Comptes Rendus Mathématique

Jean-Pierre Tignol

Invariants de Witt des involutions de bas degré en caractéristique 2

Santiago Gómez Cobos

Joel E. Restrepo

Michael Ruzhansky

Comptes Rendus
Mathématique
Santiago Gómez Cobos, Joel E. Restrepo and Michael Ruzhansky
Lp −Lq estimates for non-local heat and wave type equations on locally compact
groups
Volume 362 (2024), p. 1331-1336
Online since: 14 November 2024
https://doi.org/10.5802/crmath.643
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
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Comptes Rendus. Mathématique
2024, Vol. 362, p. 1331-1336
https://doi.org/10.5802/crmath.643
Research article / Article de recherche
Partial differential equations / Équations aux dérivées partielles
Lp −Lq estimates for non-local heat and wave
type equations on locally compact groups
Estimations Lp −Lq pour les équations non-locales de
type chaleur et de type onde sur des groupes localement
compacts
Santiago Gómez Cobos a, Joel E. Restrepo ∗, ,a,b andMichael Ruzhansky ,a,c
a Department of Mathematics: Analysis, Logic and Discrete Mathematics, Ghent University,
Krijgslaan 281, Building S8, B 9000 Ghent, Belgium
b Department of Mathematics, Cinvestav IPN, Mexico city, Mexico
c School of Mathematical Sciences, Queen Mary University of London, United Kingdom
E-mails: davidsantiago.gomezcobos@ugent.be, joel.restrepo@ugent.be, cocojoel89@yahoo.es
(J. E. Restrepo), michael.ruzhansky@ugent.be
Abstract. We prove the Lp − Lq (1 < p ⩽ 2 ⩽ q < +∞) norm estimates for the solutions of heat and wave
type equations on a locally compact separable unimodular group G by using an integro-differential operator
in time and any positive left invariant operator (maybe unbounded) on G . We complement our studies by
giving asymptotic time estimates for the solutions, which in some cases are sharp.
Résumé. On montre les estimations de norme Lp −Lq (1 < p ⩽ 2⩽ q <+∞) pour les solutions des équations
dites « de type chaleur » et « de type onde » définies sur un groupe localement compact, séparable et
unimodulaire G en utilisant un opérateur intégro-différentiel sur le temps et un opérateur positif invariant
á gauche quelconque sur G . De plus, on donne des estimations de temps asymptotiques pour ces solutions,
qui deviennent des estimations optimales dans quelques cas.
Keywords. Locally compact groups, heat type equations, wave type equations, asymptotic estimates, non-
local operators.
Mots-clés. Groupes localement compacts, équations de type chaleur, équations de type onde, estimations
asymptotiques, opérateurs non locaux.
2020 Mathematics Subject Classification. 43A15, 43A85, 45K05.
Funding. The authors were supported by the FWO Odysseus 1 grant G.0H94.18N: Analysis and Partial
Differential Equations, the Methusalem programme of the Ghent University Special Research Fund (BOF)
(Grant number 01M01021). M. R. is also supported by EPSRC grant EP/R003025/2 and FWO Senior Research
Grant G011522N.
Manuscript received 22 June 2023, accepted 21 April 2024.
∗Corresponding author
ISSN (electronic): 1778-3569 https://comptes-rendus.academie-sciences.fr/mathematique/
1332 Santiago Gómez Cobos, Joel E. Restrepo and Michael Ruzhansky
1. Introduction
As an application of some spectral multipliers results, it was shown recently in [1] that the Lp −Lq
norm estimates for the solution of the L -heat equation
∂t w(t , x)+L w(t , x) = 0, t > 0, x ∈G ,
w(t , x)|
t=0 = w0(x),
on a locally compact separable unimodular group G can be reduced to the time asymptotics of its
propagator in the noncommutative Lorentz space norm [11], which involves calculating the trace
of the spectral projections of the operator L . The considered operator can be any positive linear
left invariant operator acting on G with the possibility (generality) of having either continuous
or discrete spectrum. From convention of [1], a positive operator means nonnegative and self-
adjoint. Unfortunately, the results did not cover the Lp −Lq estimates for the case of L -wave
equations. The propagators can not be estimated by the noncommutative Lorentz space norm.
This is still an open question. In this paper, we fill this gap if we consider a non-local integro-
differential operator in time of order β (0 < β < 2), which allows us to consider L -equations
which interpolate between heat and wave types.
The type of groups that we can consider is very large. For instance, compact, semi simple,
exponential, nilpotent, some solvable ones, real algebraic, and many more.
Our results are new in this context and open a door to studying different estimates of other
types of equations which, in principle, can not be treated in the classical case but perhaps could
be carried out in the set up of non-local operators.
Thus the main results of the paper concern Lp − Lq (1 < p ⩽ 2 ⩽ q < +∞) norm estimates
for L -heat-wave type equations. Full details are given in Section 3. We then give a number of
examples by choosing concrete operators and groups. Moreover, in some cases, we are able to
claim the sharpness of the time-decay rate.
2. Von Neumann algebras and noncommutative Lorentz spaces
Let L(H ) be the set of linear operators defined on a Hilbert space H . In this context, the
concept of τ-measurability on a von Neumann algebra M (see e.g. [6, 12, 13, 17]) and the spectral
projections give us the possibility to approximate unbounded operators by bounded ones.
Let us now recall some important definitions which will be used frequently in the development
of this paper.
Definition 1. Let M ⊂ L(H ) be a semifinite von Neumann algebra acting over the Hilbert space
H with a trace τ. A linear operator L (maybe unbounded) is said to be affiliated with M, if it
commutes with the elements of the commutant M ! of M, i.e. LV =V L for all V ∈ M !.
Definition 2. A closeable operator L (maybe unbounded) is called τ-measureable if for each ϵ> 0
there exists a projection P in M such that P (H ) ⊂ D(L) and τ(I −P )⩽ ϵ, where D(L) is the domain
of L in H and M ⊂L(H ). We denote by S(M) the set of all τ-measurable operators.
Definition 3. Let L ∈ S(M), and let L =U |L| be its polar decomposition. We define the distribution
function by dγ(L) := τ
(
E(γ,+∞)(|L|)
)
for γ⩾ 0, where E(γ,+∞)(|L|) is the spectral projection of L over
the interval (γ,+∞). Also, for any t > 0, we define the generalized t−th singular numbers as
µt (L) := inf{γ⩾ 0 : dγ(L)⩽ t }.
More discussions (and properties) of the distribution function and generalized singular num-
bers can be found e.g. in [7].
We conclude this section by recalling the noncommutative Lorentz spaces associated with a
semifinite von Neumann algebra M [11].
Santiago Gómez Cobos, Joel E. Restrepo and Michael Ruzhansky 1333
Definition 4. We denote by Lp,q (M) (1 ⩽ p < +∞, 1 ⩽ q < +∞) the set of all operators L ∈ S(M)
such that
∥L∥Lp,q (M) =
(∫ +∞
0
(
t 1/pµt (L)
)q dt
t
)1/q
<+∞.
Therefore the Lp -spaces on M are given by
∥L∥Lp (M) := ∥L∥Lp,p (M) =
(∫ +∞
0
µt (L)
p dt
)1/p
.
For the case q =∞, Lp,∞(M) is the set of all operators L ∈ S(M) with
∥L∥Lp,∞(M) = sup
t>0
t 1/pµt (L).
3. Heat and wave type equations
In this section we establish for non-local heat and wave type equations the Lp−Lq norm estimates
for 1 < p ⩽ 2 ⩽ q <+∞. Indeed, we will see that the norm estimates can be reduced to the time
asymptotics of the propagator in the noncommutative Lorentz space norm. Also, we show that
the latter norm mainly involves calculating the trace of the spectral projections of the operator L .
In the presentation of the propagators of this section, frequently the two-parametric Mittag–
Leffler function will appear:
Eα,δ(z) =
+∞∑
k=0
zk
Γ(αk +δ) , z,δ ∈C, ℜ(α) > 0,
which is absolutely and locally uniformly convergent for the given parameters ([8]).
