(t ) = rn−1(t 2)−2n−1rn−1(t ) for n ≥ 2.
Based on these properties of the above linear isomorphism, we will be done by showing that
the following result is true.
Theorem 2. There are constants ck and sk such that rn(t ) =
∑
ck dk (t
sk ).
Before proceeding with the proof of Theorem 2, we need to make a clarification. Our solution
to the theorem has the numbers sk non-negative integers instead of real numbers, so we can
think of sk as s, with s ≥ 0. Then the ck are viewed as ck,s , for a more precisely indexed sum
J. Marshall Ash, Stefan Catoiu and Hajrudin Fejzić 351
∑
ck dk (t
sk ) =∑∞s=0 ∑k ck,s dk (t s ) =∑∞s=0 ∑k ck,s t−sk (t s−1)n , where the ranges for k will be clarified
later, since these are different for different cases in the proof of the theorem. Summarizing, in
order to prove Theorem 2, it suffices to show that
rn(t ) ∈Vn := span
{
t−sk (t s −1)n
∣∣∣k = (0),1, . . . , n −1, s = 1,2, . . .} ,
where (0) means that the value 0 is taken only for n even.
∗ ∗ ∗
The proof of Theorem 2 is much shorter for the variant of the GGR conjecture than it is for the
conjecture itself. For this reason, we deal with the variant first.
Proof of Theorem 2 (Variant Case)
In this case, by replacing the index of summation k with −k, the range −(n −2) ≤ k ≤ 0 becomes
0 ≤ k ≤ n −2. In this way, Vn becomes
Vn = span
{
t sk
(
t s −1)n ∣∣∣k = 0, . . . , n −2; s = 1,2, . . .} .
The following lemma provides a new set of generators for the space Vn .
Lemma 3. Vn = span{(t s −1)n+k | k = 0,1, . . . , n −2, s = 1,2, . . .}.
Proof. It suffices to show the following equality of subspaces:
span
{
t k (t −1)n
∣∣∣k = 0,1, . . . , n −2}= span {(t −1)n+k ∣∣∣k = 0,1, . . . , n −2} .
Indeed, this is the result of multiplying by (t −1)n both sides of the obvious equation
span
{
1, t , t 2, . . . , t n−2
}= span{1, t −1,(t −1)2, . . . , (t −1)n−2} . 
We are now ready to proceed with the proof of Theorem 2 in its variant case.
Proof of Theorem 2 (Variant Case). Induct on n. When n = 2, r2 = (t − 1)2 is clearly in V2.
Suppose rn ∈ Vn , for some n, n ≥ 2, and prove the same property for n + 1. By Lemma 3, rn is
a linear combination of polynomials of the form (t s − 1)n+k , where s is a positive integer and
k = 0,1, . . . ,n − 2. By the recursion, rn+1(t ) = rn(t 2) − 2nrn(t ) will be a linear combination of
polynomials (
t 2s −1)n+k −2n (t s −1)n+k , for various k and s.
By Lemma 3, these polynomials belong to Vn+1 in all cases, except for k = 0, when(
t 2s −1)n −2n (t s −1)n = (t s −1)n ((t s +1)n −2n)= (t s −1)n+1 p (t s) ,
where p is a polynomial in t of degree n − 1, so that (t s − 1)n+1p(t s ) belongs to the subspace
span{(t s −1)n+1, (t s −1)n+2, . . . , (t s −1)2n} of Vn+1. 
Proof of Theorem 2 (GGR Case)
The GGR case in the proof of Theorem 2 is similar to the variant case. The proof of the inductive
step is now split further into two subcases, n odd and n even. In both cases, following the refined
result of the GGR Theorem from [2],
Vn = span
{
t−sk
(
t s −1)n ∣∣∣k = (0),1, . . . , n −1; s = 1,2, . . .} ,
where (0) means that 0 is taken for n even, and not taken for n odd. More explicitly, this is
Vn =
{
span
{
(t −1)n , t−1(t −1)n , . . . , t−(n−1)(t −1)n , . . .} , n even,
span
{
t−1(t −1)n , t−2(t −1)n , . . . , t−(n−1)(t −1)n , . . .} , n odd,
352 J. Marshall Ash, Stefan Catoiu and Hajrudin Fejzić
where the last dots in both cases mean that the generating set also includes the previously listed
generators evaluated at t s , for all s at least 2. Let Wn be the subspace of Vn spanned by all
generators with s = 1. Then Wn has the expression
Wn =
{
span
{
(t −1)n , t−1(t −1)n , . . . , t−(n−1)(t −1)n} , n even,
span
{
t−1(t −1)n , t−2(t −1)n , . . . , t−(n−1)(t −1)n} , n odd.
The following lemma provides new sets of generators for Wn in both parity cases.
Lemma 4. With the above notation,
Wn =
{
span
{
t−n/2(t −1)n , t−1(t −1)n+1, . . . , t−(n−1)(t −1)n+1} , n even,
span
{
t−(n−1)/2(t −1)n , t−2(t −1)n+1, . . . , t−(n−1)(t −1)n+1} , n odd.
Proof. When n is even, the result follows from t−n/2(t − 1)n being one of the generators in the
definition of Wn , and t−k (t −1)n+1 = t−(k−1)(t −1)n − t−k (t −1)n , for each k = 1, . . . , n−1. The case
when n is odd has a similar proof. 
We are now ready to prove Theorem 2 in the GGR case.
Proof of Theorem 2 (Case GGR). Induct on n. When n = 1, r1 = t−1 ∈V1. We assume that rn ∈Vn
and prove that rn+1 ∈Vn+1 in two possible cases:
Case 1. When n is even,
Vn+1 = span
{
t−1(t −1)n+1, t−2(t −1)n+1, . . . , t−n(t −1)n+1, . . .} .
The inductive hypothesis and Lemma 4 imply that rn(x) is a linear combination of
t−n/2(t −1)n and t−k (t −1)n+1, for k = 1, . . . , n −1,
and their evaluations at t s , for s at least 2. Then rn+1(t ) = rn(t 2)−2nrn(t ) is a linear combination
of two kinds of polynomials and their evaluations at t s , for s ≥ 2. The first kind of polynomial has
the form t−n(t 2 −1)n −2n t−n/2(t −1)n
= t−n(t −1)n ((t +1)n −2n t n/2)= t−n(t −1)n+1p(t ),
where p(t ) is a polynomial degree n −1, hence the whole expression lives inside of
(t −1)n+1 span {t−1, t−2, . . . , t−n} ,
a subspace of Vn+1. The polynomials of the second kind are polynomials of the form
t−2k
(
t 2 −1)n+1 −2n t−k (t −1)n+1, for k = 1, . . . , n −1.
Their second term is a scalar multiple of a generator of Vn+1, while their first term is an (s = 2)-
dilation of the same generator, so all polynomials of the second kind also belong to Vn+1. We
conclude that rn+1 ∈Vn+1, as needed.
Case 2. When n is odd,
Vn+1 = span
{
(t −1)n+1, t−1(t −1)n+1, . . . , t−n(t −1)n+1, . . .} .
The inductive hypothesis and Lemma 4 imply that rn(x) is a linear combination of
t−(n−1)/2(t −1)n and t−k (t −1)n+1, for k = 2, . . . , n −1,
and their evaluations at t s , for s at least 2. Then rn+1(t ) = rn(t 2)−2nrn(t ) is a linear combination
of two kinds of polynomials and their evaluations at t s , for s ≥ 2. The first kind of polynomial is
of the form t−(n−1)(t 2 −1)n −2n t−(n−1)/2(t −1)n
= t−(n−1)(t −1)n ((t +1)n −2n t (n−1)/2)= t−(n−1)(t −1)n+1p(t ),
where p(t ) is a polynomial degree n −1, hence the above expression lives inside of
(t −1)n+1 span {1, t−1, t−2, . . . , t−(n−1)} ,
J. Marshall Ash, Stefan Catoiu and Hajrudin Fejzić 353
a subspace of Vn+1. The polynomials of the second kind are polynomials of the form
t−2k
(
t 2 −1)n+1 −2n t−k (t −1)n+1, for k = 2, . . . , n −1.
Their second term is a scalar multiple of a generator of Vn+1, while their first term is an (s = 2)-
dilation of the same generator, so all polynomials of the second kind also belong to Vn+1. We
conclude that rn+1 ∈Vn+1 in this case as well. 
References
[1] J. M. Ash, S. Catoiu, “Quantum symmetric Lp derivatives”, Trans. Am. Math. Soc. 360 (2008), no. 2, p. 959-987.
[2] J. M. Ash, S. Catoiu, “Characterizing Peano and symmetric derivatives and the GGR conjecture’s solution”, Int. Math.
Res. Not., IMRN (2022), no. 10, p. 7893-7921.
[3] J. M. Ash, S. Catoiu, H. Fejzić, “Gaussian Riemann derivatives”, https://arxiv.org/abs/2211.09209, to appear in Israel
J. Math., 23 pp., online first, 2022.
[4] ——— , “Two pointwise characterizations of the Peano derivative”, https://arxiv.org/abs/2209.04088, preprint, 2022.
[5] J. M. Ash, S. Catoiu, R. Ríos-Collantes-de Terán, “On the nth quantum derivative”, J. Lond. Math. Soc. 66 (2002), no. 1,
p. 114-130.
[6] S. Catoiu, “A differentiability criterion for continuous functions”, Monatsh. Math. 197 (2022), no. 2, p. 285-291.
[7] I. Ginchev, A. Guerraggio, M. Rocca, “Equivalence of Peano and Riemann derivatives”, in Generalized convexity and
optimization for economic and financial decisions (Verona, 1998), Pitagora Editrice, Bologna, 1998, p. 169-178.
[8] I. Ginchev, M. Rocca, “On Peano and Riemann derivatives”, Rend. Circ. Mat. Palermo 49 (2000), no. 3, p. 463-480.
[9] J. Marcinkiewicz, A. Zygmund, “On the differentiability of functions and summability of trigonometric series”,
Fundam. Math. 26 (1936), p. 1-43.
[10] G. Peano, “Sulla formula di Taylor”, Atti Acad. Sci. Torino 27 (1892), p. 40-46.
[11] B. Riemann, Über die Darstellbarkeit einer Funktion durch eine trigonometrische Reihe, Ges. Werke, 2. Aufl., Leipzig,
Dieterichschen Buchhandlung, 1892 (1867), 227-271 pages.
[12] C.-J. de la Vallée Poussin, “Sur l’approximation des fonctions d’une variable réelle et de leurs dérivées par les
pôlynomes et les suites limitées de Fourier”, Belg. Bull. Sciences 1908 (1908), p. 193-254.


