Comptes Rendus Mathématique

Nuno C. Dias

Maurice A. de Gosson

João N. Prata

On Orthogonal Projections of Symplectic Balls

William Golding

Nonlinear asymptotic stability in $L^\infty $ for Lipschitz solutions to scalar conservation laws

Szilárd Szabó

Comptes Rendus
Mathématique
Szilárd Szabó
Exponential decay of harmonic 1-forms for wild harmonic bundles on curves
Volume 362 (2024), p. 1015-1021
https://doi.org/10.5802/crmath.607
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
2024, Vol. 362, p. 1015-1021
https://doi.org/10.5802/crmath.607
Research article / Article de recherche
Complex analysis and geometry / Analyse et géométrie complexes
Exponential decay of harmonic 1-forms for
wild harmonic bundles on curves
Décroissance exponentielle des 1-formes harmoniques
pour de fibrés harmoniques sauvages sur une courbe
Szilárd Szabó a,b
a Budapest University of Technology and Economics, 1111. Budapest, Egry József utca 1. H épület, Hungary
b Rényi Institute of Mathematics, 1053. Budapest, Reáltanoda utca 13-15. Hungary
Current address: Institute of Mathematics, Faculty of Science, Eötvös Loránd University, Pázmány Péter
sétány 1/C, Budapest, Hungary, H-1117
E-mails: szilard.szabo@ttk.elte.hu, szabo.szilard@renyi.hu
Abstract. We give a slightly reorganized exposition of the proof of the exponential decay result of T. Mochizuki
for harmonic 1-forms for wild harmonic bundles on a disc.
Résumé. Nous donnons une démonstration légèrement modifiée du résultat de décroissance exponentielle
des 1-formes harmoniques pour un fibré harmonique sauvage sur un disque, dû à T. Mochizuki.
Keywords. Harmonic bundle, harmonic section, irregular singularity.
Mots-clés. fibré harmonique, section harmonique, singularité irrégulière.
2020 Mathematics Subject Classification. 53C43, 53C07, 58J37.
Funding. The author received funding from the Lendület Low Dimensional Topology grant of the Hungarian
Academy of Sciences and from the grants KKP144148 and KKP126683 of NKFIH..
Manuscript received 20 June 2023, accepted 7 January 2024.
1. Introduction
One of the key analytic results needed in the proof of the Hard Lefschetz theorem for polarized
wild pure twistor D-modules over a smooth projective curve [3, Corollary 18.1.2] is the exponen-
tial decay property of harmonic 1-forms. The purpose of the present text is to provide an alterna-
tive proof of the relevant exponential decay estimates, along the method developped in [6, Chap-
ter 2]. This method relies on the repeated application of the Weitzenböck formula and Stokes’
theorem. Moreover, the procedure takes all irregular parts of the same order in one step rather
than one by one, which perhaps makes the presentation simpler. On the other hand, the proof
relies on strong analytic results proved in [3, Section 7]. Analysis of tame harmonic bundles was
pioneered in [5]. The first analytical study of wild harmonic bundles was performed in [4].
ISSN (electronic): 1778-3569 https://comptes-rendus.academie-sciences.fr/mathematique/
1016 Szilárd Szabó
2. Statement of exponential energy decay
We consider an unramified wild harmonic bundle (E ,∂E ,θ,h) over the unit disc ∆= {|z| < 1} ⊂ C,
equipped with the Euclidean metric dz ·dz. We will denote by ∇+ the connection in E compatible
with (∂E ,h), byΦ= θ+θ†h the self-adjoint part of the associated integrable connection, and by
∆= (∂E +θ)◦ (∂E +θ)∗h + (∂E +θ)∗h ◦ (∂E +θ)
the associated Laplace operator. An L2-section τ of E⊗Ω1 is said to be harmonic if∆(τ) = 0. There
exists a finite set Irr(θ) ⊂ C[z−1] and a holomorphic decomposition
E =⊕a∈Irr(θ)Ea (1)
into eigenbundles with respect to the various irregular parts of θ, i.e. such that
θ|Ea −da IdEa
have at most a first-order pole. Here we assume that rk(Ea) ≥ 1 for every a ∈ Irr(θ). For any section
τ of E ⊗Ω1 we write
τ= ∑
a∈Irr(θ)
τa
the corresponding decomposition. For any a ∈ Irr(θ), we set
ord(a) = min{d ∈ Z | z−d ã has a removable singularity at 0}
for any lift ã of a to a meromorphic function on∆. We have ord(a) ≤ 0 and ord(a) = 0 if and only if
a= 0. For an integer j ≤ 0, let us set
τ j =
∑
ord(a)= j
τa.
We set
τ< j =
∑
j ′< j
τ j ′ , τ> j =
∑
j ′> j
τ j ′ ,
and
m = min{ j | τ j ̸= 0}.
We have the decomposition
τ=
0∑
j=m
τ j ,
which is analogous to the decomposition of the endomorphism bundle of E that appears in [2].
For any section τ of E ⊗Ω1 we introduce the norm
∥τ∥2H 1(∆) =
∫
∆
|τ|2h +|∇+τ|2h +|Φ⊗τ|2h dz dz
and the resulting Hilbert space H 1(∆) as the completion of smooth compactly supported sections
with respect to the norm ∥ ·∥H 1(∆). Following [6], given any τ ∈ H 1(∆) and r ∈ (0,1) we now define
the energy functional on H 1(∆) by
E(τ,r ) =
∫
|z|≤r
|∇+τ|2h +|Φ⊗τ|2h dz dz.
It is obvious that E(τ,r ) is non-negative, monotonically increasing in r and satisfies
E(τ,r ) ≤ ∥τ|∆(r )∥2H 1 ,
where ∆(r ) = {|z| < r } ⊂∆.
Theorem 1. Let τ satisfy ∆(τ) = 0 and τ,∇+τ,Φ⊗τ ∈ L2
h,dz·dz . For every j < 0 there exist R j ∈ (0,1)
and C j ,ε j > 0 only depending on (E ,∂E ,θ,h) such that for each 0 < R < R j we have
E(τ j ,R) ≤C j exp
(
−ε j R j
)
E(τ,R j ).
Szilárd Szabó 1017
We now turn our attention to the case j = 0. Let us denote by E0,λ the generalizedλ-eigenspace
of resz=0θ on E0 (the summand in (1) corresponding to irregular part a= 0), and set
E0,sing =⊕λ̸=0E0,λ.
We then have
E0|z=0 = E0,sing ⊕E0,0.
This direct sum extends holomorphically to an open neighborhood of z = 0. Let us write
τ0 = τ0,sing +τ0,0
with respect to this decomposition. We are ready to state the energy estimate in the case j = 0.
Theorem 2. There exists R0 ∈ (0,1) and C0,ε0 > 0 only depending on (E ,∂E ,θ,h) such that for each
0 < R < R0 we have
E(τ0,sing,R) ≤C0E(τ,R0)Rε0 .
Remark 3.
(1) These results are similar to [3, Proposition 8.4.5], up to three differences. First, in [3]
a pointwise result is proved rather than an energy bound. Second, in [3] one takes the
Poincaré metric instead of the Euclidean one. This latter point, however, makes no
difference, because the two metrics are mutually bounded by polynomial factors in |z|.
Third, in [3] L2-conditions on ∇+τ,Φ⊗τ are not assumed. For applications to the Hard
Lefschetz theorem in [3, Section 18.2], this point does not make any difference either.
Indeed, for such applications one has to deal there with certain L2-cohomology classes
(see [3, Equation (360)]), which presumes precisely L2-conditions on ∇+τ,Φ⊗τ.
(2) In the case of ramified wild harmonic bundles, similar estimates hold and can be de-
duced from the unramified version.
(3) Similar results hold for sections of associated vector bundles, for instance
endomorphism-valued 1-forms.
3. Energy estimates
