Numérisation de Documents Anciens Mathématiques

Younso, Ahmad

Comptes Rendus
Mathématique
Ahmad Younso
Consistency of the k-nearest neighbors rule for functional data
Volume 361 (2023), p. 237-242
https://doi.org/10.5802/crmath.402
This article is licensed under the
Creative Commons Attribution 4.0 International License.
http://creativecommons.org/licenses/by/4.0/
Les Comptes Rendus. Mathématique sont membres du
Centre Mersenne pour l’édition scientifique ouverte
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e-ISSN : 1778-3569
Comptes Rendus
Mathématique
2023, Vol. 361, p. 237-242
https://doi.org/10.5802/crmath.402
Statistics / Statistiques
Consistency of the k-nearest neighbors rule
for functional data
Consistance de la règle des k-plus proches voisins pour
des données fonctionnelles
Ahmad Younsoa, b
a MISTEA, Université Montpellier, INRAE, Institut Agro,Montpellier, France
b Department of mathematical statistics, Damascus university, Damascus, Syria
E-mail: ahmad.younso@inrae.fr
Abstract. The problem of nonparametric classification by k-nearest neighbors rule in a general metric space
will be considered. Consistency and strong consistency of the classifier will be established under mild
conditions.
Résumé. Le problème de la classification non paramétrique par la règle des k- plus proches voisins dans un
espace métrique général sera considéré. La consistance et la forte consistance du classifieur seront établies
sous des conditions légères.
2020 Mathematics Subject Classification. 00X99.
Manuscript received 20 April 2021, accepted 26 June 2022.
1. Introduction
Let X be a random variable defined on some probability space (Ω,A ,P) with values in a separable
metric space (F ,ρ) where F is a function space and ρ denotes the metric on F , and let Y be
a random variable with values 0 or 1. In this paper, we study the classical binary supervised
classification model for data from F . Given a new incoming observation X , our goal is to predict
its corresponding label Y . The distribution of the pair (X ,Y ) is well defined by (µ,η) where
µ(B) = P(X ∈ B), for all Borel sets B on F , and η(x) = E(Y |X = x) the regression function of Y
given X = x, for all x ∈ F . In order to predict the unknown label Y of an observation X = x, we
use a classifier that provides a decision rule for this problem. Formally, a classifier is a measurable
mapping g : F −→ {0,1}. Given a classifier g , its corresponding miss-classification error is given
by L = L(g ) = P{g (X ) 6= Y }. In practice, the best classifier is that associated with the smallest
possible error. It is well known that the Bayes classifier given by
g∗(x) =1{η(x)≥1/2},
ISSN (electronic) : 1778-3569 https://comptes-rendus.academie-sciences.fr/mathematique/
238 Ahmad Younso
where 1A denotes the indicator function of the set A, leads to the lowest possible miss-
classification error, i.e.,
L∗ = L(g∗) = inf
g :F→{0,1}
P{g (X ) 6= Y }.
Unfortunately, g∗ is not available since it depends on the distribution of (X ,Y ) which is generally
unknown. But it is often possible to construct a classifier from a set of n independent and
identically distributed copies Dn = {(X1,Y1), . . . , (Xn ,Yn)} of (X ,Y ). The set Dn is called the
training data. Among the various ways to define a classifier from a training data, one of the most
wide spread and simplest is the k-nearest neighbors (k-N N ) classifier given by
gn(x) =
{
0 if
∑n
i=1 wni Yi ≤ 1/2
1 otherwise,
(1)
where wni = wni (x;Dn) is 1/k if Xi is one of the k-nearest neighbor of x in Dn with respect to the
metric ρ, and wni is zero otherwise with k = k(n) is a sequence of positive integers satisfying
k −→∞ and k/n −→ 0 as n →∞. (2)
Ties are broken by preferring points earlier in the sequence. According to the rule (1), an unclas-
sified element is assigned to the class represented by a majority of its k-nearest neighbors in the
training set. If we let ηn(x) = ∑ni=1 wni Yi be the k-nearest neighbor estimator of η(x), the classi-
fier (1) can be re-written as follows
gn(x) =
{
0 if ηn(x) ≤ 1/2
1 otherwise.
Let Ln = L(gn) = P{gn(X ) 6= Y } be the miss-classification error of gn(x). The main challenge in
this supervised classification setting is to construct a classifier gn whose miss-classification error
will be as close as possible to L∗. In this context, the classifier gn(x) is called consistent if
ELn −→ L∗ as n →∞
and called strongly consistent if
Ln −→ L∗ with probability one as n →∞.
A classifier can be consistent for certain class of distribution of (X ,Y ), but not be consistent for
others. The classifier gn(x) is called (strongly) universally consistent, if it is (strongly) consistent
for all distribution of (X ,Y ). In finite-dimensional spaces, the k-nearest neighbors rule is univer-
sally strongly consistent under classical conditions. (see [5] and [9]). [2] prove the consistency of
the k-N N rule in separable Hilbert spaces. [3] give some examples showing that the results of [6]
on the consistency are no more valid in a general functional metric space (F ,ρ) and they estab-
lish consistency of the k-N N rule on a separable metric space. [1] show that the moving window
rule is not consistent in general metric spaces and give conditions on the space and the regres-
sion function to ensure the (strong) consistency of the estimator. More recently, [7] extend the
Stone’s seminal result to the case of metric spaces when the probability measure of the explana-
tory variables is tight. Then, under slight variations on the hypotheses, they extend the result to
some general metric measure spaces. In this paper, the consistency of the classifier (1) will be
proved under weaker assumptions than that of [3] and [7]. Furthermore, the strong consistency
of the classifier (1) will also be established under some weak conditions. Denote Bx,ε the closed
ball of radius ε > 0 and center at x ∈ F . To establish the main results, we will need the following
Besicovitch condition, for every ε> 0,
lim
δ→0
µ
{
x ∈F :
∣∣∣∣ 1µ(Bx,δ)
∫
Bx,δ
ηdµ−η(x)
∣∣∣∣> ε}= 0. (3)
Ahmad Younso 239
Note that (3) a classical assumption for this kind of results and it holds for example if η(x) is µ-
continuous (for more detail on this topic, see [3]). Besicovitch condition plays an important role
also in the consistency of kernel rules (see [1]). In finite dimension, (3) holds automatically for any
integrable function since it is just the differentiation theorem with respect to a finite measure.
2. Main results
Before we state the main results of this paper, we introduce the following lemma which is needed
in the proofs. Denote
supp(µ) = {x ∈F :µ(Bx,ε) > 0,∀ ε> 0}.
Lemma 1. If (2) is fulfilled then, for each x ∈ supp(µ), there exists a sequence an = an(x) → 0 as
n →∞ such that µ(Bx,an ) ≥
p
k/n.
For the proof of Lemma 1, the reader is referred to [7].
2.1. Consistency of the classifier
The following theorem state the consistency of the classifier (1).
Theorem 2. If (2)–(3) are fulfilled then the k-N N classifier is consistent.
Note that classical condition (2) is the same as that used by [6] in the finite dimensional case
and by [3] in the infinite dimensional case. Condition (3) is used by [3].
Proof. By Theorem 2.2 in [6], whose extension to the infinite dimensional setting is straightfor-
ward, we can write
E(Ln)−L∗ ≤ 2E
∫
F
|η(x)−ηn(x)|µ(dx). (4)
Therefore, the theorem will be proved if we show that
E
∫
F
|η(x)−ηn(x)|µ(dx) −→ 0 as n →∞. (5)
Define
η̂n(x) = 1
nµ(Bx,an )
n∑
i=1
Yi1{Xi∈Bx,an } (6)
with an → 0 is the sequence given in Lemma 1. We have
E
∫
F
|η(x)−ηn(x)|µ(dx) ≤ E
∫
F
|η(x)−Eη̂n(x)|µ(dx)+E
∫
F
|Eη̂n(x)−ηn(x)|µ(dx). (7)
Therefore, it suffices to prove that each term in the right-hand side of (7) tends to zero as n → 0.
Besicovitch condition yields as n → 0,∫
F
|η(x)−Eη̂n(x)|µ(dx) → 0. (8)
It remains to prove that as n → 0,
E
∫
F
|Eη̂n(x)−ηn(x)|µ(dx) → 0. (9)
Clearly, we have
E
∫
F
|Eη̂n(x)−ηn(x)|µ(dx) ≤ E
∫
F
|Eη̂n(x)− η̂n(x)|µ(dx)+E
∫
F
|η̂n(x)−ηn(x)|µ(dx). (10)
240 Ahmad Younso
Therefore, we will prove that each term in the right-hand side of the above inequality tends to
zero. Cauchy–Schwartz inequality yields
E|η̂n(x)−Eη̂n(x)| ≤ E((η̂n(x)−Eη̂n(x))2)1/2 ≤ (var(η̂n(x)))1/2 ≤
(
E(Y 1{X∈Bx,an })
2
n(µ(Bx,an ))
2
)1/2
.
Since |Y | ≤ 1, we obtain
E|η̂n(x)−Eη̂n(x)| ≤
(
1
nµ(Bx,an )
)1/2
. (11)
Hence, Lemma 1 yields
1
nµ(Bx,an )
≤
p
n/k
n
= 1p
kn
. (12)
By (11)–(12) together with Fubini’s theorem and the dominated convergence theorem, we get as
n →∞,
E
∫
F
|Eη̂n(x)− η̂n(x)|µ(dx) → 0. (13)
Now, let X(k)(x) be the k-nearest neighbor of x in Dn and denote rn = rn(x) = ρ(X(k)(x), x). Clearly,
ηn(x) = 1
k
n∑
i=1
Yi1{Xi∈Bx,rn }.
