Comptes Rendus Mathématique

Thierry Meyrath

Comptes Rendus
Mathématique
Thierry Meyrath
Universal radial limits of meromorphic functions in the unit disk
Volume 360 (2022), p. 893-898
https://doi.org/10.5802/crmath.352
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
2022, Vol. 360, p. 893-898
https://doi.org/10.5802/crmath.352
Complex analysis and geometry / Analyse et géométrie complexes
Universal radial limits of meromorphic
functions in the unit disk
Thierry Meyratha
a University of Luxembourg, Department of Mathematics, 6, avenue de la Fonte, 4364
Esch-sur-Alzette, Luxembourg
E-mail: thierry.meyrath@uni.lu
Abstract. We consider the space of meromorphic functions in the unit disk D and show that there exists a
dense Gδ-subset of functions having universal radial limits. Our results complement known statements about
holomorphic functions and further imply the existence of meromorphic functions having maximal cluster
sets along certain subsets of D.
2020 Mathematics Subject Classification. 30K15, 30D40, 30D35, 30D30.
Manuscript received 18 January 2022, accepted 6 March 2022.
1. Introduction
Let D denote the unit disk {z : |z| < 1} and T the unit circle {z : |z| = 1}. We further denote by C (T)
the set of continuous functions from T to C∞, where C∞ :=C∪ {∞} is equipped with the chordal
metric χ. On C (T) we consider the metric dT defined by dT( f , g ) := maxTχ( f (z), g (z)) for f , g ∈
C (T). Finally, let M(D) be the set of meromorphic functions in D, including the function f∞|D,
where f∞ ≡ ∞. Endowed with the topology of spherically uniform convergence (i.e. uniform
convergence with respect to the metric χ) on compact subsets of D, the space M(D) becomes
a completely metrizable space. By d we denote a metric on M(D) that induces its topology, such
a metric can be defined via a compact exhaustion of D (see [7, Chap. VII]).
A result of Bayart [1] from 2005 implies the existence of functions f holomorphic in D having
universal radial limits in the following sense: For every measurable function ϕ on T, there exists
an increasing sequence (ρl ) in [0,1) with ρl → 1 for l →∞, such that for every z ∈ D and almost
every ζ ∈Twe have
lim
l→∞
∣∣ f (ρl (ζ− z)+ z)−ϕ(ζ)∣∣= 0.
In a similar vein, the following result was recently proved by Charpentier [4], here, f (k) denotes
the kth antiderivative of f for k ∈Zwith k < 0, and f (0) = f (see [5, 8] for related results).
ISSN (electronic) : 1778-3569 https://comptes-rendus.academie-sciences.fr/mathematique/
894 Thierry Meyrath
Theorem ([4]). Let U denote the set of functions f holomorphic in D having the following
property: For every compact set K ⊂ T with K 6= T, every continuous function ϕ : K → C, every
compact set L ⊂D and every k ∈Z, there exists an increasing sequence (ρl ) in [0,1) with ρl → 1 for
l →∞, such that
max
ζ∈K
max
z∈L
∣∣∣ f (k)(ρl (ζ− z)+ z)−ϕ(ζ)∣∣∣→ 0 for l →∞.
Then U is a dense Gδ-subset of the space of holomorphic functions inD, endowed with the topology
of uniform convergence on compact subsets of D.
The aim of this note is to show that similar universal radial limits are possible for functions in
M(D), where, in this case, the approximation property holds on the entire boundary of D, i.e. for
K =T.
2. Main result
Using similar ideas as in [1, 4], we prove the following result:
Theorem 1. Let U denote the set of functions f ∈ M(D) having the following property: For every
function ϕ ∈ C (T) and every compact set L ⊂ D, there exists an increasing sequence (ρl ) in [0,1)
with ρl → 1 for l →∞, such that
max
ζ∈T
max
z∈L
χ( f (ρl (ζ− z)+ z),ϕ(ζ)) → 0 for l →∞.
Then U is a dense Gδ-subset of M(D).
Proof. We first note that the space C (T) is separable (this follows e.g. from [15, Thm. 13.3]
and [9, Thm. 3]) and denote by (ϕ j ) a sequence that is dense in C (T). Furthermore, we set
Ls := {z : |z| ≤ ss+1 } for s ∈ N, and consider an increasing sequence (ρl ) in [0,1) with ρl → 1 for
l →∞. For j , s, l ∈N, we define the set
U ( j , s, l ) :=
{
f ∈ M(D) : max
ζ∈T
max
z∈Ls
χ( f (ρl (ζ− z)+ z),ϕ j (ζ)) <
1
s
}
.
We then have that
U =⋂
j ,s
⋃
l
U ( j , s, l ).
Indeed, suppose that f belongs to the set on the right hand side and let be given a function
ϕ ∈ C (T), a compact set L ⊂ D and some ε > 0. There exists s0 ∈ N such that L ⊂ Ls0 and 1s0 <
ε
2 .
Furthermore, since (ϕ j ) is dense in C (T), there exists j0 ∈N such that maxTχ(ϕ j0 (ζ),ϕ(ζ)) < ε2 . By
assumption, there exists l0 ∈N such that f ∈U ( j0, s0, l0), and hence
max
ζ∈T
max
z∈Ls0
χ( f (ρl0 (ζ− z)+ z),ϕ j0 (ζ)) <
1
s0
.
It follows
max
ζ∈T
max
z∈L
χ( f (ρl0 (ζ− z)+ z),ϕ(ζ))
≤ max
ζ∈T
max
z∈Ls0
χ( f (ρl0 (ζ− z)+ z),ϕ j0 (ζ))+max
ζ∈T
χ(ϕ j0 (ζ),ϕ(ζ)) < ε,
so that f ∈U , and hence U ⊃⋂ j ,s ⋃l U ( j , s, l ). The other inclusion is obvious.
Let now j1 ∈ N and s1 ∈ N be given. We show that the set ⋃l U ( j1, s1, l ) is open and dense in
M(D). Consider therefore l ∈N and suppose that f ∈U ( j1, s1, l ). Since the set
Kl ,s1 := {ρl (ζ− z)+ z : ζ ∈T, z ∈ Ls1 }
Thierry Meyrath 895
is a compact subset of D, there then exists ε> 0 such that
max
ζ∈T
max
z∈Ls1
χ( f (ρl (ζ− z)+ z),ϕ j1 (ζ)) <
1
s1
−ε.
Furthermore, according to Lemma 1.7 in [7, Chap. VII], there exists δ > 0, such that for every
g ∈ M(D) with d(g , f ) < δ we have that
max
z∈Kl ,s1
χ(g (z), f (z)) < ε.
Hence, for g ∈ M(D) with d(g , f ) < δ, we obtain
max
ζ∈T
max
z∈Ls1
χ(g (ρl (ζ− z)+ z),ϕ j1 (ζ))
≤ max
ζ∈T
max
z∈Ls1
χ(g (ρl (ζ− z)+ z), f (ρl (ζ− z)+ z))+max
ζ∈T
max
z∈Ls1
χ( f (ρl (ζ− z)+ z),ϕ j1 (ζ))
= max
z∈Kl ,s1
χ(g (z), f (z))+max
ζ∈T
max
z∈Ls1
χ( f (ρl (ζ− z)+ z),ϕ j1 (ζ)) <
1
s1
,
so that g ∈U ( j1, s1, l ). This implies that for l ∈N, the set U ( j1, s1, l ) is open in M(D), and the same
then obviously holds for the set
⋃
l U ( j1, s1, l ).
In order to show that the set
⋃
l U ( j1, s1, l ) is dense in M(D), we consider f ∈ M(D) and ε> 0. By
Lemma 1.7 in [7, Chap. VII], there exists s2 ∈Nwith s2 ≥ s1 and δ> 0, such that for every g ∈ M(D)
with maxLs2 χ(g (z), f (z)) < δ, we have that d(g , f ) < ε. Consider now the set K := Ls2 ∪T and the
function
h(z) :=
{
f (z), for z ∈ Ls2 ,
ϕ j1 (z), for z ∈T.
Then K is a compact subset of C and C \ K has two components, so that a combination of
Mergelyan’s Theorem on rational approximation (e.g. [15, Thm. 13.3]) and [9, Thm. 3] yields the
existence of a rational function r (z), such that
max
z∈K
χ(r (z),h(z)) < min
{
δ,
1
2s1
}
,
so that we obtain
max
z∈Ls2
χ(r (z), f (z)) < δ and max
ζ∈T
χ(r (ζ),ϕ j1 (ζ)) <
1
2s1
.
Note that by the first inequality, we have that d(r, f ) < ε. It remains to show that r ∈⋃l U ( j1, s1, l ).
Clearly, r is uniformly continuous on compact subsets of C, hence there exists δs1 > 0 such that
for z1, z2 ∈D, we have that
|z1 − z2| < δs1 implies χ(r (z1),r (z2)) <
1
2s1
.
Since ρl → 1 for l →∞, there further exists l1 ∈N such that
max
ζ∈T
max
z∈Ls1
∣∣ρl1 (ζ− z)+ z −ζ∣∣= max
ζ∈T
max
z∈Ls1
∣∣ζ(ρl1 −1)− z(ρl1 −1)∣∣< δs1 .
Finally, we obtain
max
ζ∈T
max
z∈Ls1
χ(r (ρl1 (ζ− z)+ z),ϕ j1 (ζ))
≤ max
ζ∈T
max
z∈Ls1
χ(r (ρl1 (ζ− z)+ z),r (ζ))+max
ζ∈T
χ(r (ζ),ϕ j1 (ζ)) <
1
s1
,
so that r ∈U ( j1, s1, l1) and the set ⋃l U ( j1, s1, l ) is dense in M(D).
Hence, for any j , s ∈N, the set ⋃l U ( j , s, l ) is open and dense in M(D). By Baire’s Theorem, the
set
⋂
j ,s
⋃
l U ( j , s, l ) =U is thus a dense Gδ-subset of M(D), which proves the theorem. 
We explicitly state the case L = {0} in Theorem 1:
896 Thierry Meyrath
Corollary 2. There exists a dense Gδ-subset U of M(D), such that every f ∈ U has the following
property: For every function ϕ ∈C (T), there exists an increasing sequence (ρl ) in [0,1) with ρl → 1
for l →∞, such that
max
ζ∈T
χ( f (ρlζ),ϕ(ζ)) → 0 for l →∞.
3. Properties of functions f ∈U
In the following, we will study some properties of functions f ∈ U , where U is the set defined
in Theorem 1. We start by showing that in contrast to the holomorphic case, the universal
approximation property in Theorem 1 does not hold for the derivatives of f .
Proposition 3. Consider a function f ∈ M(D). Then we have f ′ ∉U .
Proof. Suppose that there exists f ∈ M(D) such that f ′ ∈ U . Then, there exists an increasing
sequence (ρl ) in [0,1) with ρl → 1 for l →∞, such that
max
ζ∈T
χ
(
f ′(ρl ζ),
1
ζ
)
→ 0 for l →∞.
Note that this implies that for l sufficiently large, the function f ′(ρl ζ) has no poles on T. Now,
since 1ζ is bounded onT, we obtain that the sequence ( f
′(ρl ζ)) converges uniformly to 1ζ onT for
l →∞, which in turn implies
1
2πi
∫
T
f ′(ρl ζ)dζ→
1
2πi
∫
T
1
ζ
dζ= 1 for l →∞.
According to the Residue Theorem, the integral on the left hand side equals 0 for every l ∈ N,
hence we get a contradiction. 
We further mention that Theorem 1 implies the existence of functions in M(D) having maximal
cluster sets along certain subsets ofD, as well as functions having no asymptotic values. We recall
the corresponding definitions. A value a ∈ C∞ is said to be an asymptotic value of f ∈ M(D), if
there exists a continuous curve γ : [0,1) →Dwith limr→1−
∣∣γ(r )∣∣= 1, such that limr→1− f (γ(r )) = a.
The curve γ is then called an asymptotic curve for f . Furthermore, for a function f ∈ M(D) and a
set A ⊂Dwith A∩T 6= ;, the cluster set C ( f , A) of f along A is defined by
C ( f , A) := {a ∈C∞ :∃ (zn) in A and ζ ∈T : zn → ζ and χ( f (zn), a) → 0 for n →∞},
and the cluster set is said to be maximal, if C ( f , A) = C∞. Given f ∈ U and setting Az,ζ :=
{ρ(ζ− z)+ z : ρ ∈ [0,1)} for z ∈D and ζ ∈T, it is easily seen that for every z ∈D and every ζ ∈T, the
radial cluster set C ( f , Az,ζ) is maximal and the curve γz,ζ : [0,1) → D with γz,ζ(r ) := r (ζ− z)+ z is
not an asymptotic curve for f . Hence, as a consequence of Theorem 1, we obtain that the set of
functions in M(D) having this property contains a dense Gδ-subset of M(D). In fact, we have the
following stronger statement (see [2, 3, 11–13] for related results):
Corollary 4. Let be given a function f ∈U . Then the following hold:
(i) the cluster set C ( f ,γ([0,1))) is maximal for every continuous curve γ : [0,1) → D with
limsupr→1−
∣∣γ(r )∣∣= 1.
(ii) the function f does not have an asymptotic value. Furthermore, f takes every value a ∈C∞
infinitely many times in D and for every ζ ∈ T and every ε > 0, we have that the set
C∞ \ f (Uε(ζ)∩D) contains at most two points, where Uε(ζ) := {z : |z −ζ| < ε}.
In particular, the set of functions in M(D) having the above properties contains a dense Gδ-subset
of M(D).
Thierry Meyrath 897
Proof. (i). Let f ∈ U and a ∈ C∞ be given and consider a continuous curve γ : [0,1) → D with
limsupr→1−
∣∣γ(r )∣∣= 1. By assumption, there exists an increasing sequence (ρl ) in [0,1) withρl → 1
for l →∞, such that
max
ζ∈T
χ
(
f (ρl ζ), a
)→ 0 for l →∞.
Since limsupr→1−
∣∣γ(r )∣∣= 1, there exists l0 ∈N such that for every l > l0, we have that γ([0,1))∩{z :
|z| = ρl } 6= ;. Thus, there exists a sequence (zs ) on γ([0,1)) with |zs | = ρl0+s for s ∈ N, and hence
|zs |→ 1 for s →∞. We finally obtain
χ( f (zs ), a) ≤ max
{z:|z|=ρl0+s }
χ( f (z), a) = max
ζ∈T
χ( f (ρl0+sζ), a) → 0 for s →∞,
which implies a ∈C ( f ,γ([0,1))), hence C ( f ,γ([0,1))) is maximal.
(ii). Since by the first statement, the cluster set C ( f ,γ([0,1))) is maximal for every continuous
curve γ : [0,1) →Dwith limr→1−
∣∣γ(r )∣∣= 1, it is clear that f can not have an asymptotic value. The
other statements then immediately follow from classical results in [6, 14]. 
The above corollary shows that f ∈ U does not have a Picard value. We finish with a slightly
stronger statement, namely that f ∈ U can not have a deficient value in the sense of Nevan-
linna. We therefore introduce some standard terminology from Nevanlinna theory, see [10] for
the corresponding definitions. For a function f ∈ M(D), a value a ∈ C∞ and r ∈ [0,1), we de-
note by m(r, a, f ) and N (r, a, f ) the proximity function and the (integrated) counting function,
respectively. We write T (r, f ) for the characteristic function of f and we recall that in case that
T (r, f ) →∞ for r → 1−, the deficiency δ(a, f ) of a is defined by
δ(a, f ) = 1− limsup
r→1−
N (r, a, f )
T (r, f )
= liminf
r→1−
m(r, a, f )
T (r, f )
,
and a is called a deficient value of f , if δ(a, f ) > 0.
Proposition 5. Consider a function f ∈U . Then f has no deficient value.
Proof. Let be given a function f ∈ U . Then, the radial limit limρ→1− f (ρζ) does not exist for any
ζ ∈T, and it is an immediate consequence of Nevanlinna’s extension of Fatou’s Theorem (e.g. [10,
Thm. 6.12]) that T (r, f ) →∞ for r → 1−. By assumption, there exists an increasing sequence (ρl )
in [0,1) with ρl → 1 for l →∞, such that
max
ζ∈T
χ
(
f (ρl ζ),0
)→ 0 for l →∞.
This implies the uniform convergence of ( f (ρl ζ)) to 0 on T for l → ∞, from which we obtain
m(ρl , a, f ) =O (1) for l →∞ and every a ∈C∞ \ {0}, and hence
δ(a, f ) = liminf
r→1−
m(r, a, f )
T (r, f )
≤ lim
l→∞
m(ρl , a, f )
T (ρl , f )
= 0.
By a very similar argumentation, we obtain an increasing sequence (ρl ) in [0,1) with ρl → 1 for
l → ∞, and m(ρl ,0, f ) = O (1) for l → ∞, which implies δ(0, f ) = 0. Hence, no value a ∈ C∞ is
deficient for f . 
References
[1] F. Bayart, “Universal radial limits of holomorphic functions”, Glasg. Math. J. 47 (2005), no. 2, p. 261-267.
[2] L. Bernal-González, M. C. Calderón-Moreno, J. A. Prado-Bassas, “Maximal cluster sets along arbitrary curves”, J.
Approx. Theory 129 (2004), no. 2, p. 207-216.
[3] ——— , “Simultaneously maximal radial cluster sets”, J. Approx. Theory 135 (2005), no. 1, p. 114-124.
[4] S. Charpentier, “Holomorphic functions with universal boundary behaviour”, J. Approx. Theory 254 (2020), article
no. 105391 (19 pages).
898 Thierry Meyrath
[5] S. Charpentier, V. Nestoridis, “On the boundary behaviour of derivatives of functions in the disc algebra”, C. R. Acad.
Sci. Paris 356 (2018), no. 7, p. 732-736.
[6] E. F. Collingwood, M. L. Cartwright, “Boundary theorems for a function meromorphic in the unit circle”, Acta Math.
(1952), no. 57, p. 86-146.
[7] J. B. Conway, Functions of one complex variable I, Graduate Texts in Mathematics, vol. 11, Springer, 1978.
[8] Y. Dupain, “Extension à la dimension n d’un théorème de Ortel et Schneider”, Math. Z. 206 (1991), no. 1, p. 71-80.
[9] P. M. Gauthier, A. Roth, J. L. Walsh, “Possibility of uniform rational approximation in the spherical metric”, Can. J.
Math. 28 (1976), p. 112-115.
[10] W. K. Hayman, Meromorphic functions, Oxford Mathematical Monographs, Clarendon Press, 1964.
[11] S. Kierst, E. Szpilrajn, “Sur certaines singularités des fonctions analytiques uniformes”, Fundam. Math. 21 (1933),
p. 276-294.
[12] G. R. MacLane, “Meromorphic functions with small characteristic and no asymptotic values”, Mich. Math. J. 8 (1961),
p. 177-185.
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1545.
[14] K. Noshiro, “Contributions to the theory of meromorphic functions in the unit circle”, J. Fac. Sci., Hokkaido Univ.,
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Universal radial limits of meromorphic functions in the unit disk