For the existence, uniqueness, representation and other properties of solutions for the consid-
ered equations below, we refer to the theory on evolutionary integral equations in [14, Chapter I]
or, alternatively, one can use the Borel functional calculus [2] to construct the propagators. More
concrete and similar studies on abstract spaces can be found in [4, Chapter 3] (see also [3]). For
the estimation of the Lp −Lq norms, we usually combine [1, Theorem 5.1] with the alternative
form provided by [1, Theorem 6.1] to calculate the noncommuative Lorentzian norm.
3.1. L -heat type equation
We consider the following heat type equation:∫ t
0
(t − s)−β
Γ(1−β) ∂s w(s, x)ds︸ ︷︷ ︸
C∂
β
t w(t ,x)
+L w(t , x) = 0, t > 0, x ∈G ,
w(t , x)|
t=0 = w0(x),
(1)
where C∂βt is the so-called Dzhrbashyan–Caputo fractional derivative, 0 < β < 1, G is a locally
compact separable unimodular group and L is any positive linear left invariant operator on G
(maybe unbounded). Our next result also covers the caseβ= 1, i.e. the classical L -heat equation,
but it is already known [1, Section 7].
Theorem 5. Let G be a locally compact separable unimodular group, 0 < β < 1 and 1 < p ⩽ 2 ⩽
q <+∞. Let L be any positive left invariant operator on G (maybe unbounded) such that
sup
t>0
sup
s>0
[
τ
(
E(0,s)(L )
)] 1
p − 1q Eβ(−tβs) <+∞.
1334 Santiago Gómez Cobos, Joel E. Restrepo and Michael Ruzhansky
If w0 ∈ Lp (G) then there exists a unique solution w ∈ C
(
[0,+∞);Lq (G)) for the L -heat type
equation (1) represented by
w(t , x) = Eβ(−tβL )w0(x), t > 0, x ∈G .
In the particular case that
τ
(
E(0,s)(L )
)
≲ sλ, s −→+∞, for some λ> 0, (2)
we get the following time decay rate for the solution of equation (1) for all t > 0,
∥w(t , ·)∥Lq (G) ⩽Cβ,λ,p,q t−βλ
(
1
p − 1q
)
∥w0∥Lp (G), 1
λ
⩾
1
p
− 1
q
.
3.2. L -wave type equation
Here we investigate the solution of the following equation, which in a sense interpolates between
wave (without being wave, β< 2) and heat types:
C∂
β
t w(t , x)+L w(t , x) = 0, t > 0, x ∈G ,
w(t , x)|
t=0 = w0(x),
∂t w(t , x)|t=0 = w1(x),
(3)
where C∂βt is the Dzhrbashyan–Caputo fractional derivative, L is any positive linear left invariant
operator on G (maybe unbounded) and 1 <β< 2.
Before giving the main result of this subsection, first we mention a technical and important
theorem whose proof will appear somewhere else. Remember that for L ∈ S(M) we denote
L =U |L| its polar decomposition.
Theorem 6. Let L be a closed (maybe unbounded) operator affiliated with a semifinite von
Neuman algebra M. Let φ be a Borel measurable function on [0,+∞). Suppose also that ψ is a
monotonically decreasing continuous function on [0,+∞) such that ψ(0) = 1, limv→+∞ψ(v) = 0
and |φ(v)|⩽ψ(v) for all v ∈ [0,+∞). Then for every 1⩽ r <∞ we have the inequality
∥φ(|L|)∥Lr,∞(M) ⩽ sup
v>0
ψ(v)
[
τ(E(0,v)(|L|))
] 1
r .
Utilizing Theorem 6 we will be able to prove and state the main result on wave type equations.
Now we need to recall an operator which is involved in the representation of the wave type
propagator. Therefore, we introduce RL0I
βu(t ) = 1Γ(β)
∫ t
0 (t − s)β−1u(s)ds, the Riemann–Liouville
fractional integral of order β> 0, which is well-defined for functions on L1(0,T ).
Theorem 7. Let G be a locally compact separable unimodular group, 1 < β < 2 and 1 < p ⩽ 2 ⩽
q < +∞. Let L be any positive left invariant operator on G (maybe unbounded) satisfying the
condition
sup
t>0
sup
s>0
[
τ
(
E(0,s)(L )
)] 1
p − 1q
1+ tβs <+∞.
If w0, w1 ∈ Lp (G) then there exists a unique solution w ∈ C
(
[0,+∞);Lq (G)) for the L -wave type
equation (3) given by
w(t , x) = Eβ(−tβL )w0(x)+RL0I 1t Eβ(−tβL )w1(x), t > 0, x ∈G .
In particular, if the condition (2) holds then for any 1 < p ⩽ 2⩽ q <+∞ such that 1λ ⩾ 1p − 1q we get
the following time decay rate for the solution of equation (3):
∥w(t , ·)∥Lq (G) ⩽Cβ,λ,p,q t−βλ
(
1
p − 1q
)(∥w0∥Lp (G) + t∥w1∥Lp (G)).
Santiago Gómez Cobos, Joel E. Restrepo and Michael Ruzhansky 1335
In Theorems 5, 7, the time decay rate for the solutions of equations (1) and (3), is predeter-
mined by the condition (2). Hence, let us mention briefly several examples of operators (on dif-
ferent groups) such that the trace of the spectral projections behave like sλ as s → +∞. In the
Euclidean space Rn , consider the Laplacian∆Rn . Here, we have such a behaviour [1, Example 7.3]
with λ= n/2. Moreover it is given in more generality since it allows one to consider not just con-
jugate exponents. Here we can recover the sharp estimate (time-decay) given in [10, Theorem 3.3,
item (i)] whenever 2n > 1p − 1q . The sub-Laplacian ∆sub on a compact Lie group [9] with λ=Q/2,
where Q is the Hausdorff dimension of G . The positive sub-Laplacian on the Heisenberg group
Hn [1, Formula (7.17)] with λ= n+1. A positive Rockland operator R of homogeneous order ν on
a graded Lie group [15, Theorem 8.2] with λ=Q/ν, where Q is the homogeneous dimension of G .
By using the latter estimate we can get sharp time decay in Theorems 5, 7. For operators of non-
Rockland-type on the Engel group B4 (λ= 3) and the Cartan group B5 (λ= 9/2) [5, Examples 2.2
and 3.2]. An m-th order weighted subcoercive positive operator on a connected unimodular Lie
group [16, Proposition 0.3] with λ = Q∗/m, where Q∗ is the local dimension of G relative to the
chosen weighted structure on its Lie algebra.
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] R. Akylzhanov and M. Ruzhansky, “Lp -Lq multipliers on locally compact groups”, J. Funct.
Anal. 278 (2020), no. 3, article no. 108324 (49 pages).
[2] W. Arveson, A short course on spectral theory, Springer, 2002, pp. x+135.
[3] E. G. Bajlekova, Fractional Evolution Equations in Banach Spaces, PhD thesis, Eindhoven
University of Technology, 2001.
[4] P. M. Carvalho-Neto, Fractional differential equations: a novel study of local and global
solutions in Banach spaces, PhD thesis, Universidade de São Paulo, São Carlos, 2013.
[5] M. Chatzakou, “A note on spectral multipliers on Engel and Cartan groups”, Proc. Am.
Math. Soc. 150 (2022), no. 5, pp. 2259–2270.
[6] J. Dixmier, Von Neumann algebras. Transl. from the French by F. Jellett, North-Holland,
1981, pp. xxxviii+437.
[7] T. Fack and H. Kosaki, “Generalized s-numbers of τ-measurable operators”, Pac. J. Math.
123 (1986), no. 2, pp. 269–300.
[8] R. Gorenflo, A. A. Kilbas, F. Mainardi and S. Rogosin, Mittag–Leffler functions, related topics
and applications, Second edition, Springer, 2020, pp. xvi+540.
[9] A. Hassannezhad and G. Kokarev, “Sub-Laplacian eigenvalue bounds on sub-Riemannian
manifolds”, Ann. Sc. Norm. Super. Pisa, Cl. Sci. 16 (2016), no. 4, pp. 1049–1092.
[10] J. Kemppainen, J. Siljander, V. Vergara and R. Zacher, “Decay estimates for time-fractional
and other non-local in time subdiffusion equations in Rd ”, Math. Ann. 366 (2016), no. 3-4,
pp. 941–979.
[11] H. Kosaki, “Non-commutative Lorentz spaces associated with a semi-finite von Neumann
algebra and applications”, Proc. Japan Acad., Ser. A 57 (1981), no. 6, pp. 303–306.
[12] F. J. Murray and J. von Neumann, “On rings of operators. II”, Trans. Am. Math. Soc. 41
(1937), no. 2, pp. 208–248.
1336 Santiago Gómez Cobos, Joel E. Restrepo and Michael Ruzhansky
[13] F. J. Murray and J. Von Neumann, “On rings of operators”, Ann. Math. 37 (1936), no. 1,
pp. 116–229.
[14] J. Prüss, Evolutionary integral equations and applications, Birkhäuser, 1993, pp. xxvi+366.
[15] D. Rottensteiner and M. Ruzhansky, “Harmonic and anharmonic oscillators on the Heisen-
berg group”, J. Math. Phys. 63 (2022), no. 11, article no. 111509 (23 pages).
[16] D. Rottensteiner and M. Ruzhansky, “An update on the Lp -Lq norms of spectral multipliers
on unimodular Lie groups”, Arch. Math. 120 (2023), no. 5, pp. 507–520.
[17] M. Terp, “Lp Spaces Associated with Von Neumann Algebras”, Copenhagen University,
1981.