Then, since |Yi | ≤ 1,
|η̂n(x)−ηn(x)| =
∣∣∣∣∣ 1nµ(Bx,an )
n∑
i=1
Yi1{Xi∈Bx,an } −
1
k
n∑
i=1
Yi1{Xi∈Bx,rn }
∣∣∣∣∣
≤
n∑
i=1
∣∣∣∣ 1nµ(Bx,an )1{Xi∈Bx,an } − 1k 1{Xi∈Bx,rn }
∣∣∣∣
≤
∣∣∣∣∣ 1nµ(Bx,an )
n∑
i=1
1{Xi∈Bx,an } −1
∣∣∣∣∣ := |η̃n(x)−Eη̃n(x)|. (14)
Observe that η̃n(x) = η̂n(x) if we let Yi = 1 for all i = 1, . . . ,n. Consequently, the proof of the limit
E
∫
F
|η̃n(x)−Eη̃n(x)|→ 0 (15)
is the same as that of (13). Finally, by (7)–(10), (13) and (14)–(15) the theorem is proved. 
2.2. Strong consistency of the classifier
The following theorem state the strong consistency of the classifier (1).
Theorem 3. Suppose that (2)–(3) are fulfilled. If in addition k/(logn) → ∞, then the k-N N
classifier is strongly consistent.
Note that the classical condition k/(logn) → ∞, used by [4], is crucial to get the strong
consistency of the classifier (2).
Proof. By Theorem 2.2 in [6], the theorem will be proved if we show that∫
F
|η(x)−ηn(x)|µ(dx) −→ 0 as n →∞ with probability one. (16)
Using the term η̂n(x) defined above in (6), we can write∫
F
|η(x)−ηn(x)|µ(dx) ≤
∫
F
|η(x)−Eη̂n(x)|µ(dx)+
∫
F
|Eη̂n(x)−ηn(x)|µ(dx). (17)
As a consequence, by (8), it suffices to show that∫
F
|Eη̂n(x)−ηn(x)|µ(dx) → 0 with probability one as n →∞. (18)
Ahmad Younso 241
Clearly, we have∫
F
|Eη̂n(x)−ηn(x)|µ(dx) ≤
∫
F
|Eη̂n(x)− η̂n(x)|µ(dx)+
∫
F
|η̂n(x)−ηn(x)|µ(dx). (19)
Hence, we will prove each term in the right-hand side of the above inequality tends to zero as
n →∞ with probability one. Let us first deal with the first term in the right-hand side of (19). If
we replace (Xi ,Yi ) by (X ′i ,Y
′
i ), suppose that the value of η̂n(x) is changed to η̂ni (x). Then,∣∣∣∫
F
|Eη̂n(x)− η̂n(x))|µ(dx)−
∫
F
|Eη̂n(x)− η̂ni (x)|µ(dx)
∣∣∣
≤
∫
F
|η̂n(x)− η̂ni (x)|µ(dx) ≤
∫
F
1
nµ(Bx,an )
µ(dx) ≤ 1p
nk
. (20)
The last inequality is due to Lemma 1. Now, for any ε> 0, and n large enough, we have for n large
enough
P
(∫
F
|Eη̂n(x)− η̂n(x)|µ(dx) > ε
)
≤P
(∣∣∣∣∫
F
|Eη̂n(x)− η̂n(x)|µ(dx −
∫
F
|Eη̂n(x)− η̂n(x)|µ(dx)
∣∣∣∣> ε/2) .
Hence, using McDiarmid’s inequality (see [8]) with taking into account (20), we get for n large
enough
P
(∫
F
|Eη̂n(x)− η̂n(x)|µ(dx) > ε
)
≤ 2exp
(
−ε
2k
2
)
Since k/(logn) →∞ by assumption, Borel–Cantelli lemma yields∫
F
|Eη̂n(x)− η̂n(x)|µ(dx) → 0 with probability one as n →∞. (21)
Let us now deal with the second term in the right-hand side of (19). By (14),
|η̂n(x)−ηn(x)| ≤ |η̃n(x)−Eη̃n(x)|, (22)
with η̃n(x) = η̂n(x) if we let Yi = 1 for all i = 1, . . . ,n. Hence, if we follow the same arguments that
use to prove (21), we get∫
F
|η̃n(x)−Eη̃n(x)|µ(dx) → 0 with probability one as n →∞. (23)
Finally, by (8), (17), (19) and (21)–(23) we get (16) and the proof is completed. 
Conclusion. The consistency result of Theorem 2 is obtained under weaker assumptions than
that of [7]. If we let (F ,ρ) = (Rd ,‖·‖) with ‖·‖ denotes the Euclidean norm, then the Stone’s
theorem is considered as a particular case of Theorem 2. Furthermore, Theorem 3 is an extension
of Theorem 11.1 of [6] to a general metric space. In the latter result the authors suppose that µ is
absolutely continuous while µ in Theorem 2 is any probability measure.
References
[1] C. Abraham, G. Biau, B. Cadre, “On the kernel rule for function classification”, Ann. Inst. Stat. Math. 58.
[2] G. Biau, F. Bunea, M. H. Wegkamp, “Functional classification in Hilbert spaces”, IEEE Trans. Inf. Theory 51 (2005),
no. 6, p. 2163-2172.
[3] F. Cérou, A. Guyader, “Nearest neighbor classification in infinite dimension”, ESAIM, Probab. Stat. 10 (2006), p. 340-
355.
[4] K. Chaudhuri, S. Dasgupta, “Rates of Convergence for Nearest Neighbor Classification”, 2014, https://arxiv.org/abs/
1407.0067.
[5] L. Devroye, L. Györfi, A. Krzyżak, G. Lugosi, “On the Strong Universal Consistency of Nearest Neighbor Regression
Function Estimates”, Ann. Stat. 22 (1994), no. 3, p. 1371-1385.
[6] L. Devroye, L. Györfi, G. Lugosi, A probabilistic Theory of Pattern Recognition, Applications of Mathematics, vol. 31,
Springer, 1996.
242 Ahmad Younso
[7] L. Forzani, R. Fraiman, P. Llop, “Consistent Nonparametric Regression for Functional Data Under the
Stone–Besicovitch Conditions”, IEEE Trans. Inf. Theory 58 (2012), no. 11, p. 6697-6708.
[8] C. McDiarmid, “On the method of bounded differences”, in Surveys of combinatorics, London Mathematical Society
Lecture Note Series, vol. 141, Cambridge University Press, 1989, p. 148-188.
[9] C. J. Stone, “Consistent nonparametric regression”, Ann. Stat. 5 (1977), no. 4, p. 595-620.


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Monneau, Régis

Comptes Rendus
Mathématique
Mohammad Al Haj and Régis Monneau
The velocity diagram for traveling waves
Volume 361 (2023), p. 777-782
Published online: 11 May 2023
https://doi.org/10.5802/crmath.433
This article is licensed under the
Creative Commons Attribution 4.0 International License.
http://creativecommons.org/licenses/by/4.0/
Les Comptes Rendus. Mathématique sont membres du
Centre Mersenne pour l’édition scientifique ouverte
www.centre-mersenne.org
e-ISSN : 1778-3569
Comptes Rendus
Mathématique
2023, Vol. 361, p. 777-782
https://doi.org/10.5802/crmath.433
Partial differential equations / Équations aux dérivées partielles
The velocity diagram for traveling waves
Mohammad Al Haja and Régis Monneau∗, b, c
a Lebanese University, Faculty of Science (section 5), Nabatiye 1700, Lebanon
b CERMICS, Ecole des Ponts ParisTech, Université Paris-Est, 6 et 8 avenue Blaise
Pascal, Cité Descartes, Champs-sur-Marne, 77455 Marne-la-Vallée Cedex 2, France
c CEREMADE, Université Paris-Dauphine, Place du Maréchal De Lattre De Tassigny,
75775 Paris Cedex 16, France
Abstract. In this Note, we consider traveling waves in a reaction-diffusion equation in dimension one.
Motivated by the motion of dislocations in crystals, we introduce an additive parameter σ in the reaction
term, which may be interpreted as an exterior force applied on the crystal. Under certain natural assumptions
and for every value of σ ∈ [σ−,σ+], we show the existence of traveling waves φ of velocity c. The range
σ ∈ (σ−,σ+) corresponds to bistable cases with a unique velocity c = c(σ). On the contrary, the case σ=σ+ is
positively monostable with a branch of velocities c ≥ c+, while the case σ = σ− is negatively monostable
with a branch of velocities c ≤ c−. This study gives rise to a natural connection between bistable cases
and monostable cases in a single velocity diagram. We also give some qualitative properties of the velocity
function σ 7→ c(σ).
Résumé. Dans cette Note, nous considérons des ondes progressives pour une équation de réaction-diffusion
en dimension un. Motivés par le mouvement de dislocations dans les cristaux, nous introduisons un para-
mètre additif σ dans le terme de réaction, qui peut être interprété comme une force extérieure appliquée au
cristal. Sous certaines hypothèses naturelles et pour chaque valeur de σ ∈ [σ−,σ+], nous montrons l’exis-
tence d’ondes progressives φ se déplaçant à la vitesse c. Le domaine σ ∈ (σ−,σ+) correspond aux cas bis-
tables avec une unique vitesse c = c(σ). Au contraire, le cas σ = σ+ est positivement monostable avec une
branche de vitesses c ≥ c+, et le cas σ=σ− est négativement monostable avec une branche de vitesse c ≤ c−.