Marian Bocea

Mihai Mihăilescu

Denisa Stancu-Dumitru

Comptes Rendus
Mathématique
Marian Bocea, Mihai Mihăilescu and Denisa Stancu-Dumitru
The Monotonicity of the Principal Frequency of the Anisotropic
p-Laplacian
Volume 360 (2022), p. 993-1000
https://doi.org/10.5802/crmath.348
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
2022, Vol. 360, p. 993-1000
https://doi.org/10.5802/crmath.348
Partial differential equations / Équations aux dérivées partielles
The Monotonicity of the Principal Frequency
of the Anisotropic p-Laplacian
Marian Boceaa, Mihai Mihăilescu∗, b, c and Denisa Stancu-Dumitrud
a Division of Mathematical Sciences, National Science Foundation, 2415 Eisenhower
Avenue, Alexandria, VA 22314 U.S.A.
b Department of Mathematics, University of Craiova, 200585 Craiova, Romania
c “Gheorghe Mihoc - Caius Iacob” Institute of Mathematical Statistics and Applied
Mathematics of the Romanian Academy, 050711 Bucharest, Romania
d Department of Mathematics and Computer Science, Politehnica University of
Bucharest, 060042 Bucharest, Romania
E-mails: mbocea@nsf.gov (M. Bocea), mmihailes@yahoo.com (M. Mihăilescu),
denisa.stancu@yahoo.com (D. Stancu-Dumitru)
Abstract. Let D > 1 be a fixed integer. Given a smooth bounded, convex domain Ω⊂RD and H :RD → [0,∞)
a convex, even, and 1-homogeneous function of class C 3,α(RD \ {0}) for which the Hessian matrix D2(H p )
is positive definite in RD \ {0} for any p ∈ (1,∞), we study the monotonicity of the principal frequency of the
anisotropic p-Laplacian (constructed using the function H) onΩwith respect to p ∈ (1,∞). As an application,
we find a new variational characterization for the principal frequency on domains Ω having a sufficiently
small inradius. In the particular case where H is the Euclidean norm in RD , we recover some recent results
obtained by the first two authors in [3, 4].
Mathematical subject classification (2010). 35P30, 47J10, 49R05, 49J40, 58C40.
Funding. The work of the first author was supported by the National Science Foundation under the IR/D
Program. The third author has been partially supported by CNCS-UEFISCDI Grant No. PN-III-P1-1.1-TE-
2019-0456.
Manuscript received 5 November 2021, accepted 26 February 2022.
1. Introduction and main results
For each positive integer D let E D be the Euclidean norm in RD . We define the set H D as follows:
if D = 1, H 1 := {E 1}; if D ≥ 2 we let H D be the family of all maps H : RD → [0,∞) which are
convex, even, 1-homogeneous, and of class C 3,α(RD \ {0}) such that the Hessian matrix D2(H p ) is
∗Corresponding author.
ISSN (electronic) : 1778-3569 https://comptes-rendus.academie-sciences.fr/mathematique/
994 Marian Bocea, Mihai Mihăilescu and Denisa Stancu-Dumitru
positive definite in RD \ {0} for all p ∈ (1,∞). For D ≥ 2 and H ∈ H D , let H◦ : RD → [0,∞) be the
polar function of H , defined by
H◦(η) := sup
ξ∈RD \{0}
〈ξ,η〉
H(ξ)
, η ∈RD .
Next, for each positive integer D and H ∈H D , define
P D,H :={
Ω⊂RD ∣∣Ω is a C 2,bounded, convex domain with nonnegative anisotropic mean curvature} .
For Ω ∈P D,H let δH ,Ω :Ω→ [0,∞) be the anisotropic distance function to the boundary of Ω,
given by
δH ,Ω(x) := inf
y ∈∂Ω
H◦(x − y), x ∈Ω.
Further, forΩ ∈P D,H and s > 0, define
P D,H (s) :=
{
Ω ∈P D,H : ∥∥δH ,Ω∥∥L∞(Ω) = s} .
Finally, forΩ ∈P D,H and p ∈ (1,∞), we define the principal Dirichlet frequency of the anisotropic
p-Laplacian by
λH
(
p;Ω
)
:= min
u∈W 1,p0 (Ω)\{0}
∫
Ω
H(∇u)p d x∫
Ω
|u|p d x
. (1)
It is known (see e.g. Belloni, Ferone & Kawohl [1] or Belloni, Kawohl & Juutinen [2]) that λH (p;Ω)
is the lowest eigenvalue λ of the problem−
D∑
i=1
∂
∂xi
[
H(∇u)p−2Ki (∇u)
]=λ|u|p−2u if x ∈Ω
u = 0 if x ∈ ∂Ω,
(2)
where Ki (ξ) := ∂∂ξi (
1
2 H(ξ)
2), for all ξ ∈ RD and i ∈ {1, · · · , D}. In the particular case when H = E D ,
the differential operator involved in the eigenvalue problem (2) reduces to the classical p-Laplace
operator ∆p . For this reason, (2) is called the eigenvalue problem for the anisotropic p-Laplacian.
The main goal of this paper is to analyze the monotonicity of the function p 7→ λH (p;Ω) with
respect to p ∈ (1,∞) for given H ∈H D andΩ ∈P D,H .
When D = 1 andΩ= (a,b) with a,b ∈R, it is well known (see [11]) that the principal frequency
of the p-Laplacian (H = E 1) is given by the explicit formula
λE 1
(
p; (a,b)
)= (p −1)( 2
b −a
)p ( π/p
sin
(
π/p
) )p .
It can be shown that when b−a2 ∈ (1,∞) there exists p? = p?( b−a2 ) ∈ (1,∞) such that
p 7→ λE 1 (p; (a,b)) is increasing on (1, p?) and decreasing on (p?,∞) (see, Kajikiya, Tanaka &
Tanaka [10, Theorem 1.1(ii)]). On the other hand, it is easy to check that if b−a2 ≤ 1 then the map
p 7→ λE 1 (p; (a,b)) is increasing on (1,∞) (see, Kajikiya, Tanaka & Tanaka [10, Theorem 1.1(i)]).
A similar result was established in the case when D ≥ 2 and H = E D by the first two authors of this
paper in [4, Theorem 1]. Our main goal here is to show that the results obtained in [4] continue
to hold in the anisotropic case where H ∈H D is a general function as described above. Our main
result is stated in the following theorem.
Theorem 1. Let D ≥ 2 and H ∈H D be fixed, and let MH be defined by
MH := sup
{
s > 0 ∣∣λH (p;Ω) <λH (q ;Ω) ∀ 1 MH there exists a domain Ω ∈ P D,H (s) for which the map (1,+∞) 3 p 7→
λH (p;Ω) is not monotone.
Next, using Theorem 1, we obtain a new variational characterization of λH (p;Ω) for domains
Ω ∈P D,H (s) with s ∈ (0, MH ], where MH is defined by (3). Precisely, we prove
Theorem 2. Let D ≥ 2 and H ∈H D be fixed. For eachΩ ∈P D,H and p ∈ (1,∞), define
ΛH (p;Ω) := inf
u∈X0\{0}
∫
Ω
[
exp
(
H(∇u)p)−1] d x∫
Ω
[
exp
(|u|p)−1] d x ,
where
X0 :=W 1,∞(Ω)∩
(
∩q >1W 1,q0 (Ω)
)
.
If ‖δH ,Ω‖L∞(Ω) > 1 then ΛH (p;Ω) = 0, while if ‖δH ,Ω‖L∞(Ω) ≤ 1 we have ΛH (p;Ω) > 0. Moreover, if
‖δH ,Ω‖L∞(Ω) ≤ MH , with MH defined by (3), then ΛH (p;Ω) =λH (p;Ω).
Note that a similar result was proved in [3, Theorem 2] in the particular case H = E D .
2. The principal frequency of the anisotropic p-Laplacian
In this section we recall a number of known properties of λH (p;Ω) (with H ∈H D , p ∈ (1,∞) and
D ≥ 2) that will be useful in the sequel. We begin with a result of Belloni, Kawohl & Juutinen [2]
(see also Juutinen, Lindqvist & Manfredi [9] or Fukagai, Ito & Narukawa [7] for the case where
H = E D ). Let λH (∞;Ω) be defined by
λH (∞;Ω) := min
{∥∥H(∇ϕ)∥∥L∞(Ω)
‖ϕ‖L∞(Ω)
∣∣∣∣∣ ϕ ∈ X0 \ {0}
}
, (4)
where X0 :=W 1,∞(Ω)∩(∩q >1W 1,q0 (Ω)). Then, the minimum in (4) is always achieved at δH ,Ω, the
anisotropic distance function to the boundary ofΩ, and
λH (∞;Ω) =
∥∥δH ,Ω∥∥−1L∞(Ω) . (5)
Moreover, by [2, Lemma 3.1], we have
lim
p →∞
p
√
λH
(
p;Ω
)=λH (∞;Ω) = ∥∥δH ,Ω∥∥−1L∞(Ω) . (6)
It is also well known that λH (∞;Ω) = RH (Ω)−1, where RH (Ω) stands for the anisotropic inradius
of Ω with respect to H ∈ H D . The following lower bound for λH (p;Ω) is due to Della Pietra, di
Blasio, and Gavitone (see [5, Theorem 5.1]).
p −1
RH (Ω)p
(
π/p
sin(π/p)
)p
≤λH (p;Ω) ∀ p > 1. (7)
Note that the left-hand side in (7) is exactly λE 1 (p; (−RH (Ω),RH (Ω))), i.e. the principal frequency
of the p-Laplacian on the interval (−RH (Ω),RH (Ω)) when D = 1. Moreover, by [5, Theorem 5.8],
equality in (7) is achieved whenΩ approaches a suitable infinite slab. More precisely, if we define,
for k > 0, Ω(k) := (−RH (Ω)(H◦(e1))−1,RH (Ω)(H◦(e1))−1)× (−k,k)D−1, then
lim
k →∞
λH
(
p;Ω(k)
)= p −1
RH (Ω)p
(
π/p
sin
(
π/p
) )p ∀ p > 1. (8)
996 Marian Bocea, Mihai Mihăilescu and Denisa Stancu-Dumitru
By [5, Proposition 2.2(iii)] (see also [6, Proposition 3.3(iii)] or [12, Theorem 3.2] for the case
H = E D )
p p
√
λH
(
p;Ω
)≤ q q√λH (q ;Ω), ∀ 1 0, considering the rescaled domain ΩR := RΩ = {Rx | x ∈ Ω}, we
have (see, e.g., [5, Proposition 2.2(i)] or [6, Proposition 3.3(v)])
λH
(
p;ΩR
)= R−pλH (p;Ω) ∀ p > 1. (10)
Moreover, it is easy to check that in this case ‖δH ,ΩR ‖L∞(ΩR ) = R‖δH ,Ω‖L∞(Ω).
3. Proof of the main results
3.1. Proof of Theorem 1
First, in view of [10, Theorem 1.1(ii)] and [5, Theorem 5.8] we have the following result.
Proposition 3. Let D ≥ 2 and H ∈ H D . For any s ∈ (1,∞), there exists a domain Ω ∈ P D,H (s) for
which the function p 7→λH (p;Ω) is not monotone on (1,∞).
Proof. We start by observing that since s > 1, (6) yields
lim
p →∞λH
(
p;Ω
)= 0, ∀ Ω ∈P D,H (s). (11)
For each k > 0, define Ωs (k) := (−s(H◦(e1))−1, s(H◦(e1))−1)× (−k,k)D−1. It is clear that Ωs (k) ∈
P D,H (s) whenever k ≥ s is sufficiently large. By (8), we know that
λE 1
(
p; (−s, s))= lim
k →∞
λH
(
p;Ωs (k)
) ∀ p > 1. (12)
Moreover, [10, Theorem 1.1(ii)] guarantees that the function λE 1 (·; (−s, s)) is not monotone.
We claim that there exists k ≥ s for which the function λH (·;Ωs (k)) is not monotone and prove
this by contradiction. Thus, let us assume thatλH (·;Ωs (k)) is monotone for every k ≥ s sufficiently
large. In view of (11), λH (·;Ωs (k)) must be non-increasing, so that
λH
(
p;Ωs (k)
)≥λH (q ;Ωs (k)) ∀ 1 s sufficiently large.
Letting k →∞ and taking (12) into account we obtain
λE 1
(
p; (−s, s))≥λE 1 (q ; (−s, s)) ∀ 1 < p < q <∞,
which contradicts the fact that λE 1 (·; (−s, s)) is not monotone. This concludes the proof of
Proposition 3. 
Proposition 4. LetΩ ∈P D,H (s) with s ∈ (0,e−1]. Then λH (p;Ω) is strictly increasing as a function
of p on (1,∞).
The proof of Proposition 4 follows from the next two lemmas.
Lemma 5. LetΩ ∈P D,H and suppose that λH (q ;Ω) ≤λH (p;Ω) for some 1 1.
Since x1/(x−1) < e, the result follows (note that the function t 7→ t 1/(t−1) is strictly decreasing on
(1,∞) and limt →1+ t 1/(t−1) = e). 
Marian Bocea, Mihai Mihăilescu and Denisa Stancu-Dumitru 997
Lemma 6. LetΩ ∈P D,H (s) with s > 0. Then
1
sp
<λH
(
p;Ω
) ∀ p > 1.
Proof. Observing that
(p −1)
(
π/p
sin(π/p)
)p
=λE 1
(
p; (−1,1)) ,
(7) can be rewritten as
λE 1
(
p; (−1,1))
sp
≤λH
(
p;Ω
) ∀ p > 1.
Hence, using the fact that λE 1 (·; (−1,1)) is strictly increasing and limp →1+ λE 1 (p; (−1,1)) = 1
(see [10, Theorem 1.1(i)] and [8, Theorem 3.3], respectively), we obtain
1
sp
< λE 1
(
p; (−1,1))
sp
≤λH
(
p;Ω
) ∀ p > 1.