From now on, unless otherwise specified, τ will stand for a harmonic section of E ⊗Ω1.
3.1. Preliminaries
We make use of the following results.
Theorem 4 (Consequence of [3, Theorem 7.2.1]). Let τ,ω be arbitrary sections of E ⊗Ω1. There
exist A′1 > 0,ε′1 > 0 only depending on (E ,∂E ,θ,h) such that for any j ′ ̸= j the following pointwise
estimate holds:
|h(ω j ′ ,τ j )| ≤ A′1 exp
(
−ε′1|z|min( j , j
′)
)
|ω j ′ |h |τ j |h .
The actual form in which we will apply this result is its combination with the inequality
between the geometric and quadratic means:
|h(ω j ′ ,τ j )| ≤
1
2
A′1 exp
(
−ε′1|z|min( j , j
′)
)(
c|ω j ′ |2h + c−1|τ j |2h
)
,
where c > 0 is an arbitrary constant.
Theorem 5 ([1, Theorem 5.4.]). We have the Weitzenböck formula
∆= (∇+)∗∇++ (Φ⊗)∗Φ⊗ .
1018 Szilárd Szabó
A straightforward modification of the argument of [6, Claim 2.10] (see also [3, Lemma 8.4.8])
then gives
Lemma 6. The subspace of smooth 1-forms that are compactly supported in ∆ \ {0} is dense in
H 1(∆).
3.2. Proof of Theorem 1
By virtue of Lemma 6, we may assume that τ is smooth and compactly supported in∆\{0}. Stokes’
theorem gives
E(τ j ,R) =
∫
|z|<R
h
(
(∇+)∗∇+τ j + (Φ⊗)∗Φ⊗τ j ,τ j
)+∫
|z|=R
h(∇+∂r τ j ,τ j ) (2)
where ∂r = ∂∂r is the unit normal vector of |z| = R. It follows from Theorem 5 that
(∇+)∗∇+τ+ (Φ⊗)∗Φ⊗τ= 0,
equivalently
(∇+)∗∇+τ j + (Φ⊗)∗Φ⊗τ j =−
∑
j ′ ̸= j
(∇+)∗∇+τ j ′ − (Φ⊗)∗Φ⊗τ j ′ .
Plugging this identity to (2) and applying Stokes’ theorem in the converse direction we find
E(τ j ,R) =−
∑
j ′ ̸= j
∫
|z|<R
h
(∇+τ j ′ ,∇+τ j )+h (Φ⊗τ j ′ ,Φ⊗τ j )+∫|z|=R h(∇+∂r τ,τ j ).
Applying Theorem 4 to this identity produces the estimate
E(τ j ,R) ≤ D exp(−ε′1R j )E(τ,R)+
∫
|z|=R
h(∇+∂r τ j ,τ j )Rdϕ+
∑
j ′ ̸= j
∫
|z|=R
h(∇+∂r τ j ′ ,τ j )Rdϕ (3)
for suitable D,ε1 > 0.
Now, we are given that there exist R j ,c > 0 only depending on (E ,θ) such that for any (not
necessarily harmonic) section ω of E j the following pointwise estimate holds for all |z| < R j :
|Φ⊗ω(z)|h > c|z| j−1|ω(z)|h .
Indeed, for any a ∈ Irr(θ) of order j < 0, we have ord(da) = j − 1, and we may choose any c > 0
such that 2c is smaller than the minimum of the absolute values of the leading-order terms of all
irregular parts a of order j appearing in (1). Importantly, notice that we may decrease c > 0 to
make it arbitrarily small (without having to modify R j ). It then follows that for every 0 < r < R j
we have
dE(τ j ,r )
dr
=
∫
|z|=r
|∇+τ j |2h +|Φ⊗τ j |2hr dϕ
≥ cr j−1
∫
|z|=r
1
cr j−1
|∇+τ j |2h + cr j−1|τ j |2hr dϕ. (4)
Similarly, we have
dE(τ,r )
dr
=
∫
|z|=r
|∇+τ|2h +|Φ⊗τ|2hr dϕ
≥ cr
j−1
m
∑
j ′ ̸= j
∫
|z|=r
1
cr j−1
|∇+τ j ′ |2h + cr j−1|τ j |2hr dϕ. (5)
On the other hand, combining (3) with the inequality
2h(a,b) ≤ 1
cr j−1
|a|2h + cr j−1|b|2h ,
Szilárd Szabó 1019
we find
E(τ j ,r ) ≤ D exp(−ε′1r j )E(τ,r )+
1
2
∫
|z|=r
1
cr j−1
|∇+τ j |2h + cr j−1|τ j |2hr dϕ
+ D
2
exp(−ε′1r j )
∑
j ′ ̸= j
∫
|z|=r
1
cr j−1
∣∣∇+τ j ′ ∣∣2h + cr j−1|τ j |2hr dϕ. (6)
Let us introduce the modified energy functional
Ẽ(τ j ,R) = E(τ j ,R)+F exp(−ε′1R j )E(τ,R),
where F > 0 is a constant to be chosen later. Then, using estimates (4), (5) and (6), for every
0 < r < R j we infer
dẼ(τ j ,r )
dr
≥ cr
j−1
m
E(τ j ,r )− cr
j−1
m
D exp(−ε′1r j )E(τ,r )
− cr
j−1
m
D
2
exp(−ε′1r j )
∑
j ′ ̸= j
∫
|z|=r
1
cr j−1
∣∣∇+τ j ′ ∣∣2h + cr j−1|τ j |2hr dϕ
−F jε′1r j−1 exp(−ε′1r j )E(τ,r )+F exp(−ε′1r j )
dE(τ,r )
dr
≥ cr
j−1
m
E(τ j ,r )− (cD +F jε′1)r j−1 exp(−ε′1r j )E(τ,R j )
+
(
F − D
2
)
exp(−ε′1r j )
dE(τ,r )
dr
.
Let us now choose F so that
F = max
( −cD
c
m + jε′1
,
D
2
)
.
Notice that if c > 0 is chosen sufficiently small, then cm + jε′1 < 0 because jε′1 < 0. Then, we have
−(cD +F jε′1) ≥ cm F and the above bound implies
dẼ(τ j ,r )
dr
≥ cr
j−1
m
Ẽ(τ j ,r ).
In different terms, this states that
dln Ẽ(τ j ,r )
dr
≥ cr
j−1
m
for all r < R j . For any 0 < R < R j , integrating this inequality from R to R j with respect to r shows
that
ln Ẽ(τ j ,R j )− ln Ẽ(τ j ,R) ≥ c
m j
(R jj −R j ).
Taking exponential of both sides gives the bound
K exp
(
c
m j
R j
)
Ẽ(τ j ,R j ) ≥ Ẽ(τ j ,R)
for
K = exp
(
− c
m j
R jj
)
.
Finally, this estimate combined with
E(τ j ,R) = Ẽ(τ j ,R)−F exp(−ε′1R j )E(τ,R j ) < Ẽ(τ j ,R)
gives the desired bound.
1020 Szilárd Szabó
3.3. Proof of Theorem 2
Similarly to (6), using that min( j ,0) = j for all j < 0, we now find
E(τ0,sing,r ) ≤ D exp(−ε′1r−1)E(τ,r )+
1
2
∫
|z|=r
r
c
|∇+τ0,sing|2h +
c
r
|τ0,sing|2hr dϕ
+ D
2
exp(−ε′1r−1)
∑
j ′ ̸=0
∫
|z|=r
r
c
∣∣∇+τ j ′ ∣∣2h + cr |τ0,sing|2hr dϕ.
It then follows that
dE(τ0,sing,r )
dr
≥ c
r m
E(τ0,sing,r )− c
r m
D exp(−ε′1r−1)E(τ,r )
− c
r m
D
2
exp(−ε′1r−1)
∑
j ′ ̸=0
∫
|z|=r
r
c
∣∣∇+τ j ′ ∣∣2h + cr |τ0,sing|2hr dϕ
≥ c
r m
E(τ0,sing,r )− c
r m
D exp(−ε′1r−1)E(τ,r )
− D
2
exp(−ε′1r−1)
dE(τ,r )
dr
.
We apply the same argument as in Theorem 1, this time with the modified energy functional
Ẽ(τ0,sing,r ) = E(τ0,sing,r )+F exp(−ε′1r−1)E(τ,r )
for some F > 0. We obtain
dẼ(τ0,sing,r )
dr
≥ c
r m
E(τ0,sing,r )+
(
ε′1
r 2
F − c
r m
D
)
exp(−ε′1r−1)E(τ,r )
+
(
F − D
2
)
exp(−ε′1r−1)
dE(τ,r )
dr
.
We infer that if F > D/2 is fixed sufficiently large then we have
dẼ(τ0,sing,r )
dr
≥ c
r m
Ẽ(τ0,sing,r ),
i.e.
dln Ẽ(τ0,sing,r )
dr
≥ c
r m
.
Integrating from R to R0 and exponentiating this estimate yields
Ẽ(τ0,sing,R) ≤ Ẽ(τ0,sing,R0)R−c/m0 Rc/m .
for every 0 < R < R0. Finally, comparing to the definition of Ẽ shows that
E(τ0,sing,R) ≤
(
E(τ0,sing,R0)+F exp(−ε′1R−10 )E(τ,R0)
)
R−c/m0 R
c/m
≤C0E(τ,R0)Rc/m .
This concludes the proof.
Acknowledgements
This manuscript is a write-up of the author’s talk at the Arbeitsgemeinschaft: Twistor D-modules
and the Decomposition Theorem held at the Mathematisches Forschungsinstitut Oberwolfach
in April 2023. The author would like to thank the Insitute and the organizers T. Mochizuki and
C. Sabbah for providing optimal working conditions and a venue for fruitful discussions.
Szilárd Szabó 1021
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] O. Biquard, “Fibrés de Higgs et connexions intégrables: le cas logarithmique (diviseur lisse)”,
Ann. Sci. Éc. Norm. Supér. 30 (1997), no. 1, pp. 41–96.
[2] O. Biquard and P. Boalch, “Wild non-abelian Hodge theory on curves”, Compos. Math. 140
(2004), no. 1, pp. 179–204.
[3] T. Mochizuki, Wild harmonic bundles and wild pure twistor D-modules, Société Mathéma-
tique de France, 2011.
[4] C. Sabbah, “Harmonic metrics and connections with irregular singularities”, Ann. Inst.
Fourier 49 (1999), no. 4, pp. 1265–1291.
[5] C. T. Simpson, “Harmonic bundles on noncompact curves”, J. Am. Math. Soc. 3 (1990), no. 3,
pp. 713–770.
[6] S. Szabó, Nahm transform for integrable connections on the Riemann sphere, Société Math-
ématique de France, 2007, pp. ii+114.