$L^p-L^q$ estimates for non-local heat and wave type equations on locally compact groups

Guillaume Aubrun

Jing Bai

Maximal exponent of the Lorentz cones

Benjamin Hinrichs

Jonas Lampart

A Lower Bound on the Critical Momentum of an Impurity in a Bose–Einstein Condensate

Mehdi Lejmi

Scott O. Wilson

Comptes Rendus
Mathématique
Mehdi Lejmi and Scott O. Wilson
Conformally symplectic structures and the Lefschetz condition
Volume 362 (2024), p. 1491-1495
Online since: 14 November 2024
https://doi.org/10.5802/crmath.660
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. 1491-1495
https://doi.org/10.5802/crmath.660
Research article / Article de recherche
Geometry and Topology / Géométrie et Topologie
Conformally symplectic structures and the
Lefschetz condition
Les structures conformément symplectiques et la
condition de Lefschetz
Mehdi Lejmi ∗, ,a and Scott O. Wilson ,b
a Department of Mathematics, Bronx Community College, City University of New
York, 2155 University Av., Bronx, NY 10453, USA.
b Department of Mathematics, Queens College, City University of New York, 65-30
Kissena Blvd., Flushing, NY 11367, USA.
E-mails: mehdi.lejmi@bcc.cuny.edu, scott.wilson@qc.cuny.edu
Abstract. This short note provides a symplectic analogue of Vaisman’s theorem in complex geometry. Namely,
for any compact symplectic manifold satisfying the hard Lefschetz condition in degree 1, every locally
conformally symplectic structure is in fact globally conformally symplectic, whenever there is a mutually
compatible almost complex structure.
Résumé. Cette courte note fournit un analogue symplectique au théorème de Vaisman en géométrie com-
plexe. Plus précisement, pour toute variété compacte symplectique satisfaisant la condition de Lefschetz en
degré 1, chaque structure localement conformément symplectique est en fait globalement conformément
symplectique, quand celle-ci et la forme symplectique sont compatibles avec la même structure presque
complexe.
Keywords. Locally conformally symplectic structures, hard Lefschetz condition, almost complex structures.
Mots-clés. Structures localement conformément symplectiques, condition de Lefschetz, structures presque
complexes.
2020 Mathematics Subject Classification. 53D05, 53B35.
Funding. The first author is supported by the Simons Foundation Grant #636075. The second author
acknowledges support provided by the Simons Foundation Award program for Mathematicians (#963783)
and a PSC-CUNY Award, jointly funded by The Professional Staff Congress and The City University of New
York (TRADB # 66358-00 54).
Manuscript received 2 May 2024, revised 5 June 2024, accepted 18 June 2024.
This short note provides a symplectic analogue of Vaisman’s foundational result for complex
manifolds, which states that if a complex manifold is Kähler, then every locally conformally
Kähler structure is indeed globally conformally Kähler [20].
Recall that a locally conformally symplectic structure is a pair (η,θ) consisting of a real closed
1-form θ, and a non-degenerate 2-form η, such that dη= θ∧η, where d is the exterior derivative.
If θ is d-exact, then (η,θ) is globally conformally symplectic. Moreover, if θ = 0, then η is a
symplectic form. For more about locally conformally symplectic structures, we refer the reader
∗Corresponding author
ISSN (electronic): 1778-3569 https://comptes-rendus.academie-sciences.fr/mathematique/
1492 Mehdi Lejmi and Scott O. Wilson
for instance to [3, 10, 16]. An almost complex structure J is compatible with η if η(J ·, J ·) = η and
η(·, J ·) defines a Riemannian metric. The induced Riemannian metric is called an almost-Kähler
metric if η is a symplectic form.
Eliashberg and Murphy [11, Theorem 1.11] used the h-principle to prove that compact almost-
complex manifolds with a non-zero degree one cohomology class admit locally conformally sym-
plectic structures which are not globally conformally symplectic. In addition to this, symplectic
manifolds can have locally conformally symplectic structures which are not globally conformally
symplectic, even in the presence of mutually compatible almost complex structures, cf. Example
3 below (we also refer the reader to [2, 4, 5, 18]). A sufficient condition for every locally confor-
mally symplectic structure to be globally conformally symplectic, in the presence of a compatible
almost complex structure, is given by the hard Lefschetz condition in degree 1.
Theorem 1. If a compact symplectic manifold (M ,ω) satisfies hard Lefschetz in degree 1, then
any locally conformally symplectic structure (η,θ) on (M ,ω) is necessarily globally conformally
symplectic whenever η and ω share a compatible J .
Recall that on a symplectic manifold (M ,ω) of dimension 2n the hard Lefschetz condition
holds in degree 1 if
Ln−1 : H 1d(M) −→ H 2n−1d (M)
is an isomorphism, where Lα=ω∧α. We begin with a lemma, after establishing some notation.
For an almost complex manifold, we let dc := J−1 dJ , where J is extended as an algebra
automorphism, and for an odd operator δ define its Laplacian ∆δ = δ∗δ+δ∗δ with harmonics
H k
δ
= Ker∆δ∩Ωk , whereΩk is the space of real differential k-forms.
Lemma 2. Let (M ,ω) be a compact symplectic manifold of dimension 2n. The following are
equivalent:
(1) The hard Lefschetz condition holds in degree 1.
(2) For any compatible almost-Kähler structure, H 1d = H 1dc , i.e. every d-harmonic 1-form is
d c -harmonic (and vice versa).
(3) For any compatible almost-Kähler structure, every d-harmonic 1-form is the sum of d-
harmonic forms of pure bi-degree,
H 1d =H 1,0d ⊕H
0,1
d .
(4) For any compatible almost-Kähler structure,
H 1d =H 1∂ ∩H
1
µ ,
where d =µ+∂+∂+µ, with bidegrees
|µ| = (−1,2), |∂| = (0,1), |∂| = (1,0), and |µ| = (2,−1).
A proof that the first condition implies the second is given in [14, Proposition 3.3], which we
streamline below, using the same essential idea. We note that in [19, Theorem 5.4], Tomassini and
Wang show that the second condition implies first (in fact, in all degrees). Namely, for an almost-
Kähler manifold, the space Hd satisfies the hard Lefschetz condition whenever Hd =Hd c (in all