Cette étude met en évidence un lien naturel entre les cas bistables et les cas monostables au sein d’un unique
diagramme en vitesse. On donne aussi des propriétés qualitatives de la fonction vitesse σ 7→ c(σ).
Manuscript received 2 August 2022, revised and accepted 28 September 2022.
1. Introduction
In this Note, we consider particular solutions u(t , x) to the standard reaction-diffusion equation
with an additional exterior parameter σ ∈R
u(t , x) =φ(x + ct ) satisfying ut = uxx + f (u)+σ for (t , x) ∈R×R (1)
where φ is a traveling wave moving with velocity c. Such a model is for instance inspired from
the dynamics of a dislocation defect in a crystal where σ is the exterior shear stress applied on
∗Corresponding author.
ISSN (electronic) : 1778-3569 https://comptes-rendus.academie-sciences.fr/mathematique/
778 Mohammad Al Haj and Régis Monneau
the crystal. From this point of view, equation (1) can be seen as an approximation of the fully
overdamped Frenkel–Kontorova model (see [8]). We assume that the function f satisfies
(Regularity) f is Lipschitz-continuous
(Periodicity) f (v +1) = f (v) for all v ∈R
(Decreasing) f decreasing on (θ,1) for θ ∈ (0,1)
(Stricly increasing) f ′(v) ≥ g (v) > 0
for almost every v ∈ (0,θ) with g : (0,θ) →R continuous
(2)
The graph of f is represented on Figure 1.
0 θ 1
Figure 1. Nonlinearity f
Setting
σ+ :=−min f , σ− :=−max f
we define for each σ ∈ [σ−,σ+], the unique roots mσ and bσ solutions of
f (mσ)+σ= 0, mσ ∈ [θ−1,0] and f (bσ)+σ= 0, bσ ∈ [0,θ]
Here the nonlinearity f σ := f +σ falls in one the following three cases
bistable case σ ∈ (σ−,σ+), f σ < 0 on (mσ,bσ) and f σ > 0 on (bσ,1+mσ)
positive monostable case σ=σ+, f σ+ > 0 on (mσ+ ,1+mσ+ ) with bσ+ = mσ+ = 0
negative monostable case σ=σ−, f σ− < 0 on (mσ− ,1+mσ− ) with bσ− = 1+mσ− = θ
Our goal is to connect bistable and monostable types as the parameter σ varies, as it is very
natural from the point of view of the motion of a dislocation defect under the exterior force σ. To
this end, we now consider traveling wavesφ of velocity c associated to the parameterσ ∈ [σ−,σ+],
solutions of
cφ′ =φ′′+ f (φ)+σ on R with φ(−∞) = mσ, φ(+∞) = 1+mσ (3)
Recall here some known results. When f ∈ C 1(R) with f ′(mσ) < 0, the uniqueness of c is known
in [6] in the bistable case. In the positive monostable case, when f ∈C 1([0,1]) with f ′(0) > 0, the
existence of a branch of velocities c ≥ c+ is also known (see [3, 7, 9]).
Then our result is the following velocity diagram.
Theorem 1 (Velocity diagram). Assume that f satisfies (2).
(i) (Bistable case σ ∈ (σ−,σ+)) For every σ ∈ (σ−,σ+), there exists a unique velocity c = c(σ)
and a unique (up to translations) traveling waveφ solution of (3). Moreoverφ is increasing
and the map σ 7→ c(σ) is continuous increasing for σ ∈ (σ−,σ+) with finite limits
c− := lim
σ−<σ→σ− c(σ), c
+ := lim
σ+>σ→σ+
c(σ)
There exists also δ> 0 such that 0 < δ≤ c(σ2)−c(σ1)σ2−σ1 for all σ1,σ2 ∈ (σ−,σ+) with σ1 ̸=σ2.
If moreover f ∈C 1,1(θ,1) with f ′ < 0 on (θ,1), then the mapσ 7→ c(σ) is locally Lipschitz-
continuous inside the interval (σ−,σ+).
(ii) (Positive monostable case σ = σ+) For every c ≥ c+, there exists a unique (up to transla-
tions) traveling wave φ solution of (3). Moreover φ is increasing. For all c < c+, there are
no traveling waves solutions of (3).
Mohammad Al Haj and Régis Monneau 779
(iii) (Negative monostable case σ = σ−) For every c ≤ c−, there exists a unique (up to transla-
tions) traveling wave φ solution of (3). Moreover φ is increasing. For all c > c−, there are
no traveling waves solutions of (3).
Figure 2. Schematic velocity diagram: the velocity function c(σ) with vertical branches at
σ=σ±.
The result of Theorem 1 is presented in the velocity diagram in Figure 2, and there are cases
where dcdσ blows up at σ = σ± (see Remark 2). This result allows to understand the branch of
solutions c ≥ c+ in the positive monostable case as a sort of continuation of the increasing
velocity σ 7→ c(σ) for σ ∈ (σ−,σ+). This phenomenon is also related to the fact that for σ ∈
R \ [σ−,σ+], there are no roots of f +σ = 0 and then no possible traveling waves. We see that
Theorem 1 gives the whole picture of the velocity diagram which connects bistable to monostable
cases.
Let us also mention that part of the results stated in Theorem 1 stays true (loosing possibly
the strict monotonicity of the velocity function σ 7→ c(σ) and loosing possibly the uniqueness of
the profile (up to translations)) for traveling waves of discrete reaction-diffusion equations of the
type
u(t , x) =φ(x + ct ) satisfying ut = F (u(t , x + r0), . . . ,u(t , x + rN ))
where r0 = 0 and the ri ∈ R are discrete shifts for i = 1, . . . , N with N ≥ 1, under certain period-
icity and monotonicity assumptions on F (in order to insure a comparison principle, but pos-
sibly loosing the strong comparison principle) joint to the previous assumption (2) on f (v) :=
F (v, . . . , v). The results are given in [2].
2. Sketch of the proof of the theorem
The originality of this Note is probably more in the statement of the theorem than in the proof
itself. Part of the arguments of the proof are classical. Some of those arguments are for instance
780 Mohammad Al Haj and Régis Monneau
developed in [2] for discrete reaction-diffusion equations, and are simply adapted here. For those
reasons, we only indicate here the sketch of the proof of the theorem.
Step 1: existence and uniqueness for σ ∈ (σ−,σ+). A standard way to build a solution (c,φ) con-
sists to use the method of sub/supersolutions to build a solution on a large finite interval [−R,R]
and to adjust the velocity c in such a way that the transition arises in the middle of the interval for
instance, and finally to pass to the limit as R →+∞ (like in [4] for ignition type nonlinearities).
A different method is used in [1] and is more generally useful to get results on the velocity in the
case of degenerate equations without strong maximum principle. This method consists to solve
the cell problem for homogenization (similarly to what is done in [8]) finding solutions (λp ,hp )
with λp ∈R, associated to a fixed slope p > 0
λp h
′
p = h′′p + f (hp )+σ, hp
(
x + 1
p
)
= hp (x)+1, h′p ≥ 0
and to define (c,φ) as a limit of (λp ,hp ) as p → 0 (similarly to what is done in [1, 2]). The
strict increasing property of assumption (2) on f insures that the root bσ is unstable, and then
that the limit profile φ does not split in two profiles, one in the range (mσ,bσ) and one in the
range (bσ,1+mσ). This insures that the profile φ is a true transition in the range (mσ,1+mσ).
The decreasing property of assumption (2) insures the comparison principle at infinity. This
allows to prove uniqueness of the velocity denoted c = c(σ). Using moreover the standard sliding
method [5] (based on the strong maximum principle), this allows to get uniqueness (up to
translations) and the monotonicity of the profile φ=:φσ.
Step 2: continuity and monotonicity of c(σ) for σ ∈ (σ−,σ+). The continuity of the map σ 7→
c(σ) follows from the uniqueness of the velocity. The fact that the maps σ 7→ mσ and σ 7→ bσ are
increasing, insures the comparison at infinity of the associated profiles for two values σ1 < σ2,
and this shows that the map σ 7→ c(σ) is nondecreasing. Moreover the strong comparison princi-
ple implies that the map σ 7→ c(σ) is increasing.
Step 3: vertical branch of solutions for large velocities c >> 1 for σ=σ+. Using the solution g
to the ODE g ′ = f (g ) +σ+ ≥ 0, say with g (0) = δ > 0, and using the Lipschitz regularity of f ,
we deduce that g ε(x) := g (εx) satisfies the supersolution inequality for all ε> 0
cεg
′
ε ≥ g ′′ε + f (g ε)+σ+ with cε =
1
ε
+ε| f ′|L∞(R)
Using the fact that g
ε
:= δ is a subsolution on (0,+∞), we can then build a solution gε,δ on (0,+∞).
Up to translate gε,δ, we can pass to the limit δ→ 0 and get a solution φ := gε associated to the
velocity c := cε, which works for all velocities c ≥ 2
√| f ′|L∞(R).
Step 4: definition of c±. If (c,φ) is a solution for the parameter σ+, then we can compare it to a
solution of velocity c(σ) for σ ∈ (σ−,σ+), and get c(σ) ≤ c. This implies that limσ+>σ→σ+ c(σ) =:
c+ ≤ c. In particular from Step 3, we deduce that c+ < +∞, and that there are no solutions (c,φ)
with c < c+ for the parameter σ+. We get similar results for c−.