Exponential decay of harmonic $1$-forms for wild harmonic bundles on curves

Louis Boutet de Monvel

Yves Colin de Verdière

Comptes Rendus
Mathématique
Louis Boutet de Monvel and Yves Colin de Verdière
The action of pseudo-differential operators on functions harmonic outside a
smooth hyper-surface
Volume 362 (2024), p. 1037-1041
https://doi.org/10.5802/crmath.626
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
2024, Vol. 362, p. 1037-1041
https://doi.org/10.5802/crmath.626
Research article / Article de recherche
Partial differential equations / Équations aux dérivées partielles
The action of pseudo-differential operators on
functions harmonic outside a smooth
hyper-surface
L’action d’opérateurs pseudo-différentiels sur les
fonctions harmoniques en dehors d’une hypersurface
lisse
Louis Boutet deMonvel ∗∗ and Yves Colin de Verdière ∗,a
a Institut Fourier, Unité mixte de recherche CNRS-UJF 5582, BP 74, 38402-Saint
Martin d’Hères Cedex, France
E-mail: yves.colin-de-verdiere@univ-grenoble-alpes.fr
Dedicated to the memory of Louis Boutet de Monvel.
Abstract. The goal of this note is to describe the action of pseudo-differential operators on the space of square
integrable functions which are harmonic outside a smooth closed hyper-surface of a compact Riemannian
manifold.
Résumé. Dans cette note, nous étudions un opérateur du type B = P ∗AP où A est un opérateur pseudo-
différentiel et P l’opérateur de Poisson bilatéral associé à une hypersurface. Nous montrons que, sous
certaines conditions, B est un opérateur pseudo-différentiel sur cette hypersurface dont nous calculons le
symbole principal.
Keywords. Pseudo-differential operators, Dirichlet-to-Neumann map, Poisson operator.
Mots-clés. Opérateurs pseudos-différentiels, application Dirichlet-to-Neumann, Opérateurs de Poisson.
2020 Mathematics Subject Classification. 35S05, 35S15.
Manuscript received 13 January 2024, accepted 12 February 2024.
The goal of this note is to describe the action of pseudo-differential operators on the space
H of L2 functions which are harmonic outside a smooth closed hyper-surface Z of a compact
Riemannian manifold without boundary (X , g ) and whose traces from both sides of Z coïncide.
We will represent these L2 harmonic functions as harmonic extensions of functions in the Sobolev
space H−1/2(Z ) by a Poisson operator P . The main result says that, if A is a pseudo-differential
operator of degree d < 3 on X , the operator
B =P ⋆ ◦ A ◦P
is a pseudo-differential operator on Z of degree d −1 whose principal symbol of degree d −1 can
be computed by integration of the principal symbol of A on the co-normal bundle of Z .
∗∗Deceased
∗Corresponding author
ISSN (electronic): 1778-3569 https://comptes-rendus.academie-sciences.fr/mathematique/
1038 L. Boutet de Monvel and Y. Colin de Verdière
These “bilateral” extensions are simpler (at least for the Laplace operator) than the “unilateral”
ones whose study is the theory of pseudo-differential operators on manifolds with boundary (see
[1–5, 7]).
1. Symbols
The following classes of symbols are defined in the books [5, Section 7.1] and in [6, Section 18.1].
A symbol of degree d on Ux×Rnξ where U is an open set inRN is a smooth complex valued function
a(x,ξ) on U ×Rn which satisfies the following estimates: for any multi-indices (α,β), there exists
a constant Cα,β so that
|Dαx Dβξ a(x;ξ)| ≤Cα,β(1+∥ξ∥)d−|β| .
The symbol a is called classical if a admits an expansion a ∼∑∞l=0 ad−l where a j is homogeneous
of degree j ( j an integer) for ξ ∈Rn large enough; more precisely, for any J ∈N, a −∑Jj=0 ad− j is a
symbol of degree d − J −1.
We will need the
Lemma 1. If a(x;ξ,η) is a symbol of degree d < −1 defined on Ux ×
(
Rn
ξ
×Rη
)
, b(x;ξ) =∫
R a(x;ξ,η)dη is a symbol of degree d+1 defined on Ux×Rnξ . Moreover, if a is classical, b is also clas-
sical and the homogeneous components of b are given for l ≤ d +1, by bl (x;ξ) =
∫
R al−1(x;ξ,η)dη
2. A general reduction Theorem for pseudo-differential operators
We choose local coordinates in some neighborhood of a point in Z denoted x = (z, y) ∈Rd−1 ×R,
so that Z = {y = 0}. We denote by (Ω j , j = 1, . . . , N ) a finite cover of Z by such charts and denote by
Ω0 an open set disjoint from Z so that X =⋃Nj=0Ω j . We choose the chartsΩ j so that the densities
|dz| and |dx| are the Lebesgue measures.
If X is a smooth manifold, we denote by D′(X ) the space of generalized functions on X of
which the space of smooth functions on X is a dense subspace. We assume that X and Z are
equipped with smooth densities |dx| and |dz|. This allows to identify generalized functions with
Schwartz distributions, i.e. linear functionals on test functions; this duality extending the L2
product is denoted by 〈 | 〉. We introduce the extension operator E : D′(Z ) → D′(X ) sending the
distribution f to the distribution f δ(y = 0) defined
〈 f δ(y = 0) |φ(z, y)〉 = 〈 f |φ(z,0)〉
and its adjoint, the trace T : C∞(X ) →C∞(Z ) defined by φ→φ|Z . Let A be a pseudo-differential
operator on X : let us call A j the restriction of A to test functions compactly supported in Ω j . We
will work with one of the A j ’s given by the following “quantization” rule
A j u(z, y) = 1
(2π)d
∫
R2d
ei(〈z−z
′|ζ〉+(y−y ′)η) a j (z, y ;ζ,η)u(z ′, y ′)dz ′ dy ′ dζdη .
So we have formally, using the facts that the densities on X and Z are given by the Lebesgue
measures in these local coordinates:
T ◦ A j ◦E v(z) = 1
(2π)d
∫
R2d−1
ei〈z−z
′|ζ〉 a j (z,0;ζ,η)v(z ′)dz ′ dζdη ,
which we can rewrite
T ◦ A j ◦E v(z) = 1
(2π)d−1
∫
R2d
ei〈z−z
′|ζ〉 b j (z;ζ)v(z ′)dz ′ dζ ,
with
b j (z;ζ) = 1
2π
∫
R
a j (z,0;ζ,η)dη . (1)
We have the
L. Boutet de Monvel and Y. Colin de Verdière 1039
Theorem 2. If A is a pseudo-differential operator on X of degree m <−1 whose full symbol in the
chart Ω j is a j , then the operator T ◦ A ◦E is a pseudo-differential operator on Z of degree m +1
whose symbol is given in the chartsΩ j ∩Z by Equation (1).
This is proved by looking at the actions on test functions compactly supported in the chart
Ω j , j ≥ 1: then we use Lemma 1.
Remark 3. The principal symbol can be described in a more intrinsic way: let z ∈ Z be given,
from the smooth densities on Tz X and on Tz Z given by |dx| and |dz|, we get, using the Liouville
densities, densities on the dual bundles T⋆z Z and T
⋆
z X . Let us denote byΩ
1(E) the 1-dimensional
space of densities on the vector space E . From the exact sequence
0 −→ N⋆z Z −→ T⋆z X −→ T⋆z Z −→ 0 ,
we deduce
Ω1(T⋆X ) ≡Ω1(N⋆Z )⊗Ω1(T⋆Z )
and a canonical density dm(z) in Ω1(N⋆z Z ). The principal symbol of B = T ◦ A ◦E is given in
coordinates by b(z,ζ) = (1/2π)∫N⋆z Z a(z;ζ,η)dm(η).
3. The “bilateral” Dirichlet-to-Neumann operator
We will assume that the local coordinates x = (z, y) along Z are chosen so that g (z,0) = h(dz)+dy2
and the Riemannian volume along Z is |dx|g = |dz|h |dy |. We will choose the associated densities
on X and Z . We will denote by ∆g the Laplace–Beltrami operator on (X , g ) as defined by
Riemannian geometers (i.e. with a minus sign in front of the second order derivatives).
If f is given on Z , let us denote by DN ( f ) minus the sum of the interior normal derivatives
on both sides of Z of the harmonic extension F of f ; this always makes sense, even if the normal
bundle of Z is not orientable. We have the
Lemma 4. The distributional Laplacian of the harmonic extension F of a smooth function f on
Z is ∆g F = E (DN ( f )).
Proof. The proof is a simple application of the Green’s formula: by definition of the action of the
Laplacian on distributions, if φ is a test function on X , 〈∆g F | φ〉 := 〈F | ∆gφ〉. We can compute
the righthandside integral as an integral on X \ Z using Green’s formula.∫
X \Z
(F∆gφ−φ∆g F )|dx|g =
∫
Z
(Fδφ−φδF )|dz|h
where δ is the sum of the interior normal derivatives from both sides of Z . Using the fact that
∆g F = 0 in X \ Z and δφ= 0, we get the result. □
Denoting by ∆−1g the “quasi-inverse” of ∆g defined by ∆−1g φ j = λ−1j φ j for the eigenfunc-
tions φ j of ∆g with non-zero eigenvalue λ j and ∆−1g 1 = 0, we have f =
(
T ◦∆−1g ◦E
)
◦
DN ( f ) (mod constants). By Theorem 2, the operator B = T ◦∆−1g ◦E is an elliptic self-adjoint
pseudo-differential operator on Z . The operator DN is a right inverse of B modulo smoothing
operators and hence also a left inverse modulo smoothing operators. So that DN = B−1 is an
elliptic self-adjoint of principal symbol the inverse of
1
2π
∫
R
(∥ζ∥2h +η2)−1dη=
1
2∥ζ∥h
,
namely 2∥ζ∥h . Hence
Theorem 5. The bilateral Dirichlet-to-Neumann operator DN is a self-adjoint elliptic pseudo-
differential operator of degree 1 on L2(Z , |d z|) and of principal symbol 2∥ζ∥h . The kernel of DN is
the space of constant functions.
1040 L. Boutet de Monvel and Y. Colin de Verdière
The full symbol of DN can be computed in a similar way from the full symbol of the resolvent
∆−1g along Z .
4. The Poisson operator
Let A be an pseudo-differential operator on X of principal symbol a. We are interested to
the restriction to the space H of the quadratic form Q A(F ) = 〈AF |F 〉 associated to A. We
will parametrize H as harmonic extensions of functions which are in H−
1
2 (Z ) by the so-called
Poisson operator denoted by P ; the pull-back RA of Q A on L2(Z ) is defined by
RA( f ) = 〈AP f |P f 〉 = 〈P ⋆AP f | f 〉 .
The goal of this section is to compute the operator B = P ⋆AP associated to the quadratic
form RA .
From Lemma 4, we have, modulo smoothing operators,
P =∆−1g ◦E ◦DN .
Hence
B =DN ◦
[
T ◦
(
∆−1g ◦ A ◦∆−1g
)
◦E
]
◦DN .
The operator ∆−1g ◦ A ◦∆−1g is a pseudo-differential operator of principal symbol a/(∥ζ∥2h +η2)2
near Z .
Applying Theorem 2 to the inner bracket and Theorem 5, we get the:
Theorem 6. If A is a pseudo-differential operator of degree d < 3 on X and P the Poisson operator
associated to Z , the operator B = P ⋆AP is a pseudo-differential operator of degree d −1 on Z of
principal symbol
b(z,ζ) = 2
π
∥ζ∥2h
∫
R
a(z,0;ζ,η)
(∥ζ∥2h +η2)2
dη .
Remark 7. Note that if A is a pseudo-differential operator without the transmission property, the
operator A ◦P may be ill-behaved and have disagreeable singularities along Z ; however P ∗AP
is always a good pseudo-differential operator on Z .
Note
This note was written with Louis in 2012. We had the project to publish it as an Appendix to a
work still in progress with Gregory Berkolaiko. Finally, I decided to publish it independently and
to dedicate it to the memory of Louis.
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.
L. Boutet de Monvel and Y. Colin de Verdière 1041
References
[1] L. Boutet de Monvel, “Comportement d’un opérateur pseudo-différentiel sur une variété à
bord. I. La propriété de transmission.”, J. Anal. Math. 17 (1966), pp. 241–253.
[2] L. Boutet de Monvel, “Comportement d’un opérateur pseudo-différentiel sur une variété à
bord. II. Pseudo-noyaux de Poisson”, J. Anal. Math. 17 (1966), pp. 255–304.
[3] L. Boutet de Monvel, “Boundary problems for pseudo-differential operators”, Acta Math.
126 (1971), no. 1-2, pp. 11–51.
[4] A.-P. Calderón, “Boundary value problems for elliptic equations”, in Outlines Joint Sym-
pos. Partial Differential Equations (Novosibirsk, 1963), Academy of Sciences of the USSR,
Siberian Branch, Moscow, 1963, pp. 303–304.
[5] G. Grubb, Distributions and operators, Springer, 2009, pp. xii+461.
[6] L. Hörmander, The analysis of linear partial differential operators. III Pseudodifferential
operators, Springer, 1985, pp. viii+525.
[7] M. E. Taylor, Partial differential equations. II Qualitative studies of linear equations, Springer,
1996, pp. xxii+528.


The action of pseudo-differential operators on functions harmonic outside a smooth hyper-surface