A new proof of the GGR conjecture

Nicolau Saldanha

Boris Shapiro

Michael Shapiro

Finiteness of rank for Grassmann convexity

Guillaume Ferriere

Matthieu Hillairet

Unbounded-energy solutions to the fluid+disk system and long-time behavior for large initial data

Pierre Bizeul

Comptes Rendus
Mathématique
Pierre Bizeul
Entropy and Information jump for log-concave vectors
Volume 361 (2023), p. 487-493
Published online: 1 February 2023
https://doi.org/10.5802/crmath.390
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
2023, Vol. 361, p. 487-493
https://doi.org/10.5802/crmath.390
Functional analysis, Probability theory / Analyse fonctionnelle, Probabilités
Entropy and Information jump for
log-concave vectors
Pierre Bizeul∗, a
a Institut de Mathématiques de Jussieu-Paris Rive Gauche, Sorbonne Université, 4
place de Jussieu 75005 Paris, France
E-mail: pbizeul@imj-prg.fr
Abstract. We extend the result of Ball and Nguyen on the jump of entropy under convolution for log-concave
random vectors. We show that the result holds for any pair of vectors (not necessarily identically distributed)
and that a similar inequality holds for the Fisher information, thus providing a quantitative Blachmann–Stam
inequality.
2020 Mathematics Subject Classification. 94A17.
Manuscript received 4 May 2022, accepted 2 June 2022.
1. Introduction
Let X be a random vector distributed according to a measure µ in Rd , with density f with respect
to the Lebesgue measure. We will denote this by X ∼ µ = f dλ. If ∫ f | log f | < +∞, we define its
entropy by
EntL(µ) = EntL(X ) =−
∫
Rd
f log f ,
where the subscript L stands for “Lebesgue”.
It should be noted that this entropy can be either positive or negative, and that for any
invertible matrix A, EntL(AX ) = EntL(X )+ log(|det A|). Entropy is also translation invariant, and
it is classical that, when the covariance matrix is fixed, the Gaussian distribution maximizes
entropy. It will be useful to normalize vectors so that they are centered, and have covariance
matrix identity. Such a vector, as well as its distribution, is called isotropic.
The classical Shannon–Stam inequality asserts that taking a convolution increases entropy: for
two iid random vectors X1 and X2
EntL(X1) ≤ EntL
(
X1 +X2p
2
)
.
Moreover, there is equality if and only if X has a Gaussian distribution. Now, one can wonder
if that equality case is stable, meaning if the entropy jump EntL(
X1+X2p
2
)−EntL(X ) is small, does
∗Corresponding author.
ISSN (electronic) : 1778-3569 https://comptes-rendus.academie-sciences.fr/mathematique/
488 Pierre Bizeul
it imply that X is almost Gaussian? The general answer is no, as one can convince himself by
considering a well chosen double-bump Gaussian [4].
However, when the distribution of X admits a spectral gap, excluding double-bumped type
distributions, some positive answers exist. Recall that X is said to have a spectral gap, or equiva-
lently satisfy a Poincaré inequality, if there exists a constant c > 0 such that for any smooth enough
function f , the variance of f (X ) can be controlled in terms of the euclidean norm of ∇ f (x) as fol-
lows:
var f (X ) ≤ c E
[∣∣∇ f (X )∣∣2 ] .
The smallest such constant c will be denoted cX and called the Poincaré constant of X . Under a
spectral gap assumption, it was proven by Ball, Barthe and Naor in [2] that for a one dimensional
isotropic random variable X ,
EntL
(
X1 +X2p
2
)
−EntL(X1) ≥ 1
2(1+ cX ) (
EntL(G)−EntL(X1)) ,
where G is a standard Gaussian.
We can rewrite the right-hand side as a Kullback–Leibler divergence. Recall that, if X is
isotropic,
EntL(G)−EntL(X ) = D(X ||G) =
∫
Rd
fγ log
(
fγ
)
dγ≥ 0,
where fγ is the relative density of X with respect to the Gaussian measure γ, that is X ∼ fγdγ
and G ∼ γ. In the sequel we shall use the notation D(X ) = D(X ||G). This is a strong measure of
closeness to the Gaussian; for instance the Pinsker–Csiszar–Kullback inequality states that(∫
Rd
∣∣ f − g ∣∣)2 ≤ 1
2
D(X ||G),
where f and g are the density of X and G , respectively. In 2012, Ball and Nguyen generalized the
result to arbitrary dimension, assuming log-concavity of X ([3]). They use a semigroup approach,
differentiating twice the entropy along the Ornstein–Uhlenbeck semigroup.
The Fisher information of a random vector with smooth density f is
IL(X ) =
∫ ∣∣∇ f ∣∣2
f
= 4
∫ ∣∣∣∇(√ f )∣∣∣2 ,
whenever those integrals are well defined. As for the entropy, for a fixed covariance matrix, the
Gaussian is extremal; in this case, it has the smallest information. Information is classically the
derivative of the entropy along the semi-group. In the spirit of the Shannon–Stam Inequality, the
Blachman–Stamn inequality asserts that taking a convolution decreases the information:
IL(X1) ≥ IL
(
X1 +X2p
2
)
.
As before, we can define a relative information, notably to the Gaussian measure dγ. If X ∼
fγdγ is a random vector with smooth density, we will denote its relative information to dγ by
I (X ||G) = Iγ(X ) =
∫ ∣∣∇ fγ∣∣2
fγ
dγ.
When X is isotropic, integrating by parts yields:
I (X ||G) = IL(X )−d = IL(X )− IL(G).
Consequently, for a measure µ on Rd , we write D(µ||γ) = D(X ||G), IL(µ) = IL(X ) and I (µ||γ) =
I (X ||G) where X ∼µ is a random vector distributed according to µ
Pierre Bizeul 489
In this note, we use the same strategy as in [3], but improve their result in two directions. First
we generalize it to non identically distributed pairs of random vectors. For two measures µ and ν
define
δE ,λ(µ,ν) = EntL(Xλ)− (1−λ)EntL(X0)−λEntL(X1),
where X0 and X1 are independent random vectors distributed according to µ and ν respectively,
and Xλ =
p
1−λX0 +
p
λX1. The Shannon–Stam inequality asserts that δλ(µ,ν) ≥ 0 and this
quantity is precisely the deficit in the Shannon–Stam inequality.
Theorem 1 (Quantitative Shannon–Stam). Let µ,ν be two log-concave isotropic measures with
Poincaré constant respectively c0 and c1, and λ ∈ [0,1]. Then,
δE ,λ(µ,ν) ≥
λ(1−λ)
4max(c0,c1)
(
D
(
µ||γ)+D (ν||γ)) .
This should be compared with a recent result of Eldan and Mikulincer ([5, Theorem 3]). They
get a more general result, allowing µ and ν to have different covariance matrices, but in the case
where µ and ν have the same covariance matrix, they get a worst dependence on the Poincaré
constant.
Secondly, we get a same kind of inequality for the information, yielding a stability result for
the Blachman-Stam inequality. Define this time the information deficit of a pair of measures by
δI ,λ(µ,ν) = (1−λ)IL(X0)+λIL(X1)− IL(Xλ)
where X0 and X1 are independent random vectors distributed according to µ and ν respectively,
and Xλ =
p
1−λX0 +
p
λX1
Theorem 2 (Quantitative Blachman–Stam). Let µ,ν be two log-concave isotropic measures with
Poincaré constant respectively c0 and c1, and λ ∈ [0,1]. Then,
δI ,λ(µ,ν) ≥
λ(1−λ)
4max(c0,c1)
(
I (µ||dγ)+ I (ν||dγ)) .
In the sequel, quantities computed with respect to the Lebesgue measure have a subscript “L"
while quantities that are computed with respect to the Gaussian measure have none.
2. A lemma of Ball–Nguyen
Let X be a random vector with smooth density f = e−ψ with respect to the Lebesgue measure. We
define σL(X ) to be the random matrix σL(X ) :=∇2(ψ)(X ) and, we denote
KL(X ) = E
[∥σL(X )∥2] ,
where ∥.∥ denotes the Hilbert–Schmidt norm on matrices and the subscripts L yet again stands
for “Lebesgue”, which we take temporarily as the reference measure. Understanding this quantitiy
will prove to be important later on, as it will appear in the second derivative of the entropy along
the Ornstein–Uhlenbeck semigroup.
We recall a lemma of Ball–Nguyen [3], for which we provide a simple proof.