Having proven Lemmas 5 and 6, the conclusion of Proposition 4 is now immediate. Indeed,
assume by contradiction that there exists 1 e−1. This concludes the proof.
Lemma 7. If for some r ∈ (0,1] and any domainΩ ∈P D,H (r ) we have
λH
(
p;Ω
)≤λH (q ;Ω) ∀ 1 < p < q <∞,
then for any s ∈ (0,r ) and anyΩ ∈P D,H (s) we also have
λH
(
p;Ω
)<λH (q ;Ω) ∀ 1 < p < q <∞.
Proof. Indeed, if Ω ∈ P D,H (r ) then for each R ∈ (0,1) we have ‖δH ,ΩR ‖L∞(ΩR ) = Rr < r and, in
view of (10), we get that
λH
(
p;ΩR
)= 1
Rp
λH
(
p;Ω
) ∀ 1 < p <∞.
Now, fix s ∈ (0,r ) and take R := s/r ∈ (0,1). IfΩ ∈P D,H (s) thenΩR−1 ∈P D,H (r ) and, consequently,
λH
(
p;Ω
)= 1
Rp
λH
(
p;ΩR−1
) ∀ 1 < p <∞.
But
λH
(
p;ΩR−1
)≤λH (q ;ΩR−1) ∀ 1 < p < q <∞,
and since R ∈ (0,1), we deduce that
1
Rp
λH
(
p;ΩR−1
)< 1
Rq
λH
(
q ;ΩR−1
) ∀ 1 < p < q <∞.
Equivalently, λH (p;Ω) <λH (q ;Ω) ∀ 1 < p < q <∞. 
We are now ready to complete the proof of Theorem 1. Let D ≥ 2 be a fixed integer, H ∈ H D ,
and MH be defined by (3). In view of Proposition 4, we have that MH ∈ [e−1,1]. If MH = 1,
then by Proposition 3 and the definition of MH it follows that for any s > MH = 1 there exists
a domain Ω ∈ P D,H (s) for which the function λH (p;Ω) is not monotone in p on the interval
(1,∞). This conclusion is still valid in the case where MH ∈ [e−1,1). Indeed, if MH ∈ [e−1,1) and
s ∈ (MH ,1] then, if λH (·;Ω) were monotone for every Ω ∈ P D,H (s), one would have (noting that
λH (p;Ω) → ∞ as p → ∞, since s ≤ 1) that λH (·;Ω) must be nondecreasing (but not necessarily
998 Marian Bocea, Mihai Mihăilescu and Denisa Stancu-Dumitru
increasing). Hence, by fixing r ∈ (MH , s) and applying Lemma 7, one can show that λH (·;Ω) is
strictly increasing for everyΩ ∈P D,H (r ). This contradicts the definition of MH .
Up to this point we have shown that for any r ∈ (0, MH ) and any domain Ω ∈P D,H (r ) we have
λH (p;Ω) < λH (q ;Ω) ∀ 1 < p < q <∞. To finish the proof of Theorem 1 it remains to prove that
we still have λH (p;Ω) ≤λH (q ;Ω) ∀ 1 < p < q <∞ whenΩ ∈P D,H (MH ). To justify this, note that
for any R ∈ (0,1), ‖δH ,ΩR ‖L∞(ΩR ) = RMH ∈ (0, MH ) and hence λH (p;ΩR ) < λH (q ;ΩR ) ∀ 1 < p <
q <∞, or, by virtue of (10),
1
Rp
λH
(
p;Ω
)< 1
Rq
λH
(
q ;Ω
) ∀ 1 1 thenΛH (p;Ω) = 0 for all p ∈ (1,∞).
Proof. Let ε0 > 0 and ω be an open subset of Ω having positive Lebesgue measure |ω| > 0, such
that δH ,Ω(x) ≥ 1+ε0 for any x ∈ω. Since δH ,Ω ∈ X0 \ {0},
ΛH (p;Ω) ≤
∫
Ω
[
exp
(
H
(∇(nδH ,Ω))p)−1] d x∫
Ω
[
exp
((
nδH ,Ω
)p)−1] d x ∀ n ≥ 1. (13)
Taking into account the fact that H(∇δH ,Ω(x)) = 1 for a.e. x ∈Ω, we have∫
Ω
[
exp
(
H
(∇(nδH ,Ω))p)−1] d x∫
Ω
[
exp
((
nδH ,Ω
)p)−1] d x =
|Ω|[exp(np )−1]∫
Ω
[
exp
(
npδH ,Ω(x)
p)−1] d x
≤ |Ω|
[
exp(np )−1]∫
ω
[
exp
(
npδH ,Ω(x)
p)−1] d x
≤ |Ω|
[
exp(np )−1]
|ω|[exp(np (1+ε0)p ))−1]
for every integer n ≥ 1. Letting n →∞ in (13) givesΛH (p;Ω) = 0. 
Lemma 9. If ‖δH ,Ω‖L∞(Ω) ∈ (0,1] thenΛH (p;Ω) > 0 for all p ∈ (1,∞).
Proof. First, we claim that
ΛH (p;Ω) ≥ inf
k ∈N\{0}
λH (kp;Ω). (14)
Indeed, recall that the definition of λH (kp;Ω) implies that if u ∈ X0 \ {0} (which, in particular,
means that u ∈W 1,q0 (Ω) \ {0} for any q > 1) then
∞∑
k=1
1
k !
∫
Ω
H (∇u)kp d x
∞∑
k=1
1
k !
∫
Ω
|u|kp d x
≥
∞∑
k=1
λH
(
kp;Ω
)
k !
∫
Ω
|u|kp d x
∞∑
k=1
1
k !
∫
Ω
|u|kp d x
≥ inf
k ∈N\{0}
λH
(
kp;Ω
)
.
Passing to the infimum over all u ∈ X0 \ {0} gives (14).
Marian Bocea, Mihai Mihăilescu and Denisa Stancu-Dumitru 999
Next, in view of (7), and since RH (Ω) = ‖δH ,Ω‖L∞(Ω), we arrive at
q −1∥∥δH ,Ω∥∥qL∞(Ω)
(
π/q
sin
(
π/q
) )q ≤λH (q ;Ω) ∀ q ∈ (1,∞). (15)
Further, recall the fact that the function
(1,∞) 3 p 7→ (p −1)( π/p
sin
(
π/p
) )p ,
is increasing (see, e.g. [10, Theorem 1.1(i)] for the proof). Taking into account (15) and the fact
that ‖δH ,Ω‖L∞(Ω) ∈ (0,1], we obtain
0 < (p −1)( π/p
sin
(
π/p
) )p ≤λH (kp;Ω) ∀ k ∈N\ {0} and p ∈ (1,∞).
Thus, by (14),ΛH (p;Ω) > 0 for any p ∈ (1,∞). This concludes the proof of Lemma 9. 
Lemma 10. If MH is defined by (3) and ‖δH ,Ω‖L∞(Ω) ∈ (0, MH ], then ΛH (p;Ω) = λH (p;Ω) for all
p ∈ (1,∞).
Proof. First, we show thatΛH (p;Ω) ≤λH (p;Ω) for all p ∈ (1,∞). To this aim, first note that
ΛH
(
p;Ω
)≤
∫
Ω
[
exp
(
H (∇(tu))p)−1] d x∫
Ω
[
exp
(|tu|p)−1] d x ∀ u ∈C∞0 (Ω) \ {0} ⊂ X0 \ {0} and t ∈ (0,1).
Hence,
ΛH
(
p;Ω
)≤
∞∑
k=1
∫
Ω
H (∇(tu))kp
k !
d x
∞∑
k=1
∫
Ω
|tu|kp
k !
d x
=
∫
Ω
H (∇u)p d x +
∞∑
k=2
t (k−1)p
∫
Ω
H (∇u)kp
k !
d x∫
Ω
|u|p d x +
∞∑
k=2
t (k−1)p
∫
Ω
|u|kp
k !
d x
for any u ∈C∞0 (Ω) \ {0} and t ∈ (0,1). Letting t → 0+ in the above inequality we get
ΛH
(
p;Ω
)≤
∫
Ω
H (∇u)p d x∫
Ω
|u|p d x
∀ u ∈C∞0 (Ω) \ {0}.
We obtain
ΛH
(
p;Ω
)≤λH (p;Ω) ∀ p ∈ (1,∞), (16)
as claimed.
Next, taking into account that ‖δH ,Ω‖L∞(Ω) ∈ (0, MH ] by Theorem 1 we deduce that λH (p;Ω) ≤
λH (q ;Ω) whenever 1 < p < q <∞. Combining this with (14) we are led to
λH
(
p;Ω
)≤ΛH (p;Ω) ∀ p ∈ (1,∞). (17)
The conclusion of Lemma 10 now follows from (16) and (17). 
References
[1] M. Belloni, V. Ferone, B. Kawohl, “Isoperimetric inequalities, Wulff shape and related questions for strongly nonlinear
elliptic operators”, Z. Angew. Math. Phys. 54 (2003), no. 5, p. 771-783.
[2] M. Belloni, B. Kawohl, P. Juutinen, “The p-Laplace eigenvalue problem as p →∞ in a Finsler metric”, J. Eur. Math.
Soc. 8 (2006), no. 1, p. 123-138.
[3] M. Bocea, M. Mihăilescu, “Minimization problems for inhomogeneous Rayleigh quotients”, Commun. Contemp.
Math. 20 (2018), no. 7, article no. 1750074 (13 pages).
1000 Marian Bocea, Mihai Mihăilescu and Denisa Stancu-Dumitru
[4] ——— , “On the monotonicity of the principal frequency of the p-Laplacian”, Adv. Calc. Var. 14 (2021), no. 1, p. 147-
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[5] F. Della Pietra, G. di Blasio, N. Gavitone, “Sharp estimates on the first Dirichlet eigenvalue of nonlinear elliptic
operators via maximum principle”, Adv. Nonlinear Anal. 9 (2020), p. 278-291.
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operator”, Bull. Sci. Math. 155 (2019), p. 10-32.
[7] N. Fukagai, M. Ito, K. Narukawa, “Limit as p →∞ of p-Laplace eigenvalue problems and L∞-inequality of Poincaré
type”, Differ. Integral Equ. 12 (1999), no. 2, p. 183-206.
[8] Y. X. Huang, “On the eigenvalues of the p-Laplacian with varying p”, Proc. Am. Math. Soc. 125 (1997), no. 11, p. 3347-
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[9] P. Juutinen, P. Lindqvist, J. J. Manfredi, “The ∞-eigenvalue problem”, Arch. Ration. Mech. Anal. 148 (1999), no. 2,
p. 89-105.
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The Monotonicity of the Principal Frequency of the Anisotropic $p$-Laplacian