Arthur Danielyan

The radial limits and boundary uniqueness

Xing-Wang Jiang

Ya-Li Li

Comptes Rendus
Mathématique
Xing-Wang Jiang and Ya-Li Li
On the cardinality of subsequence sums II
Volume 362 (2024), p. 1279-1285
Online since: 14 November 2024
https://doi.org/10.5802/crmath.613
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
2024, Vol. 362, p. 1279-1285
https://doi.org/10.5802/crmath.613
Research article / Article de recherche
Number theory / Théorie des nombres
On the cardinality of subsequence sums II
Sur le cardinal de sommes de sous-suites II
Xing-Wang Jiang a and Ya-Li Li b
a Department of Mathematics, Luoyang Normal University, Luoyang 471934, P. R. China
b School of Mathematics and Statistics, Henan University, Kaifeng 475001, P. R. China
E-mails: xwjiangnj@sina.com, njliyali@sina.com
Abstract. Let A = (a1, . . . , a1︸ ︷︷ ︸
r1
, a2, . . . , a2︸ ︷︷ ︸
r2
, . . . , ak , . . . , ak︸ ︷︷ ︸
rk
) be a finite sequence of integers with a1 < a2 < ·· · < ak
and ri ≥ 1 (1 ≤ i ≤ k). The sum of all terms of a subsequence B of A is called a subsequence sum of A and we
denote it by σ(B). For 0 ≤α≤∑ki=1 ri , let Σα(A) = {σ(B)|B is a subsequence of A of length ≥α}. In this paper,
we completely settle the two problems posed by Bhanja and Pandey about finding the optimal lower bound
of |Σα(A)| and determining the structure of the sequence A for which the lower bound of |Σα(A)| is optimal.
Résumé. Soit A = (a1, . . . , a1︸ ︷︷ ︸
r1
, a2, . . . , a2︸ ︷︷ ︸
r2
, . . . , ak , . . . , ak︸ ︷︷ ︸
rk
)) une suite finie d’entiers avec a1 < a2 < ·· · < ak et ri ≥
1 (1 ≤ i ≤ k). La somme de tous les termes d’une sous-suite B de A est appelée somme de sous-suite de A et
nous la désignons par σ(B). Pour 0 ≤α≤∑ki=1 ri , soit Σα(A) = {σ(B)|B une sous-suite de A de longueur ≥α}.
Dans cet article, nous résolvons complètement les deux problèmes posés par Bhanja et Pandey concernant
la recherche de la borne inférieure de |Σα(A)| et la détermination de la structure de la suite A pour laquelle la
borne inférieure de |Σα(A)| est atteinte.
Keywords. Subsequence sums, direct problem, inverse problem.
Mots-clés. Sommes de sous-suites, problème direct, problème inverse.
2020 Mathematics Subject Classification. 11P70, 11B25.
Funding. This work is supported by the National Natural Science Foundation of China (Grant Nos.
12201281, 12171243, 11901156) and the Natural Science Youth Foundation of Henan Province (Grant No.
222300420245).
Manuscript received 5 July 2023, revised 22 October 2023, accepted 15 January 2024.
1. Introduction
Let k ≥ 1 and r = (r1,r2, . . . ,rk ) with ri ≥ 1 (1 ≤ i ≤ k). And let
A = (a1, . . . , a1︸ ︷︷ ︸
r1
, a2, . . . , a2︸ ︷︷ ︸
r2
, . . . , ak , . . . , ak︸ ︷︷ ︸
rk
)
be a finite sequence of integers, where a1 < a2 < ·· · < ak and ai repeats ri times. For convenience,
we denote this sequence by A = (a1, a2, . . . , ak )r . For a given subsequence B of A, the sum σ(B) :=∑
b∈B b is called a subsequence sum of A. Here, we assume that σ(;) = 0. For 0 ≤α≤
∑k
i=1 ri , let
Σα(A) be the set of all subsequence sums corresponding to the subsequences of A that are of the
ISSN (electronic): 1778-3569 https://comptes-rendus.academie-sciences.fr/mathematique/
1280 Xing-Wang Jiang and Ya-Li Li
size at least α and Σα(A) be the set of all subsequence sums corresponding to the subsequences
of A that are of the size at most
∑k
i=1 ri −α, that is,
Σα(A) =
{
σ(B)
∣∣B is a subsequence of A of length ≥α} ,
Σα(A) =
{
σ(B)
∣∣∣∣∣B is a subsequence of A of length ≤ k∑
i=1
ri −α
}
.
Briefly, we writeΣ1(A) asΣ(A). Obviously, |Σα(A)| = |Σα(A)| and 0 ∈Σα(A). When ri = 1 (1 ≤ i ≤ k),
A is a set of integers, and now we call the subsequence sums as subset sums. The subset and
subsequence sums are fundamental objects in additive number which is very useful in some
other combinatorial problems such as the zero-sum problems. The direct problem for the
subsequence sums Σα(A) is to find the optimal lower bound of |Σα(A)| and the inverse problem
for Σα(A) is to determine the structure of A for which |Σα(A)| is minimal.
In 1995, Nathanson [8] firstly studied the direct and inverse problems for Σ(A) for sets A
of integers. In 2015, Mistri, Pandey and Prakash [7] generalized Nathanson’s results to the
subsequence sums Σ(A) for sequences A which contains either only nonnegative integers or
only nonpositive integers. Later, the present author and Li [5] settled the remaining case, i.e.,
the sequence A contains positive integers, negative integers, and/or zero. In 2020, Bhanja and
Pandey [2] considered the direct and inverse problems for subsequence sums Σα(A) when the
sequence A contains nonnegative or nonpositive integers. Some conclusions are as follows.
Theorem A ([2, Theorem 3.1]). Let k ≥ 1 be an integer and A = (a1, a2, . . . , ak )r be a finite sequence
of integers, where 0 < a1 < a2 < ·· · < ak and r = (r1,r2, . . . ,rk ) with ri > 0 for 1 ≤ i ≤ k. Let
0 ≤α<∑ki=1 ri . Then there exists an integer m ∈ [1,k] such that ∑m−1i=1 ri ≤α<∑mi=1 ri and
|Σα(A)| ≥
k∑
i=1
i ri −
m∑
i=1
i ri +m
(
m∑
i=1
ri −α
)
+1.
Theorem B ([2, Theorem 3.2 and Remark 3.1]). Let k ≥ 1 and r = (r1,r2, . . . ,rk ) with ri > 0 for
1 ≤ i ≤ k. For 0 ≤ α ≤ ∑ki=1 ri −2, let m ∈ [1,k] be the integer such that ∑m−1i=1 ri ≤ α < ∑mi=1 ri . If
A = (a1, a2, . . . , ak )r is a finite sequence of integers with 0 < a1 < a2 < ·· · < ak and
|Σα(A)| =
k∑
i=1
i ri −
m∑
i=1
i ri +m
(
m∑
i=1
ri −α
)
+1,
then
A = (a1,2a1, . . . ,ka1)r ,
except that A = (a1, a2)r with r = (1,r2) and A = (a1, a2, a1+a2)r with r = (1,1,r3), where a1, a2 are
two arbitrary positive integers.
Recently, Bhanja and Pandey [3] posed the following open problems.
Problem 1 ([3, Problem (1)]). It is important to find the optimal lower bound of |Σα(A)| for
arbitrary finite sequence of integers.
Problem 2 ([3, Problem (2)]). It is also an important problem to study the structure of the
sequence A for which the lower bound for |Σα(A)| is optimal.
They [3] settled Problem 1 for the special case ri = r for all 1 ≤ i ≤ k. For more related results,
one may refer to [1, 4, 6, 9–11]. Problem 1 is the direct problem of the subsequence sums and
Problem 2 is the inverse problem of the subsequence sums. In this paper, we solve completely
the above two problems. In Section 2, Theorem 1 and Corollary 2 give the answer to Problem 1.
In section 3, Theorem 3 and Corollary 4 give the answer to Problem 2.
Xing-Wang Jiang and Ya-Li Li 1281
Next, we give some definition and notations. Define
∑v
i=u f (i ) = 0 if u > v . For any sequences
A and B , we denote by A ∪B the sequence obtained by merging the sequences A and B and
reordering the terms in nondecreasing order. For an integer a, a + A and a ∗ A are defined
respectively to be the sequences as follows:
a + A = (a +a1, a +a2, . . . , a +ak )r , a ∗ A = (a ∗a1, a ∗a2, . . . , a ∗ak )r .
For integers l < k, we write (l , l +1, . . . ,k)r as [l ,k]r briefly.
2. Direct problem
Firstly, we consider the direct problem of the sequence A does not contain zero terms.
Theorem 1. Let l ,k ≥ 1 be two integers and A = (a−l , a−l+1, . . . , a−1)r ∪ (a1, a2, . . . , ak )r ′ be a
finite sequence of integers, where a−l < a−l+1 < ·· · < a−1 < 0 < a1 < a2 < ·· · < ak and r =
(r−l ,r−l+1, . . . ,r−1), r ′ = (r1,r2, . . . ,rk ) with ri > 0 for i ∈ [−l ,k]\{0}. For 0 ≤α<
∑l
i=1 r−i +
∑k
i=1 ri , let
Lα =

∑l
i=1 i r−i +
∑k
i=1 i ri +1, if α≤ min
{∑l
i=1 r−i ,
∑k
i=1 ri
}
;∑l
i=1 i r−i +
∑k
i=1 i ri −
∑m
i=1 i ri
+m (∑li=1 r−i +∑mi=1 ri −α)+1, if ∑li=1 r−i <α≤∑ki=1 ri ;∑l
i=1 i r−i +
∑k
i=1 i ri −
∑m′
i=1 i r−i
+m′
(∑k
i=1 ri +
∑m′
i=1 r−i −α
)
+1, if ∑ki=1 ri <α≤∑li=1 r−i ;∑l
i=1 i r−i +
∑k
i=1 i ri −
∑m
i=1 i ri +m
(∑l
i=1 r−i +
∑m
i=1 ri −α
)
−∑m′i=1 i r−i +m′(∑ki=1 ri +∑m′i=1 r−i −α)+1, if α> max{∑li=1 r−i ,∑ki=1 ri },
where m, m′ are integers which satisfy
∑m−1
i=1 ri ≤ α−
∑l
i=1 r−i <
∑m
i=1 ri and
∑m′−1
i=1 r−i ≤ α−∑k
i=1 ri <
∑m′
i=1 r−i , respectively. Then
|Σα(A)| ≥ Lα.
Moreover, the lower bound is optimal.
Proof of Theorem 1. Let
I =
l∑
i=1
r−i a−i , J =
k∑
i=1
ri ai
and
A1 = (a−l , a−l+1, . . . , a−1)r , A2 = (a1, a2, . . . , ak )r ′ .
Clearly, −Σβ(A1) =Σβ(−A1) for any nonnegative integer β, and the terms in Σβ(−A1) are positive.
Let
A0 = [−l ,−1]r ∪ [1,k]r ′ .
If α≤ min
{∑l
i=1 r−i ,
∑k
i=1 ri
}
, then I + J −Σ(A1), I + J −Σ(A2) and {I + J } are included in Σα(A)
and it is clear that they are pairwise disjoint. Thus,
|Σα(A)| ≥ |I + J −Σ(A1)|+ |I + J −Σ(A2)|+ |{I + J }|
= |Σ(−A1)|+ |Σ(A2)|+1
≥
l∑
i=1
i r−i +
k∑
i=1
i ri +1 = Lα, (1)
1282 Xing-Wang Jiang and Ya-Li Li
where the third inequality comes from Theorem A. Moreover,
Σα(A0) ⊆
[
l∑
i=1
(−i )r−i ,
k∑
i=1
i ri
]
.
Then |Σα(A0)| ≤ Lα. Thus, |Σα(A0)| = Lα. Therefore, at this point, Lα is optimal.
If
∑l
i=1 r−i < α ≤
∑k
i=1 ri , then
∑m−1
i=1 ri ≤ α−
∑l
i=1 r−i := α1 <
∑m
i=1 ri . At this point, it is clear
that I +Σα1 (A2) and I + J −Σ(A1) are included in Σα(A). Since
max
{
I +Σα1 (A2)
}= I + J < min{I + J −Σ(A1)}= I + J −a−1,
these two sets are disjoint. Thus,
|Σα(A)| ≥ |I +Σα1 (A2)|+ |I + J −Σ(A1)|
= |Σα1 (A2)|+ |Σ(−A1)|
≥
k∑
i=1
i ri −
m∑
i=1
i ri +m
(
m∑
i=1
ri −α1
)
+1+
l∑
i=1
i r−i = Lα, (2)
where the third inequality comes from Theorem A. Moreover,
Σα(A0) ⊆
[
l∑
i=1
(−i )r−i +
m∑
i=1
i ri −m
(
m∑
i=1
ri −α1
)
,
k∑
i=1
i ri
]
.
Then |Σα(A0)| ≤ Lα. Thus, |Σα(A0)| = Lα. Therefore, at this point, Lα is optimal.
If
∑k
i=1 ri <α≤
∑l
i=1 r−i , then by |Σα(A)| = |Σα(−A)| and similar to the above argument for −A,
we can obtain that
|Σα(A)| = |Σα(−A)| ≥
l∑
i=1
i r−i +
k∑
i=1
i ri −
m′∑
i=1
i r−i +m′
(
k∑
i=1
ri +
m′∑
i=1
r−i −α
)
+1 = Lα.
Moreover, we can get that |Σα(A0)| = Lα and so at this point, Lα is optimal.
If α> max{∑li=1 r−i ,∑ki=1 ri }, then there exist integers m,m′ such that
m−1∑
i=1
ri ≤α−
l∑
i=1
r−i :=α1 <
m∑
i=1
ri ,
m′−1∑
i=1
r−i ≤α−
k∑
i=1
ri :=α2 <
m′∑
i=1
r−i .
At this point, I +Σα1 (A2) and I + J −Σα2 (A1) are included in Σα(A). It follows from
max
{
I +Σα1 (A2)
}= I + J = min{I + J −Σα2 (A1)}
that
|Σα(A)| ≥ |I +Σα1 (A2)|+ |I + J −Σα2 (A1)|− |{I + J }|
= |Σα1 (A2)|+ |Σα2 (−A1)|−1
≥
k∑
i=1
i ri −
m∑
i=1
i ri +m
(
m∑
i=1
ri −α1
)
+
l∑
i=1
i r−i −
m′∑
i=1
i r−i +m′
(
m′∑
i=1
r−i −α2
)
+1
= Lα, (3)
where the third inequality comes from Theorem A. Moreover,
Σα(A0) ⊆
[
l∑
i=1
(−i )r−i +
m∑
i=1
i ri −m
(
m∑
i=1
ri −α1
)
,
k∑
i=1
i ri +
m′∑
i=1
(−i )r−i − (−m′)
(
m′∑
i=1
r−i −α2
)]
.
Then |Σα(A0)| ≤ Lα. Thus, |Σα(A0)| = Lα. Therefore, at this point, Lα is optimal.
Thus, we complete the proof of Theorem 1. □
Xing-Wang Jiang and Ya-Li Li 1283
If zero is a term in A, one could easily get the following result.
Corollary 2. Let l ,k ≥ 1 be two integers and A = (a−l , a−l+1, . . . , ak )r be a finite sequence of integers,
where a−l < a−l+1 < ·· · < a−1 < 0 = a0 < a1 < a2 < ·· · < ak and r = (r−l ,r−l+1, . . . ,rk ) with ri > 0 for
i ∈ [−l ,k]. Let 0 ≤α<∑ki=−l ri and
L′α=