Lemma 3 ((Ball-Nguyen)). Let X be a random vector in Rd with smooth density, E ⊂ Rd be any
subspace, pE be the orthogonal projection onto E and XE = pE (X ). Then
σL(XE ) ≤ pEE [σL(X ) |XE ] p∗E , a.s,
for the partial order on symmetric matrices.
490 Pierre Bizeul
Proof. Let ψE : E −→R be such that XE ∼ e−ψE (x)d x. We have for x ∈ E
ψE (x) =− ln
∫
E⊥
e−ψ(x,y)d y.
Then, setting dνx = e−ψ(x,y)∫
E⊥ e
−ψ(x,y)d y d y , a straightforward computation shows that:
∀ x ∈ E , ∇2ψE (x) =
∫
E⊥
∇2xxψdνx −covdνx
(∇xψ) .
In particular,
∀ x ∈ E , ∇2ψE (x) ≤
∫
E⊥
∇2xxψdνx ,
which is the desired result. □
If X0 and X1 are two independent random vectors in Rd and λ ∈ [0,1], applying the previous
lemma to the random vector (X0, X1) and the projection p(x, y) =
p
1−λx +pλy yields:
Lemma 4. For any independent random vectors X0, X1 in Rd , with smooth densities, and any
λ ∈ [0,1]
σL(Xλ) ≤ E [(1−λ)σL(X0)+λσL(X1) |Xλ] a.s,
where Xλ =
p
1−λX0 +
p
λX1.
If X0 and X1 are log-concave and independent, then so is Xλ, by Prékopa’s theorem. Thus
σL(X0),σL(X1) and σ(Xλ) are positive matrices, so as in Ball–Nguyen’s article, the inequality
above translates to an inequality on their norm.
Lemma 5. For any log-concave independent random vectors X0, X1 in Rd , with smooth densities,
and any λ ∈ [0,1]
(1−λ)KL(X0)+λKL(X1)−KL(Xλ) ≥λ(1−λ)E
[∥σL(X1)−σL(X0)∥2] .
Remark 6. In particular, we have:
KL(Xλ) ≤ (1−λ)KL(X0)+λKL(X1),
which can be seen as a second-order Blachman–Stam inequality.
Proof. As explained, the matrices being positive, the inequality in Lemma 4 implies that:
∥σL(Xλ)∥ ≤ ∥E [(1−λ)σL(X0)+λσL(X1)|Xλ]∥ .
Taking the expectation of the square and using Jensen’s inequality then implies:
KL(Xλ) ≤ E
[∥(1−λ)σL(X0)+λσL(X1)∥2]
= (1−λ)2KL(X0)+λ2KL(X1)+2λ(1−λ)E [〈σL(X0),σL(X1)〉]
= (1−λ)KL(X0)+λKL(X1)−λ(1−λ)E
[∥σL(X0)−σL(X1)∥2]
since X0 and X1 are independent. □
Now we want to translate this result to the Gaussian setting. Assuming that X has density
fγ = e−ϕ with respect to the Gaussian measure, we similarly introduce:
σ(X ) :=σγ(X ) :=∇2(ϕ)(X )
K (X ) :=Kγ(X ) = E
[∥σ(X )∥2]
I (X ) =Iγ(X ) = E
[∥∥∇ϕ(X )∥∥2] .
Note that the definition of I is consistent with the one given in the introduction. Then Lemma 5
becomes:
Pierre Bizeul 491
Lemma 7. For any log-concave independent isotropic random vectors X0, X1 in Rd , with smooth
densities, and any λ ∈ [0,1], we have
(1−λ)M(X0)+λM(X1)−M(Xλ) ≥λ(1−λ)E
[∥σ(X1)−σ(X0)∥2] , (1)
where M(X ) = K (X )+2I (X ).
Proof. Let X be an isotropic log-concave random vector with density e−ϕ with respect to the
Gaussian measure, and e−ψ with respect to the Lebesgue measure. By definition we have:
σ(X ) =σL(X )− I d .
Hence,
K (X ) = KL(X )−2E [tr(σL(X ))]+n.
Now, by integration by parts:
E [tr(σL(X ))] =
∫
Rn
div
(∇ψ) (x)e−ψ(x)d x
=
∫
Rn
∇ψ(x) ·∇ψ(x)e−ψ(x)d x
= E
[∥∥∇ψ(X )∥∥2]= E[∥∥∇ϕ(X )+X ∥∥2]= I (X )+n.
The lemma follows. □
The next lemma provides a lower bound for the right-hand side in the inequality (1).
Lemma 8. For any log-concave independent isotropic random vectors X0, X1 in Rd with smooth
densities we have
(1−λ)M(X0)+λM(X1)−M(Xλ) ≥
λ(1−λ)
2max(c0,c1)
(
I (X0)+K (X0)+ I (X1)+K (X1)
)
,
where c0,c1 are the Poincaré constants of X0 and X1, respectively.
Proof. We denote by c0 the Poincaré constant of X0. We condition on X1 and apply the Poincaré
inequality for X0 to the function ∇ϕ0(X0)−∇2ϕ1(X1)X0 which is centered and we use the fact that
X0 and X1 are isotropic:
E
[∥∥∇2ϕ0(X0)−∇2ϕ1(X1)∥∥2]≥ 1
c0
E
[∥∥∇ϕ0(X0)−∇2ϕ1(X1)X0∥∥2]
= 1
c0
E
[∥∥∇ϕ0(X0)∥∥2 +∥∥∇2ϕ1(X1)∥∥]
= 1
c0
(I (X0)+K (X1)).
By symmetry we get:
E
[∥∥σγ(X1)−σγ(X0)∥∥2]≥ 1
2max(c0,c1)
(
I (X0)+K (X0)+ I (X1)+K (X1)
)
.
Plugging this into Lemma 7 concludes the proof. □
3. The Ornstein–Uhlenbeck process and proof of the theorems
Let X be a random vector in Rd with density f with respect to the Gaussian measure γ. Let Lγ be
the diffusion operator defined by Lγ f (x) =∆ f (x)−∇ f (x) ·x. The differential equation associated
to Lγ is the modified heat equation:
∂ ft
∂t
= Lγ ft ; f0 = f .
492 Pierre Bizeul
Its solution ft is the relative density of the random vector X t = e−t X +
p
1−e−2t G , where G
is a standard Gaussian independent of X , with respect to the Gaussian measure. From this
description of X t it is clear that X t has a C ∞ density, with integrability properties as good as
f , and that the process commutes with convolutions, in the sense that, with the notations of the
previous sections, for all independent X0, X1 random vectors, and any λ ∈ [0,1], we have
(Xλ)t = (X t )λ in law. (2)
It is also useful to note that if X is such that cX ≥ 1, in particular if X is isotropic, then for all t ≥ 0:
cX t ≤ cX .
Indeed, if X and Y are independent random vectors satisfying a Poincaré inequality andλ ∈ [0,1],
then using the conditional variance formula, a few computations show that
cpλX+p1−λY ≤λcX + (1−λ)cY ,
which in our case yields cX t ≤ e−2t cX + (1−e−2t )×1 ≤ cX .
We denote by (Pt )t ≥0 the semi-group defined by Pt ( f ) = ft . The following computations are
standard (see [1]): if X has finite entropy, then for t > 0
∂
∂t
Ent(X t ) = I (X t ), ∂I (X t )
∂t
=−2I (X t )−2K (X t ) =−M(X t )−K (X t ).
As a consequence, a linear inequality on the information can be integrated along the semi-group
to get the same inequality for the entropy. We also get that ∂I (X t )∂t ≤ −2I (X t ) which implies that
I (X t ) ≤ e−2t I (X0). Moreover, the control role of L comes from the observation that
e2t
∂
(
e−2t I (X t )
)
∂t
=−2M(X t ). (3)
Now we are in position to prove Theorem 2, from which Theorem 1 will be an immediate
corollary.
Let X0 and X1 be two isotropic random log-concave vectors in Rd and λ ∈ [0,1], and Xλ =p
1−λX0+
p
λX1. Denote by (X0)t , (X1)t and (Xλ)t their evolution along the Ornstein–Uhlenbeck
semi-group. Further define:
I0(t ) = I ((X0)t ) , I1(t ) = I ((X1)t ) , Iλ(t ) = I ((Xλ)t ) ,
and similarly, define K0(t ),K1(t ),Kλ(t ), M0(t ), M1(t ), Mλ(t ); using (3), the commutation prop-
erty (2), the observation that the Poincaré constants only decrease along the semi-group and
Lemma 8, we get the following:
Lemma 9. With the previous notations, for all t ≥ 0,
− ∂
∂t
(
(1−λ)e−2t I0(t )+λe−2t I1(t )−e−2t Iλ(t )
)
≥ λ(1−λ)e
−2t
max(c0(t ),c1(t ))
(I0(t )+K0(t )+ I1(t )+K1(t ))
≥− λ(1−λ)
2max(c0,c1)
e−2t
∂
∂t
(I0(t )+ I1(t )) .
Proofs of Theorem 1 and Theorem 2. Integrating the inequality of Lemma 9 from 0 to ∞, we get:
(1−λ)I (X0)+λI (X1)− I (Xλ) ≥−
λ(1−λ)
2max(c0,c1)
∫ +∞
0
e−2t
d
d t
(I0(t )+ I1(t )) d t
= λ(1−λ)
2max(c0,c1)
(
I (X0)+ I (X1)−2
∫ ∞
0
e−2t (I0(t )+ I1(t ))d t
)
≥ λ(1−λ)
4max(C0,C1)
(I (X0)+ I (X1)) ,
Pierre Bizeul 493
where in the last inequality, we used the fact that I1(t ) ≤ e−2t I (X1) and I0(t ) ≤ e−2t I (X0). This
proves Theorem 2.
Integrating Theorem 2 along the semi-group yields Theorem 1. □
References
[1] D. Bakry, I. Gentil, M. Ledoux, Analysis and geometry of Markov diffusion operators, Grundlehren der Mathematischen
Wissenschaften, vol. 348, Springer, 2014.
[2] K. Ball, F. Barthe, A. Naor, “Entropy jumps in the presence of a spectral gap”, Duke Math. J. 119 (2003), no. 1, p. 41-63.
[3] K. Ball, V. H. Nguyen, “Entropy jumps for isotropic log-concave random vectors and spectral gap”, Stud. Math. 213
(2012), no. 1, p. 81-96.
[4] T. A. Courtade, M. Fathi, A. Pananjady, “Quantitative stability of the entropy power inequality”, IEEE Trans. Inf. Theory
64 (2018), no. 8, p. 5691-5703.
[5] R. Eldan, D. Mikulincer, “Stability of the Shannon-Stam inequality via the Föllmer process”, Probab. Theory Relat.
Fields 177 (2020), no. 3-4, p. 891-922.