Simon Raulot

Comptes Rendus
Mathématique
Simon Raulot
Nonexistence of DEC spin fill-ins
Volume 360 (2022), p. 1049-1054
https://doi.org/10.5802/crmath.366
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
2022, Vol. 360, p. 1049-1054
https://doi.org/10.5802/crmath.366
Mathematical physics / Physique mathématique
Nonexistence of DEC spin fill-ins
Non-existence de remplissages spinoriels satisfaisant la
condition d’énergie dominante
Simon Raulota
a Laboratoire de Mathématiques R. Salem UMR 6085 CNRS-Université de Rouen
Avenue de l’Université, BP.12 Technopôle du Madrillet 76801
Saint-Étienne-du-Rouvray, France.
E-mail: simon.raulot@univ-rouen.fr
Abstract. In this note, we show that a closed spin Riemannian manifold does not admit a spin fill-in satisfying
the dominant energy condition (DEC) if a certain generalized mean curvature function is point-wise large.
Résumé. Dans cette note, on montre qu’une variété riemannienne fermée munie d’une structure spin
n’admet pas de remplissage spinoriel satisfaisant la condition d’énergie dominante (DEC) si une certaine
fonction, généralisant la courbure moyenne, est suffisamment grande.
Manuscript received 5 April 2022, accepted 15 April 2022.
Given a closed n-dimensional Riemannian manifold (Σ,γ), it is a very interesting question
to know whether there exists an (n +1)-dimensional compact Riemannian manifold (Ω, g ) with
nonnegative scalar curvature (NNSC) whose boundary is isometric to (Σ,γ). If so, the Riemannian
manifold is called a fill-in of (Σ,γ) and the set of all such fill-ins, referred to as NNSC fill-ins of
(Σ,γ), is denoted by F (Σ,γ). The problem of the existence of such fill-ins has recently been solved
by Shi, Wang and Wei [13] where it is shown that if Σ is the boundary of an (n +1)-dimensional
compact manifold Ω then, any metric γ on Σ can be extended to a Riemannian metric g on Ω
with positive scalar curvature.
One can also try to find a fill-in whose mean curvature is prescribed by a smooth function H
on Σ. This problem is tightly related to the Bartnik quasi-local mass [2] and a triplet (Σ,γ, H) is
then usually called a Bartnik data.
In [10], Miao proved that if Σ is the boundary of some compact (n +1)-dimensional manifold
Ω, then given any Riemannian metric γ on Σ, there exists a constant H0, depending on γ and Ω,
such that, if minΣ H ≥ H0, there does not exist NNSC fill-ins of (Σ,γ, H). The proof makes use of
the work of Shi, Wang and Wei [13] and of Schoen and Yau’s results on closed manifolds [11, 12].
Such a result is also obtained for fill-ins with a negative scalar curvature lower bound.
This fact was previously demonstrated by Gromov [7] for spin manifolds. More precisely, he
showed that if (Ω, g ) is a NNSC spin fill-in of (Σ,γ, H), then
min
Σ
H ≤ n
Rad(Σ,γ)
ISSN (electronic) : 1778-3569 https://comptes-rendus.academie-sciences.fr/mathematique/
1050 Simon Raulot
where Rad(Σ,γ) is a constant only depending on (Σ,γ) and known as the hyperspherical radius
of (Σ,γ). The main remark here, which is the key point of our argument, is that the nonexistence
of NNSC spin fill-ins can be obtained by taking another route. Indeed, it can be deduced from
an eigenvalue estimate on the first eigenvalue λ1(Σ,γ) of the Dirac operator of (Σ,γ) proved by
Hijazi, Montiel and Zhang [9] and which states that
min
Σ
H ≤ 2|λ1(Σ,γ)|. (1)
Since λ1(Σ,γ) depends only on Σ, γ and the involved spin structure, the nonexistence of NNSC
spin fill-ins is a direct consequence of (1). This inequality is sharp since round balls in the
Euclidean space satisfy the equality case. Note that a similar result can be deduced for spin fill-ins
with a negative scalar curvature lower bound from [8] as in [7].
In this note, we use the aforementioned observation to generalize this result in the context of
spin fill-ins satisfying the dominant energy condition (DEC). Following [3], a 5-uple (Σ,γ, H ,α,h)
is called a spacetime Bartnik data set if (Σ,γ) is an oriented, closed Riemannian manifold, H and
h are smooth functions on Σ and α is a smooth 1-form on Σ. In this situation, a triplet (Ω, g ,k) is
a fill-in of such a spacetime Bartnik data if
(1) (Ω, g ,k) is a compact initial data set, that is (Ω, g ) is an (n + 1)-dimensional compact
Riemannian manifold with boundary and k is a smooth symmetric (0,2)-tensor field on
Ω,
(2) there exists an isometry f : (Σ,γ) → (∂Ω, g |∂Ω) such that
(a) f ∗Hg = H , where Hg is the mean curvature of ∂Ω in (Ω, g ) with respect to the
outward unit normal ν̃,
(b) f ∗
(
k(ν̃, · )T )=α,
(c) f ∗
(
Trg|∂Ω k
)= h.
Here Trg|∂Ω denotes the trace operator on ∂Ω andω
T is the tangent part toΣ of a 1-formω defined
along Σ. In the following, we will omit the isometry f in the identification between Σ and ∂Ω.
Then, a fill-in (Ω, g ,k) of (Σ,γ, H ,α,h) satisfies the dominant energy condition, or is a DEC fill-
in, if
µ≥ |J |g
where µ and J are respectively the mass density and the current density defined by
µ= 1
2
(
Rg + (Trg k)2 −|k|2g
)
and
J = divg
(
k − (Trg k)g
)
.
Here Rg and Trg denote respectively the scalar curvature and the trace operator of (Ω, g ). In this
situation, a natural generalization of the mean curvature is given by the function
Hg := Hg −
√
|k(ν̃, · )T |2g + (Trg|∂Ω k)2
which corresponds, for a spacetime Bartnik data, to
H := f ∗Hg = H −
√
|α|2γ+h2. (2)
When Σ is endowed with a spin structure, we will say that (Ω, g ,k) is a DEC spin fill-in of the
spacetime Bartnik data (Σ,γ, H ,α,h) if (Ω, g ,k) is a DEC fill-in and if Ω is a spin manifold which
induces the given spin structure on Σ. We then have the following result.
Theorem 1. Given any Riemannian metric γ on an n-dimensional spin manifold Σ, there exists a
constant H0, depending only on Σ, γ and the spin structure of Σ, such that, if minΣH ≥H0, there
do not exist DEC spin fill-ins of (Σ,γ, H ,α,h).
Simon Raulot 1051
This question was tackled by Tsang in [14] where several partial results are proved. We remark
that if k = 0 then α= 0 and h= 0, and the DEC gives the nonnegativity of the scalar curvature so
that a DEC fill-in of (Σ,γ, H) corresponds to a NNSC fill-in. In a same way, if k = cg , c 6= 0, then
α = 0 and h = nc, and the DEC condition implies that the scalar curvature of (Ω, g ) is bounded
from below by a negative constant, namely Rg ≥ −n(n + 1)c2. These remarks imply that our
result covers both of these cases. As mentioned above, the proof is a direct consequence of an
eigenvalue estimate for the first eigenvalue of the Dirac operator of a spacetime Bartnik spin data
which admits a DEC spin fill-in. This lower bound is stated as follow.
Theorem 2. If a spacetime Bartnik spin data (Σ,γ, H ,α,h) admits a DEC spin fill-in (Ω, g ,k) with
H > 0, then the first eigenvalue λ1(Σ,γ) of the Dirac operator of (Σ,γ) satisfies
min
Σ
H ≤ 2|λ1(Σ,γ)|.
One can consider Fspi n(Σ,γ), the set of the DEC spin fill-ins of the Riemannian spin manifold
(Σ,γ) without specifying the data H , α and h. Then Theorem 2 implies that if Fspi n(Σ,γ) 6= ;, it
holds that
sup
(Ω,g ,k)∈Fspi n (Σ,γ)
min
Σ
H ≤ 2|λ1(Σ,γ)| <∞.
The proof of Theorem 2 relies on spin geometry and we refer especially to [4–6] for more details on
this subject. Let us briefly recall what we need here. Since (Ω, g ) is a Riemannian spin manifold,
there exists a smooth Hermitian vector bundle over Ω, the spinor bundle, denoted by SΩ, whose
sections are called spinor fields. The Hermitian scalar product is denoted by 〈 , 〉. Moreover, the
tangent bundle TΩ acts on SΩ by Clifford multiplication X ⊗ψ 7→ c(X )ψ for any tangent vector
fields X and any spinor fieldsψ. On the other hand, the Riemannian Levi-Civita connection∇ lifts
to the so-called spin Levi-Civita connection, also denoted by ∇, and defines a metric covariant
derivative on SΩ that preserves the Clifford multiplication. A quadruplet (SΩ,c,〈 , 〉,∇) which
satisfies the previous assumptions is usually referred to as a Dirac bundle. The Dirac operator
is then the first order elliptic differential operator acting on SΩ defined by D := c ◦∇. The spin
structure on Ω induces, via a choice of an unit normal field to ∂Ω ' Σ, a spin structure on Σ.