∑l
i=1ir−i +
∑k
i=1iri +1, if α≤min
{∑l
i=0 r−i ,
∑k
i=0 ri
}
;∑l
i=1ir−i +
∑k
i=1iri −
∑m
i=1iri +m
(∑l
i=0 r−i +
∑m
i=1 ri −α
)+1, if ∑li=0 r−i <α≤∑ki=0 ri ;∑l
i=1ir−i +
∑k
i=1iri −
∑m′
i=1ir−i +m′
(∑k
i=0 ri +
∑m′
i=1 r−i −α
)
+1, if ∑ki=0 ri <α≤∑li=0 r−i ;∑l
i=1ir−i +
∑k
i=1iri −
∑m
i=1iri +m
(∑l
i=0 r−i +
∑m
i=1 ri −α
)
−∑m′i=1ir−i +m′ (∑ki=0 ri +∑m′i=1 r−i −α)+1, if α> max{∑li=0 r−i , ∑ki=0 ri },
where m, m′ are integers which satisfy
∑m−1
i=1 ri ≤ α−
∑l
i=0 r−i <
∑m
i=1 ri and
∑m′−1
i=1 r−i ≤ α−∑k
i=0 ri <
∑m′
i=1 r−i , respectively. Then
|Σα(A)| ≥ L′α.
Moreover, the lower bound is optimal.
Proof. The proof is clear. Here, we only give the outline of the proof. Let
A′ = (a−l , a−l+1, . . . , a−1)r ′ ∪ (a1, a2, . . . , ak )r ′′ ,
where r ′ = (r−l ,r−l+1, . . . ,r−1) and r ′′ = (r1,r2, . . . ,rk ). Since 0 repeats r0 times, it is easy to see that
when 0 ≤ α < r0, we have |Σα(A)| ≥ |Σ1(A′)|; when α ≥ r0, we have |Σα(A)| ≥ |Σα−r0 (A′)|. When
0 ≤α< r0, by Theorem 1 we have
|Σα(A)| ≥ |Σ1(A′)| ≥
l∑
i=1
i r−i +
k∑
i=1
i ri +1.
When α≥ r0, by Theorem 1 for |Σα−r0 (A′)| we can easily get the desired conclusion. □
3. Inverse problem
Corresponding to Theorem 1, we firstly deal with the case in which the sequence A does not
contain zero terms.
Theorem 3. Let A be a sequence which is the same as A defined in Theorem 1. Let 0 ≤ α ≤∑l
i=1 r−i +
∑k
i=1 ri −2 and Lα, m, m′ be the integers defined in Theorem 1. If |Σα(A)| = Lα, then
A = a1 ∗ ([−l ,−1]r ∪ [1,k]r ′ ).
Proof. Let A1 and A2 be the sequences defined in the proof of Theorem 1. Since |Σα(A)| = Lα, all
inequalities of (1), (2) and (3) become equalities. By Theorem B for sequences −A1 and A2, we
have
A1 =−a−1 ∗ [−l ,−1]r
except that A1 = (a−2, a−1)r with r = (r−2,1) and A1 = (a−2 +a−1, a−2, a−1)r with r = (r−3,1,1);
A2 = a1 ∗ [1,k]r ′
except that A2 = (a1, a2)r ′ with r ′ = (1,r2) and A2 = (a1, a2, a1 +a2)r ′ with r ′ = (1,1,r3).
1284 Xing-Wang Jiang and Ya-Li Li
Next, it only suffices to prove that a−1 =−a1, a−2 =−2a1 for l ≥ 2 and a2 = 2a1 for k ≥ 2. Since
all inequalities of (1), (2) and (3) become equalities, we know that there is exactly one term I + J
in Σα(A) between I + J −a1 and I + J −a−1. Noting that I + J −a1 −a−1 ∈Σα(A) and
I + J −a1 < I + J −a1 −a−1 < I + J −a−1,
we have I+J = I+J−a1−a−1, and so a−1 =−a1. If l ≥ 2, then one could get similarly that I+J−a−1
is the only one term in Σα(A) with
I + J < I + J −a−1 < I + J −a−2.
However, in view of a−1 =−a1 > a−2, we have I + J −a1 −a−2 ∈Σα(A) and
I + J < I + J −a1 −a−2 < I + J −a−2,
which shows that I + J − a1 − a−2 = I + J − a−1, that is a−2 = −a1 + a−1 = −2a1. If k ≥ 2, then by
similar argument, one could obtain that a2 = 2a1. Therefore,
A = a1 ∗ ([−l ,−1]r ∪ [1,k]r ′ ).
This completes the proof. □
If zero is a term in A, by Theorem 3, one could get easily the following result.
Corollary 4. Let A be a sequence which is the same as A defined in Corollary 2. Let 0 ≤ α ≤∑k
i=−l ri −2 and L′α, m, m′ be the integers defined in Corollary 2. If |Σα(A)| = L′α, then
A = a1 ∗ [−l ,k]r .
Remark 5. It is not difficult to see that for any sequence A = (a1, a2, . . . , ak )r of integers with
a1 < a2 < ·· · < ak and r = (r1,r2, . . . ,rk ), where ri ≥ 1 (1 ≤ i ≤ k), |Σα(A)| = 1 for α =
∑k
i=1 ri and
|Σα(A)| = k +1 for α=∑ki=1 ri −1.
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] J. Bhanja, “A note on sumsets and restricted sumsets”, J. Integer Seq. 24 (2021), no. 4, article
no. 21.4.2 (9 pages).
[2] J. Bhanja and R. K. Pandey, “Inverse problems for certain subsequence sums in integers”,
Discrete Math. 343 (2020), no. 12, article no. 112148 (11 pages).
[3] J. Bhanja and R. K. Pandey, “On the minimum size of subset and subsequence sums in
integers”, C. R. Math. Acad. Sci. Paris 360 (2022), pp. 1099–1111.
[4] H. K. Dwivedi and R. K. Mistri, “Direct and inverse problems for subset sums with certain
restrictions”, Integers 22 (2022), article no. A112 (13 pages).
[5] X.-W. Jiang and Y.-L. Li, “On the cardinality of subsequence sums”, Int. J. Number Theory
14 (2018), no. 3, pp. 661–668.
[6] R. K. Mistri and R. K. Pandey, “A generalization of sumsets of set of integers”, J. Number
Theory 143 (2014), pp. 334–356.
[7] R. K. Mistri, R. K. Pandey and O. Prakash, “Subsequence sums: direct and inverse prob-
lems”, J. Number Theory 148 (2015), pp. 235–256.
Xing-Wang Jiang and Ya-Li Li 1285
[8] M. B. Nathanson, “Inverse theorems for subset sums”, Trans. Am. Math. Soc. 347 (1995),
no. 4, pp. 1409–1418.
[9] M. B. Nathanson, Additive number theory. Inverse problems and the geometry of sumsets,
Springer, 1996, pp. xiv+293.
[10] V. H. Vu, “Some new results on subset sums”, J. Number Theory 124 (2007), no. 1, pp. 229–
233.
[11] Q.-H. Yang and Y.-G. Chen, “On the cardinality of general h-fold sumsets”, Eur. J. Comb. 47
(2015), pp. 103–114.