Entropy and Information jump for log-concave vectors

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Juliette Venel

On the square-root approximation finite volume scheme for nonlinear drift-diffusion equations

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Daniele Cannarsa

Valentina Franceschi

Mario Sigalotti

Local controllability does imply global controllability

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Colin Faverjon

A new proof of Nishioka’s theorem in Mahler’s method

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Fractional Gagliardo–Nirenberg interpolation inequality and bounded mean oscillation

Nathan Bonin

Valentin Ovsienko

Comptes Rendus
Mathématique
Nathan Bonin and Valentin Ovsienko
A shadow Markov equation
Volume 361 (2023), p. 1483-1489
https://doi.org/10.5802/crmath.496
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
2023, Vol. 361, p. 1483-1489
https://doi.org/10.5802/crmath.496
Combinatorics / Combinatoire
A shadow Markov equation
Nathan Bonin a and Valentin Ovsienko ∗,a
a Laboratoire de Mathématiques de Reims, UMR9008 CNRS, Université de Reims Champagne-Ardenne,
U.F.R. Sciences Exactes et Naturelles, Moulin de la Housse - BP 1039, 51687 Reims cedex 2, France
E-mails: nathan.bonin1@etudiant.univ-reims.fr, valentin.ovsienko@univ-reims.fr
Abstract. We introduce an analogue of the classical Markov equation that involves dual numbers a+αε with
ε2 = 0. This equation characterizes the “shadow Markov numbers” recently considered by one of us. We show
that this equation is characterized by invariance by cluster algebra mutations.
Funding. This project was partially supported by the ANR project ANR-19-CE40-0021.
Manuscript received 9 December 2022, revised 2 March 2023 and 8 March 2023, accepted 12 March 2023.
1. Introduction
The classical Markov (or Markoff) equation [6] is the Diophantine equation
a2 +b2 + c2 = 3abc.
A positive integer a is called a Markov number if it can be completed to a triplet of positive in-
tegers (a,b,c) satisfying the Markov equation. Markov numbers attracted much interest in many
branches of mathematics, from number theory to topology, combinatorics, and mathematical
physics.
In this note we will consider pairs of positive integers, (a,α) ∈ Z×Z, that will be written in a
form
A = a +αε, (1)
where ε is a formal parameter such that ε2 = 0. Formal sums (1) are called dual numbers. The
notion of dual number goes back to Clifford and is used in various situations in algebra, geometry,
and mathematical physics. A dual number (1) is called a positive integer if a,α ∈Z>0.
Our goal is to introduce and study a deformation of the Markov equation
A2 +B 2 +C 2 = (3−2ε) ABC . (2)
Solutions (A,B ,C ) to this equation are triples of dual numbers. We call a dual Markov number a
positive integer dual number A that can be completed to a triplet (A,B ,C ) of positive integer dual
numbers satisfying (2).
∗Corresponding author.
ISSN (electronic) : 1778-3569 https://comptes-rendus.academie-sciences.fr/mathematique/
1484 Nathan Bonin and Valentin Ovsienko
The following triplets of dual numbers
(1,1+ε,1+ε), (1+ε,1+ε,2+4ε), (1+ε,2+4ε,5+13ε)
are examples of triplets of dual Markov numbers.
We study several properties of the deformed Markov equation (2). The main property is that it
is preserved by cluster algebra mutations, similarly to the classical case [11]. We will show that it
is characterized by this mutation invariance.
The notion of “shadow” sequences of integers appeared in [8], and was tested on the sequence
of Markov numbers. Every Markov number, a, is accompanied by another positive integer, α,
called the shadow of a. For a definition, see [8] and Section 2. It is a priori not clear, and was
asked in [8], if the sequence of shadows of Markov numbers satisfy any equation. It turns out that
the classical Markov numbers organized together with their shadows in the form of dual numbers
are solutions to (2).
Theorem 1. Given a Markov number a and its shadow α, the corresponding dual number A =
a +αε is a dual Markov number.
We do not have a classification of dual Markov numbers. A classical theorem Markov implies
that every solution (A,B ,C ) to (2) can be obtained by a sequence of transformations called
mutations (see Section 2) from a triplet of the form
A′ = 1+α′ε, B ′ = 1+β′ε, C ′ = 1+γ′ε,
with some α′,β′,γ′ ∈ Z, but it is an open problem to determine the conditions that guarantee
positivity of α,β,γ.
Another open problem is to understand whether the equation (2) is connected to supergeom-
etry. We believe that (2) should be considered as a “superanalogue” of the classical Markov equa-
tion and is related to the supergroup OSp(1|2,Z); see [1]. However, we were unable to prove this
so far. One also notices a certain resemblance of (2) with the super-Markov equation of [5]; see
Section 3.2.
Although the proofs of the statements of this note are rather elementary, we hope that equa-
tion (2) can be useful for what is sometimes called in mathematical physics “super number the-
ory”; see [1,12]. The idea of shadow integer sequences recently initiated in [10] and [8] was further
developed in [1, 4, 14] and generated various unexpected connections.
2. Markov numbers and their shadows
In this section we briefly explain the connection of Markov numbers with Fomin–Zelevinsky
cluster algebras [2], that was proposed in [11]. We also recall the construction of the shadow
Markov sequence introduced in [8].
2.1. Markov numbers and quiver mutations
It is well-known and due to Markov [6], that every positive integer solution to the Markov equation
can be obtained from (1,1,1) by a sequence of transformations (a,b,c) 7→ (a′,b,c), where
a′ = b
2 + c2
a
, (3)
and permutations of a,b, and c. It is easy to check that (a′,b,c) remains a solution if (a,b,c) is a
solution.
Nathan Bonin and Valentin Ovsienko 1485
It was noticed in [11], that the maps (3) can be understood as instances of cluster algebra
mutations in the sense of Fomin–Zelevinsky cluster algebra [2]. The quiver associated with the
sequence of Markov numbers is as follows:
b