This allows to define the extrinsic spinor bundle S/ := SΩ|Σ over Σ on which there exists a Clifford
multiplication c/ and a metric covariant derivative ∇/. The quadruplet (S/,c/,〈 , 〉,∇/) is thus endowed
with a Dirac bundle structure. Similarly, the extrinsic Dirac operator is defined by taking the
Clifford trace of the covariant derivative ∇/ that is D/ := c/ ◦∇/. It is by now well-known that this
operator can be expressed using the Dirac operator DΣ of (Σ,γ) endowed with the induced spin
structure. What is important to us here is that the first nonnegative eigenvalue of the extrinsic
Dirac operator D/ corresponds to |λ1(Σ,γ)|, the absolute value of the first eigenvalue of DΣ and so
it only depends on (Σ,γ) and the spin structure on Σ.
Proof of Theorem 2. Let (Ω, g ,k) be a DEC spin fill-in of the spacetime Bartnik spin data
(Σ,γ, H ,α,h) and consider the modified spin covariant derivatives defined by
∇±Xψ :=∇Xψ±
i
2
c
(
k(X )
)
ψ (3)
for X ∈ Γ(TΩ) andψ ∈ Γ(SΩ). The associated Dirac-type operators given by D± := c ◦∇± are easily
seen to satisfy
D±ψ= Dψ∓ i
2
(Trg k)ψ. (4)
These are first order elliptic differential operators whose formal adjoints, with respect to the L2
scalar product on SΩ, are D∓ as deduced from the following integration by parts formulae∫
Ω
〈D±ψ,ϕ〉dµ=
∫
Ω
〈ψ,D∓ϕ〉dµ−
∫
Σ
〈c(ν)ψ,ϕ〉dσ (5)
1052 Simon Raulot
for all smooth spinor fields ψ, ϕ onΩ. Here dµ (resp. dσ) denotes the Riemannian volume (resp.
area) element of (Ω, g ) (resp. (Σ,γ)) and ν is the inner unit normal to Σ in (Ω, g ). In a same way, a
straightforward computation implies that(∇±)∗∇±ψ=−n+1∑
j=1
∇∓e j ∇±e jψ (6)
where
(∇±)∗ denote the formal adjoints of the modified connection ∇± and {e1, · · · ,en+1} is a local
g -orthonormal frame of TΩ. In particular, the Stokes formula leads to∫
Ω
〈(∇±)∗∇±ψ,ψ〉dµ= ∫
Ω
|∇±ψ|2dµ+
∫
Σ
〈∇±νψ,ψ〉dσ. (7)
Then, it follows from the fact that
(
D±)∗ = D∓, from (6) and the classical Schrödinger–
Lichnerowicz formula
D2ψ=∇∗∇ψ+ Rg
4
ψ
that (
D±
)∗D±ψ= (∇±)∗∇±ψ+ 1
2
(
µψ± i c(J )ψ) (8)
for allψ ∈ Γ(SΩ). This is the (n+1)-dimensional Riemannian counterpart of the formula obtained
by Witten [15] in his proof of the positive energy theorem. Now observe that since the point-wise
symmetric endomorphism J± :=±i c(J ) satisfies (J±)2ψ= |J |2gψ it holds that
〈J±ψ,ψ〉 ≥−|J |g |ψ|2
so that the DEC and (8) imply the point-wise inequalities
〈(D±)∗D±ψ,ψ〉 ≥ 〈(∇±)∗∇±ψ,ψ〉
for all ψ ∈ Γ(SΩ). Now integrating by parts these estimates on Ω using (5) and (7) leads to the
following important integral inequalities∫
Ω
(
|∇±ψ|2 −|D±ψ|2
)
dµ≤−
∫
Σ
〈∇±νψ+ c(ν)D±ψ,ψ〉dσ. (9)
From the very definitions (3) and (4) of the modified covariant derivatives and the associated
Dirac operators, we compute that
−∇±νψ− c(ν)D±ψ= D/ψ−
1
2
(
Hψ∓ i c(V )ψ) (10)
where
V :=α]+hν ∈ Γ(TΩ|Σ)
since (Ω, g ,k) is a fill-in of the data (Σ,γ, H ,α,h). Here ] : T ∗Ω → TΩ denotes the classical
musical isomorphism between the cotangent bundle and the tangent bundle. Observe that
the endomorphisms V ± := ±i c(V ) of S/ is point-wise symmetric with respect to the Hermitian
structure and satisfies (V ±)2ψ= |V |2ψ in such a way that
〈V ±ψ,ψ〉 ≥−|V |g |ψ|2 =−
√
h2 +|α|2γ |ψ|2 (11)
for all ψ ∈ Γ(S/). Combining (9), (10) and (11) yields the following integral inequalities∫
Ω
(
|∇±ψ|2 −|D±ψ|2
)
dµ≤
∫
Σ
〈D/ψ− 1
2
Hψ,ψ〉dσ. (12)
which hold for all ψ ∈ Γ(SΩ) and where H is the generalized mean curvature function defined
in (2).
Now we are going to show that one can extend any spinor fields onΣ in a suitable way. For this,
we recall that the mapχ := i c(ν) is a boundary chirality operator (in the sense of [1, Example 7.26])
and so it is an orthogonal involution of S/ which induces an orthogonal splitting S/ = S/+ ⊕ S/−
into the eigenbundles of χ for the eigenvalues ±1. The associated projection maps P± : S/ → S/±
Simon Raulot 1053
define elliptic local boundary conditions for the Dirac-type operators D± (see [1, Corollary 7.23]
for example). This implies that the operators
D+± :
{
ψ ∈ H 1(SΩ)/P±ψ|Σ = 0
}−→ L2(SΩ)
are of Fredholm type and that if Φ ∈ Γ(SΩ) and Ψ ∈ Γ(S/) are smooth spinor fields, any solutions
ψ ∈ Γ(SΩ) of the boundary problem {
D+ψ=Φ onΩ
P±ψ|Σ = P±Ψ on Σ
(13)
are smooth. The same holds for the operators D−±. It turns out that these operators are isomor-
phisms. To prove this fact, we notice that it is enough to show that D+± and D
−
± are one-to-one
since it follows from the integration by parts formulae (5) that the adjoint of D+± is D
−
∓. So take
ψ±, non trivial, in the kernel of D+±, that is ψ± ∈ Γ(SΩ) satisfies (13) with Φ= 0 and Ψ= 0. In par-
ticular, ψ± is smooth on Ω. On the other hand, from the self-adjointness of the Dirac operator D/
and the fact that D/χ=−χD/, we get that∫
Σ
〈D/ψ,ψ〉dσ= 2
∫
Σ
Re〈D/(P±ψ),P∓ψ〉dσ (14)
for all ψ ∈ Γ(SΩ). Using this formula for ψ=ψ±, we deduce from (12) that
0 ≤
∫
Ω
|∇±ψ±|2dµ≤−1
2
min
Σ
H
∫
Σ
|ψ±|2dσ.
Since we assumed that H is positive on Σ, we conclude that ψ± is zero on Σ and ∇±ψ± = 0. This
leads to a contradiction since this last property implies that ψ± is nowhere vanishing on Ω. The
same holds for D−±.
Now takeΨ1 ∈ Γ(S/) an eigenspinor for the operator D/ associated with the eigenvalue |λ1(Σ,γ)|.
From the previous discussion, there exists an unique smooth solution ϕ ∈ Γ(SΩ) satisfying{
D+ϕ= 0 onΩ
P+ϕ|Σ = P+Ψ1 on Σ.
Taking ψ=ϕ in (12) and using the fact that
2Re〈P−Ψ1,P−ϕ〉 ≤ |P−Ψ1|2 +|P−ϕ|2,
finally lead to
0 ≤
(
|λ1(Σ,γ)|− 1
2
min
Σ
H
)∫
Σ
|ϕ|2dσ
which implies the estimate of Theorem 2. To get this last inequality, we implicitly used the fact
that
|λ1(Σ,γ)|
∫
Σ
|P−Ψ1|2dσ=
∫
Σ
〈D/(P+ϕ),P−Ψ1〉dσ
=
∫
Σ
〈P+ϕ,D/(P−Ψ1)〉dσ
=
∫
Σ
〈P+ϕ,P+(D/Ψ1)〉dσ
= |λ1(Σ,γ)|
∫
Σ
|P+ϕ|2dσ
which follows from the identity (14) and the self-adjointness of D/. 
We conclude this note by noticing that this method can be generalized to other situations (like
the Einstein–Maxwell equations with nonpositive cosmological constant).
1054 Simon Raulot
References
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differential geometry. Vol. XVII, Surveys in Differential Geometry, vol. 17, International Press, 2012, p. 1-78.
[2] R. Bartnik, “New definition of quasilocal mass”, Phys. Rev. Lett. 62 (1989), no. 20, p. 2346-2348.
[3] ——— , “Energy in general relativity”, in Tsing Hua lectures on geometry & analysis (Hsinchu, 1990–1991), Interna-
tional Press, 1997, p. 5-27.
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conformal geometry, EMS Monographs in Mathematics, European Mathematical Society, 2015, ix+452 pages.
[5] T. Friedrich, “On the spinor representation of surfaces in Euclidean 3-space”, J. Geom. Phys. 28 (1998), no. 1-2, p. 143-
157.
[6] N. Ginoux, The Dirac spectrum, Lecture Notes in Mathematics, vol. 1976, Springer, 2009, xvi+156 pages.
[7] M. Gromov, “Scalar curvature of manifolds with boundaries: natural questions and artificial constructions”, https:
//arxiv.org/abs/1811.04311, 2018.
[8] O. Hijazi, S. Montiel, A. Roldán, “Dirac operators on hypersurfaces of manifolds with negative scalar curvature”, Ann.
Global Anal. Geom. 23 (2003), no. 3, p. 247-264.
[9] O. Hijazi, S. Montiel, X. Zhang, “Dirac operator on embedded hypersurfaces”, Math. Res. Lett. 8 (2001), no. 1-2, p. 195-
208.
[10] P. Miao, “Nonexistence of NNSC fill-ins with large mean curvature”, Proc. Am. Math. Soc. 149 (2021), no. 6, p. 2705-
2709.
[11] R. Schoen, S.-T. Yau, “On the structure of manifolds with positive scalar curvature”, Manuscr. Math. 28 (1979), no. 1-3,
p. 159-183.
[12] ——— , “Positive scalar curvature and minimal hypersurface singularities”, in Surveys in differential geometry
2019. Differential geometry, Calabi-Yau theory, and general relativity, Surveys in Differential Geometry, vol. 24,
International Press, 2019, p. 441-480.
[13] Y. Shi, W. Wang, G. Wei, “Total mean curvature of the boundary and nonnegative scalar curvature fill-ins”, J. Reine
Angew. Math. 784 (2022), p. 215-250.
[14] T.-Y. Tsang, “On a spacetime positive mass theorem with corners”, https://arxiv.org/abs/2109.11070, 2022.
[15] E. Witten, “A new proof of the positive energy theorem”, Commun. Math. Phys. 80 (1981), no. 3, p. 381-402.