degrees). The remaining statements are included to give additional characterizations in terms of
decompositions of d-harmonic forms.
Proof. To prove the first implication, assume there’s a d-harmonic 1-form α so that Jα is not d-
harmonic. Since J is orthogonal, we may as well assume α is d-harmonic, but Jα is orthogonal
to the d-harmonics. Then
Ln−1α= 1
(n −1)! ⋆ Jα.
Mehdi Lejmi and Scott O. Wilson 1493
Since star is an isometry, Ln−1α is orthogonal to the d-harmonics, and therefore [Ln−1α] is zero
in cohomology. This shows Ln−1 has a non-zero kernel.
Using
dc =−iµ+ i∂− i∂+ iµ
we have
H 1d ∩H 1dc =H 1,0d ⊕H
0,1
d ⊂H 1d .
This proves the second condition implies the third.
Next,
H 1,0d ⊕H
0,1
d ⊂H 1∂ ∩H
1
µ ⊂H 1d
where the last containment follows from the almost-Kähler identity [8],
∆
∂
+∆µ =∆∂+∆µ.
This proves the third condition implies the fourth.
Finally, by [8, Theorem 5.1], the last condition implies the first, since the space H
∂
∩Hµ
always has the Lefschetz sl(2)-representation, and so satisfies hard Lefschetz on H 1d(M)
∼=H 1d (M)
if H 1
∂
∩H 1µ =H 1d (M). □
The proof of Theorem 1 is now essentially the same as in Vaisman [20], see also Ornea and
Verbitsky [16].
Proof. Let (η,θ) be a locally conformally symplectic structure. Suppose there is a symplectic
structure (M ,ω) satisfying hard Lefschetz in degree 1, and let
θ = h +dγ
be the Hodge decomposition of θ, with h ∈H 1d . By the conformal change η 7→ η′ = e−γη, we have
dη′ = θ′∧η′
and θ′ = h satisfies dθ′ = 0. So, we may as well assume that dη= θ∧η with θ d-harmonic.
By the Lemma’s first implication, dcθ = 0 for any compatible almost Kähler structure, so that
ddcηn−1 =−(n −1)2θ∧ Jθ∧ηn−1,
since dcη= J−1dη= J−1(θ∧η) =−Jθ∧η. Therefore,
0 =
∫
M
ddcηn−1 =−(n −1)2
∫
M
θ∧ Jθ∧ηn−1.
Since J is compatible with η, we conclude that θ = 0, so that η is symplectic. □
Example 3. Two examples show the necessity of the assumptions in the theorem.
In [18, Example 4.1] the authors show that the Kodaira–Thurston manifold supports both a
symplectic structureω and a locally conformally symplectic structure (η,θ), which is not globally
conformally symplectic, while η and ω share a compatible J . In this case the hard Lefschetz
condition fails in degree 1.
On the other hand, a locally conformally symplectic structure (η,θ) on a symplectic manifold
(M ,ω) can fail to be globally conformally symplectic if ω and η do not share a compatible J , even
if the hard Lefschetz condition holds in degree 1. For example, the solvmanifold with structure
equations dx1 = x1x3, dx2 =−x2x3, and dx3 = dx4 = 0, with symplectic form ω= x1x2 + x3x4 has
(θ = −x3,η = x1x4 + x2x3) which is locally conformally symplectic, but not globally conformally
symplectic.
From the theorem we deduce several corollaries, and emphasize that those statements con-
cerning homogeneous spaces apply to all symplectic and locally conformally symplectic struc-
tures (even inhomogeneous structures).
1494 Mehdi Lejmi and Scott O. Wilson
Corollary 4. Let G be a simply connected 4-dimensional Lie group admitting a uniform lattice
Γ. Suppose M = Γ\G admits a symplectic form ω that satisfies hard Lefschetz in degree 1. Let
(η,θ) be a locally conformally symplectic structure on M. Suppose that there is an almost complex
structure J compatible with η that is compatible with some symplectic form. Then, (η,θ) is a
globally conformal symplectic structure.
Proof. By hypothesis, ω satisfies hard Lefschetz in degree 1, and therefore all degrees. By [17,
Theorem 4.2], any other symplectic form satisfies hard Lefschetz as well, so the corollary follows
then from Theorem 1. □
Corollary 5. Let M be a 2n-dimensional torus or a hyperelliptic surface. Let (η,θ) be a locally
conformally symplectic structure on M. Suppose that there is an almost complex structure J
compatible with η that is compatible with some symplectic form. Then, (η,θ) is a globally
conformal symplectic structure.
Proof. Any symplectic form on a 2n-dimensional torus or a hyperelliptic surface satisfies hard
Lefshetz (for the torus see for instance [17, Proposition 4.1], we also refer the reader to [6, 7, 9,
13]). The corollary then follows from Theorem 1. □
Tan and Tomassini studied in [17] the hard Lefschetz condition on some 6-dimensional com-
pact completely solvable manifolds namely Nakamura manifolds N 6 [15], and M 6(c), N 6(c),P 6(c)
(we refer the reader to [1, 12, 17] for their definition). We remark that the manifolds M 6(c), N 6(c),
and P 6(c) do not admit any Kähler metric [12].
Corollary 6. Let M be a Nakamura manifold N 6 or the compact completely solvable manifold
M 6(c) or N 6(c) or P 6(c). Let (η,θ) be a locally conformally symplectic structure on M. Suppose
there is an almost complex structure J compatible with η that is compatible with some symplectic
form. Then, (η,θ) is a globally conformal symplectic structure.
Proof. It follows from [17, Theorem 5.1, Theorem 6.1, Remark 6.2] that any symplectic form on
M satisfies hard Lefschetz. The corollary then follows from Theorem 1. □
Acknowledgements
The authors are grateful to Adriano Tomassini and Nicolleta Tardini for pointing out their exam-
ple in [18], and to Giovanni Bazzoni, Liviu Ornea, and Baptiste Chantraine for helpful discussions.
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
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compact symplectic and complex solvmanifolds”, Rend. Mat. Appl. 12 (1992), no. 1, pp. 59–
67.
[2] D. Angella, G. Bazzoni and M. Parton, “Structure of locally conformally symplectic Lie
algebras and solvmanifolds”, Ann. Sc. Norm. Super. Pisa, Cl. Sci. 20 (2020), no. 1, pp. 373–
411.
Mehdi Lejmi and Scott O. Wilson 1495
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(2018), no. 1-2, pp. 129–154.
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conformal Kähler metrics”, Complex Manifolds 4 (2017), no. 1, pp. 172–178.
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Japan Acad. 48 (1972), pp. 447–449.
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Conformally symplectic structures and the Lefschetz condition