OO OO
a coooo
and (3) are precisely the exchange relations for the mutations associated to this quiver.
Integrality of a′ in (3) follows from the Laurent phenomemon [3], and can also be obtained
directly from an equivalent (and more frequently used) form of (3): a′ = 3bc −a.
2.2. The construction of [8]
The notion of shadow Markov sequence [8] is based on transformations (3). The shadow se-
quence is constructed recurrently without any equation on it.
Choose the initial triplet of dual numbers (A1,B1,C1) with
A1 = 1+α1ε, B1 = 1+β1ε, C1 = 1+γ1ε, (4)
where α1,β1,γ1 are arbitrary integers. Define an infinite sequence of transformations (A,B ,C ) 7→
(A′,B ,C ) by the same formula as (3):
A′ := B
2 +C 2
A
. (5)
More explicitly (5) reads
a′ = b
2 + c2
a
, α′ = 2bβ+2c γ−a
′α
a
(6)
and therefore determines both, the real and nilpotent parts of A′ = a′+α′ε. Furthermore, α′ will
remain integer, which follows from the version of Laurent phenomenon proved in [9].
2.3. The Markov tree and its shadow
Similarly to the classical Markov numbers, their shadows can be organized with the help of an
infinite binary tree. The version that we use is the following.
The binary tree is drawn in the plane cutting it into infinitely many regions, and every region
is labeled by a Markov number. Locally the picture is this
A B
•
C
and the transformations (5) correspond to the following branchings:
B
A • • A′
C
The most natural choice of the initial values (α1,β1,γ1) is
(α1,β1,γ1) = (0,1,1). (7)
1486 Nathan Bonin and Valentin Ovsienko
For details; see [8] and Section 4. The tree of Markov numbers with the initial triplet (7) is
presented below together with their shadow:
1 0 • 1 1
•
1 1 2 4
•
• 5 13 •
• 13 40 • • 29 117 •
• 34 • • 194 • • 433 • • 169 •
120 976 2592 921
89 1325 7561 2897 6466 37666 14701 985
354 7875 56287 20226 51320 352360 129640 6761
. . . . . . . . .
Remark. The simplest subsequence of Markov numbers (left branch of the above tree) contains
the odd Fibonacci numbers 1,2,5,13,34,89, . . .. It was noticed in [8], that the sequence
1,4,13,40,120,354,1031,2972,8495, . . .
appearing as the shadow of the odd Fibonacci branch, is the sequencel A238846 (see [13]) which
is the convolution of two bisections of the Fibonacci sequence, F2n+1 and F2n . We refer to [7, 8]
for various ε-deformations of Fibonacci numbers.
The combinatorial nature of other shadow sequence appearing in the above tree is
not understood. For instance, it would be interesting to know properties of the sequence
1,13,117,921,6761, . . . appearing as the shadow of 1,5,29,169,985, . . . which is a bisection of the
classical Pell numbers.
3. Mutation stability
In this section, we relax the positivity condition for the nilpotent part α of dual numbers A =
a +αε. We consider an arbitrary choice of the initial triplet (4) and give a slightly more general
form of the deformed Markov equation. We also obtain several properties of this equation.
3.1. Proof of Theorem 1
The following statement is a more precise and general reformulation of Theorem 1.
Proposition 2.
(i) Every triplet of dual numbers (A,B ,C ) obtained from (4) by a sequence of transforma-
tions (5) satisfies the equation
A2 +B 2 +C 2 = (3−σε) ABC , (8)
where σ=α1 +β1 +γ1.
(ii) Conversely, every triplet of solutions (A,B ,C ) to (8) with positive a,b,c can be obtained
from a triple of the form (4) by a sequence of transformations (5) mixed with permutations
of A,B and C .
Nathan Bonin and Valentin Ovsienko 1487
Proof. To prove Part (i), let us first check that the initial triplet (4) satisfies (8). Indeed, substitut-
ing (4) to the left-hand-side of the equation gives 3+2(α1+β1+γ1)ε, while in the right-hand-side
one obtains 3+3(α1 +β1 +γ1)ε−σε.
Let us check that (8) is stable with respect to the transformations (5).
Lemma 3. The equation
A2 +B 2 +C 2 = X ABC , (9)
where X is a formal parameter, is preserved by the transformations (5).
Proof of the lemma. Let us show that, if (A,B ,C ) is a solution to (9), then (A′,B ,C ), with A′ as
in (5), is also a solution to the same equation.
Assume that (A,B ,C ) is a solution to (9). Then A′ given by (5) can be rewritten:
A′ = X BC − A. (10)
Substituting this expression for A′ to the left-hand-side of (9), one obtains
A′2 +B 2 +C 2 = X 2 B 2C 2 −2X ABC + A2 +B 2 +C 2
= X 2 B 2C 2 −X ABC ,
while in the right-hand-side, one has
X A′BC = X (X BC − A)BC
= X 2 B 2C 2 −X ABC .
We conclude that the triplet (A′,B ,C ) satisfies exactly the same equation, namely A′2 +B 2 +C 2 =
X A′BC . □
The assumption that the initial triplet (A1,B1,C1) satisfies (9) implies that
X = A
2
1 +B 21 +C 21
A1B1C1
,
that gives X = 3− (α1 +β1 +γ1)ε.
Proposition 2(i) follows.
To prove Part (ii), we use the classical theorem that any positive solution (a,b,c) to the Markov
equation can be reduced to (1,1,1) by a sequence of transformations (3). We conclude following
the same sequence of transformations (5). □
3.2. Uniqueness
Let us show that (8) is the only ε-deformation of the Markov equation which is preserved by the