Nonexistence of DEC spin fill-ins

Jagannath Bhanja

Ram Krishna Pandey

On the minimum size of subset and subsequence sums in integers

Etienne Ghys

Jean-Michel Coron

Laure Saint-Raymond

Comptes Rendus
Mathématique
Etienne Ghys, Jean-Michel Coron et Laure Saint-Raymond
Jean-Pierre Demailly nous a quittés
Volume 360 (2022), p. i-iii
<https://doi.org/10.5802/crmath.368>
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
Comptes Rendus
Mathématique
2022, Vol. 360, p. i-iii
https://doi.org/10.5802/crmath.368
Éditorial / Editorial
Jean-Pierre Demailly nous a quittés
Jean-Pierre Demailly passed away
Etienne Ghys∗, a, Jean-Michel Corona et Laure Saint-Raymonda
a France
Courriels : etienne.ghys@ens-lyon.fr, jean.m.coron@gmail.com,
laure.saint-raymond@ens-lyon.fr
Manuscrit reçu et accepté le 19 avril 2022.
English version follows French version.
Nous avons la grande tristesse de vous informer du décès de Jean-Pierre Demailly, survenu le
17 mars 2022 à l’âge de soixante-quatre ans.
Jean-Pierre était l’un des rédacteurs en chef de la série Mathématique des Comptes Rendus de
l’Académie des sciences. Il participait à notre revue depuis son élection comme membre corres-
pondant en 1994 avant son élection comme membre en 2007. Depuis longtemps, il manifestait
un enthousiasme pour tout ce qui concernait la science ouverte. Son activité éditoriale ne se li-
mitait pas aux Comptes Rendus et il avait travaillé également pour les Annales de l’Institut Fou-
rier, Inventiones Mathematicae, le Journal de Crelle, le Journal of Geometric Analysis et le Journal
de Mathématiques Pures et Appliquées.
Son livre Complex analytic and differential geometry, référence aujourd’hui incontournable a
toujours été disponible librement sur sa page internet. Il est un des fondateurs d’Episciences,
plateforme permettant d’apporter la caution scientifique d’un comité éditorial à des articles
diffusés dans des archives ouvertes. Également très impliqué dans le mouvement des logiciels
libres, Jean-Pierre a contribué à leur diffusion.
Il s’est beaucoup investi dans les activités de notre Académie, par exemple au sein du comité
énergie auquel il participait avec conviction.
Il a dirigé une vingtaine de thèses et l’école qu’il a créée est d’une diversité thématique im-
pressionnante, allant de l’analyse complexe à la géométrie algébrique et la théorie des repré-
sentations. Partant des travaux de Kodaira sur l’amplitude des fibrés en droites de courbure
positive, il a été le pionnier de l’étude des différents cônes de diviseurs, qu’il a abordée par la
∗Auteur correspondant.
ISSN (electronic) : 1778-3569 https://comptes-rendus.academie-sciences.fr/mathematique/
ii Etienne Ghys, Jean-Michel Coron et Laure Saint-Raymond
théorie des courants. Ses résultats sur le cône kählérien d’une variété kählérienne compacte et le
cône pseudoeffectif d’une variété projective complexe sont d’une importance majeure. Il a éga-
lement étudié d’une façon très fine le problème de l’annulation de la cohomologie des fibrés vec-
toriels holomorphes qui ne sont pas à courbure positive (inégalités de Morse holomorphes). Il a
reçu de nombreux prix, depuis la médaille de bronze du CNRS en 1981 jusqu’au prix Heinz-Hopf
en 2021, en passant par les prix Carrière et Mergier Bourdeix de notre Académie. Ses travaux lui
ont également valu d’être invité deux fois au Congrès international des mathématiciens, en 1994
et 2006.
Jean-Pierre a effectué l’essentiel de sa carrière à l’université Joseph Fourier de Grenoble dont
il était une figure majeure. Il était un enseignant exceptionnel, admiré et estimé par tous ses
collègues ainsi que par ses élèves. Il accordait d’ailleurs une très grande importance à tout ce
qui concernait l’enseignement, à tous les niveaux. Il a par exemple présidé le SLECC (Savoir Lire
Écrire Compter Calculer) qui travaille dans les classes primaires.
Il a joué un rôle fondamental au sein de la communauté mathématique, tant au niveau
national qu’international.
Nous perdons un confrère admirable et un ami.
Jean-Michel Coron, Étienne Ghys, Laure Saint-Raymond
Version anglaise
It is with great sadness that we inform you of the death of Jean-Pierre Demailly, who passed away
on March 17, 2022 at the age of sixty-four.
Jean-Pierre was one of the chief editors of the Comptes Rendus Mathématique de l’Académie
des sciences. He participated in our journal since his election as a corresponding member in 1994
before his election as a full member in 2007. He had long been enthusiastic about everything
related to open science. His editorial activity was not limited to the Comptes Rendus and he had
also worked for the Annales de l’Institut Fourier, Inventiones Mathematicae, the Journal of Crelle,
the Journal of Geometric Analysis and the Journal of Pure and Applied Mathematics.
His book Complex analytic and differential geometry, an essential reference today, has always
been freely available on his web page. He is one of the founders of Episciences, a platform
allowing to bring the scientific guarantee of an editorial committee to articles published in open
archives. Also very involved in the free software movement, Jean-Pierre has contributed to its
diffusion.
He was very involved in the activities of our Academy, for example in the energy committee in
which he participated with conviction.
He advised about twenty PhD thesis and his school is impressive by its thematic breadth,
from complex analysis to algebraic geometry and representation theory. Building on the Kodaira
work on the ampleness of holomorphic line bundles of positivecurvature, he pioneered the
study of the various différents cones of divisors, that he attacked by the theory of currents. His
results on the Kähler cone of a compact Kähler manifold kählérienne and the pseudoeffective
cone of a projective complex manifold are of major importance. He also studied in a refined
way the vanishing problem for the cohomology of holomorphic vector bundles which do not
admit metrics of a positive curvature (holomorphic Morse inequalities). He has received many
awards, from the bronze medal of the CNRS in 1981 to the Heinz-Hopf prize in 2021, including
the Carrière and Mergier Bourdeix prizes of our academy. He has been invited twice to the
International Congress of Mathematicians, in 1994 and 2006.
Etienne Ghys, Jean-Michel Coron et Laure Saint-Raymond iii
Jean-Pierre spent most of his career at the Joseph Fourier University of Grenoble, of which he
was major figure. Jean-Pierre was also an exceptional teacher, admired by all his students and
colleagues. He attached great importance to everything that concerned teaching, at all levels.
For example, he chaired the SLECC (Savoir Lire Écrire Compter Calculer) which works in primary
classes.
He played a fundamental role in the mathematical community, both nationally and interna-
tionally.
We will miss an admirable colleague and a friend.
Jean-Michel Coron, Étienne Ghys, Laure Saint-Raymond