Samir Karaa

Comptes Rendus
Mathématique
Samir Karaa
Positivity of convolution quadratures generated by nonconvex sequences
Volume 362 (2024), p. 1627-1634
Online since: 25 November 2024
https://doi.org/10.5802/crmath.669
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. 1627-1634
https://doi.org/10.5802/crmath.669
Research article / Article de recherche
Numerical analysis / Analyse numérique
Positivity of convolution quadratures
generated by nonconvex sequences
Positivité des quadratures de convolution générées par
des séquences non convexes
Samir Karaa a
a Department of Mathematics, Sultan Qaboos University, Al-Khod 123, Muscat, Oman
E-mail: skaraa@squ.edu.om
Abstract. The positive definiteness of real quadratic forms of convolution type plays an important role in
the stability analysis of time-stepping schemes for nonlocal models. Specifically, when these quadratic
forms are generated by convex sequences, their positivity can be verified by applying a classical result due
to Zygmund. The primary focus of this work is twofold. We first improve Zygmund’s result and extend its
validity to sequences that are almost convex. Secondly, we establish a more general inequality applicable to
nonconvex sequences. Our results are then applied to demonstrate the positive definiteness of commonly
used approximations for fractional integral and differential operators, including the convolution quadrature
generated by the BDF2 formula. To conclude, we show that the stability of some simple time-fractional
schemes can be obtained in a straightforward way.
Résumé. Le caractère défini positif des formes quadratiques réelles de type convolution joue un rôle impor-
tant dans l’analyse de stabilité des schémas en temps pour les modèles non locaux. Plus précisément, lorsque
ces formes quadratiques sont générées par des séquences convexes, leur positivité peut être vérifiée en ap-
pliquant un résultat classique dû à Zygmund. L’objectif principal de ce travail est double. Nous améliorons
d’abord le résultat de Zygmund et étendons sa validité aux séquences presque convexes. Deuxièmement,
nous établissons une inégalité plus générale applicable aux séquences non convexes. Nos résultats sont en-
suite appliqués pour démontrer le caractère défini positif des approximations couramment utilisées pour
les opérateurs intégraux et différentiels fractionnaires, y compris le quadrature de convolution générée par
la formule BDF2. Pour conclure, nous montrons que la stabilité de certains schémas fractionnaires simples
peut être obtenue de manière simple.
Keywords. Convolution quadrature, positive definiteness, nonconvex sequence, minimal convex sequence.
Mots-clés. Quadrature de convolution, définie positive, suite non convexe, suite convexe minimale.
2020 Mathematics Subject Classification. 65R20, 65M12.
Funding. This work is supported by Sultan Qaboos University under Grant IG/SCI/MATH/24/02.
Manuscript received 19 April 2024, revised 11 July 2024, accepted 16 July 2024.
ISSN (electronic): 1778-3569 https://comptes-rendus.academie-sciences.fr/mathematique/
1628 Samir Karaa
1. Introduction
Let {β j }∞j=0 be a sequence of real numbers. The discrete convolution operator
qn(ϕ) =
n∑
j=0
βn− jϕ j
for ϕ := (ϕ0,ϕ1, . . . ,ϕn) ∈Rn+1, is positive definite if
Qn(ϕ) :=
n∑
k=0
k∑
j=0
βk− jϕ jϕk ≥ c0
n∑
j=0
ϕ2j (1)
for some c0 > 0 and all n,ϕ. The constant c0 is optimal if (1) does not hold with any constant
greater than c0. The operator qn is called weakly positive if c0 = 0. The positive character of
Qn is a key ingredient in analyzing the stability and convergence of of numerical schemes incor-
porating quadrature rules for integro-differential equations or time-fractional partial differential
equations (see, e.g., [2, 4, 6–8, 10]). In [1], the author presented several conditions on {β j } which
ensure the positivity property (1). A sharp result is obtained in the case of completely monotone
sequences. The objective of the work is to extend these results to cover nonconvex sequences.
Recall that a sequence {β j }∞j=0 is said to be convex if ∆β j := β j−1 −2β j +β j+1 ≥ 0 for all j ≥ 1.
The sequence is called minimal if it becomes nonconvex when its first term β0 is replaced by any
smaller number. In other words, the sequence is minimal if ∆β1 = 0. Additionally, we introduce
the notion of a nearly convex sequence, defined as a sequence where, for some small integer
M ≥ 1, the subsequence {β j }∞j=M is convex.
Consider the Fourier transform of {β j }∞j=0 given by β̂(θ) =
∑∞
j=0β j e
i jθ , and assume that
β̂ ∈ L1(0,2π). Then, by means of Parseval’s identity, we have (with ϕ j = 0 for j > n)
Qn(ϕ) = 1
2π
∫ 2π
0
β̂(θ)|ϕ̂(θ)|2 dθ = 1
2π
∫ 2π
0
Reβ̂(θ)|ϕ̂(θ)|2 dθ, (2)
where the last equality holds because Qn is real-valued. Thus, qn is weakly positive if Reβ̂(θ) :=∑∞
j=0β j cos( jθ) is nonnegative on (0,2π). To ensure positive definiteness, one needs to show that
Re β̂(θ) is bounded below by a positive constant. This task is in general delicate as β̂(θ) might not
be explicitly known or might be difficult to analyze.
For convex sequences, a sufficient condition that ensures (1) is due to Zygmund [11, Theo-
rem (1.5), p. 183]: if the sequence {β j }∞j=0 is convex and β j → 0, then
1
2
β0 +
∞∑
j=1
β j cos( jθ) ≥ 0 for 0 < θ < 2π. (3)
That is,
Qn(ϕ) ≥ β0
2
n∑
j=0
ϕ2j . (4)
A similar inequality was derived in [4], and related results can be found in [1, 6]. We notice that the
condition β j → 0 can be replaced by {β j }∞j=0 is nonnegative and bounded (see [8, Lemma 4.3]).
In this work, we derive an alternative to (4), which is applicable to nearly convex sequences.
As an illustration, we shall apply our results to the following generating sequences
γ j = (−1) j
(−α
j
)
, j ≥ 0, (5)
ω j = (( j +1)1−α− j 1−α)/Γ(2−α), j ≥ 0, (6)
ρ0 = 1
Γ(2+α) , ρ j = [( j +1)
1+α−2 j 1+α+ ( j −1)1+α]/Γ(2+α), j ≥ 1, (7)
Samir Karaa 1629
and
ξ j = (−1) j
(
2
3
)α j∑
l=0
3−l
( −α
j − l
)(−α
l
)
, j ≥ 0, (8)
arising from the approximations of fractional integral and differential operators of orderα ∈ (0,1).
The terms in (5) and (8) are the weights of convolution quadrature rules generated by Euler and
BDF2 methods, respectively, for approximating a Riemann–Liouville fractional integral, see [5,
7]. On the other hand, (6) corresponds to the L1-method for approximating a Caputo fractional
derivative [3]. The sequence (7) results from an integral average formula of a Riemann–Liouville
fractional integral [7]. We note that γ0 = 1, ω0 = 1/Γ(2−α) and ξ0 = (2/3)α.
While {γ j }∞j=0 and {ω j }
∞
j=0 are convex for all α in (0,1), see e.g., [1], the last two sequences
{ρ j }∞j=0 and {ξ j }
∞
j=0 are convex only for some range of α in (0,1). For instance, {ρ j }
∞
j=0 is convex
only when α ∈ (0,b1], where b1 ≈ 0.5546 is the root of the equation 6 − 23+α + 31+α = 0. For
this case, one can also verify that the subsequence {ρ j }∞j=1 is convex for all α in (0,1). The
analysis of the convexity of {ξ j }∞j=0 is quite complex. A comprehensive set of tests showed that
the sequence is convex whenever ∆ξ1 ≥ 0, leading to the conclusion that {ξ j }∞j=0 is convex only
when α ∈ (0,b2], where b2 = (19−
p
73)/16 ≈ 0.6534. In Figure 1, we show the first few terms ξ j
and the corresponding terms 2∆ξ j for j = 1, . . . ,15 to check for convexity for different values of α.
Noting that Zygmund’s inequality (4) does not apply to the sequences {ρ j }∞j=0 and {ξ j }
∞
j=0, one
might question the feasibility of deriving a more suitable inequality to cover these cases. The aim
of this study is to rigorously establish the positivity of quadratic forms generated by nearly convex
sequences, including {ρ j }∞j=0 and {ξ j }
∞
j=0. The results allow to overcome difficulties in analyzing
the stability of numerical schemes for solving integro-differential and time-fractional phase-field
equations.
Figure 1. The terms ξ j in (8) for j = 0, . . . ,15 and 2∆ξ j for j = 1, . . . ,15; α= 0.6,0.7,0.8,0.9.
1630 Samir Karaa
2. Nonoptimality of Zygmund’s Inequality
Adding a positive constant a0 to the initial term of a convex sequence {β j }∞j=0 preserves its
convexity. Consequently, if we apply (3) to the new sequence, we notice that the zero lower bound
in the resulting inequality can be replaced by a0/2, indicating that (4) is not optimal. Based on
this observation, we derive in the next theorem an improved lower bound, which is found to be
valid for a class of nonconvex sequences.
Theorem 1. Let {β j }∞j=0 be a sequence such that {β j }
∞
j=1 is nonnegative, bounded and convex.
Then
n∑
k=0
k∑
j=0
βk− jϕ jϕk ≥
1
2
(
β0 +∆β1
) n∑
j=0
ϕ2j (9)
for all n ≥ 0 and ϕ ∈Rn+1.
Proof. Consider the sequence {a j }∞j=0 defined by a0 = 2β1 −β2 and a j = β j for j ≥ 1. Clearly