mutations (5).
Consider the most general ε-deformed Markov equation, it can be written in the form
A2 +B 2 +C 2 = 3ABC +P (A,B ,C )ε, (11)
where P is an arbitrary polynomial
P (A,B ,C ) = ∑
i , j ,k≥0
Pi j k A
i B j C k .
Theorem 4. Equation (8) is the only equation among the family (11) preserved by the transforma-
tions (5).
1488 Nathan Bonin and Valentin Ovsienko
Proof. Let us show that, if (11) is preserved by the transformations (5), then all the coefficients
Pi j k of the polynomial P vanish, except perhaps P111, which is arbitrary. Indeed, (5) can be
rewritten
A′ = 3BC − A+ P (A,B ,C )
A
ε
so that the left-hand-side of (11) after the transformation becomes
A′2 +B 2 +C 2 = 9B 2C 2 −6ABC + A2 +B 2 +C 2 +
(
6BC
A
P (A,B ,C )−2P (A,B ,C )
)
ε
= 9B 2C 2 −3ABC +
(
6BC
A
P (A,B ,C )−P (A,B ,C )
)
ε.
The right-hand-side of (11) is
3A′BC +P (A′,B ,C )ε= 9B 2C 2 −3ABC +
(
3BC
A
P (A,B ,C )+P (A′,B ,C )
)
ε.
The assumption the equation is preserved by the mutations then reads
P (A′,B ,C )ε=
(
3BC
A
−1
)
P (A,B ,C )ε,
or with more details ∑
i , j ,k≥0
Pi j k (3BC − A)i B j C k =
(
3BC
A
−1
) ∑
i , j ,k≥0
Pi j k A
i B j C k .
(Note that these terms appear with multiple ε, so that we omit P (A,B ,C )A ε in the left-hand-side
since ε2 = 0.)
The latter equation implies that Pi j k = 0 for i > 1, and similarly for j and k > 1. Indeed, taking
the maximal i , for which Pi j k ̸= 0, if i ≥ 2, all of the terms in the right-hand-side contain A,
while the left-hand-side contain the term Pi j k 3
i B i+ j C i+k . Therefore, the equation (11) cannot
be invariant in this case.
The theorem follows. □
Remark. Another version of super-Markov equation was suggested by Huang, Penner, and
Zeitlin [5]:
a2 +b2 + c2 + (ab +bc +ac)ε= 3(1+ε) abc.
For a discussion and a related conjecture; see [7]. This equation has interesting geometric
properties, albeit it is not preserved by the mutations (3).
4. Initial conditions
Let us end the note with a brief discussion of the choice of initial triplet α1,β1,γ1 in (4). The
choice of [8], which is (α1,β1,γ1) = (0,1,1) is not unique, but can be justified by the following.
A simple property of the equation (8) is that the shadow parts of its solutions linearly depend
on the choice of the initial triplet α1,β1,γ1. This readily follows from the second equation in (6).
Therefore, the shadow part of every solution to (8) is a linear combination of those with one of
the following initial triplets
(0,1,1), (1,1,1), (0,1,0),
that form a basis in the space of initial conditions. However, choosing (1,1,1), one obtains
the shadow part that coincides with the Markov numbers, namely the solutions of the form
(A,B ,C ) = (a + aε,b + bε,c + cε), which is not an interesting case. The choice (0,1,0) produces
negative integers. As mentioned in the introduction, it would be interesting to characterize the
initial triplets that produce only positive shadow parts.
Nathan Bonin and Valentin Ovsienko 1489
Acknowledgements
We would like to thank Philippe Larchevêque, Sophie Morier-Genoud, Julien Rouyer, Michael
Shapiro, Sergei Tabachnikov, Alexey Ustinov, and Alexander Veselov for fruitful discussions. We
are indebted to the anonymous referee for many helpful remarks and suggestions.
References
[1] C. H. Conley, V. Ovsienko, “Shadows of rationals and irrationals: supersymmetric continued fractions and the super
modular group”, 2022, https://arxiv.org/abs/2209.10426.
[2] S. Fomin, A. Zelevinsky, “Cluster algebras. I. Foundations”, J. Am. Math. Soc. 15 (2002), no. 2, p. 497-529.
[3] ——— , “The Laurent phenomenon”, Adv. Appl. Math. 28 (2002), no. 2, p. 119-144.
[4] A. N. W. Hone, “Casting light on shadow Somos sequences”, 2021, https://arxiv.org/abs/2111.10905.
[5] Y. Huang, R. C. Penner, A. M. Zeitlin, “Super McShane identity”, to appear in J. Differ. Geom., 2019, https://arxiv.org/
abs/1907.09978.
[6] A. Markov, “Sur les formes quadratiques binaires indéfinies”, Math. Ann. 15 (1879), p. 381-406.
[7] G. Musiker, N. Ovenhouse, S. W. Zhang, “Double Dimer Covers on Snake Graphs from Super Cluster Expansions”,
2021, https://arxiv.org/abs/2110.06497.
[8] V. Ovsienko, “Shadow sequences of integers, from Fibonacci to Markov and back”, to appear in Math. Intell., 2021,
https://arxiv.org/abs/2111.02553.
[9] V. Ovsienko, M. Shapiro, “Cluster algebras with Grassmann variables”, Electron. Res. Announc. Math. Sci. 26 (2019),
p. 1-15.
[10] V. Ovsienko, S. Tabachnikov, “Dual numbers, weighted quivers, and extended Somos and Gale-Robinson sequences”,
Algebr. Represent. Theory 21 (2018), no. 5, p. 1119-1132.
[11] J. Propp, “The combinatorics of frieze patterns and Markoff numbers”, Integers 20 (2020), article no. A12 (38 pages).
[12] J. M. Rabin, “Super elliptic curves”, J. Geom. Phys. 15 (1995), no. 3, p. 252-280.
[13] N. J. A. Sloane, “The On-Line Encyclopedia of Integer Sequences”, 2010, http://oeis.org.
[14] A. Veselov, “Conway’s light on the shadow of Mordell”, 2022, https://arxiv.org/abs/2208.14184.