Step 5: full vertical branch of solutions for c ≥ c+. Using (c(σ),φσ) the solution given in Step 1
for σ ∈ (σ−,σ+), we can pass to the limit and get φσ→φ+ as σ→σ+, which is a traveling wave of
velocity c+ for σ=σ+. Then for every c ≥ c+, we see that (c,φ+) is a supersolution of the equation
for σ = σ+. Then using a construction as in Step 3, we get the existence of a solution (c,φ) for
σ = σ+, for each c ≥ c+. Finally the decreasing property of f in (2) implies the comparison
principle at x =+∞. Let us consider intervals (aδ,+∞) where φ>φ(aδ) = δ> 0. Using the sliding
method for all δ > 0 small enough with mσ+ = 0, we can easily deduce the monotonicity and
uniqueness of the profile φ (up to translations).
Mohammad Al Haj and Régis Monneau 781
Step 6: full vertical branch of solutions for c ≤ c−. Proceeding as in Step 5, we get a full branch
of solutions (c,φ) for all c ≤ c−, and no solutions for c > c−.
Step 7: bounds on dcdσ . Recall that for any σ ∈ (σ−,σ+), we have c− ≤ c(σ) ≤ c+, and the interior
regularity theory for the elliptic equation satisfied by the profile φ=φσ gives a bound |φ′σ|L∞(R) ≤
δ−1 uniformly in σ ∈ (σ−,σ+), for some δ > 0. Consider σ1,σ2 ∈ (σ−,σ+) with σ1 < σ2, and the
associated solutions (ci ,φi ) for i = 1,2 with ci = c(σi ), φi =φσi and mi = mσi .
Step 7.1: bound from below. We deduce that
c2φ
′
1 ≤φ′′1 + f (φ1)+σ2 with c2 := c1 +δ(σ2 −σ1)
The comparison implies (up to an initial shift of the profiles) that φ2(x + c2t ) ≥ φ1(x + c2t ) and
then the fact that φ2(−∞) < φ1(+∞) implies that c2 ≥ c1 +δ(σ2 −σ1) which implies the bound
from below on dcdσ .
Step 7.2: bound from above. For c = c(σ), let us consider the equation cζ′ = ζ′′+ f ′(φσ)ζ satisfied
by ζ := φ′σ. Using the fact that f ′ < −µ < 0 in a neighborhood of mσ, we can introduce the roots
of λ2 − cλ−µ = 0 which are λ± := c±
p
c2+4µ
2 and show by comparison that ζ(y) ≤ ζ(x)eλ+(y−x)
for y ≤ x and ζ(x) small enough. This gives by integration for φσ(x) − mσ =
∫ x
−∞ dy ζ(y) that
φ′σ ≥λ+(φσ−mσ) whereφσ−mσ is small enough (using for instance interior estimates for elliptic
equations). Now forσ1 <σ2, we setφ2 :=ψ+m2 withψ :=φ1−m1. Using the fact that f (m2+ψ)−
f (m2)−
{
f (m1 +ψ)− f (m1)
}= ∫ 10 dt ∫ 10 ds { f ′′(tψ+m1 + s(m2 −m1))− f ′′(tψ+m1)}·ψ(m2−m1)
and f ′(mσ) dmσdσ =−1 with f ′(mσ) ≤−µ< 0 and λ+ ≥ λ+ > 0 for σ ∈ [σ1,σ2], we get on an interval
(−∞,−R] where ψ is small enough
c+2φ
′
2 ≥φ
′′
2 + f (φ2)+σ2 on (−∞,−R], with c+2 := c1 +
{
| f ′′|L∞(R)
µλ+
}
(σ2 −σ1)
We get a similar inequality on [R,+∞) where 1−ψ is small enough, with some velocity c−2 . Using
Harnack inequality, we can show that there exists some KR > 0 such that φ′1 ≥ (KR )−1 on [−R,R],
which implies
c02φ
′
2 ≥φ
′′
2 + f (φ2)+σ2 on [R,R], with c02 := c1 +KR
{
1+ | f
′|L∞(R)
µ
}
(σ2 −σ1)
Using the fact that all the constants can be taken uniformly forσ ∈ [σ1,σ2] ⊂⊂ [σ−,σ+], we deduce
that c2 ≤ max(c−2 ,c02,c+2 ) which implies the bound from above on dcdσ . This ends the sketch of the
proof of the theorem. □
Remark 2. Choosing a normalization like for instance φσ(0) = θ, we can show that ψ := ddσ (φσ−
mσ) formally satisfies an equation that we can multiply by e−cxφ′σ(x) and integrate by parts to get
with I1 :=
∫
R e
−cxφ′2σdx =
∫
R e
−cx (φσ−mσ)( f (φσ)− f (mσ))dx and I2 =
∫
R e
−cx
{
1− f ′(φσ)f ′(mσ)
}
φ′σdx,
the relation(
dc
dσ
)
· I1 = I2
with I2 :=
{
c
− f ′(mσ)
∫
R e
−cx { f (φσ)− f (mσ)− (φσ−φσ(∓∞)) f ′(mσ)}dx if ± c > 0
1 if c = 0
When f ∈ C 1,1(θ,1) and f ′ < 0 on (θ,1), then we can justify the above computation at least at
every point of differentiability of c. This is the case for instance when we consider the 1-periodic
function f defined by f (v) = 12 −|v − 12 | for v ∈ [0,1]. Then θ = 12 and c+ = 2
√
f ′(0+). Moreover we
can then show that the integral I2 blows up much faster than I1 as σ→ σ+, which implies that
dc
dσ is not bounded as σ→ σ+ (and by symmetry also as σ→ σ−). For general nonlinearities f ,
782 Mohammad Al Haj and Régis Monneau
it would be interesting to refine the analysis on the behaviour of c at σ = σ± which is out of the
scope of this Note.
References
[1] M. Al Haj, N. Forcadel, R. Monneau, “Existence and uniqueness of traveling waves for fully overdamped Frenkel-
Kontorova models”, Arch. Ration. Mech. Anal. 210 (2013), no. 1, p. 45-99.
[2] M. Al Haj, R. Monneau, “The velocity diagram of traveling waves for discrete reaction-diffusion equations”, https:
//hal.science/hal-03777384, 2022.
[3] D. G. Aronson, H. F. Weinberger, “Multidimensional nonlinear diffusion arising in population genetics”, Adv. Math. 30
(1978), p. 33-76.
[4] H. Berestycki, B. Nicolaenko, B. Scheurer, “Traveling wave solutions to combustion models and their singular limits”,
SIAM J. Math. Anal. 16 (1985), no. 6, p. 1207-1242.
[5] H. Berestycki, L. Nirenberg, “On the method of moving planes and the sliding method”, Bol. Soc. Bras. Mat. 22 (1991),
no. 1, p. 1-37.
[6] P. C. Fife, Dynamics of internal layers and diffusive interfaces, CBMS-NSF Regional Conference Series in Applied
Mathematics, vol. 53, Society for Industrial and Applied Mathematics, 1988.
[7] R. A. Fisher, “The wave of advance of advantageous genes”, Ann. Eugenics, London 7 (1937), p. 335-369.
[8] N. Forcadel, C. Imbert, R. Monneau, “Homogenization of fully overdamped Frenkel-Kontorova models”, J. Differ.
Equations 246 (2009), no. 3, p. 1057-1097.
[9] A. N. Kolmogorov, I. G. Petrovsky, N. S. Piskunov, “Etude de l’équation de la diffusion avec croissance de la quantité
de matière et son application à un problème biologique”, Bull. Univ. État Moscou, Sér. Int., Sect. A: Math. et Mécan. 1
(1937), no. 6, p. 1-26.


The velocity diagram for traveling waves

Walker, Christoph

Comptes Rendus
Mathématique
Christoph Walker
A Remark on a Nonlocal-in-Time Heat Equation
Volume 361 (2023), p. 825-831
Published online: 11 May 2023
https://doi.org/10.5802/crmath.443
This article is licensed under the
Creative Commons Attribution 4.0 International License.
http://creativecommons.org/licenses/by/4.0/
Les Comptes Rendus. Mathématique sont membres du
Centre Mersenne pour l’édition scientifique ouverte
www.centre-mersenne.org
e-ISSN : 1778-3569
Comptes Rendus
Mathématique
2023, Vol. 361, p. 825-831
https://doi.org/10.5802/crmath.443
Partial differential equations / Équations aux dérivées partielles
A Remark on a Nonlocal-in-Time Heat
Equation
ChristophWalkera
a Leibniz Universität Hannover, Institut für Angewandte Mathematik, Welfengarten 1,
30167 Hannover, Germany
E-mail: walker@ifam.uni-hannover.de
Abstract. Schauder’s fixed point theorem is used to derive the existence of solutions to a semilinear heat
equation. The equation features a nonlinear term that depends on the time-integral of the unknown on the
whole, a priori given, interval of existence.
Keywords. Semilinear heat equation, nonlocal in time, existence of global solutions.
Mathematical subject classification (2010). 35K58.
Manuscript received 12 October 2022, revised 22 December 2022, accepted 23 December 2022.
1. Introduction
This note is dedicated to the nonlocal problem
∂t u −div
(
d(x)∇u)+ϕ(∫ ∞
0
a(s, x)u(s, x)ds
)
u = f (t , x) , (t , x) ∈ (0,∞)×Ω , (1a)
u(0, · ) = u0 , u|∂Ω = 0 (1b)
on a bounded smooth domain Ω ⊂ Rn with d ∈ C 1(Ω) such that d(x) ≥ d0 > 0 for x ∈ Ω. The
potential ϕ, the weight a, the initial datum u0, and the right-hand side f are suitable given
functions.