{a j }∞j=0 is nonnegative, bounded and convex. Hence, by (4),
n∑
k=0
k∑
j=0
ak− jϕ jϕk ≥
a0
2
n∑
j=0
ϕ2j (10)
for all n ≥ 0 and ϕ ∈Rn+1. Adding (β0 −a0)
n∑
j=0
ϕ2j to both sides of (10) yields
n∑
k=0
k∑
j=0
βk− jϕ jϕk ≥
(
β0 −β1 + β2
2
) n∑
j=0
ϕ2j , (11)
which is the desired inequality. □
When {β j }∞j=0 is convex, ∆β1 ≥ 0, and therefore (9) is an improvement over Zygmund’s
inequality (4). The results coincide when ∆β1 = 0, i.e., for minimal sequences. We verify that
the bound β0 +∆β1 is positive if β0 >β1 −β2/2 (or β0 > a0/2), and vanishes when β0 =β1 −β2/2.
It is worth noting that within the range β1 −β2/2 <β0 < 2β1 −β2, while the convexity property of
convexity is violated, the positivity of the corresponding quadratic form is preserved.
In Figure 2, we compare Zygmund’s constant (CZ := β0/2), the new in (9), and the optimal
constants
Cγ = 1
2α
and Cω = 2Liα−1(−1)
(−1)Γ(2−α) ,
obtained in [1] for the sequences {γ j }∞j=0 and {ω j }
∞
j=0, respectively. Here, Lim(z) :=
∑∞
j=1
z j
j m
represents the polylogarithm function. The Figure shows that the current bound is remarkably
close to the optimal bound for all values of α, particularly as α approaches 0 and 1. In contrast,
there exists a significant disparity between Zygmund’s bound and the optimal bound, particularly
on one end of the interval.
Theorem 1 yields an immediate result for symmetric Toeplitz matrices, as follows.
Corollary 2. Let {β j }∞j=0 be a sequence such that {β j }
∞
j=1 is nonnegative, bounded and convex. Let
ρN denote the spectral radius of the N ×N symmetric Toeplitz matrix
β0 β1 β2 · · · βN−1
β0 β1 · · · βN−2
. . .
. . .
...
s ym β0 β1
β0
 .
Then, ρN ≥∆β1 for all N ≥ 3.
Samir Karaa 1631
Figure 2. Comparison of three bounds for {γ j }∞j=0 (left) and {ω j }
∞
j=0 (right); α ∈ (0,1).
Zygmund (solid line), current (dashed line) and optimal (dot-dashed line).
The Corollary provides a simple tool for checking the positive definiteness of a symmetric
Toeplitz matrix. To illustrate, consider a 6 × 6 symmetric Toeplitz matrix A whose first row is
[4 2 1 1 1 1]. Clearly, A is not diagonally dominant, and the Gershgorin circle theorem does not
yield useful information about its positive definiteness. Nevertheless, applying Corollary 2 yields
ρ(A) ≥ 1. We verify that ρ(A) ≈ 1.198, which confirms the accuracy of the inequality (9) in the
present example.
3. Nearly convex sequences
Let {β j }∞j=0 be such that {β j }
∞
j=M is nonnegative, bounded and convex for some M ≥ 1. Let
f be a convex real-valued function defined on the interval [0,∞) that interpolates the points
( j ,β j ), j ≥ M . Define the sequence {a j }∞j=0 by a j = f ( j ) for j ≥ 0. Since {a j }∞j=0 is nonnegative,
bounded and convex, Zygmund’s inequality implies Re â(θ) ≥ a0/2 for all θ ∈ (0,2π). Adding∑M−1
j=0 (β j −a j )cos( jθ) to both sides of this inequality yields
Re β̂(θ) ≥β0 − a0
2
+min
θ
M−1∑
j=1
(β j − f ( j ))cos( jθ). (12)
This step simplifies the task of finding a lower bound for Re β̂(θ) as the last term in (12) (involving
a finite sum) is more manageable compared to the original infinite series.
The distribution of the first terms of the nonconvex sequences illustrated in Figure 1 has served
as our main motivation to study the case where {β j }∞j=M is convex for some M ≥ 1. A noteworthy
observation is that all the points ( j ,β j ), 0 < j < M , lie below the line passing through the points
(M ,βM ) and (M +1,βM+1). We proceed to present our findings in the subsequent theorem.
Theorem 3. Let {β j }∞j=0 be a sequence such that, for some M ≥ 1, the subsequence {β j }∞j=M is
nonnegative, bounded and convex. Let P (t ) denote the linear function whose graph interpolates
the points (M ,βM ) and (M +1,βM+1). Assume β j ≤ P ( j ) for j = 1, . . . , M −1. then
n∑
k=0
k∑
j=0
βk− jϕ jϕk ≥CM
n∑
j=0
ϕ2j (13)
for all n ≥ 0 and ϕ ∈Rn+1, where
CM = 1
2
(
β0 +
M∑
j=1
j 2∆β j
)
. (14)
1632 Samir Karaa
Proof. Clearly P (t ) = (M − t )(βM −βM+1)+βM . Consider the sequence {a j }∞j=0 defined by
a j = P ( j ) = (M − j )(βM −βM+1)+βM , 0 ≤ j ≤ M −1, (15)
and a j =β j for j ≥ M . Then, {a j }∞j=0 is nonnegative, bounded and convex. Hence, Re â(θ) ≥ a0/2
for all θ ∈ (0,2π), that is,
Re â(θ) ≥
(
M
2
+ 1
2
)
βM − M
2
βM+1.
Referring to (12) and using the fact that β j ≤ a j for j = 1, . . . , M −1, we deduce
Re β̂(θ) ≥
M−1∑
j=0
β j −
M−1∑
j=0
a j +
(
M
2
+ 1
2
)
βM − M
2
βM+1.
Using (15), we readily obtain
M−1∑
j=0
a j =
(
M 2
2
+ 3M
2
)
βM −
(
M 2
2
+ M
2
)
βM+1,
and as a result
CM =
M−1∑
j=0
β j −
(
M 2
2
+M − 1
2
)
βM + M
2
2
βM+1. (16)
Now we show that the right hand side of (16) coincides with that of (14). Indeed, we have
C1 = 12 (β0 +∆β1) and we verify that
CM −CM−1 = 1
2
M 2∆βM , M ≥ 2,
which completes the proof. □
A comparison between the bound derived in Theorem 3 and one by Zygmund is presented
in Figure 3, for {ρ j }∞j=0 and {ξ j }
∞
j=0. Note that Zygmund’s bound is only applicable for convex
sequences. Further details on the convexity of {ξ j }∞j=0 and the integer M are given in Table 3.
Figure 3. Comparison of two bounds for {ρ j }∞j=0 (left) and {ξ j }
∞
j=0 (right); α ∈ (0,1). Zyg-
mund (solid line) and CM in (14) (dashed line).
Table 1. Comparison of CM in (14) with Zygmund’s constant CZ for {ξ j }∞j=0.
α 0.4 0.5 0.6 0.7 0.8 0.9 0.95
M 0 0 0 1 2 3 4
CZ = ξ0/2 0.425 0.408 0.392 – – – –
CM in (14) 0.552 0.476 0.413 0.361 0.308 0.210 0.138
Samir Karaa 1633
4. Numerical Riemann–Liouville integral
We now apply our results to investigate the stability of an implicit numerical scheme for a time-
fractional equation involving a fractional Riemann–Liouville integral. Recall the definition of the
Riemann–Liouville integral operator with order α> 0 (0 <α< 1), (see, e.g., [9])
Iαt ϕ(t ) = (ωα∗ϕ)(t ) =
∫ t
0
ωα(t − s)ϕ(s)ds,
where ωα(t ) = tα−1/Γ(α) and Γ( · ) is the usual Gamma function. In the following, Ω ⊂ Rd
(d = 1,2,3) is a bounded domain with a boundary ∂Ω. We denote by ( · , ·) the usual L2(Ω) inner
product with induced norm ∥ · ∥. For a fixed time T > 0, we divide the time interval [0,T ] into a
partition 0 = t0 < t1 < ·· · < tN = T with a uniform time step τ= T /N .
As an example, we consider the (modified) time-fractional diffusion-wave model equation
∂tφ+I γt φ−κIαt ∆φ= g (x, t ), (x, t ) ∈Ω× (0,T ], φ(x,0) =φ0(x), (17)
where γ ∈ (0,1), κ > 0 is a constant and g ∈ L1(0,T ;L2(Ω)). We assume that φ( · , t ) satisfies a
homogeneous Dirichlet boundary condition. We consider the implicit backward time-stepping
scheme
∂τφ
n +I γτ φn −κIατ ∆φn = g n , n ≥ 1, (18)
where g n = g (·, tn), ∂τφn = (φn −φn−1)/τ, and I aτ φn is an appropriate approximation of I at φ(tn)
of the form
I aτ φ
n = τα
n∑
j=0
βn− jφ j ,
with the property that the weights β j satisfy (1). Taking the inner product of both sides of (18)
with φn , and using Green’s formula, we get
(∂τφ
n ,φn)+ (I γτ φn ,φn)+ (κIατ ∇φn ,∇φn) = (g n ,φn), n ≥ 1.
Summing over n and applying property (1), we easily obtain the following stability result
∥φn∥ ≤ ∥φ0∥+2τ
n∑
k=1
∥g k∥.
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
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equations”, SIAM J. Numer. Anal. 59 (2021), no. 4, pp. 2040–2053.
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the variable-step L1-type approximations of convolution operators”, Sci. China, Math. 67
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[3] Y. Lin and C. Xu, “Finite difference/spectral approximations for the time-fractional diffu-
sion equation”, J. Comput. Phys. 225 (2007), no. 2, pp. 1533–1552.
[4] J. C. López Marcos, “A difference scheme for a nonlinear partial integrodifferential equa-
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[5] C. Lubich, “Convolution quadrature and discretized operational calculus. I”, Numer. Math.
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1634 Samir Karaa
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approximation”, IMA J. Numer. Anal. 42 (2022), no. 2, pp. 1831–1865.
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wave equation”, Numer. Math. 105 (2007), no. 3, pp. 481–510.
[8] W. McLean and V. Thomée, “Numerical solution of an evolution equation with a positive-
type memory term”, J. Aust. Math. Soc., Ser. B 35 (1993), no. 1, pp. 23–70.
[9] I. Podlubny, Fractional differential equations. An introduction to fractional derivatives,
fractional differential equations, to methods of their solution and some of their applications,
Academic Press Inc., 1999, pp. xxiv+340.
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fractional phase-field equations”, Commun. Comput. Phys. 33 (2023), no. 4, pp. 962–991.
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University Press, 2002, pp. xiii+364. with a foreword by Robert A. Fefferman.