A shadow Markov equation

Phuong Le

Comptes Rendus
Mathématique
Phuong Le
Uniqueness of bounded solutions to p-Laplace problems in strips
Volume 361 (2023), p. 795-801
Published online: 11 May 2023
https://doi.org/10.5802/crmath.442
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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e-ISSN : 1778-3569
Comptes Rendus
Mathématique
2023, Vol. 361, p. 795-801
https://doi.org/10.5802/crmath.442
Partial differential equations / Équations aux dérivées partielles
Uniqueness of bounded solutions to
p-Laplace problems in strips
Phuong Le a, b
a Faculty of Economic Mathematics, University of Economics and Law, Ho Chi Minh
City, Vietnam
b Vietnam National University, Ho Chi Minh City, Vietnam
E-mail: phuongl@uel.edu.vn
Abstract. We consider a p-Laplace problem in a strip with two-constant boundary Dirichlet conditions. We
show that if the width of the strip is smaller than some d0 ∈ (0,+∞], then the problem admits a unique
bounded solution, which is strictly monotone. Hence this unique solution is one-dimensional symmetric and
belongs to the C 2 class. We also show that the problem has no bounded solution in the case that d0 < +∞
and the width of the strip is larger than or equal to d0. An analogous rigidity result in the whole space was
obtained recently by Esposito et al. [8]
Keywords. p-Laplace equation, uniqueness, monotonicity, 1D symmetry.
Mathematical subject classification (2010). 35J92, 35A01, 35A02, 35B06.
Manuscript received 26 September 2022, revised 31 October 2022, accepted 9 November 2022.
1. Introduction
Given N ≥ 1, p > 1, a < b, Ω = RN−1 × (0,h) and f is a locally Lipschitz continuous function, we
study the uniqueness, symmetry and monotonicity of solutions to the problem
−∆p u = f (u) inΩ,
u = a on RN−1 × {0},
u = b on RN−1 × {h},
(1)
where ∆p u = div(|∇u|p−2∇u) is the p-Laplacian of u. When p = 2, this is a classical topic which
was investigated in the works of Berestycki, Caffarelli and Nirenberg [2,3], Angenent [1], Clément
and Sweers [4] for various types of domains.
One difficulty that arises in the case p ̸= 2 is that the p-Laplacian is nonlinear. Moreover, this
operator is singular or degenerate depending on 1 < p < 2 or p > 2, respectively. Hence, solutions
to (1) should be understood in the weak sense, see [6, 10]. Throughout our paper, solutions to
p-Laplace problems with boundary conditions will be always understood in the following weak
sense.
ISSN (electronic) : 1778-3569 https://comptes-rendus.academie-sciences.fr/mathematique/
796 Phuong Le
Definition 1. Let f :R→R and g : ∂U →R be continuous functions, where U is a domain of RN .
A function u ∈W 1,ploc (U )∩C (U ) is called a (weak) solution to the problem{
−∆p u = f (u) in U ,
u = g on ∂U
if
u(x) = g (x) for all x ∈ ∂U
and ∫
U
|∇u|p−2∇u ·∇ψ=
∫
U
f (u)ψ for all ψ ∈C 10 (U ). (2)
Throughout this paper, we denote
Z f := {z ∈ [a,b] | f (z) = 0}.
Under some mild assumptions on f , we find a necessary and sufficient condition on h to
ensure that problem (1) has a bounded solution (by bounded solution, we mean weak solutions
in L∞(Ω)). We also show that the solution is unique and monotone increasing in the xN -direction.
Our main result is the following.
Theorem 2. Suppose that f is a locally Lipschitz continuous function such that f (a) = f (b) = 0,
f (s) > 0 for s < a, f (s) < 0 for s > b, and
liminf
t→z+
f (t )
(t − z)p−1 >−∞, limsupt→z−
f (t )
(z − t )p−1 <+∞ for all z ∈ Z f . (3)
Then problem (1) has a bounded solution if and only if
h <
(
p −1
p
) 1
p
∫ b
a
d s[
max[a,b] F −F (s)
] 1
p
, (4)
where F (s) = ∫ sa f (t )dt . Moreover, such a solution u is unique and ∂xN u > 0 in Ω. Consequently,
u ∈C 2(Ω) and u depends only on xN , i.e., u(x) = u(xN ).
If we further assume that f (t ) =− f (a +b − t ) for t ∈ [a, a+b2 ], then
u(xN ) = a +b −u(h −xN ) for xN ∈
[
0,
h
2
]
.
Remark 3. Locally Lipschitz continuous functions satisfy (3) automatically when 1 < p ≤ 2.
Notice also that the integral in (4) may be infinity. In particular, this happens to the Allen–Cahn
nonlinearity f (t ) = t − t 3 when (a,b) = (−1,1) and 1 < p ≤ 2.
Theorem 2 is motivated by a recent result by Esposito et al. [8], where they studied the problem
−∆p u = f (u) in RN ,
|u| ≤ 1 in RN ,
limxN→±∞ u(x ′, xN ) =±1 uniformly in x ′ ∈RN−1,
(5)
under assumptions 2N+2N+2 < p < 2 and f ∈ C 1([−1,1]) such that f (−1) = f (1) = 0, f ′+(−1) < 0,
f ′−(1) < 0 and Z f is finite. They prove that every solution u of (5) depends only on xN and ∂u∂xN > 0
in RN .
The proof in [8] is based on the method of moving planes, in which various types of compar-
ison principles for the p-Laplacian are exploited. This in turn requires the technical assumption
2N+2
N+2 < p < 2 and some restrictions on f .
The moving plane method can also be applied to problem (1), which can be regarded as a
finite version of problem (5), to yield the one-dimensional symmetry of solutions under similar
restrictions on p and f . To cover the full range 1 < p <+∞ and to deal with weaker assumptions
Phuong Le 797
on f in Theorem 2, we follow another approach based on the ideas in [5, 7]. First, we construct
the minimal and maximal solutions to the problem. These solutions depend only on xN . Then
a careful ODE analysis of (1) with N = 1 reveals that the problem has a unique solution in
dimension one, and it is monotone increasing. This implies that solutions in higher dimensions
are also unique, one-dimensional symmetric and monotone increasing in xN .
In Section 2, we prove Theorem 2 for dimension one. In Section 3, we prove it for more
general cases. We always use C to denote a generic positive constant whose concrete value is
not important.
2. ODE analysis
In this section, we study the ODE problem
−(|u′|p−2u′)′ = f (u) in (0,h),
a ≤ u ≤ b in (0,h),
u(0) = a, u(h) = b.
(6)
Theorem 4. Let f be a locally Lipschitz continuous function such that f (a) = f (b) = 0 and (3)
holds. Then problem (6) has a solution if and only if (4) holds. Moreover, such a solution u is
unique, u′ > 0 in (0,h) and u ∈ C 2((0,h)). If we further assume that f (t ) = − f (a + b − t ) for
t ∈ [a, a+b2 ], then
u(t ) = a +b −u(h − t ) for t ∈
[
0,
h
2
]
. (7)
Proof. Assume that problem (6) admits a solution u. By Hopf’s lemma [11], we have u′(0) > 0. Let
(0, t0) be the largest interval such that u′(t ) > 0 for all t ∈ (0, t0). By the standard regularity, u is C 2
in (0, t0) and in this interval
(p −1)|u′|p−2u′′ =− f (u). (8)
We show that t0 = h.
Suppose on the contrary that t0 < h. Then u′(t0) = 0.
First, we claim that f (u(t0)) > 0. Indeed, if f (u(t0)) = 0, then we obtain from (3) that
−∆p (u(t0)−u(t ))+C (u(t0)−u(t ))p−1 =− f (u(t ))+C (u(t0)−u(t ))p−1 ≥ 0
for some constant C > 0 and all t < t0 close to t0. We can now apply Hopf’s lemma [11] to conclude
that u′(t0) > 0, a contradiction to u′(t0) = 0. Now we assume f (u(t0)) < 0, then from (8) we obtain
u′′(t ) > 0 for all t < t0 and close to t0. It follows that u′(t0) > u′(t ) > 0 for t < t0 and close to t0. This
contradiction shows that f (u(t0)) > 0.
By continuity, we have f (u(t )) > 0 for t ∈ [t0, t0 +ε] for some small ε> 0.