Equation (1) is used in the modeling of a biological nanosensor in the chaotic dynamics of a
polymer chain in an aqueous solution and has been introduced and considered in [5–8]. It can
also be seen as a toy model for equilibrium states in age-structured diffusive populations (with t
referring to the age of individuals), see [9, 10] for instance.
Note that the unknown weighted time-integral
u =
∫ ∞
0
a(s)u(s)ds
depends on the whole, a priori given interval of existence (0,∞) (the case of a bounded time
interval (0,T ) is included in (1), of course). Hence, only global solutions are of interest. Moreover,
(1) is no usual evolution problem satisfying a Volterra property since solutions at a time instant
depend also on future time instants. For the homogeneous version of (1) on a bounded interval
ISSN (electronic) : 1778-3569 https://comptes-rendus.academie-sciences.fr/mathematique/
826 Christoph Walker
time (0,T ) with vanishing right-hand side f and without weight a, existence of weak solutions
was derived in [5, 6] and strong solutions in [11]. We also refer to [4] for existence of weak
solutions when the Laplacian is replaced by an integrodifferential operator of Lévy type. The
non-homogeneous problem (1) on a bounded interval (0,T ) was investigated in [7] where, for
unbounded potentials ϕ, a truncation approach and weak compactness methods were used to
prove the existence of weak solutions under fairly general conditions.
The purpose of this work is to propose an alternative approach to (1) for deriving the existence
of mild and strong solutions under slightly different conditions. This approach has been used
in [11] and may also be a template for other nonlocal problems. More precisely, we shall use the
fact that solutions to (1) may be written as mild solutions in the form
u(t ) = e t A(u)u0 +
∫ t
0
e(t−s)A(u) f (s)ds , t ≥ 0, (2)
where (e t A(u))t≥0 is the contraction semigroup on Lp (Ω) generated by the operator
A(u) := [w 7→ div(d(x)∇w)−ϕ(u)w]
subject to Dirichlet boundary conditions (see below for details). Integrating the representation (2)
yields the equivalent fixed point equation
u =
∫ ∞
0
a(t )e t A(u)u0 dt +
∫ ∞
0
a(t )
∫ t
0
e(t−s)A(u) f (s)ds dt (3)
for u. We then shall focus on this fixed point equation and prove, in particular, that the right-
hand side of (3) enjoys suitable compactness properties with respect to u that allow us to apply
Schauder’s theorem leading to the following existence result:
Theorem 1. Let a ∈ L1(R+,L∞(Ω)), ϕ ∈ C (R,R+), and f ∈ L1(R+,L∞(Ω))∩L∞,loc (R+,Lp (Ω)) for
some p ∈ (n/2∨1,∞) and let u0 ∈ L∞(Ω). Then there is a mild solution u ∈C
(
R+,Lp (Ω)
)
to (1) such
that
∥u(t )∥∞ ≤ ∥u0∥∞+∥ f ∥L1((0,t ),L∞(Ω)) , t ∈R+ .
If f ∈Cθ(R+,Lq (Ω))+C (R+,W θq (Ω)) with θ > 0 and q ∈ (1,∞), then u is a strong solution with
u ∈C(R+,Lq (Ω))∩C 1(Ṙ+,Lq (Ω))∩C(Ṙ+,W 2q (Ω)) ,
with Ṙ+ := (0,∞).
Some of the assumptions may be weaken, e.g. for linearly bounded ϕ or smoother a.
In Section 2 we prove Theorem 1. The crucial compactness properties of the integral terms
appearing on the right-hand side of (3) are postponed to Section 3. The proofs there are inspired
by the works [2, 3] and may be extended to more general frameworks than the one considered
herein for (1), e.g. to other semilinear and possibly quasilinear equations (see Remark 5 in this
regard).
2. Proof of Theorem 1
Notation and Preliminaries
We use the notation
W αp,D (Ω) :=
{
{u ∈W αp (Ω) ; u = 0 on ∂Ω} if 1p <α≤ 2,
W αp (Ω) if 0 ≤α< 1p ,
and we write A ∈H (W 2p,D (Ω),Lp (Ω)) if A ∈L (W 2p,D (Ω),Lp (Ω)) generates an analytic semigroup
(e tA )t≥0 on Lp (Ω). Recall that[
w 7→ div(d(x)∇w)] ∈H (W 2p,D (Ω),Lp (Ω))
Christoph Walker 827
provided d ∈C 1(Ω)) with d(x) ≥ d0 > 0 for x ∈Ω.
Since ϕ is uniformly continuous and bounded on bounded sets, it follows that (considered as
Nemytskii operator)
ϕ ∈C(L∞(Ω),L∞(Ω)) is bounded on bounded sets . (4)
Given R0 > 0 denote by
X :=BL∞(Ω)(0,R0)
the closed ball in L∞(Ω) of radius R0 centered at the origin. Recall that p ∈ (n/2∨1,∞) and note
that, given any u ∈ X , the mapping ϕ(u) := [w 7→ϕ(u)w] ∈L (Lp (Ω)) satisfies
∥ϕ(u)∥L (Lp (Ω)) ≤ ∥ϕ(u)∥∞ ≤ max[−R0,R0]ϕ , u ∈ X .
Setting
A(u)w := div(d∇w)−ϕ(u)w , w ∈W 2p,D (Ω) ,
it then follows from standard perturbation results that
A(u) ∈H (W 2p,D (Ω),Lp (Ω)) , u ∈ X .
In fact, since ϕ is nonnegative, (e t A(u))t≥0 is a positive contraction semigroup on each Lq (Ω) for
q ∈ (1,∞] (which, however, is not strongly continuous for q =∞), hence
∥e t A(u)∥L (Lq (Ω)) ≤ 1, t ≥ 0, q ∈ (1,∞] . (5)
Moreover, we have s(A(u)) ≤ s0 < 0 for its spectral bound with s0 denoting the spectral bound of
the operator
[
w 7→ div(d∇w)]. It then follows from [1, II.Lemma 5.1.3] that there is ν > 0 and,
given 2θ ∈ [0,2] \ {1/p}, there is M(R0) ≥ 1 such that
∥e t A(u)∥L (Lp (Ω),W 2θp,D (Ω)) ≤ M(R0)e
−νt t−θ , t > 0, u ∈ X . (6)
In the following we fix 2θ ∈ (n/p,2) and note the compact embedding
W 2θp,D (Ω)
c
,→ L∞(Ω) ,→ Lp (Ω) . (7)
Let us also observe that, given t > 0 and u, v ∈ X , we have
e t A(u) −e t A(v) =−
∫ t
0
d
ds
e(t−s)A(u)e s A(v) ds =−
∫ t
0
e(t−s)A(u)
(
ϕ(u)−ϕ(v))e s A(v) ds . (8)
We then use (6) and (7) to get
∥e t A(u) −e t A(v)∥L (Lp (Ω),L∞(Ω)) ≤ c ∥e t A(u) −e t A(v)∥L (Lp (Ω),W 2θp,D (Ω))
≤
∫ t
0
∥e(t−s)A(u)∥L (Lp (Ω),W 2θp,D (Ω)) ∥ϕ(u)−ϕ(v)∥L (Lp (Ω)) ∥e
s A(v)∥L (Lp (Ω)) ds
≤ c(R0)e−νt t 1−θ ∥ϕ(u)−ϕ(v)∥∞ . (9)
We are now in a position to provide the proof of Theorem 1.
Proof of Theorem 1. We show that the mapping
Φ(u) :=
∫ ∞
0
a(t )e t A(u)u0 dt +
∫ ∞
0
a(t )
∫ t
0
e(t−s)A(u) f (s)ds dt , u ∈ X =BL∞(Ω)(0,R0) ,
with
R0 := ∥a∥L1(R+,L∞(Ω))
(∥u0∥∞+∥ f ∥L1(R+,L∞(Ω)))
828 Christoph Walker
has a fixed point. Note first from (5) that ∥Φ(u)∥∞ ≤ R0 for u ∈ X , hence Φ : X → X . Next, due
to (9) and (7) we obtain, for u, v ∈ X ,
∥Φ(u)−Φ(v)∥∞ ≤
∫ T
0
∥a(t )∥∞ ∥e t A(u) −e t A(v)∥L (Lp (Ω),L∞(Ω)) ∥u0∥p dt
+
∫ T
0
∥a(t )∥∞
∫ t
0
∥e(t−s)A(u) −e(t−s)A(v)∥L (Lp (Ω),L∞(Ω)) ∥ f (s)∥p ds dt
≤ c(R0)
∫ ∞
0
∥a(t )∥∞ e−νt t 1−θ dt ∥u0∥p ∥ϕ(u)−ϕ(v)∥∞
+ c(R0)
∫ ∞
0
∥a(t )∥∞
∫ t
0
e−ν(t−s) (t − s)1−θ ∥ f (s)∥p ds dt ∥ϕ(u)−ϕ(v)∥∞
and hence, using ν> 0,
∥Φ(u)−Φ(v)∥∞ ≤ c(R0)∥a∥L1(R+,L∞(Ω))
(∥u0∥p +∥ f ||L1(R+,Lp (Ω)))∥ϕ(u)−ϕ(v)∥∞ . (10)
Therefore, Φ ∈ C (X , X ) according to (4). From Proposition 3 and Proposition 4 we deduce that
Φ ∈C (X , X ) has precompact image so that Schauder’s fixed point theorem yields u ∈ X such that
u =Φ(u). We may then define u by (2) in order to obtain a mild solution to (1) which belongs to
C (R+,Lp (Ω)) due to [1, II.Theorem 5.3.1] since f ∈ L∞,loc (R+,Lp (Ω)).