Positivity of convolution quadratures generated by nonconvex sequences

Seong-Hun Paeng

Comptes Rendus
Mathématique
Seong-Hun Paeng
A simple generalization of a Willmore-type inequality
Volume 362 (2024), p. 1793-1799
Online since: 25 November 2024
https://doi.org/10.5802/crmath.684
This article is licensed under the
Creative Commons Attribution 4.0 International License.
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C EN T R E
MER S ENN E
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. 1793-1799
https://doi.org/10.5802/crmath.684
Research article / Article de recherche
Geometry and Topology / Géométrie et Topologie
A simple generalization of a Willmore-type
inequality
Une généralisation simple d’une inégalité de type
Willmore
Seong-Hun Paeng a
a Department of Mathematics, Konkuk University, 120 Neungdong-ro, Gwangjin-gu, Seoul 05029, Korea
E-mail: shpaeng@konkuk.ac.kr
Abstract. We give a simple generalization of the Willmore-type inequality in [1] without assuming that Ricci
curvature is nonnegative everywhere.
Résumé. Nous donnons une généralisation simple de l’inégalité de type Willmore dans [1] sans supposer que
la courbure de Ricci est partout non négative.
Keywords. Willmore-type inequality, Ricci curvature.
Mots-clés. Inégalité de type Willmore, courbure de Ricci.
2020 Mathematics Subject Classification. 53C20, 53C42.
Funding. This work was supported by the National Research Foundation of Korea (NRF) grant funded by the
Korea government (MSIT)(No. RS-2023-00242367).
Manuscript received 22 September 2022, revised 23 June 2023 and 30 January 2024, accepted 3 September 2024.
1. Introduction
The classical Willmore inequality for a bounded domainΩ of R3 with smooth boundary says that∫
∂Ω
H 2 dvol∂Ω ≥ 16π,
where H is the mean curvature of ∂Ω and dvol∂Ω is the volume form of ∂Ω. In [1], Agostiniani,
Fogagnolo, and Mazzieri obtained the following Willmore-type inequality for an n-dimensional
complete noncompact Riemannian manifold (M , g ) with the nonnegative Ricci curvature and
Euclidean volume growth: ∫
∂Ω
∣∣∣ H
n −1
∣∣∣n−1 dvol∂Ω ≥ AVR(g )|Sn−1|, (1)
where |Sn−1| is the volume of the standard sphere and AVR(g ) is the asymptotic volume ratio, i.e.
AVR(g ) = lim
r→∞
n vol(B(p,r ))
r n |Sn−1| ,
ISSN (electronic): 1778-3569 https://comptes-rendus.academie-sciences.fr/mathematique/
1794 Seong-Hun Paeng
where B(p,r ) is the r -ball centered at p ∈ M . They used the monotonicity-rigidity properties of
the function Uβ(t ) which is defined as follows:
Uβ(t ) = t−β(
n−1
n−2 )
∫
{u=t }
|∇u|β+1,
where u is the harmonic function which vanishes at infinity and u = 1 on ∂Ω.
In this paper, we will generalize the inequality without assuming Ric ≥ 0 everywhere. The
condition Ric ≥ 0 is essential in [1] to apply results on harmonic functions.
Let ρ(q) = max{(−Ric(v, v))+ | |v | = 1, v ∈ Tq M }, where f+ = max{0, f }. We define integral
norms Rn−1 as follows:
Rn−1 =
∫
M
ρn−1 dV ,
where dV is the volume form of M .
Theorem 1. Let (M , g ) be an n-dimensional complete noncompact Riemannian manifold with
Euclidean volume growth and Ric ≥ 0 outside B(p,R0). If Ω ⊂ M is a bounded and open subset
with smooth boundary, then(∫
∂Ω
∣∣∣∣ Hn −1
∣∣∣∣n−1 dvol∂Ω)
1
n−1
≥ (AVR(g )|Sn−1|) 1n−1 − (2R0) n−1n−2 (Rn−1) 1n−1
n −1 .
(2)
If Ric ≥ 0, then Rn−1 = 0, so we obtain (1).
2. Proof of Theorem 1
We will use the following notations. Let
Ωt = {x ∈ M | d(x,Ω) ≤ t }
∂Ωt = {x ∈ M | d(x,Ω) = t }.
Let γq be the outward normal geodesic such that γq (0) = q and γ′q (0) is perpendicular to ∂Ω for
q ∈ ∂Ω. Let
tq = max{t | d(γq (t ),∂Ω) = t }.
Then we have
M \Ω= ⋃
q∈∂Ω
{γq (t ) | t ≤ tq }.
We denote by g t the induced metric of ∂Ωt from the metric g of M . Let dvolt be the volume
form of ∂Ωt induced from g t . Then the volume form dvol∂Ω of ∂Ω is dvol0 and the volume form
of M satisfies dV = dt ∧dvolt . By identifying γq (t ) ∈ ∂Ωt with q ∈ ∂Ω for t ≤ tq , we define ω(t , q)
and h(t , q) as follows:
dvolt =ω(t , ·)dvol∂Ω (3)
∂
∂t
ω(t , q) = h(t , q)ω(t , q), (4)
where h(t , ·) is the mean curvature of ∂Ωt . So H = h(0, ·). We abbreviateω(t , q) and h(t , q) toω(t )
and h(t ), respectively. For t < tq , h satisfies
h′+ h
2
n −1 ≤−Ric
(
∂
∂t
,
∂
∂t
)
, (5)
where ∂∂t = γ′q (t ) and | ∂∂t | = 1. For t > tq , we let ω(t ) = 0. Then ω(t ) is defined for all t > 0 and
vol(Ωt \Ω) =
∫ t
0
∫
∂Ωω(s, q)dvol∂Ω ds. Furthermore, ω
′(t ) = h(t )ω(t ) for t < tq and ω′(t ) = 0 for
t > tq . We denote left and right limits limδ→0− f (t +δ) and limδ→0+ f (t +δ) by f (t−) and f (t+),
respectively.
Seong-Hun Paeng 1795
Let
ψ(t ) =