Jean-Pierre Demailly nous a quittés

Marianna Chatzakou

Vishvesh Kumar

Comptes Rendus
Mathématique
Marianna Chatzakou and Vishvesh Kumar
Lp -Lq Boundedness of Spectral Multipliers of the Anharmonic Oscillator
Volume 360 (2022), p. 343-347
<https://doi.org/10.5802/crmath.290>
© Académie des sciences, Paris and the authors, 2022.
Some rights reserved.
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Creative Commons Attribution 4.0 International License.
http://creativecommons.org/licenses/by/4.0/
Les Comptes Rendus. Mathématique sont membres du
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Comptes Rendus
Mathématique
2022, 360, p. 343-347
https://doi.org/10.5802/crmath.290
Harmonic analysis / Analyse harmonique
Lp-Lq Boundedness of Spectral Multipliers of
the Anharmonic Oscillator
Lp-Lq continuité des multiplicateurs spectraux de
l’oscillateur anharmonique
Marianna Chatzakoua and Vishvesh Kumara
a Department of Mathematics: Analysis Logic and Discrete Mathematics, Ghent
University, Belgium
E-mails: Marianna.Chatzakou@UGent.be (M. Chatzakou),
Vishvesh.Kumar@UGent.be, vishveshmishra@gmail.com (V. Kumar)
Abstract. In this note we study the Lp − Lq boundedness of Fourier multipliers of anharmonic oscillators,
and as a consequence also of spectral multipliers, for the range 1 < p ≤ 2 ≤ q < ∞. The underlying Fourier
analysis is associated with the eigenfunctions of an anharmonic oscillator in some family of differential
operators having derivatives of any order. Our analysis relies on a version of the classical Paley-type inequality,
introduced by Hörmander, that we extend in our nonharmonic setting.
Résumé. Dans cette note, nous étudions la Lp -Lq continuité des multiplicateurs de Fourier des oscillateurs
anharmoniques, et par conséquent des multiplicateurs spectraux également, pour 1 < p ≤ 2 ≤ q < ∞.
L’analyse de Fourier sous-jacente est associée aux fonctions propres d’un oscillateur anharmonique dans
certaines familles d’opérateurs différentiels ayant des dérivées d’ordre quelconque. Notre analyse s’appuie
sur une version de l’inégalité classique de type Paley, introduite par Hörmander, que nous étendons dans
notre cadre non harmonique.
2020 Mathematics Subject Classification. 42B15, 58J40, 47B10, 47G30, 35P10.
Funding. The authors are supported by the FWO Odysseus 1 grant G.0H94.18N: Analysis and Partial Differ-
ential Equations and by the Methusalem programme of the Ghent University Special Research Fund (BOF)
(Grant number 01M01021).
Manuscript received 10th May 2021, revised and accepted 24th October 2021.
1. Introduction
The aim of this note is to study the Lp −Lq boundedness conditions for the Fourier and spectral
multipliers where the pertinent Fourier analysis is developed with respect to a member of the
family of anharmonic oscillators on Rn . Such operators were considered in [6] as “prototype” for
the spectral analysis that is performed there.
ISSN (electronic) : 1778-3569 https://comptes-rendus.academie-sciences.fr/mathematique/
344 Marianna Chatzakou and Vishvesh Kumar
The boundedness of Fourier multipliers is a central topic in harmonic analysis and its ori-
gin can be traced back to Hörmander’s seminal paper [10] in 1960. The results concerning
to the boundedness of multipliers are of Lp -type, or Lp − Lq -type. In the first case, Mihlin–
Hörmander [11] or Marcinkiewicz theorems [5] come into play providing conditions on the reg-
ularity of the symbol, while in the second case, the sharp decay of the spectral information is
analysed.
Regarding the Lp − Lq boundedness that we study here, let us recall Hörmander’s Fourier
multiplier theorem as in [10]: For 1 < p ≤ 2 ≤ q <∞, the Fourier multiplier Tσ : S (Rn) →S ′(Rn)
associated with symbol σ :Rn →C defined by
F
(
Tσ f
)
(ξ) =σ(ξ)F ( f ) (ξ)
for ξ ∈Rn , where F denotes the Euclidean Fourier transform of f , has a bounded extension from
Lp (Rn) to Lq (Rn) provided that the symbol satisfies the condition
sup
s>0
s
(∫
ξ∈Rn : |σ(ξ)|≥ s
dξ
) 1
p − 1q <+∞ . (1)
The Lp − Lq boundedness of spectral and Fourier mutlipliers has been studied in different
settings; we refer for instance to the works [8] on symmetric spaces, and [1–3] on groups. In [2, 3]
the authors suggest an approach that is also effective for our purposes: to establish the Hausdorff–
Young–Paley inequality by interpolating between the Hausdorff–Young inequality and Paley-type
inequality. However, the machinery we use here differs from the one in [3] and relies on the
underlying Fourier analysis, and so on the spectral properties of our “model operator”, and on
the Sobolev estimates associated with it as acquired in [6].
Precisely, the Fourier analysis we present here is the natural analogue of the one developed
in [12] in the general setting of nonharmonic analysis associated with a “model operator”.
A concrete example of the latter is our choice of the anharmonic oscillator with a potential that
is a homogeneous polynomial of a certain degree, and with derivatives of any order; in particular
the operator we consider is of the form 1
(−∆)l +|x|2k +1, x ∈Rn ,
where l ,k ≥ 1 are some fixed integers, | · | stands for the Euclidean norm on Rn , and ∆ is the usual
Laplace operator onRn . Under such analysis a Fourier multiplier is Lp−Lq bounded if a condition
similar to (1) is satisfied for a symbol of the multiplier adapted to our setting.
2. Notation and preliminaries
In order to present our results, let us briefly recall the necessary notation and concepts involved
in our analysis. For a more detailed discussion on concepts related to the spectral analysis of the
anharmonic oscillator we refer to [6, 7], and for those related to the associated Fourier analysis
we refer to [7], or, for a more general consideration, to [12].
2.1. The anharmonic oscillator and its spectrum
The anharmonic oscillator is a differential operator on Rn that is served as a generalisation of
the harmonic oscillator. Here we consider an anharmonic oscillator in the family of operators
(Ak, l )k, l , for different values of the integer parameters k, l ≥ 1, defined as follows
Ak, l := (−∆)l +|x|2k +1, x ∈Rn .
1The addition of the constant 1 to the potential is required for the invertibility of the operator and for controlling the
lower bound of the spectrum of the operator.
C. R. Mathématique — 2022, 360, 343-347
Marianna Chatzakou and Vishvesh Kumar 345
Here and thereafter we assume that the parameters k, l are fixed, and the notation will be
simplified to A := Ak, l .
The operator A is densely defined on its domain S (Rn) ⊂ L2(Rn) and admits a (unique) self-
adjoint extension on L2(Rn). Its spectrum is purely discrete and lies in the interval [1,∞). If we
keep the same notation for its self adjoint extension, then by the functional calculus we have, for
m ∈R,
Dom
(
Am
)={u ∈ L2 (Rn) : ∞∑
j=1
λ j
2m ∣∣〈u j ,u〉L2 ∣∣2 <∞
}
,
where the set (u j ) j , with ‖u j ‖L2 = 1, is the set of normalized eigenfunctions of A, and (λ j ) j is the
corresponding set of eigenvalues. Thus, for u ∈ Dom (Am), we have
Amu =
∞∑
j=1
λmj
〈
u j ,u
〉
L2 u j .
Despite the intensive research on the eigenvalue problem associated with the anhramonic oscil-
lator, the exact solution of it, even in the quartic case, is still explicitly unknown. We refer to [6]
for a recent exposition of the related works. Let us summarise the spectral information that is
known in the case of the anharmonic oscillator A: For the inverse operator A−1 we know that
A−1 ∈ Sr (L2(Rn)) for r > (k+l )n2kl , where by Sr (L2(Rn)) we have denoted the r th Schatten–von Neu-
mann class of operators on the Hilbert space L2(Rn), see [6, Corollary 5.3]. As a consequence, for
the eigenvalue counting function N (λ) := #{ j :λ j <λ}, we get the following asymptotic estimate
N (λ).λn
( 1
2k + 12l
)
, as λ→∞ , (2)
where we have assumed that the eigenvalues (λ j ) j are arranged in the decreasing order; see [6,
Remark 5.7], or [9, Theorem 5.5].
2.2. Fourier analysis associated with the anharmonic oscillator
For the setup of this section we follow [12].
The space of functions
C∞A
(
Rn
)
:=∩∞m=1 Dom
(
Am
)
,
will be the domain of the Fourier operator FA in our setting. The space C∞A is a Fréchet space if
endowed with the topology induced by the family of norms∥∥ f ∥∥C mA := maxk ≤m
∥∥∥Ak f ∥∥∥
L2(Rn )
, m ∈N0, f ∈C∞A
(
Rn
)
.
Then, defining the A-Fourier transform by(
FA f
)(
j
)
:=
∫
Rn
f (x)u j (x)d x ,
we see that FA(C∞A ) ⊂S (N), where S (N) is the space of rapidly decreasing functions φ :N→ C,
i.e., for any N <∞, there exists a constant Cφ,N > 0 such that∣∣φ( j )∣∣≤Cφ, N 〈 j〉−N for all j ∈N .
The Fréchet topology of S (N) is induced by the family of semi-norms {pr }r ∈N given by
pr
(
φ
)
:= sup
j ∈N
〈
j
〉r ∣∣φ( j )∣∣ .
With the topological structures as described above, the map FA becomes a bijective homeomor-
phism form C∞A to S (N), and the Fourier inversion formula becomes available in our setting.
C. R. Mathématique — 2022, 360, 343-347
346 Marianna Chatzakou and Vishvesh Kumar
Moreover, for f ∈ L2, the operator FA becomes a bijective isomorphism onto the Hilbert space l 2A
defined as the space of sequences f = ( f j ) j ∈N under the norm
‖ f ‖l 2A :=
( ∑
j ∈N
∣∣ f j ∣∣2
) 1
2
.
The latter reads as the Plancherel identity in our setting, i.e., in symbols: ‖ f ‖L2 = ‖FA( f )‖l 2A .
3. Main results
The Hausdorff–Young inequality in our setting is derived verbatim as a particular case of the
Fourier analysis associated with a “model operator” [12]. The Paley-type inequality is more
involved and its proof can be found in the preprint [7, Theorem 4.1]. Then, by means of the
Stein–Weiss interpolation theorem, we can extend the Hausdorff–Young–Paley inequality in our
setting [7, Theorem 4.4] which is the key for proving our result on the boundedness of Fourier
multipliers as follows. We refer to [7] for the detailed proofs of the results below.
First, let us recall that Tσ is an A-Fourier multiplier, if it satisfies the identity
FA (Tσ)
(
j
)=σ( j )FA f ( j ) , j ∈N,
whereσ :N→C is a function that can be served as the symbol of the multiplier Tσ for our analysis.
Theorem 1. Let 1 0
s
 ∑
j ∈N
|σ( j )|≥ s
∥∥u j ∥∥2L∞(Rn )