On the cardinality of subsequence sums II

Christian Bär

Bernhard Hanke

<i>K</i>-cowaist of manifolds with boundary

Gustavo Jasso

Comptes Rendus
Mathématique
Gustavo Jasso
On a theorem of B. Keller on Yoneda algebras of simple modules
Volume 362 (2024), p. 1449-1453
Online since: 14 November 2024
https://doi.org/10.5802/crmath.655
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
2024, Vol. 362, p. 1449-1453
https://doi.org/10.5802/crmath.655
Research article / Article de recherche
Representation theory / Théorie des représentations
On a theorem of B. Keller on Yoneda algebras
of simple modules
Sur un théorème de B. Keller sur les algèbres de Yoneda
de modules simples
Gustavo Jasso a
a Lund University, Centre for Mathematical Sciences, Box 118, 22100 Lund, Sweden
Current address: Mathematisches Institut, Universität zu Köln, Weyertal 86-90, 50931 Cologne, Germany
E-mail: gjasso@math.uni-koeln.de
Abstract. A theorem of Keller states that the Yoneda algebra of the simple modules over a finite-dimensional
algebra is generated in cohomological degrees 0 and 1 as a minimal A∞-algebra. We provide a proof of an
extension of Keller’s theorem to abelian length categories by reducing the problem to a particular class of
Nakayama algebras, where the claim can be shown by direct computation.
Résumé. Un théorème de Keller stipule que l’algèbre de Yoneda des modules simples sur une algèbre de di-
mension finie est générée en degrés cohomologiques 0 et 1 comme une A∞-algèbre minimale. Nous prou-
vons une extension du théorème de Keller aux catégories de longueur abélienne en réduisant le problème à
une classe particulière d’algèbres de Nakayama, où l’affirmation peut être démontrée par un calcul direct.
Keywords. Yoneda algebras, simple modules, Nakayama algebras, A∞-algebras.
Mots-clés. Algèbres de Yoneda, modules simples, algèbres de Nakayama, A∞-algèbres.
2020 Mathematics Subject Classification. 18G70, 16G20, 16G70.
Funding. The author’s research was supported by the Swedish Research Council (Vetenskapsrådet) Research
Project Grant 2022-03748 ‘Higher structures in higher-dimensional homological algebra.’
Manuscript received 23 February 2024, revised 15 May 2024, accepted 3 June 2024.
We work over an arbitrary field k. Let A be a finite-dimensional algebra and S the direct sum of
a complete set of representatives of the simple (right) A-modules. The Yoneda algebra Ext∗A(S,S),
as a graded algebra, does not determine the algebra A up to Morita equivalence, as the following
example shows (see [8, Section 2.1] or [13, Example B.2.2]): Let A = k[x]/(xℓ), ℓ≥ 3; then
Ext∗A(S,S) ∼= k[u, v]/(u2), |u| = 1, |v | = 2, (1)
does not depend on ℓ. On the other hand, the Yoneda algebra of an arbitrary finite-dimensional
algebra inherits, via Kadeishvili’s Homotopy Transfer Theorem [5, 15], the structure of a mini-
mal1 A∞-algebra since the Yoneda algebra is the cohomology of the differential graded algebra
RHomA(S,S). Endowed with this additional A∞-structure the Yoneda algebra does determine the
algebra A up to Morita equivalence [7, Section 7.8]. The purpose of this short article is to provide
a proof of a minor extension of a theorem of Keller [8, Section 2.2, Proposition 1(b)] that is stated
1Recall that an A∞-algebra is minimal if its underlying complex has vanishing differential.
ISSN (electronic): 1778-3569 https://comptes-rendus.academie-sciences.fr/mathematique/
1450 Gustavo Jasso
below. A proof of Keller’s theorem that utilises the calculus of Massey products was announced
by Minamoto in [16]; our proof should have some similarities with his.
Theorem (Keller). Let A be a k-linear abelian length category [10] with only finitely many
pairwise non-isomorphic simple objects, for example the category of finite-dimensional modules
over a finite-dimensional algebra. Let S1, . . . ,Sn be a complete set of representatives of the simple
objects in A and set S := S1 ⊕·· ·⊕Sn . Then, the Yoneda algebra Ext∗A (S,S) is generated by its
homogeneous components of cohomological degrees 0 and 1 as an A∞-algebra.
Remark. Since the abelian category A is not assumed to have any non-zero projective or injec-
tive objects, we interpret the Yoneda algebra Ext∗
A
(S,S) in terms of Yoneda equivalence classes of
exact sequences [19]. The results in [18, Section III.3] (see in particular paragraph III.3.3.2 therein)
readily imply that we can identify the Yoneda algebra with the graded algebra
⊕
k≥0 Hom(S,S[k])
of endomorphisms of S in the derived category of A . Thus, we may and we will identify minimal
A∞-algebra structures on the Yoneda algebra with those of the latter graded algebra.
Remark. The isomorphism (1) shows that the conclusion of the theorem may fail if the Yoneda
algebra Ext∗
A
(S,S) is considered as a graded algebra only.
Proof of the theorem. We use freely the theory of A∞-categories [7, 12], as well as the theory
of differential graded (=DG) categories and their derived categories [6, 9]. We also assume
familiarity with the Auslander–Reiten (=AR) theory of Nakayama algebras, see for example [1,
Chapter V].
Let d ≥ 1 and δ ∈ Extd+1
A
(S,S) a Yoneda class represented by an exact sequence
0 −→ Sa −→ M1 −→ M2 −→ ·· · −→ Md+1 −→ Sb −→ 0 (2)
between some simple objects; notice that we identify Extk
A
(S,S) =⊕ni , j=1 ExtkA (Si ,S j ). Below we
prove that δ is generated by Yoneda classes of degrees 1 and d under the operations mAk with
k ≤ len(M1). The theorem then follows by induction on d ≥ 1.
A straightforward inductive argument shows that the exact sequence (2) can be extended, by
first choosing composition series of Ni := img(Mi → Mi+1), to a commutative diagram of the
following form in which all squares are bicartesian, the apparent sequences at the bottom are
short exact, and all of the objects along the bottom of the diagram are simple:
Sa
•
•
•
•
M1
•
•
•
•
N1
•
•
•
•
•
M2
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•
Sc
•
•
•
•
•
M ′2
•
•
•
•
•
N2
•
•
•
•
· · ·
•
•
•
•
•
•
•
•
· · ·
•
•
•
•
•
•
•
· · ·
•
•
•
•
•
•
· · ·
•
•
•
•
•
· · ·
•
•
•
•
. . .
•
•
•
•
•
•
•
•
•
Nd
Md+1
•
•
•
•
•
•
•
•
•
•
•
•
•
•
Sb
ℓ= len(M1)
Consider now the Nakayama algebra B = kQB /I with Gabriel quiver
QB : 1 → 2 → . . . → p, p =∑d+1i=1 len(Mi )−∑dj=1 len(N j ) ,
Gustavo Jasso 1451
and Kupisch series
(1,2, . . . , len(M1) , len(N1)+1, len(N1)+2, . . . , len(M2) ,
. . . , len(Nd−1)+1, len(Nd−1)+2, . . . , len(Md ) , len(Md+1)).
Observe that the above commutative diagram is indexed by the AR quiver of the category modB
of finite-dimensional (right) B-modules. In other words, we can extend the exact sequence (2) to
an exact functor F : modB → A that sends the simple B-modules to simple objects of A . The
exact functor F lifts to an A∞-functor
F̃ : Db(modB)dg −→ Db(A )dg
between the corresponding bounded derived DG categories, for example because Db(modB)dg is
the pre-triangulated hull of modB [3, Theorem 6.1 and Example 6.2]. The many-objects version
of the Homotopy Transfer Theorem yields minimal models of Db(modB)dg and D
b(A )dg so that
we may consider the induced A∞-functor
H∗(Db(modB)dg) H∗(Db(A )dg)
Db(modB)dg D
b(A )dg
F
≀
F̃
≀
H∗(Db(modB)dg) H∗(Db(A )dg)
H∗(Db(modB)dg) H∗(Db(A )dg)
F1
id
H∗(F̃ )
id
We obtain, in particular, an A∞-morphism
F : Ext∗B (S,S) −→ Ext∗A (S,S) ,
where the Yoneda algebras are now endowed with minimal A∞-algebra structures.
We claim that the underlying morphism of graded algebras
F1 : Ext
∗
B (S,S) −→ Ext∗A (S,S) , γ 7−→ δ, (3)
maps the Yoneda class γ of an augmented minimal projective resolution of the simple B-module
concentrated at the vertex p of the quiver QB to the class δ. Indeed, γ = αd+1α2 · · ·α1 is the
product of Yoneda classes of short exact sequences between certain indecomposable B-modules
and, therefore,
F1(γ) =F1(αd+1αd · · ·α1) =F1(αd+1)F1(αd ) · · ·F1(α1) = δ.
Here, the classes F1(αi ), i = 1, . . . ,d +1, satisfy the equality on the right-hand side by construc-
tion since the restriction of the graded functor F1 : H∗(Db(modB)dg) → H∗(Db(A )dg) to the
graded subcategories spanned by modB and A , respectively, is induced by the exact functor
F : modB →A above.
Since Nakayama algebras are monomial algebras, an explicit description of a minimal model
of the Yoneda algebra Ext∗B (S,S) is available due to independent work of Chuang and King
(unpublished) and of Tamaroff [17, Theorem 4.9].2 In particular, we can assume that3
mBℓ (η1 ⊗η2 ⊗·· ·⊗ηℓ−1 ⊗η) = γ, ℓ= len(M1) , (4)
where ηi ∈ Ext1B (Si+1,Si ) ∼= k is the Yoneda class of an AR sequence and η ∈ ExtdB
(
Sp ,Sℓ
)∼= k is the
Yoneda class of an exact sequence of the form
0 −→ Sℓ −→ M ′2 −→ ·· · −→ Md+1 −→ Sp −→ 0,
2The computation of the explicit product that we need is straightforward using the HTT; see also [4].
3Compare with [16, Theorem 1] where the equivalent of the Massey product 〈η1, . . . ,ηℓ−1,η〉 ∋ ±γ appears.
1452 Gustavo Jasso
with M ′2 the injective hull of the simple B-module Sℓ and Md+1 the projective cover of Sp , as
indicated in the above diagram that we now interpret as the AR quiver of modB . Moreover, an
elementary computation shows that
mBj−i+1(ηi ⊗ηi+1 ⊗·· ·⊗η j ) ∈ Ext2B
(
S j+1,Si
)= 0 (5)
whenever 1 ≤ i < j < ℓ, and also
mBℓ−i+1(ηi ⊗ηi+1 ⊗·· ·⊗ηℓ−1 ⊗η) ∈ Ext∗B
(
Sp ,Si
)= 0 (6)
whenever 1 < i < ℓ. Here we use that the operation mBk has degree 2−k.
Finally, the A∞-morphism F : Ext∗B (S,S) → Ext∗A (S,S) satisfies in particular an equation of the
form
0 = ∂(Fℓ) =
∑
r+1+t=k
r+s+t=ℓ
±Fk ◦ (1⊗r ⊗mBs ⊗1⊗t )−
∑
2≤k≤ℓ
i1+···+ik=ℓ
±mAk ◦ (Fi1 ⊗·· ·⊗Fik ),
where ∂= 0 since the A∞-algebras involved are minimal. Therefore, using equations (3) and (4),4
δ=F1(γ) =F1(mBℓ (η1 ⊗·· ·⊗ηℓ−1 ⊗η))
= mAℓ (F1(η1)⊗·· ·⊗F1(ηℓ−1)⊗F1(η)︸ ︷︷ ︸
∈Ext1,d
A
(S,S)
)+ω,
where
ω= ∑
r+1+t=k>1
r+s+t=ℓ
±Fk ( · · ·︸ ︷︷ ︸
=0
)+ ∑
2≤k<ℓ
i1+···+ik=ℓ
±mAk ( · · ·︸ ︷︷ ︸
∈Ext1,d
A
(S,S)
).
To argue the vanishing of the first term we use that the inputs involve one the higher products
(5)-(6) that we know vanish, and for the condition on the degrees of the inputs in the second term
we note that
F j−i+1(ηi ⊗·· ·⊗η j ) ∈ Ext1A (S,S) 1 ≤ i < j < ℓ,
Fℓ−i+1(ηi ⊗·· ·⊗ηℓ−1 ⊗η) ∈ ExtdA (S,S) 1 < i < ℓ,
since the component Fk is a morphism of degree 1−k. This finishes the proof. □
Remark. In a MathOverflow post [14], Madsen gives an outline of Keller’s original proof of the
theorem, which is unpublished.5 Keller’s proof uses the description of the category of finite-
dimensional modules over a finite-dimensional algebra A as the category of twisted stalks on
the Yoneda algebra Ext∗A(S,S) with its minimal A∞-algebra structure. In fact, one may replace
the ambient abelian category by the exact subcategory of objects filtered by a finite collection of
objects, with Ext∗ now understood in this exact subcategory. Our proof, which can be adapted
to this more general setting with the appropriate modifications, avoids making explicit use of
the category of twisted stalks by instead reducing the problem to a computation with Nakayama
algebras.
Acknowledgements
The author thanks Julian Külshammer for informing him of Madsen’s MathOverflow post [14]
and for interesting discussions on possible extensions of Keller’s theorem. The author also thanks
Bernhard Keller for clarifications on his original proof of the theorem.
4Compare with [2, Theorem A(ii)] and see also Section 3 therein.
5But see [11, Theorem 3.22].
Gustavo Jasso 1453
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] I. Assem, D. Simson and A. Skowroński, Elements of the representation theory of associative
algebras. Vol. 1. Techniques of representation theory, Cambridge University Press, 2006,
pp. x+458.
[2] U. Buijs, J. M. Moreno-Fernández and A. Murillo, “A∞ structures and Massey products”,
Mediterr. J. Math. 17 (2020), no. 1, article no. 31 (15 pages).
[3] X. Chen, “On exact dg categories”, 2023. https://arxiv.org/abs/2306.08231.
[4] E. Herscovich, “A simple note on the Yoneda (co)algebra of a monomial algebra”, Ukraïn.
Mat. Zh. 73 (2021), no. 2, pp. 275–277.
[5] T. V. Kadeishvili, “The algebraic structure in the homology of an A(∞)-algebra”, Soobshch.
Akad. Nauk Gruzin. SSR 108 (1982), no. 2, pp. 249–252.
[6] B. Keller, “Deriving DG categories”, Ann. Sci. Éc. Norm. Supér. 27 (1994), no. 1, pp. 63–102.
[7] B. Keller, “Introduction to A-infinity algebras and modules”, Homology Homotopy Appl. 3
(2001), no. 1, pp. 1–35.
[8] B. Keller, “A-infinity algebras in representation theory”, in Representations of algebra. Vol.
I (D. Happel and Y. B. Zhang, eds.), Beijing Norm. Univ. Press, 2002, pp. 74–86.
[9] B. Keller, “On differential graded categories”, in International Congress of Mathematicians.
Vol. II , European Mathematical Society, 2006, pp. 151–190.
[10] H. Krause and D. Vossieck, “Length categories of infinite height”, in Geometric and topo-
logical aspects of the representation theory of finite groups, Springer, 2018, pp. 213–234.
[11] J. Külshammer and V. Miemietz, “Uniqueness of exact Borel subalgebras and bocses”, 2021.
https://arxiv.org/abs/2109.03586.
[12] K. Lefèvre-Hasegawa, “Sur les A-infini catégories”, 2003. https : / / arxiv. org / abs / math /
0310337.
[13] D.-O. Madsen, Homologcal aspects in representation theory, PhD thesis, Norges teknisk-
naturvitenskapelige universitet (NTNU), 2002.
[14] D.-O. Madsen, “Ext-algebra generated by Hom and Ext1 as A∞-algebra?”, MathOverflow,
2015. https://mathoverflow.net/q/215918 (version: 2015-08-29).
[15] M. Markl, “Transferring A∞ (strongly homotopy associative) structures”, Rend. Circ. Mat.
Palermo 79 (2006), pp. 139–151.
[16] H. Minamoto, “Higher products on Yoneda Ext algebras”, in Proceedings of the 48th Sym-
posium on Ring Theory and Representation Theory, Symp. Ring Theory Represent. Theory
Organ. Comm., Yamanashi, 2016, pp. 92–95.
[17] P. Tamaroff, “Minimal models for monomial algebras”, Homology Homotopy Appl. 23
(2021), no. 1, pp. 341–366.
[18] J.-L. Verdier, Des catégories dérivées des catégories abéliennes, Société Mathématique de
France, 1996, pp. xii+253. With a preface by Luc Illusie, Edited and with a note by Georges
Maltsiniotis.
[19] N. Yoneda, “On Ext and exact sequences”, J. Fac. Sci. Univ. Tokyo, Sect. I 8 (1960), pp. 507–
576.