h+(t ) if t < tq
h+(t−) if t = tq
0 if t > tq ,
(6)
where h+(t ) = max{h(t ),0}. Then we have
ω′(t ) ≤ψ(t )ω(t ) (7)
for t < tq . Also ψ′ = h′ on {h(t ) > 0} and ψ′ = 0 for t > tq .
Although both ω and ψ are continuous along γq (t ) for t ̸= tq , they are left continuous for all
t > 0, i.e. ω(t−) =ω(t ) and ψ(t−) =ψ(t ) for all t > 0. Even if ω is not continuous at t = tq , the left
derivative limδ→0− ω(t+δ)−ω(t )δ is well defined for any t > 0. On the other hand, the right derivative
limδ→0+ ω(t+δ)−ω(t )δ is not well defined, i.e. it can be −∞. For a while, we will abbreviate the left
derivative limδ→0−
f (t+δ)− f (t )
δ to f
′(t ) for simplicity.
The left derivative ω′(t ) is bounded on [0,T ] × ∂Ω for T > 0 by the Jacobi equation J ′′ =
R( ∂∂t , J )
∂
∂t , where R is the curvature tensor and J is the Jacobi field. Also (ψ
n−1)′(t ) is bounded
on [0,T ]×∂Ω since h′(t ) ̸= −∞ (i.e. γq (t ) is not a focal point) at t when h(t ) ≥ 0.
Since ρ = max{(−Ric(v, v))+ | |v | = 1, v ∈ Tq M }, (5) becomes
ψ′+ ψ
2
n −1 ≤ ρ, (8)
where we also let ρ(γq (t )) = 0 if t > tq . We let
Hn−1(t ) =
∫
∂Ωt
ψn−1dvolt =
∫
∂Ω
ψn−1ω.
Since (ψn−1ω)(t−) = (ψn−1ω)(t ) for every t , Hn−1(t ) is left continuous, i.e.
Hn−1(t−) = lim
δ→0−
∫
∂Ω
(ψn−1ω)(t +δ) =
∫
∂Ω
(ψn−1ω)(t ) =Hn−1(t ) (9)
by the dominated convergence theorem.
Since (ψ
n−1ω)(t+δ)−(ψn−1ω)(t )
δ ≤ 0 for t > tq and δ< 0, we obtain that
∫
∂Ω
ψn−1(t +δ)ω(t +δ)−ψn−1(t )ω(t )
δ
≤
∫
{q∈∂Ω | t≤tq }
(ψn−1(t +δ)−ψn−1(t ))ω(t +δ)+ψn−1(t )(ω(t +δ)−ω(t ))
δ
−→
∫
{q∈∂Ω | t≤tq }
(ψn−1)′ω+ψn−1ω′ =
∫
∂Ω
(ψn−1)′ω+ψn−1ω′ (10)
as δ→ 0− by the dominated convergence theorem for t ≤ tq and ω(t ) =ψ(t ) = 0 for t > tq .
From now on, we redefine f ′(t ) = ddt f (t ) as follows:
f ′(t ) := limsup
δ→0−
f (t +δ)− f (t )
δ
.
1796 Seong-Hun Paeng
We call f ′(t ) the left derivative of f for simplicity. Then the left derivative H ′n−1(t ) satisfies
d
dt
∫
∂Ωt
ψn−1dvolt = d
dt
∫
∂Ω
ψn−1ω
= limsup
δ→0−
(∫
∂Ω
ψn−1(t +δ)ω(t +δ)−ψn−1(t )ω(t )
δ
)
≤
∫
∂Ω
(ψn−1)′ω+ψn−1ω′
≤
∫
∂Ω
ψn−2((n −1)ψ′+ψ2)ω
≤ (n −1)
∫
∂Ωt
ψn−2ρdvolt
≤ (n −1)
(∫
∂Ωt
ψn−1dvolt
) n−2
n−1
(∫
∂Ωt
ρn−1dvolt
) 1
n−1
(11)
by (8), (10).
Proposition 2. For sufficiently large t , we have(∫
∂Ω
|H |n−1 dvol∂Ω
) 1
n−1 ≥Hn−1(t )
1
n−1 − (2R0)
n−1
n−2 (Rn−1)
1
n−1 . (12)
Proof. Since
∫
∂Ω |H |n−1 ≥Hn−1(0), we only need to show that
Hn−1(t )
1
n−1 −Hn−1(0)
1
n−1 ≤ (2R0)
n−1
n−2 (Rn−1)
1
n−1 .
By (11),
H ′n−1(t )
(n −1)Hn−1(t ) n−2n−1
≤
(∫
∂Ωt
ρn−1dvolt
) 1
n−1
(13)
if Hn−1(t ) ̸= 0.
If g (x) = x 1n−1 and f is left continuous, then the left derivative g ( f )′ satisfies that
g ( f )′(a) = limsup
δ→0−
(
g ( f (a +δ))− g ( f (a))
f (a +δ)− f (a)
)(
f (a +δ)− f (a)
δ
)
= g ′( f (a)) f ′(a)
(14)
if f (a) ̸= 0. By (13) and (14), the left derivative ddt Hn−1(t )
1
n−1 satisfies that
d
dt
(
Hn−1(t )
1
n−1
)
≤
(∫
∂Ωt
ρn−1 dvolt
) 1
n−1
(15)
if Hn−1(t ) ̸= 0. If Hn−1(t ) = 0, then the left derivative (Hn−1(t )) 1n−1 )′ ≤ 0 since Hn−1(s) 1n−1 ≥ 0 for
any s. Thus (15) holds for any t .
Lemma 3. Let f and g ≥ 0 be left continuous on [a,b] and f (t ) ≥ f (t+) for all t ∈ [a,b]. If the left
derivative f ′ satisfies that f ′(t ) ≤ g (t ) for any t and g is bounded, then
f (b)− f (a) ≤
∫ b
a
g .
Proof. For η > 0, let I = {x ∈ [a,b] | f (b)− f (x) ≤ ∫ bx (g (t )+η)d t } and A = inf I . Then it is clear
that b ∈ I . First, we show that A ∈ I . If not, there is a sequence an ∈ I such that an → A. Since
f (A) ≥ f (A+), we have
f (b)− f (A) ≤ f (b)− f (A+) ≤ lim
n→∞
∫ b
an
(g (t )+η)dt =
∫ b
A
(g (t )+η)dt .
Hence we obtain that A ∈ I .
Seong-Hun Paeng 1797
Now we show that A = a. If not, for s satisfying a ≤ s < A,
f (b)− f (s) >
∫ b
s
(g (t )+η)dt .
Since f (b)− f (A) ≤ ∫ bA (g (t )+η)dt , we have
f (A)− f (s) >
∫ A
s
(g (t )+η)dt .
Then
f ′(A) = limsup
s→A−
f (A)− f (s)
A− s ≥ limsups→A−
1
A− s
∫ A
s
(g (t )+η)dt = g (A)+η
since g is left continuous. But this is a contradiction to our assumption. So we obtain that
A = a, i.e. f (b)− f (a) ≤ ∫ ba (g (t )+η)dt . Since η is arbitrarily chosen, letting η→ 0, we obtain
Lemma 3. □
By the same reason that Hn−1 is left continuous,
(∫
∂Ωt
ρn−1 dvolt
) 1
n−1 is left continuous.
Hence, by (15) and Lemma 3 with f =Hn−1 and g =
(∫
∂Ωt
ρn−1 dvolt
) 1
n−1 ,
Hn−1(t )
1
n−1 −Hn−1(0)
1
n−1 ≤
∫ t
0
(∫
∂Ωs
ρn−1dvols
) 1
n−1
ds. (16)
Since the diameter of B(p,R0) is not larger than 2R0, B(p,R0) \Ω ⊂Ωt0+2R0 \Ωt0 for some t0 ≥ 0.
For t > t0 +2R0,
Hn−1(t )
1
n−1 −Hn−1(0)
1
n−1 ≤
∫ t0+2R0
t0
(∫
∂Ωs
ρn−1 dvols
) 1
n−1
ds
≤
(∫
M
ρn−1 dV
) 1
n−1
(2R0)
n−2
n−1 ,
(17)
so we have Proposition 2. □
Now we calculate Hn−1(t ) with vol(∂Ωt ). Let A(t ) = volt (∂Ωt ) =
∫
∂Ωω(t , ·). Then A(t ) is left
continuous similarly as in (9). Similarly as in (11),
A′(t ) ≤
∫
∂Ω
ω′
≤
∫
∂Ωt
ψdvolt
≤
(∫
∂Ωt
ψn−1 dvolt
) 1
n−1
(∫
∂Ωt
dvolt
) n−2
n−1
≤Hn−1(t )
1
n−1 A(t )
n−2
n−1 .
(18)
by (7). By (14), (18), and the left continuity of A(t ),
(n −1)(A 1n−1 (t ))′ ≤Hn−1(t ) 1n−1 . (19)
From Proposition 2, we obtain that(∫
∂Ω
|H |n−1 dvol∂Ω
) 1
n−1 ≥ (n −1)(A 1n−1 (t ))′− (2R0) n−1n−2 (Rn−1) 1n−1 . (20)
Now we calculate the left derivative
(
A
1
n−1 (t )
)′ with AVR(g ).
Lemma 4. Let f : [0,∞) → R have left derivative and satisfy f (t+) ≤ f (t ) for all t . If
limsupt→∞
f (t )
t = a > 0, then the left derivative f ′(t ) satisfies
limsup
t→∞
f ′(t ) ≥ a. (21)
1798 Seong-Hun Paeng
Proof. Assume limsupt→∞ f ′(t ) < a −2ϵ < a. Then there exists r0 > 0 such that f ′(t ) < a − ϵ for
t ≥ r0. Take a large R > r0. By Lemma 3,
f (R)− f (r0) ≤
∫ R
r0
(a −ϵ) = (a −ϵ)(R − r0), (22)
which implies that
lim sup
t→∞
f (t )
t
≤ a −ϵ.
It is a contradiction to our condition. Hence,
limsup
t→∞
f ′(t ) ≥ a, (23)
which completes the proof of Lemma 4. □
We let f (t ) = vol(Ωt \Ω) 1n , where vol(Ωt \Ω) =
∫ t
0
∫
∂Ωω(s, q)dvol∂Ω ds. Although vol(Ωt \Ω)
may not be differentiable since ω(t ) is not continuous at tq , f is continuous. We can obtain the
left derivative as follows:
d
dt
vol(Ωt \Ω) =
∫
∂Ω
ω(t , q)dvol∂Ω = A(t ). (24)
Since vol(Ωt \Ω) is continuous, the left derivative satisfies
f ′(t ) = 1
n
A(t )
vol(Ωt \Ω)
n−1
n
by (14).
Since vol(ΩR \Ω)vol(B(p,R)) → 1 as R →∞, we have
lim
t→∞
f (t )
t
=
(
AVR(g )|Sn−1|
n
) 1
n
. (25)
From f ′(t ) = 1n A(t )vol(Ωt \Ω) n−1n
and Lemma 4, we obtain that
(
AVR(g )|Sn−1|
n
) 1
n
≤ limsup
t→∞
f ′(t ) = limsup
t→∞
1
n
A(t )
vol(Ωt \Ω)
n−1
n
= limsup
t→∞
1
n
A(t )
vol(B(p, t ))
n−1
n
.
(26)
Although A
1
n−1 (t ) may not be continuous, we have A
1
n−1 (t+) ≤ A 1n−1 (t ) since ω(t ) ≥ω(t+). By
Lemma 4 and (25), (26),
limsup
t→∞
(A
1
n−1 (t ))′ ≥ limsup
t→∞
A
1
n−1 (t )
t
≥ n 1n−1 lim
t→∞
(
AVR(g )|Sn−1|
n
) 1
n(n−1) vol(B(p, t ))
1
n
t
=
(
AVR(g )|Sn−1|
) 1
n−1
.
Consequently, from (20),(∫
∂Ω
( |H |
n −1
)n−1) 1n−1
≥
(
AVR(g )|Sn−1|
) 1
n−1 − 1
n −1 (2R0)
n−1
n−2 (Rn−1)
1
n−1 ,
which completes the proof of Theorem 1.
Seong-Hun Paeng 1799
Acknowledgements
The author would like to express his gratitude to the referee for their helpful comments, which
improve this paper very much.
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] V. Agostiniani, M. Fogagnolo and L. Mazzieri, “Sharp geometric inequalities for closed
hypersurfaces in manifolds with nonnegative Ricci curvature”, Invent. Math. 222 (2020),
no. 3, pp. 1033–1101.


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