Finally, if f ∈ Cθ(R+,Lq (Ω)) + C (R+,W θq (Ω)) for some θ > 0 and q ∈ (1,∞), then u is a
strong solution with the regularity properties stated in Theorem 1, see [1, II.Theorem 1.2.1,
II.Theorem 1.2.2]. This proves Theorem 1. □
Remark 2. If ϕ is locally Lipschitz continuous, then one may derive the existence and unique-
ness of a solution using Banach’s fixed point argument for Φ : X → X provided that
R0 = ∥a∥L1(R+,L∞(Ω))
(∥u0∥∞+∥ f ∥L1(R+,L∞(Ω)))
is small enough.
3. Compactness Properties
We provide the compactness results used in the proof of Theorem 1. This section relies on the
papers [2, 3] and adapts these ideas to our setting.
We first consider the non-homogeneous part.
Proposition 3. Let a ∈ L1(R+,L∞(Ω)) and f ∈ L1(R+,L∞(Ω)) ∩ L∞,loc (R+,Lp (Ω)) for some p ∈
(n/2∨1,∞). Define
F (u) :=
∫ ∞
0
a(t )
∫ t
0
e(t−s)A(u) f (s)ds dt , u ∈ X .
Then the set
{
F (u) ; u ∈ X } is precompact in L∞(Ω).
Proof. Given T > 0 introduce
XT :=C ([0,T ],L∞(Ω))
and
F (u)(t ) :=
∫ t
0
e(t−s)A(u) f (s)ds , t ∈ [0,T ] , u ∈ X =BL∞(Ω)(0,R0) .
It suffices to prove that F ∈ C (X ,XT ) is compact for every T > 0 since the assertion then follows
by a diagonal sequence argument and the assumption a ∈ L1(R+,L∞(Ω)).
(i). Since f ∈ L∞,loc (R+,Lp (Ω)), we infer from [1, II.Theorem 5.3.1] that F (u) ∈ C ([0,T ],W 2θp (Ω))
and hence F (u) ∈XT for u ∈ X by (7). Moreover, given u, v ∈ X and t ∈ [0,T ], we have, as in (10),
∥F (u)(t )−F (v)(t )∥∞ ≤ c(R0)∥ f ∥L1(R+,Lp (Ω)) ∥ϕ(u)−ϕ(v)∥∞ .
Therefore, F ∈C (X ,XT ) owing to (4).
Christoph Walker 829
(ii). In order to prove that F ∈C (X ,XT ) has precompact image we use an idea inspired by [2, 3]:
for fixed λ> 0 we first show that{
eλA(u)F (u) ; u ∈ X
}
is precompact in XT . (11)
To this end, we note from (6) and (5) that, for u ∈ X and t ∈ [0,T ],
∥eλA(u)F (u)(t )∥W 2θp,D (Ω) ≤ c(R0)λ
−θ ∥F (u)(t )∥p ≤ c(R0)λ−θ∥ f ∥L1(R+,Lp (Ω)) . (12)
That is, invoking (7), the set{
eλA(u)F (u)(t ) ; u ∈ X
}
is precompact in L∞(Ω) (13)
for every t ∈ [0,T ]. Before proceeding let us note that, given u ∈ X , δ> 0 and h ≥ 0,
e(δ+h)A(u) −eδA(u) =
∫ h
0
d
ds
e(δ+s)A(u) ds =
∫ h
0
e(
δ
2 +s)A(u) A(u)e
δ
2 A(u) ds
so that, using (6),
∥e(δ+h)A(u) −eδA(u)∥L (Lp (Ω),W 2θp,D (Ω))
≤
∫ h
0
∥e( δ2 +s)A(u)∥L (Lp (Ω),W 2θp,D (Ω)) ∥A(u)∥L (W 2p,D (Ω),Lp (Ω)) ∥e
δ
2 A(u)∥L (Lp (Ω),W 2p,D (Ω)) ds
≤ c(R0)e−νδ
(
δ
2
)−1 ∫ h
0
(
δ
2
+ s
)−θ
ds .
Consequently, invoking (7),
∥e(δ+h)A(u) −eδA(u)∥L (Lp (Ω),L∞(Ω)) ≤ c(R0)e−νδδ−1−θh , h ≥ 0, δ> 0, u ∈ X . (14)
Next, given u ∈ X and 0 ≤ t ≤ t +h ≤ T we have, using (7),
∥eλA(u)F (u)(t +h)−eλA(u)F (u)(t )∥∞
≤ c
∫ t+h
t
∥e(λ+t+h−s)A(u)∥L (Lp (Ω),L∞(Ω)) ∥ f (s)∥p ds
+ c
∫ t
0
∥e(λ+t+h−s)A(u) −e(λ+t−s)A(u)∥L (Lp (Ω),L∞(Ω)) ∥ f (s)∥p ds .
We use (6)-(7) once more along with (14) to get
∥eλA(u)F (u)(t +h)−eλA(u)F (u)(t )∥∞
≤ c(R0)
∫ t+h
t
(λ+ t +h − s)−θ ∥ f (s)∥p ds + c(R0)h
∫ t
0
(λ+ t − s)−1−θ ∥ f (s)∥p ds
≤ c(R0)λ−θ
∫ t+h
t
∥ f (s)∥p ds + c(R0)hλ−1−θ ∥ f ∥L1(R+,Lp (Ω)) .
Therefore, since f ∈ L1(R+,Lp (Ω)) we deduce that
lim
h→0
sup
u∈X
∥eλA(u)F (u)(·+h)−eλA(u)F (u)∥XT = 0, λ> 0. (15)
Gathering (15) and (13) we conclude that
{
eλA(u)F (u) ; u ∈ X } is indeed precompact in XT due to
the Arzelà-Ascoli Theorem.
(iii). Next, we claim that
lim
λ→0
sup
u∈X
∥eλA(u)F (u)−F (u)∥XT = 0. (16)
830 Christoph Walker
Let δ ∈ (0,T ). Using (5) we have, for 0 ≤ t ≤ δ,
∥eλA(u)F (u)(t )−F (u)(t )∥∞ ≤ 2∥F (u)(t )∥∞ ≤ 2
∫ δ
0
∥ f (s)∥∞ ds . (17)
On the other hand, for δ≤ t ≤ T , we use (5) to get
∥eλA(u)F (u)(t )−F (u)(t )∥∞ ≤ ∥eλA(u)F (u)(t )−e(λ+δ)A(u)F (u)(t −δ)∥∞
+∥e(λ+δ)A(u)F (u)(t −δ)−eδA(u)F (u)(t −δ)∥∞
+∥eδA(u)F (u)(t −δ)−F (u)(t )∥∞
≤ 2∥eδA(u)F (u)(t −δ)−F (u)(t )∥∞
+∥e(λ+δ)A(u)F (u)(t −δ)−eδA(u)F (u)(t −δ)∥∞ .
For the first term on the right-hand side we use (5) to estimate
∥eδA(u)F (u)(t −δ)−F (u)(t )∥∞ =
∥∥∥∥∫ t
t−δ
e(t−s)A(u) f (s)ds
∥∥∥∥∞ ≤
∫ t
t−δ
∥ f (s)∥∞ ds
while we use (14) and (12) for the second term to obtain
∥e(λ+δ)A(u)F (u)(t −δ)−eδA(u)F (u)(t −δ)∥∞ ≤ c(R0)δ−1−θλ∥F (u)(t −δ)∥p
≤ c(R0)δ−1−θλ∥ f ∥L1(R+,Lp (Ω)) .
Gathering these estimates we derive, for δ≤ t ≤ T ,
∥eλA(u)F (u)(t )−F (u)(t )∥∞ ≤ 2
∫ t
t−δ
∥ f (s)∥∞ ds + c(R0)δ−1−θλ∥ f ∥L1(R+,Lp (Ω)) . (18)
Since f ∈ L1(R+,L∞(Ω)) we may first choose δ > 0 small enough and then let λ tend to zero to
conclude from (17) and (18) that (16) indeed holds true.
(iv). Let ε> 0 be arbitrary. Then, due to (16), there is λ0 > 0 such that
∥eλ0 A(u)F (u)−F (u)∥XT ≤
ε
3
, u ∈ X , (19)
while (11) yields finitely many u1, . . . ,uN ∈ X such that for every u ∈ X there exists k ∈ 1, . . . , N
such that
∥eλ0 A(u)F (u)−eλ0 A(uk )F (uk )∥XT ≤
ε
3
. (20)
Hence ∥F (u) − F (uk )∥XT ≤ ε so that
{
F (u) ; u ∈ X } is totally bounded in XT . This proves the
assertion. □
We prove a compactness result for the part involving the initial condition:
Proposition 4. Let a ∈ L1(R+,L∞(Ω)). Given u0 ∈ L∞(Ω) define
G (u) :=
∫ ∞
0
a(t )e t A(u) u0 dt , u ∈ X =BL∞(Ω)(0,R0) .
Then the set
{
G (u) ; u ∈ X } is precompact in L∞(Ω).
Proof. Let λ> 0 and set
Gλ(u) :=
∫ ∞
0
a(t )e(λ+t )A(u) u0 dt , u ∈ X .
Similarly as in Proposition 3 we infer that{
Gλ(u) ; u ∈ X
}
is precompact in L∞(Ω) . (21)
Christoph Walker 831
Taking δ> 0 and using (5), (7), and (14) we then get
∥Gλ(u)−G (u)∥∞ ≤ 2
∫ δ
0
∥a(t )∥∞ dt ∥u0∥∞+
∫ ∞
δ
∥a(t )∥∞
∥∥e(λ+t )A(u)u0 −e t A(u)u0∥∞ dt
≤ 2
∫ δ
0
∥a(t )∥∞ dt ∥u0∥∞+ c(R0)δ−1−θ ∥a∥L1(R+,L∞(Ω)) ∥u0∥p λ .