On a theorem of B. Keller on Yoneda algebras of simple modules

Alexei Tsygvintsev

Comptes Rendus
Mathématique
Alexei Tsygvintsev
Velocity Syzygies and Bounding Syzygy Moments in the Planar Three-Body Problem
Volume 362 (2024), p. 1785-1791
Online since: 25 November 2024
https://doi.org/10.5802/crmath.683
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
2024, Vol. 362, p. 1785-1791
https://doi.org/10.5802/crmath.683
Research article / Article de recherche
Dynamical systems / Systèmes dynamiques
Velocity Syzygies and Bounding Syzygy
Moments in the Planar Three-Body Problem
Syzygies de vitesse et encadrement des instants de
syzygies dans le problème plan des trois corps
Alexei Tsygvintsev ,a
a ENS de Lyon, UMPA, 46 allée d’Italie, 69364 Lyon Cedex 07, France
E-mail: alexei.tsygvintsev@ens-lyon.fr
Abstract. We consider the Newtonian planar three-body problem, defining a syzygy (velocity syzygy) as a
configuration where the positions (velocities) of the three bodies become collinear. We demonstrate that
if the total energy is negative, every collision-free solution has an infinite number of velocity syzygies.
Specifically, the velocities of the three bodies become parallel within every interval of time containing three
consecutive syzygies. Using comparison theory for matrix Riccati equations, we derive new upper and lower
bounds on the moments when syzygies occur.
Résumé. Nous considérons le problème newtonien des trois corps dans le plan, en définissant une syzy-
gie (syzygie de vitesse) comme une configuration où les positions (vitesses) des trois corps deviennent co-
linéaires. Nous démontrons que si l’énergie totale est négative, toute solution sans collision présente un
nombre infini de syzygies de vitesse. Plus précisément, les vitesses des trois corps deviennent parallèles dans
chaque intervalle de temps contenant trois syzygies consécutives. En utilisant la théorie de comparaison
pour les équations de Riccati matricielles, nous dérivons de nouvelles bornes supérieures et inférieures sur
les instants où les syzygies se produisent.
Keywords. dynamical systems, celestial mechanics, three-body problem, syzygies.
Mots-clés. Systèmes dynamiques, mécanique céleste, problème à trois corps, Syzygies.
2020 Mathematics Subject Classification. 92B05.
Manuscript received 16 July 2024, accepted 3 September 2024.
1. Introduction and Preliminary Results
The Newtonian planar three-body problem has been a subject of extensive study in celestial me-
chanics. It involves predicting the motion of three bodies moving under their mutual gravita-
tional attraction. Understanding syzygies, where the positions or velocities of the three bodies
become collinear, provides significant insight into the dynamics of the system.
Let P1, P2, and P3 be three points in the plane with strictly positive masses m1,m2,m3 and
Euclidean coordinates (xi , yi ) ∈ R2, i = 1,2,3. The Newtonian three-body problem [7] can be
formulated as follows:
z̈1 = m2 z21|z21|3
−m3 z13|z13|3
, z̈2 = m3 z32|z32|3
−m1 z21|z21|3
, z̈3 = m1 z13|z13|3
−m2 z32|z32|3
, (1)
where zk = xk + i yk ∈C, k = 1,2,3 and zkl = zk − zl .
ISSN (electronic): 1778-3569 https://comptes-rendus.academie-sciences.fr/mathematique/
1786 Alexei Tsygvintsev
Given that the total linear momentum (which is conserved) is zero, one can always set∑
k
mk żk = 0,
∑
k
mk zk = 0, (2)
by placing the center of mass of the bodies at the origin.
Let t 7→ zi (t ), i = 1,2,3 be any collision-free solution of equations (1) defined for t ∈ I = [0, a),
a > 0, and determined by a set of initial conditions (zi (0), żi (0)), i = 1,2,3.
Definition 1. We say that the three bodies P1,P2,P3 form a syzygy (velocity syzygy) at the moment
t0 ∈ I if the complex triplet (z1, z2, z3)(t0) ((ż1, ż2, ż3)(t0)) is collinear.
Our first result concerns the existence of velocity syzygies.
Theorem 2. Let t 7→ (z1(t ), z2(t ), z3(t )), t ∈ [0,+∞), be a zero angular momentum, collision-free
solution to the three-body problem (1) with negative energy. Then, it has an infinite number of
velocity syzygies.
Proof. After introducing the new variables wi = mi zi , i = 1,2,3, the relations (2) yield∑
i
wi =
∑
i
ẇi = 0. (3)
Writing
wk = Xk + i Yk , Xk = mk xk , Yk = mk yk , k = 1,2,3 , (4)
and using (3), one derives from the equations (1) the following 2×2 matrix equation:
Ẍ = AX , X =
(
X1 Y1
X2 Y2
)
, A =
(−m2ρ3 −m13ρ2 m1ρ32
m2ρ31 −m1ρ3 −m32ρ1
)
, (5)
where
ρ1 = 1|z32|3
, ρ2 = 1|z13|3
, ρ3 = 1|z21|3
, mi j = mi +m j , ρi j = ρi −ρ j . (6)
We define the determinants
∆1(t ) =
∣∣∣∣ X1 Y1X2 Y2
∣∣∣∣ (t ), ∆2(t ) = ∣∣∣∣ Ẋ1 Ẏ1Ẋ2 Ẏ2
∣∣∣∣ (t ), t ∈ [0,+∞) . (7)
It is sufficient to show that ∆2 has an infinite number of zeros for t ≥ 0. As demon-
strated by Montgomery [4], and later independently by the author in [5, 6], the solution t 7→
(z1(t ), z2(t ), z3(t )), for t ∈ [0,+∞), has an infinite number of consecutive syzygies. That is, there
exist ti ≥ 0, i = 1,2, . . . , with ti+1 > ti such that ∆1(ti ) = 0. As shown in [5, p. 6833], in the zero
angular momentum case:
∆̇1(t )
2 −4∆1(t )∆2(t ) ≥ 0, ∀t ≥ 0, (8)
where equality occurs only if the positions zi (t ) and velocities żi (t ) for i = 1,2,3 are parallel.
Let ti < ti+1 < ti+2 be three consecutive syzygy moments. By Rolle’s theorem, there exist
τ ∈ (ti , ti+1) and η ∈ (ti+1, ti+2) such that τ < η and ∆̇1(τ) = ∆̇1(η) = 0. Without loss of generality,
we can assume that ∆1(τ) > 0 and ∆1(η) < 0. Indeed, if for some i ≥ 0, ∆1(ti ) = ∆̇1(ti ) = 0, then
according to (8), the corresponding solution is a straight-line one (with all positions and velocities
lying on the same line) and will result in a triple collision since the energy is negative.
Thus, by the Intermediate Value Theorem and (8),∆2 has a zero in the interval [τ,η]. The proof
is complete. □
Alexei Tsygvintsev 1787
2. Bounds on Syzygy Moments in the Zero Angular Momentum Case
In the author’s previous works [5, 6], an upper bound was determined for the moment when the
very first syzygy occurs, expressed as a function of the energy value and the masses alone. In
this section, we refine our results by providing both upper and lower bounds that depend on the
initial positions and velocities of the bodies, resulting in significantly more precise estimates.
First, we present a preliminary result from the comparison theory for matrix Riccati equations,
developed by Eschenburg and Heintze in 1990 [3], which will be utilised later in the proof of our
main result.
Let E be a finite-dimensional real vector space equipped with an inner product ( · , · ), and let
S(E) denote the space of self-adjoint linear endomorphisms of E . Consider the Riccati differential
equation with a smooth coefficient curve R : (0, t0) → S(E):
Ḃ +B 2 +R = 0, (9)
where B : (0, t0) → S(E) is the solution.
We define R1 ≥ R2, where R1,R2 ∈ S(E), if the difference R1 −R2 is positive semidefinite. With
this definition, one can compare the solutions B1 and B2 of the following Riccati equations:
Ḃ1 +B 21 +R1 = 0, (10)
and
Ḃ2 +B 22 +R2 = 0, (11)
subject to appropriate initial conditions.
Theorem 3 ([3]). Let R1,R2 : (0, t0) → S(E) be smooth coefficient curves with R1 ≥ R2. For j = 1,2,
let B j : (0, t j ) → S(E) be the solution to the Riccati equation corresponding to R j , with maximal
t j ∈ (0,+∞]. If the difference U := B2 −B1 has a continuous extension to 0 with U (0) ≥ 0, then
t1 ≤ t2 and B1 ≤ B2 on (0, t1).
We now assume that at the instant t = 0, the configuration of the three-body problem is
different from a syzygy, i.e., the positions of the three bodies are not parallel. In this case, the
matrix C0 = Ẋ0X −10 , where X0 = X (0) and Ẋ0 = Ẋ (0), is well-defined since det(X0) ̸= 0. In the
zero angular momentum case, as shown in [5], both eigenvalues of C0 are real. We denote the
spectrum of C0 as:
Spec(C0) = {λ1,λ2} ⊂R , (12)
and define
πs = min{λ1,λ2} . (13)
Finally, we suppose that during the motion, all mutual distances of the bodies are bounded
above and below by the constants α,β> 0, with α<β, i.e.,
α≤ |zi j (t )| ≤β, ∀ i < j , (14)
and we define
θα =
p
M
α3/2
, θβ =
p
M
β3/2
, M = m1 +m2 +m3 . (15)
Theorem 4. Let Ts > 0 be the moment in time when the first syzygy occurs. Then the following
bounds hold:
1
θα
arccot
(
−πs
θα
)
≤ Ts ≤ 1
θβ
arccot
(
−πs
θβ
)
. (16)
1788 Alexei Tsygvintsev
Proof. Let t 7→φi (t ), i = 1,2,3, be arbitrary positive smooth functions on [0,+∞). We consider a
linear system of second order
Z̈ =A Z , Z =
(
Z11 Z12
Z21 Z22
)
, A (φ1,φ2,φ3) =
(−m2φ3 −m13φ2 m1φ32
m2φ31 −m1φ3 −m32φ1
)
, (17)
where φi j =φi −φ j , mi j = mi +m j .
If (Z11, Z12) = (X1,Y1), (Z21, Z22) = (X2,Y2), and φi = ρi for i = 1,2,3, then the equations (17)
and (5) coincide.