We claim that u′(t ) ≤ 0 or all t ∈ (t0, t0 + ε). Otherwise we can find r1 ∈ (t0, t0 + ε) such that
u′(r1) > 0. Consider the maximal interval (r2,r1] ⊂ (t0,r1] such that
u′(t ) > 0 in (r2,r1], u′(r2) = 0.
From the standard elliptic regularity, we know that u is C 2 in (r2,r1]. Hence (8) holds in this
interval and u′′(t ) < 0 for t ∈ (r2,r1]. This implies u′(r2) > u′(r1) > 0, which is a contradiction.
We further claim that u′(t ) < 0 or all t ∈ (t0, t0 +ε). Suppose on the contrary that u′(r1) = 0 for
some r1 ∈ (t0, t0 + ε). If u′(t ) = 0 for all t ∈ [t0,r1] then from the equation we deduce f (u(t )) = 0
in this interval, a contradiction. Hence there exists r2 ∈ (t0,r1) such that u′(r2) < 0. Let (r2,r3) ⊂
(r2,r1) be the maximal interval such that
u′(t ) < 0 in [r2,r3), u′(r3) = 0.
798 Phuong Le
Now, (8) holds in [r2,r3), which implies u′′(t ) < 0 in this interval. Therefore, u′(r2) > u′(r3) = 0, a
contradiction.
From (8), we have
p −1
p
|u′|p +F (u) = p −1
p
|u′(0)|p for t ∈ [0, t0]. (9)
Taking t = t0 we deduce F (u(t0)) = p−1p |u′(0)|p > 0. From this and the fact that u′ > 0 in (0, t0),
formula (9) becomes
p −1
p
(u′)p = F (u(t0))−F (u),
or equivalently,
u′
[F (u(t0))−F (u)]
1
p
=
(
p
p −1
) 1
p
.
It follows that u(t ) is uniquely determined in (0, t0) by∫ u(t )
a
ds
[F (u(t0))−F (s)]
1
p
=
(
p
p −1
) 1
p
t for all t ∈ [0, t0]. (10)
Now let (t0, t1) ⊂ (t0,h) be the maximal interval such that u′(t ) < 0 or all t ∈ (t0, t1). We show
that t1 ≥ 2t0 and u(t ) = u(2t0 − t ) for t ∈ [0, t0]. Clearly, (9) holds in [t0, t1] and we deduce∫ u(t0)
u(t )
ds
[F (u(t0))−F (s)]
1
p
=
(
p
p −1
) 1
p
(t − t0) for all t ∈ [t0, t1].
Combining this with (10), we have∫ u(t )
a
ds
[F (u(t0))−F (s)]
1
p
=
(
p
p −1
) 1
p
(2t0 − t ) for all t ∈ [t0, t1],
or equivalently, ∫ u(2t0−t )
a
ds
[F (u(t0))−F (s)]
1
p
=
(
p
p −1
) 1
p
t for all t ∈ [2t0 − t1, t0],
Comparing this with (10) we immediately obtain t1 ≥ 2t0 and u(2t0 − t ) = u(t ) in [0, t0]. Hence
u′(2t0) =−u′(0) < 0. This implies u(t ) < a for t > 2t0 and close to 2t0, a contradiction.
Therefore, we must have t0 = h. By Hopf’s lemma [11], we also have u′(h) > 0.
Hence u′(t ) > 0 for all t ∈ [0,h] and we have
p −1
p
|u′|p +F (u) = p −1
p
|u′(0)|p for t ∈ [0,h]. (11)
This implies p−1p |u′(0)|p > max[a,b] F and∫ u(t )
a
ds[
p−1
p |u′(0)|p −F (s)
] 1
p
=
(
p
p −1
) 1
p
t for all t ∈ [0,h].
Choosing t = h, we have ∫ b
a
ds[
p−1
p |u′(0)|p −F (s)
] 1
p
=
(
p
p −1
) 1
p
h.
Therefore, the necessary condition for the existence of a solution of (6) is that (4) holds.
Conversely, if (4) holds, then every solution u to (6) satisfies∫ u(t )
a
ds
[c −F (s)] 1p
=
(
p
p −1
) 1
p
t for all t ∈ [0,h], (12)
Phuong Le 799
where c > max[a,b] F is such that ∫ b
a
ds
[c −F (s)] 1p
=
(
p
p −1
) 1
p
h.
This implies uniqueness. It also implies
u′(t ) =
(
p
p −1
) 1
p
[c −F (u(t ))] 1p .
Hence u′ > 0 in [0,h], c = p−1p |u′(0)|p and (11) holds. This means that u defined by (12) is actually
a solution to problem (6). Property (7) can easily be checked using the above formulae. This
completes the proof. □
3. Higher dimensions
This section is devoted to the proof of our main result.
Lemma 5. Under assumptions of Theorem 2, we have
a ≤ u ≤ b inΩ.
Proof. We follow an idea in [9]. Assume by contradiction that supu = M > b. Let γ ∈ (b, M) and
ϕR ∈C∞c (RN ) be a standard cut-off function. That is, 0 ≤ϕ≤ 1 and
ϕR = 1 in BR ,
ϕR = 0 in RN−1 \ B2R ,
|∇ϕR | ≤ 2R in B2R \ BR .
We denote a+ := max{a,0} for a ∈ R. By approximation argument, we can use ψ = (u −γ)+ϕpRχΩ
as a test function in (2) to obtain
−
∫
{u>γ}
f (u)(u −γ)ϕpR =−
∫
{u>γ}
|∇u|pϕpR −p
∫
{u>γ}
|∇u|p−2(u −γ)ϕp−1R ∇u ·∇ϕR
≤−
∫
{u>γ}
|∇u|pϕpR +p
∫
{u>γ}
|∇u|p−1(u −γ)ϕp−1R |∇ϕR |
≤ −
∫
{u>γ}
|∇u|pϕpR +
∫
{u>γ}
{(
|∇u|p−1ϕp−1R
) p
p−1 +Cp
[
(u −γ)|∇ϕR |
]p}
≤Cp
∫
{u>γ}
[(u −γ)]p |∇ϕR |p .
Taking into account that u is bounded and p > 1, we have
C0
∫
{u>γ}
(u −γ)ϕpR ≤Cp
∫
{u>γ}
[(u −γ)]p |∇ϕR |p ≤C1R−p
∫
{u>γ}∩B2R
(u −γ), (13)
where C0 = min[γ,M ](− f ) > 0. By setting
L (R) =
∫
{u>γ}∩BR
(u −γ),
then L (R) ≤C2RN . Moreover, from (13), there exists R0 > 0 such that
L (R) ≤ 2−N−1L (2R) for all R > R0.
Hence Lemma 2.1 in [9] implies L (R) ≡ 0, which means u ≤ γ in Ω. This contradicts the
assumption supu = M > γ.
Therefore, u ≤ b inΩ. Similarly, we can show that u ≥ a inΩ. □
800 Phuong Le
Proof of Theorem 2. The existence follows from Theorem 4.
Let us assume that problem (1) has a bounded solution u. By Lemma 5,
a ≤ u ≤ b inΩ.
Since f (a) = f (b) = 0 and by the boundary conditions, we can apply the strong maximum
principle [11] to deduce that
a < u < b inΩ.
We show that there is a maximal solution u∗ in the order interval [u,b] in the sense that any
solution v with u ≤ v ≤ b satisfies v ≤ u∗. To this end, for each n > 1 we define
Ωn = {(x ′, xN ) ∈Ω | |x ′| < n}.
Setting
φn(x) =
{
b on ∂Ωn \ {xN = 0},
a + (b −a)(|x ′|−n +1)+ on ∂Ωn ∩ {xN = 0}.
We now consider the auxiliary problem{
−∆p u = f (u) inΩn ,
u =φn on ∂Ωn .
(14)
Obviously, u|Ωn is a lower solution and b is an upper solution to (14). Hence by standard upper
and lower solution argument, problem (14) has a maximal solution un in [u,b]. Since un+1|Ωn is a
lower solution to (14) in this interval, we have un ≥ un+1 inΩn . By a standard regularity argument,
u∗ = limn→∞ un is well defined inΩ and it is a solution of (1).
Clearly, u∗ ∈ [u,b]. If v is any solution of (1) in [u,b], then it is evident that v |Ωn is a lower
solution of (14) and it follows that un ≥ v (since the standard upper and lower solution argument
implies the existence of a solution in [v,b] which is less than or equal to un). It follows that v ≤ u∗.
Thus u∗ is the maximal solution in [u,b].
Since the equation is invariant under translations in x ′, the maximality implies that u∗ is a
function of xN only. Indeed, we assume by contradiction that there exist two points x = (x0, l )
and y = (y0, l ), a < l < b, such that u∗(x) ̸= u∗(y). We define ũ∗(x ′, y) = u∗(x ′+ y0 − x0, xN ). Then
ũ∗ is a solution to (1) and v(x) := max{u∗(x), ũ∗(x)} ≥ u∗(x). Moreover, v ̸≡ u∗ and v is a lower
solution to (1) satisfying v ≤ b. Hence (1) has a solution v∗ in the order interval [v,b]. It follows
that v∗ is a solution in [u,b] which satisfies v∗ ≥ u∗ and v∗ ̸≡ u∗. This contradicts the maximality
of u∗.
Thus u∗ is a function of xN only and hence it is a solution of (6).
In the same way, we can construct a minimal solution u∗ in [a,u] by considering the auxiliary
problem {
−∆p u = f (u) inΩn ,
u =ψn on ∂Ωn ,
(15)
where
ψn(x) =
{
a on ∂Ωn \ {xN = h},
b + (a −b)(|x ′|−n +1)+ on ∂Ωn ∩ {xN = h}.
Moreover, the minimality of u∗ implies that u∗ is a function of xN only. Thus u∗ is a solution
of (6).
By Theorem 4, we necessarily have u∗ = u∗ = U , where U is the unique solution of (6).
Moreover, condition (4) holds. Since u∗ ≤ u ≤ u∗, this implies that u(x) ≡U (xN ). The conclusion
then follows from the properties of U obtained in Theorem 4. □
Phuong Le 801
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Uniqueness of bounded solutions to $p$-Laplace problems in strips

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