Since a ∈ L1(R+,L∞(Ω)) we may choose δ> 0 small to make the first term small and then let λ go
to zero to conclude
lim
λ→0
sup
u∈X
∥Gλ(u)−G (u)∥∞ = 0. (22)
Combining (21) and (22) we deduce that
{
G (u) ; u ∈ X } is precompact in L∞(Ω). □
Remark 5. The compactness results of this section rely on properties (5), (6), and (7) and thus
may also be derived for truly quasilinear operators A(u) based on the stability estimates of [1,
II.Section 5]. One has, however, to replace the set X ⊂ L∞(Ω) by a subset of W 2θp,D (Ω) with
θ ∈ (0,2) \ {1/p} and to impose suitable assumptions on the data (e.g. the weight a has to be
sufficiently smooth in x).
References
[1] H. Amann, Linear and Quasilinear Parabolic Problems. Volume I: Abstract Linear Theory, Monographs in Mathemat-
ics, vol. 89, Birkhäuser, 1995.
[2] P. Baras, “Compacité de l’opérateur f 7→ u solution d’une équation non linéaire (du/d t )+ Au ∋ f ”, C. R. Acad. Sci.
Paris 286 (1978), no. 23, p. 1113-1116.
[3] P. Baras, J.-C. Hassan, L. Véron, “Compacité de l’opérateur définissant la solution d’une équation d’évolution non
homogène”, C. R. Math. Acad. Sci. Paris 284 (1977), no. 14, p. 799-802.
[4] J.-D. Djida, G. F. Foghem Gounoue, Y. K. Tchaptchié, “Nonlocal complement value problem for a global in time
parabolic equation”, J. Elliptic Parabol. Equ. 8 (2022), no. 2, p. 767-789.
[5] V. N. Starovoitov, “Initial boundary value problem for a nonlocal in time parabolic equation”, Sib. Èlektron. Mat. Izv.
15 (2018), p. 1311-1319.
[6] ——— , “Boundary value problem for a global in time parabolic equation”, Math. Methods Appl. Sci. 44 (2021), no. 1,
p. 1118-1126.
[7] ——— , “Weak solvability of a boundary value problem for a parabolic equation with a global-in-time term that
contains a weighted integral”, J. Elliptic Parabol. Equ. 7 (2021), no. 2, p. 623-634.
[8] V. N. Starovoitov, B. Syarovoitova, “Modeling the dynamics of polymer chains in water solution. Application to sensor
design”, J. Phys., Conf. Ser. 894 (2017), no. 1, article no. 012088.
[9] C. Walker, “On positive solutions of some system of reaction-diffusion equations with nonlocal initial conditions”, J.
Reine Angew. Math. 660 (2011), p. 149-179.
[10] ——— , “Some results based on maximal regularity regarding population models with age and spatial structure”, J.
Elliptic Parabol. Equ. 4 (2018), no. 1, p. 69-105.
[11] ——— , “Strong solutions to a nonlocal-in-time semilinear heat equation”, Q. Appl. Math. 79 (2021), no. 2, p. 265-272.


A Remark on a Nonlocal-in-Time Heat Equation

Andrade, Jose A. A.

Rathie, Pushpa

Exact Posterior distribution of risk ratio in the Kumaraswamy–Binomial model

Acosta Babb, Ryan Luis

Comptes Rendus
Mathématique
Ryan Luis Acosta Babb
Remarks on the Lp convergence of Bessel–Fourier series on the disc
Volume 361 (2023), p. 1075-1080
Published online: 24 October 2023
https://doi.org/10.5802/crmath.464
This article is licensed under the
Creative Commons Attribution 4.0 International License.
http://creativecommons.org/licenses/by/4.0/
Les Comptes Rendus. Mathématique sont membres du
Centre Mersenne pour l’édition scientifique ouverte
www.centre-mersenne.org
e-ISSN : 1778-3569
Comptes Rendus
Mathématique
2023, Vol. 361, p. 1075-1080
https://doi.org/10.5802/crmath.464
Harmonic analysis, Partial differential equations / Analyse harmonique, Équations aux
dérivées partielles
Remarks on the Lp convergence of
Bessel–Fourier series on the disc
Remarques sur la convergence Lp des séries de
Bessel–Fourier sur le disque
Ryan Luis Acosta Babba
a University of Warwick, United Kingdom
E-mail: r.acosta-babb@warwick.ac.uk
Abstract. The Lp convergence of eigenfunction expansions for the Laplacian on planar domains is largely
unknown for p ̸= 2. After discussing the classical Fourier series on the 2-torus, we move onto the disc, whose
eigenfunctions are explicitly computable as products of trigonometric and Bessel functions. We summarise
a result of Balodis and Córdoba regarding the Lp convergence of the Bessel–Fourier series in the mixed norm
space L
p
rad(L
2
ang) on the disk for the range
4
3 < p < 4. We then describe how to modify their result to obtain
Lp (D,r dr dt ) norm convergence in the subspace L
p
rad(L
q
ang) (
1
p + 1q = 1) for the restricted range 2 ≤ p < 4.
Résumé. La convergence Lp des développements en fonctions propres du Laplacien dans des domaines du
plan est largement inconnue lorsque p ̸= 2. Après avoir discuté des séries de Fourier classiques sur le tore,
nous passons au disque, dont les fonctions propres sont explicitement calculables comme étant le produit
des fonctions trigonométriques et de Bessel. Nous résumons un résultat de Balodis et Córdoba concernant
la convergence Lp de la série de Bessel–Fourier dans l’espace de norme mixte L
p
rad(L
2
ang) dans le disque
pour l’intervalle 43 < p < 4. Nous décrivons ensuite comment on peut modifier leur résultat pour obtenir
la convergence dans la norme Lp (D,r dr dt ) dans le sous-espace L
p
rad(L
q
ang) (
1
p + 1q = 1) pour l’intervalle
2 ≤ p < 4.
2020 Mathematics Subject Classification. 42-02, 42C05, 33C10, 34L10.
Funding. This work was conducted thanks to the EPSRC studentship 2443915 under the project
EP/V520226/1.
Manuscript received 25 February 2022, revised 10 May 2022, accepted 10 January 2023.
1. Introduction
For a function f ∈ L2(Tn), we can truncate its Fourier series by “spherical modes”
SN f :=
∑
|k|≤N
f̂ (k)e2πik·x , (1)
ISSN (electronic) : 1778-3569 https://comptes-rendus.academie-sciences.fr/mathematique/
1076 Ryan Luis Acosta Babb
or by “cubic modes”
S[N ] f :=
∑
|k j |≤N
f̂ (k)e2πik·x , (2)
where
k = (k1, . . . ,kn) ∈Zn and |k|2 =
n∑
j=1
|k j |2.
It is well known that SN f from (1) fails in general to converge to f in Lp when p ̸= 2. This follows,
by standard transference arguments (see [7]), from Fefferman’s result [6] that the indicator
function of the ball is not an Lp -bounded Fourier multiplier for any p ̸= 2. (See [8] for a detailed
discussion and related results.) On the other hand, the square truncations from (2) are perfectly
well-behaved for all 1 < p <∞ (see again [7]).
This behaviour is not restricted to the torus and related domains. For consider the diskD⊂R2.
The eigenfunctions for the Laplacian on D are of the form
e2πiθm Jm( j
n
mr ) for (r,θ) ∈ [0,1]2, (m,n) ∈Z×N,
corresponding to the respective eigenvalues 4π2m2 + ( j mn )2. Here Jm := J|m| denotes a Bessel
function of the first kind and j mn := j |m|n its non-negative zeros (see [10]).
Consider now the function f (r ) = r−3/2, which lies in the space Lp ([0,1],r dr ) for 1 ≤ p < 4/3.
Wing [11] proved that, for any choice of Jm , the 1-dimensional Bessel series of f fails to converge
in Lp ([0,1],r dr ). By letting g (r, t ) := f (r ) in Lp (D), it follows that the 2-dimensional Bessel–
Fourier series of g is ∑
m∈Z
∑
n∈N
am,n Jm( j
m
n r )e
2πimt = ∑
n∈N
an Jm( j
m
n r )
and so does not converge to g for any m ≥ 0, whether we truncate the 2-dimensional series by
cubic or spherical modes.
Recently, Fefferman et al. [5] have asked whether, given a differential operator with an or-
thonormal family wk of eigenfunctions, there is a choice {w ∈ EN : N ∈N} of eigenfunctions such
that the “truncations”
SEN f =
∑
w∈EN
〈 f , w〉w
are “well-behaved” in Lp for all 1 < p < ∞. To our knowledge, this conjecture is still open. For
instance, Wing’s counterexample could be summable if we were to find a more cunning grouping
of terms to exploit further cancellations to which the coarser truncations are not sensitive. We
have, however, obtained some new results for a certain class of triangular domains in [1].
Returning to the disk, Wing’s result tells us that we should expect restrictions on the range of
p. A natural range is 4/3 < p < 4, since this is precisely the range that works for the 1-dimensional
Bessel series [11]. (It is instructive to compare this to the ranges of Lp convergence for the
Bochner–Riesz means on R2; see [7] for details. We will return to this question later.)
The best result known at this time is due to Balodis and Córdoba [2], who reduced the
problem of convergence on the disc to extant results on the convergence of Fourier and 1-
dimensional Bessel series, albeit with a modified norm. We will exploit their argument to obtain
Lp convergence in a certain subspace of Lp .