It is a straightforward computation to verify that for any solution Z of the system (17), the
following identity holds:
1
m1
∣∣∣∣Z11 Z12Ż11 Ż12
∣∣∣∣+ 1m2
∣∣∣∣Z21 Z22Ż21 Ż22
∣∣∣∣+ 1m3
∣∣∣∣Z11 +Z21 Z12 +Z22Ż11 + Ż21 Ż12 + Ż22
∣∣∣∣= k, k ∈R , (18)
which is an analog of the angular momentum conservation law in the three-body problem
written in the form (5) (recall that replacing (Zi 1, Zi 2) by (Xi ,Yi ), i = 1,2, in formula (18) the third
determinant is nothing but det(X3,Y3; Ẋ3, Ẏ3)).
The matrix B = Ż Z−1 is a solution of the following Riccati equation:
Ḃ +B 2 +R = 0, R =−A , (19)
an equation utilized in the study of the three-body problem in [6].
We introduce the matrices
I2 =
(
1 0
0 1
)
, Ã1 =
( m32
2 0
−m2 −m322
)
, Ã2 =
(−m132 −m1
0 m132
)
. (20)
Then, if k = 0,
B = δ̇
2δ
I2 + b
m2δ
Ã1 − a
m1δ
Ã2 = 1
δ
[
βa +γb + δ̇2 a
−b −βa −γb + δ̇2
]
, (21)
where
β= 1
2
(
m3
m1
+1
)
, γ= 1
2
(
m3
m2
+1
)
, (22)
and
δ=
∣∣∣∣Z11 Z12Z21 Z22
∣∣∣∣ , a = ∣∣∣∣Z11 Z12Ż11 Ż12
∣∣∣∣ , b = ∣∣∣∣Z21 Z22Ż21 Ż22
∣∣∣∣ . (23)
The equality (21) follows directly from formulas (2.21), (2.32), (2.35)–(2.37) of [6] in which one
should replace X = Z and ∆1 = δ.
In order to apply Theorem 3 to equation (19), the solution B and the matrix R should be
symmetric. This can be achieved by the linear transformation B̃ = P−1BP with the invertible
matrix P defined by:
P =
(
− m1m13
1
m13
√
m1m3M
m2
1 0
)
, M = m1 +m2 +m3 . (24)
Indeed, it is easy to check that the matrices P−1 Ãi P , i = 1,2, and R̃ = P−1RP are symmetric.
Since B̃ is a linear combination of I2 and P−1 Ãi P , it is also symmetric. Thus, equation (19)
becomes:
˙̃B + B̃ 2 + R̃ = 0, (25)
and Theorem 3 can be applied.
This fact also follows directly from the work in [1], where the same matrix A (up to reduction,
transposition, and some scaling factor) was introduced in relation to the Wintner–Conley endo-
morphism, which encodes the forces. Specifically, it was demonstrated that in the case of zero
angular momentum, this matrix is always conjugate to a symmetric matrix, simultaneously with
B = Ż Z−1.
Alexei Tsygvintsev 1789
Lemma 5. The matrix Ã (φ1,φ2,φ3) = P−1A (φ1,φ2,φ3)P is negative semidefinite.
Proof. Since Ã is symmetric, it is sufficient to show that the eigenvalues of A are negative. We
have:
det(A ) = M(m3φ1φ2 +m2φ1φ3 +m1φ2φ3) ≥ 0, Tr(A ) =−
(
m32φ1 +m13φ2 +m21φ3
)≤ 0, (26)
since φi ≥ 0 for i = 1,2,3.
Thus, Ã is negative semidefinite. □
Let us consider two Riccati equations:
˙̃B1 + B̃ 21 + R̃1 = 0, R̃1 =−P−1A (ρ1,ρ2,ρ3)P , (27)
and
˙̃B2 + B̃ 22 + R̃2 = 0, R̃2 =−P−1A
(
1
β3
,
1
β3
,
1
β3
)
P (28)
subject to the same initial conditions
B̃ 1(0) = B̃ 2(0) = P−1 Ẋ0X −10 P . (29)
By the linearity of A as a function of φi (i = 1,2,3), we have
R̃2 − R̃1 = P−1A
(
ρ1 − 1
β3
,ρ2 − 1
β3
,ρ3 − 1
β3
)
P,
and ρi − 1β3 ≥ 0 for i = 1,2,3 by the definition of β.
According to Lemma 5, it follows that R̃1 ≥ R̃2.
Equation (28) can be easily solved by setting R̃2 = Ẏ Y −1. This leads to the equivalent equation:
Ÿ =−θ2βY , θβ =
p
M
β3/2
. (30)
The solution to the Cauchy problem defined by (30) and the initial conditions Y (0) = Y0 = P−1X0,
Ẏ (0) = Ẏ0 = P−1 Ẋ0 is:
Y (t ) = cos(θβt )Y0 +
1
θβ
sin(θβt )Ẏ0 , t ≥ 0. (31)
Therefore, the corresponding maximal solution of (28) is:
B̃ 2(t ) = Ẏ (t )Y (t )−1, t ∈ [0, t2) , (32)
where t2 is the first positive zero of t 7→ det(Y (t )).
One has
det(Y (t2)) = 0 ⇐⇒ det(Ẋ0X −10 +θβ cot(θβt2)I2) = 0, (33)
since X0 and P are invertible matrices. As a consequence, we obtain:
−θβ cot(θβt2) ∈ Spec(C0), C0 = Ẋ0X −10 , (34)
and t2 = 1
θβ
arccot
(
−πs
θβ
)
. (35)
Let Ts = t1 > 0 be the first syzygy moment for the solution of the three-body problem t 7→ zi (t ),
i = 1,2,3. Then, using a similar argument, we show that the maximal solution of the Cauchy
problem defined by (27) and (29) is defined in the interval [0, t1). Therefore, according to
Theorem 3, Ts ≤ t2, and the upper bound in (16) is proven.
To prove the lower bound, consider the Riccati equations:
˙̃B1 + B̃ 21 + R̃1 = 0, R̃1 =−P−1A
(
1
α3
,
1
α3
,
1
α3
)
P , (36)
and ˙̃B2 + B̃ 22 + R̃2 = 0, R̃2 =−P−1A (ρ1,ρ2,ρ3)P , (37)
where R̃1 ≥ R̃2. Applying the same arguments as before, the proof of Theorem 4 is complete. □
1790 Alexei Tsygvintsev
Remark 6. The uniform bounds |zi j | ≥α, ∀ i < j have a very natural astrophysical interpretation:
such motion corresponds to the collision-free movement of three rigid planets, each having the
same radius R = α/2. As seen from the proof of Theorem 4, the uniform bounds (14) and the
absence of collisions are required only for the period of time preceding the first syzygy.
3. Conclusion: Numerical Validation and Open Questions
To numerically verify our findings, we will consider the figure-eight periodic solution with equal
masses m1 = m2 = m3 = 1 and zero angular momentum, as described in [2]. The initial positions
and velocities of the bodies are set as follows:
z1(0) = 1.08075− i 0.0126893, z2(0) =−0.570154+ i 0.350807, z3(0) =−z1(0)− z2(0), (38)
and
ż1(0) = 0.0193421+ i 0.467219, ż2(0) = 1.0852− i 0.174718, ż3(0) =−ż1(0)− ż2(0) . (39)
Numerically, we can find that the first syzygy occurs at Ts = 0.55431 (see Figure 1). The
constants α and β are estimated to be α= 0.690526 and β= 2. Using formulas (15), we determine
θα = 3.01849 and θβ = 0.612372.
-1.0 -0.5 0.5 1.0
-0.3
-0.2
-0.1
0.1
0.2
0.3
Figure 1. The figure shows the trajectory of the figure-eight solution, as described in [2].
The initial positions of the bodies at t = 0 are represented by filled points. Additionally, the
configuration of the first syzygy, which occurs at t = 0.55431, is indicated by unfilled points.
Using the initial conditions (38) and (39), we compute:
X0 =
(
0.734528 1.35841
0.755791 −0.470708
)
, Spec(X0) = {1.31082,−1.047}, πs =−1.047, (40)
and the bounds (16) give us:
0.409781 ≤ Ts ≤ 0.864231, (41)
which are quite satisfactory.
We would like to highlight several open questions. It would be interesting to find bounds
analogous to (16) for the velocity syzygy moments using similar ideas based on the comparison
of matrix Riccati equations. It is important to note that the estimates provided by our Theorem 4
are valid only if the initial configuration at t = 0 is not a syzygy. However, with some effort and
by employing Theorem 3, one could establish bounds similar to (16) even in the case where the
initial configuration is a syzygy.
Finally, generalising to the case of non-zero angular momentum is an intriguing and impor-
tant challenge.
Alexei Tsygvintsev 1791
Acknowledgments
I would like to express my special gratitude to Richard Montgomery for useful suggestions and
valuable remarks and to Jean-Claude Sikorav for attracting my attention to work [3]. I also extend
my appreciation to the anonymous reviewer for their valuable feedback, which has significantly
enhanced the quality of this manuscript.
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] A. Albouy and A. Chenciner, “Le problème des n corps et les distances mutuelles”, Invent.
Math. 131 (1998), no. 1, pp. 151–184.
[2] A. Chenciner and R. Montgomery, “A remarkable periodic solution of the three-body prob-
lem in the case of equal masses”, Ann. Math. 152 (2000), no. 3, pp. 881–901.
[3] J.-H. Eschenburg and E. Heintze, “Comparison theory for Riccati equations”, Manuscr.
Math. 68 (1990), no. 2, pp. 209–214.
[4] R. Montgomery, “The zero angular momentum, three-body problem: all but one solution
has syzygies”, Ergodic Theory Dyn. Syst. 27 (2007), no. 6, pp. 1933–1946.
[5] A. Tsygvintsev, “On some collinear configurations in the planar three-body problem”, Non-
linearity 36 (2023), no. 12, pp. 6827–6838.
[6] A. Tsygvintsev, “On the existence of generalised syzygies in the planar three-body problem”,
C. R. Math. Acad. Sci. Paris 361 (2023), pp. 331–335.
[7] A. Wintner, The Analytical Foundations of Celestial Mechanics, Princeton University Press,
1941, pp. xii+448.


Velocity Syzygies and Bounding Syzygy Moments in the Planar Three-Body Problem

Aidan Hennessey

Constructing many-twist Möbius bands with small aspect ratios

Not a member yet

On Orthogonal Projections of Symplectic Balls

Nonlinear asymptotic stability in $L^\infty$ for Lipschitz solutions to scalar conservation laws

Exponential decay of harmonic $1$ -forms for wild harmonic bundles on curves

The action of pseudo-differential operators on functions harmonic outside a smooth hyper-surface

The radial limits and boundary uniqueness

On the cardinality of subsequence sums II

<i>K</i>-cowaist of manifolds with boundary

On a theorem of B. Keller on Yoneda algebras of simple modules

Velocity Syzygies and Bounding Syzygy Moments in the Planar Three-Body Problem

Constructing many-twist Möbius bands with small aspect ratios

5,369

6,067