2. Mixed-norm convergence
Define the space Lprad(L
2
ang) by the inequality
∥∥ f ∥∥p,2 :=
[∫ 1
0
(∑
m
| fm(r )|2
)p/2
r dr
]1/p
≡ ∥∥∥∥( fm(r ))∥∥ℓ2∥∥Lp (r dr ) <∞, (3)
Ryan Luis Acosta Babb 1077
where fm(r ) are the Fourier coefficients of the angular function t 7→ f (r, t ):
fm(r ) :=
∫ 1
0
f (r,θ)e−2πimθ dθ.
Denote by SN ,M f the partial sums of the Bessel–Fourier series of f :D→C:
SN ,M f (r, t ) :=
M∑
m=−M
N∑
n=1
am,n Jm( j
m
n r )e
2πimt . (4)
We drop the superscript “(d)” present in [2], since d = 2 will remain fixed in our discussion.
This simplification is imposed by our use of the Hausdorff–Young inequality below. In higher
dimensions, the expansion of the spherical part involves spherical harmonics, so the device of
the Hausdorff–Young inequality is no longer available to us.
Theorem 1 ([2]). The operators SN ,M are uniformly bounded on L
p
rad(L
2
ang) if, and only if,
4
3 < p < 4
when N ≥ AM +1 for an absolute constant A > 0.
The norm convergence of the series to f follows by the usual uniform boundedness argument.
(See [2]; cf. the analogous Fourier series argument in [7].)
To attack the proof of Theorem 1, they exploited the presence of the Fourier coefficients in the
norm (3). Indeed, SN ,M f is a trigonometric polynomial whose mthFourier mode (|m| ≤ M) is
SN ,m fm(r ) ≡
N∑
n=1
am,n Jm( j
m
n r ), (5)
which is precisely the 1-dimensional Bessel series summation operator for the radial function
r 7→ fm(r ) in terms of the mthorder Bessel function Jm . Thus,
∥∥SN ,M f ∥∥p,2 =
[∫ 1
0
(
M∑
m=−M
|SN ,m fm(r )|2
)p/2
r dr
]1/p
= ∥∥(SN ,m fm)m∥∥Lp (r dr ;ℓ2) ,
so the boundedness of SN ,M in L
p
rad(L
2
ang) is reduced to a uniform bound for vector-valued
inequalities. Such bounds must be independent of the length, 2M +1, of the vector(
SN ,m f−m , . . . ,SN ,m fm
)
.
Note that SN ,−m = SN ,m by our convention that Jm = J|m| for m ∈ Z. The functions fm and f−m ,
however, are distinct in general, as they correspond to distinct Fourier coefficients.
Let us now turn to Lp convergence on the disc, where the relevant norm is∥∥ f ∥∥Lp (D) = ∥∥∥∥∥ f (r, t )∥∥Lp (dt )∥∥∥Lp (r dr ) .
For p ̸= 2 we cannot replace the inner “angular” Lp norm by a sum of Fourier coefficients (see [9,
Chapter IV]). However, for p ≥ 2 we may use the following “Reverse” Hausdorff–Young Inequality:
if p ≥ 2 and 1p + 1q = 1, then ∥∥g∥∥Lp (T) ≤ ∥∥(ĝ (k))k∥∥ℓq (Z) .
for all g ∈ Lp (T). To see this, simply interpolate between Plancherel’s identity and the trivial
bound ∥∥g∥∥L∞(T) ≤ ∥∥(ĝ (k))k∥∥ℓ1(Z) .
1078 Ryan Luis Acosta Babb
We therefore have ∥∥ f ∥∥Lp (D) = [∫ 1
0
∥∥ f (r, t )∥∥pLp (T,dt ) r dr ]1/p
≤
[∫ 1
0
(∑
k
| fk (r )|q
)p/q
r dr
]1/p
=: ∥∥ f ∥∥p,q .
Using this norm, we define the space
Lprad(ℓ
q
ang) :=
{
f ∈ Lp (D) : ∥∥ f ∥∥p,q <∞} .
Careful inspection of the proofs in [2] shows that the space ℓ2 can be replaced by ℓq through-
out when
1
p
+ 1
q
= 1 and 2 ≤ p < 4.
The kernels of the 1-dimensional summation operators SN ,m in (5) are controlled by weighted,
vector-valued norms on the operators∫ 1
0
f (t )
2−x − t dt and
∫ 1
0
f (t )
x + t dt , (6)
the Hilbert Transform and the Hardy–Littlewood Maximal Functional. The weight r 1−p/2 satisfies
the 1-dimensional Muckenhoupt Ap condition if, and only if,
−1 < 1− p
2
< p −1 that is 4
3
< p < 4
(see [7, Example 7.1.7]) and, when this is the case, we have the inequalities∥∥(M fk )k∥∥Lp (r 1−p/2;ℓq ) +∥∥(H fk )k∥∥Lp (r 1−p/2;ℓq ) ≲p,q ∥∥( fk )k∥∥Lp (r 1−p/2;ℓq ) .
(See [4, (B), p. 25].) Furthermore, the kernels of the operators in (6) are nice enough that both are
bounded on the space Lp ([0,1],r 1−p/2 dr ) and, since they are positive, they admit vector-valued
extensions to ℓq too [7, Theorem 5.5.10].
There is one more operator to consider in [2, p. 280]:
TN ,m f (x) :=
p
x fν(A
ν
N x)
∫ 1
0
p
t fν(A
ν
N t ) f (t )dt
with fν as in [2, Lemma 1]. For the stated range of p, p/q ≥ 1, so we can apply Jensen’s Inequality
as in [2, p. 280], and we obtain the Lprad(ℓ
q
ang)-boundeness of TN ,ν too.
Theorem 2. The operators SN ,M are uniformly bounded on L
p
rad(ℓ
q
ang) if 2 ≤ p < 4 and 1p + 1q = 1,
when N ≥ AM +1 for an absolute constant A > 0.
Corollary 3. Let 2 ≤ p < 4 and 1p + 1q = 1. Then, if Nk , Mk are sequences of natural numbers such
that Nk ≥ AMk +1 and Mk →∞, we have
lim
k→∞
∥∥SNk ,Mk f − f ∥∥p,q = 0
for all f ∈ Lprad(ℓ
q
ang). In particular,
lim
k→∞
∥∥SNk ,Mk f − f ∥∥Lp (D) = 0
for all f ∈ Lprad(ℓ
q
ang).
Ryan Luis Acosta Babb 1079
Note, however, that this method does not allow us to conclude Lp norm convergence for all
f ∈ Lp , but only for the smaller space Lprad(ℓ
q
ang) ⊊ Lp (D). The following function lies in Lp (since
it is continuous [12]), but its Fourier coefficients are not ℓq summable for any q > 2:
g (t ) :=
∞∑
k=2
ei k logkp
k(logk)2
e2πit .
In other words, g ∈ Lp (D) \ Lprad(ℓ
q
ang).
The above example, which is a counterexample to Plancherel’s theorem in Lp , hints at the
underlying issue: we did not obtain Lp bounds for the partial sum operators, so we cannot apply
the proof of Corollary 3 to Lp (D) directly.
3. Concluding remarks
To summarise, we have the following pieces of the convergence puzzle:
Table 1. Lp convergence of Bessel–Fourier series for various ranges of p.
p range [1,4/3) [4/3,2) 2 (2,4) 4 (4,∞)
∥·∥Lp (D)-convergence No ? Yes f ∈ Lprad(ℓ
q
ang) ? No
In light of the existing results [2, 3, 11], we might offer the following conjecture.
Conjecture 4. For all 4/3 < p < 4 and f ∈ Lp (D),
lim
k→∞
∥∥SNk ,Mk f − f ∥∥Lp (D) = 0,
for some appropriate choice of Nk , Mk ∈N, k ≥ 1. For p outside of this range, convergence fails in
general.
However, it is not completely clear that we can expect this. Córdoba [4] proved that the ball
multiplier is bounded on the mixed norm space Lprad(L
2
ang) in the range 2n/(n+1) < p < 2n/(n−1)
that was originally conjectured for Lp (Rn), but disproved by Fefferman [6]. As we saw above, the
Lprad(L
2
ang) argument works by essentially “eliminating” the angular eigenfunctions and reducing
the problem to bounds on the one-dimensional Bessel–Fourier series. This “trick” is not available
in Lp (D), owing to the failure of Plancherel’s Theorem for p ̸= 2, so obtaining uniform bounds on
the operators SN ,M is considerably more difficult.
Acknowledgments
I owe a special thanks to my doctoral advisor, Prof James C. Robinson, for many insightful
discussions and his helpful criticism of early drafts. I also wish to thank an anonymous reviewer
for their suggestions.
References
[1] R. L. Acosta Babb, “The Lp convergence of Fourier series on triangular domains”, Proc. Edinb. Math. Soc. 66 (2023),
no. 2, p. 453-474.
[2] P. Balodis, A. Córdoba, “The convergence of multidimensional Fourier–Bessel series”, J. Anal. Math. 77 (1999), no. 1,
p. 269-286.
[3] A. Benedek, R. Panzone, “Mean convergence of series of Bessel functions”, Rev. Unión Mat. Argent. 26 (1972/73),
p. 42-61.
[4] A. Córdoba, “The disc multiplier”, Duke Math. J. 58 (1989), no. 1, p. 21-29.
1080 Ryan Luis Acosta Babb
[5] C. L. Fefferman, K. W. Hajduk, J. C. Robinson, “Simultaneous approximation in Lebesgue and Sobolev norms via
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