1
p − 1q
. (3)
Let us mention that (3) is well-defined since for the L∞-estimate of eigenfunctions we have:
‖u j ‖L∞ ≤C
√
λ j , for some C > 0 and for all j ; see [7, Lemma 2.2].
We note here that Theorem 1 can be thought as a particular case of the recently obtained
result [13, Theorem 4.8] in a more general setting of [12].
As a consequence of Theorem 1 and of the spectral decay of the anharmonic oscillator A in (2)
we derive the following result on the boundedness of the spectral multipliers in our setting.
Theorem 2. Let 1 < p ≤ 2 ≤ q < ∞, and let ϕ be any decreasing function on [1,∞) such that
limu→∞ϕ(u) = 0. Then, ∥∥ϕ(A)∥∥Lp (Rn )→Lq (Rn ) . sup
u≥1
ϕ(u)
(
u1+
(k+l )n
2kl
) 1
p − 1q
.
Let us summarise some applications of Theorem 2.
Corollary 3. Let 1 < p ≤ 2 ≤ q <∞. Then, the operator A−m is bounded from Lp (Rn) to Lq (Rn) for
m ≥ (1+ (k+l )n2kl )( 1p − 1q ).
Corollary 3 implies in particular that A−m is bounded from L2(Rn) to L2(Rn) for any m ≥ 0. This
is known in more generality by the general theory of pseudo-differential operators, see Beals [4,
Theorem 3.1], which gives us A−m ∈L (Lp ), for any m > 0, and for any p ∈ (1,∞). However, using
Corollary 3 we can study the Lp -Lq boundedness for p 6= q .
Corollary 4. If v = v(x, t ) is the solution to the heat equation associated to A, i.e., if we have
∂t v + Av = 0, v(0) = v0 ,
C. R. Mathématique — 2022, 360, 343-347
Marianna Chatzakou and Vishvesh Kumar 347
then the time decay rate for v is estimated as:∥∥v(t , ·)∥∥Lq (Rn ) . t−(1+ (k+l )n2kl )( 1p − 1q ) ‖v0‖Lq (Rn ) ,
where 1 < p ≤ 2 ≤ q <∞.
Corollary 5. Let 1 < p ≤ 2 ≤ q <∞. Then, for m ≥ (1+ (k+l )n2kl )( 1p − 1q ) we obtain the Sobolev-type
estimate:
‖ f ‖Lq (Rn ) .
∥∥Am f ∥∥Lp (Rn ) .
4. Acknowledgements
The authors would like to thank the anonymous reviewers who read and commented on this
note. We are grateful to Prof. Micheal Ruzhansky for valuable suggestions.
References
[1] R. K. Akylzhanov, E. Nursultanov, M. Ruzhansky, “Hardy–Littlewood–Paley inequalities and Fourier multipliers on
SU (2)”, Stud. Math. 236 (2016), no. 1, p. 1-29.
[2] ——— , “Hardy–Littlewood, Hausdorff–Young–Paley inequalities, and Lp -Lq Fourier multipliers on compact homo-
geneous manifolds”, J. Math. Anal. Appl. 479 (2019), no. 2, p. 1519-1548.
[3] R. K. Akylzhanov, M. Ruzhansky, “Lp −Lq multipliers on locally compact groups”, J. Funct. Anal. 278 (2019), no. 3,
article no. 108324.
[4] R. Beals, “Lp Hölder estimates for pseudo-differential operators: sufficient conditions”, Ann. Inst. Fourier 29 (1979),
no. 3, p. 239-260.
[5] A. Carbery, A. Seeger, “H p and Lp -Variants of multiparameter Calderón–Zygmund theory”, Trans. Am. Math. Soc.
334 (1992), no. 2, p. 719-747.
[6] M. Chatzakou, J. Delgado, M. Ruzhansky, “On a class of anharmonic oscillators”, https://arxiv.org/abs/1811.12566v3,
to appear in Journal de Mathématiques Pures et Appliquées, 2020.
[7] M. Chatzakou, V. Kumar, “Lp −Lq boundedness of Fourier multipliers associated with the anharmonic Oscillator”,
https://arxiv.org/abs/2004.07801v1, 2020.
[8] M. G. Cowling, S. Giulini, S. Meda, “Lp −Lq estimates for functions of theLaplace–Beltrami operator on noncompact
symmetric spaces. I”, Duke Math. J. 72 (1993), no. 1, p. 109-150.
[9] B. Helffer, D. Robert, “Comportement asymptotique précise du spectre d’opérateurs globalement elliptiques dans
Rn ”, in Goulaouic-Meyer-Schwartz Seminar, 1980–1981, École Polytech., Palaiseau, 1981.
[10] L. Hörmander, “Estimates for translation invariant operators in Lp spaces”, Acta Math. 104 (1960), p. 93-140.
[11] S. G. Mikhlin, “On multipliers of Fourier integrals”, Dokl. Akad. Nauk SSSR 109 (1956), p. 73-84.
[12] M. Ruzhansky, N. Tokmagambetov, “Nonharmonic analysis of boundary value problems”, Int. Math. Res. Not. 12
(2016), p. 3548-3615.
[13] D. C. Sánchez, V. Kumar, M. Ruzhansky, N. Tokmagambetov, “Lp −Lq boundedness of pseudo-differential operators
on smooth manifolds and its applications to nonlinear equations”, https://arxiv.org/abs/2005.04936, 2020.
C. R. Mathématique — 2022, 360, 343-347

$L^p$-$L^q$ Boundedness of Spectral Multipliers of the Anharmonic Oscillator

Ramiro Miguel Acevedo Martínez

Christian Camilo Gómez Mosquera

Finite element error estimates for a mixed degenerate parabolic model

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Completeness of coherent state subsystems for nilpotent Lie groups

Mitia Duerinckx

Antoine Gloria

A short proof of Gevrey regularity for homogenized coefficients of the Poisson point process

Yong-Gao Chen

Yuchen Ding

Comptes Rendus
Mathématique
Yong-Gao Chen and Yuchen Ding
On a conjecture of Erdős
Volume 360 (2022), p. 971-974
https://doi.org/10.5802/crmath.345
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
2022, Vol. 360, p. 971-974
https://doi.org/10.5802/crmath.345
Number theory / Théorie des nombres
On a conjecture of Erdős
Yong-Gao Chena and Yuchen Ding∗, b
a School of Mathematical Sciences, Nanjing Normal University, Nanjing 210023,
People’s Republic of China
b School of Mathematical Sciences, Yangzhou University, Yangzhou 225002, People’s
Republic of China
E-mails: ygchen@njnu.edu.cn, ycding@yzu.edu.cn
Abstract. In this note, we confirm an old conjecture of Erdős.
Mathematical subject classification (2010). 11A41, 11A67.
Funding. The first author was supported by the National Natural Science Foundation of China, Grant No.
12171243. The second author was supported by the Natural Science Foundation of Jiangsu Province of China,
Grant No. BK20210784. He was also supported by the foundations of the projects “Jiangsu Provincial Double–
Innovation Doctor Program”, Grant No. JSSCBS20211023 and “Golden Phoenix of the Green City–Yang Zhou”
to excellent PhD, Grant No. YZLYJF2020PHD051.
Manuscript received 24 January 2022, revised 10 February 2022, accepted 11 February 2022.
1. Introduction
Let P denote the set of all primes. In 1950, Erdős [5] made the following anecdotal conjecture:
Conjecture 1 (Erdős Conjecture.). Let c be any constant and x sufficiently large,
a1 < a2 < · · · < at É x, t > log x.
Then there exists an integer n so that the number of solutions of n = p + ai (p ∈ P ,1 ≤ i ≤ t ) is
greater than c.
Erdős [5] himself proved this conjecture for the case ai = 2i , which gives an affirmative answer
to a question of Turán. In a former note [3], the second author proved this conjecture for the
case ai | ai+1 with its quantitative form, which is a slight generalization of Erdős’ result. In a
subsequent note, the second author and Zhou [4] proved the conjecture for the case ai = 2pi ,
where pi is the i -th prime. This case was conjectured by the first author [1] years ago. Shortly
after, the authors of the present note recognized that the complete proof of Erdős’ conjecture
actually follows directly from a new achievement of the distributions of the primes established
by Maynard–Tao [7, 8]. We keep record here as the closure of this longstanding conjecture.
∗Corresponding author.
ISSN (electronic) : 1778-3569 https://comptes-rendus.academie-sciences.fr/mathematique/
972 Yong-Gao Chen and Yuchen Ding
In this note, the following general results are proved. The Erdős Conjecture follows from
Corollary 3.
Theorem 2. For any ` distinct integers a1, . . . , a`, there are infinitely many positive integers n such
that the number of solutions of n = p +ai (p ∈P ,1 ≤ i ≤ `) is greater than
1
8
log`−1.6.
From Theorem 2, we immediately have the following corollaries:
Corollary 3. Let x ≥ 2 and
a1 < a2 < · · · < at É x, t > log x.
Then there exist infinitely many integers n so that the number of solutions of n = p +ai (p ∈P ,1 ≤
i ≤ t ) is greater than
1
8
loglog x −1.6.
Corollary 4. Let A = {ai }∞i=1 be an infinite set of integers and let
fA (n) = #
{(
p, a
)
: n = p +a, p ∈P , a ∈A } ,
then
limsup
n→+∞
fA (n) =+∞.
2. Proofs
A set {b1, . . . , bk } is called an admissible set if there is no a fixed integer d > 1 such that d |
(n + b1) · · · (n + bk ) for all integers n. It is equivalent that for any prime p, {b1, . . . , bk } modulo
p occupies at most p −1 residues. We begin with the following deep result for the distribution of
the primes due to Maynard–Tao ( [8, Theorem 16]). We will use the following quantitative result
which was given by Granville [6] basing on Maynard [7].
Lemma 5. [6, Theorem 6.2] For any given integer m Ê 2, let k be a positive integer with k logk >
e8m+4. For any admissible set {b1, . . . , bk }, there are infinitely many integers n such that at least m
of n +b1, . . . , n +bk are prime numbers.
Lemma 6 ( [2, Lemma 3]). We have∏
3≤p ≤x
(
1− 1
p
)−1
≤ 0.923log x, x ≥ 74,
where the product is taken over all primes p with 3 ≤ p ≤ x.
Proof of Theorem 2. If `≤ e12, then
1
8
log`−1.6 ≤ 0,
and Theorem 2 is trivial. In the following, we assume that `> e12.
Let pi be the i th prime. Assume that a1, . . . , a` are ` distinct integers. For p1, one of residues
modulo p1 contains at most b`/p1c of a1, . . . , a`. So at least `−b`/p1c of a1, . . . , a` occupy at most
p1 −1 residues modulo p1. Let `0 = ` and `1 = `−b`/p1c. Without loss of generality, we assume
that a1, . . . , a`1 occupy at most p1−1 residues modulo p1. Similarly, without loss of generality, we
may assume that a1, . . . , a`2 occupy at most p2 −1 residues modulo p2, where `2 = `1 −b`1/p2c.
Continuing this process, at the t th step, we may assume that a1, . . . , a`t occupy at most pt − 1
residues modulo pt , where `t = `t−1−b`t−1/pt c. Since `≥ `1 ≥ ·· · and p1 < p2 < ·· · , there exists t
with `t < pt+1. Let s be the least integer with `s < ps+1. It is clear that {a1, . . . , a`s } is an admissible
Yong-Gao Chen and Yuchen Ding 973
set. Let m be largest integer with `s log`s > e8m+4. If m ≥ 2, then by Lemma 5, there are infinitely
many integers n such that at least m of n − a1, . . . , n − a`s are prime numbers. Since there are
infinitely many primes, it follows that there are infinitely many integers n such that at least one
of n −a1, . . . , n −a`s is prime number. So the conclusion is also true for m ≤ 1.
Now we establish an explicit relation between ` and m.
Since
`i+1 = `i −
⌊
`i
pi+1
⌋
≥ `i − `i
pi+1
= `i
(
1− 1
pi+1
)
, i = 0,1, . . . ,
it follows from the definition of s that
ps+1 > `s ≥ `s−1
(
1− 1
ps
)
≥ ·· · ≥ `
(
1− 1
p1
)
· · ·
(
1− 1
ps
)
> e12
(
1− 1
p1
)
· · ·
(
1− 1
ps
)
.
This cannot hold for s ≤ 100. So ps ≥ p100 = 541. Thus, by Lemma 6,
`s ≥ `
(
1− 1
p1
)
· · ·
(
1− 1
ps
)
≥ `
2
· 1
0.923log ps
= `
1.846log ps
.
By the definition of s, ps ≤ `s−1. Thus,
`s ≥
(
1− 1
ps
)
`s−1 ≥
(
1− 1
ps
)
ps = ps −1.
It follows that
`s ≥ `
1.846log ps
≥ log540
1.846log541
`
log
(
ps −1
) > 0.54`
log`s
.
So `s log`s ≥ 0.54`. In view of the definition of m,
e8m+12 ≥ `s log`s ≥ 0.54`.
So
m ≥ 1
8
log`− 12
8
+ log0.54
8
> 1
8
log`−1.6.
This completes the proof of Theorem 2. 
Proof of Corollary 3. Assume that
1 ≤ a1 < ·· · < at ≤ x, t > log x.
By Theorem 2, there are infinitely many positive integers n such that the number of solutions of
n = p +ai (p ∈P ,1 ≤ i ≤ t ) is greater than
1
8
log t −1.6 > 1
8
loglog x −1.6.
This completes the proof of Corollary 3. 
Proof of Corollary 4. By Theorem 2, there is a positive integer n such that the number of solu-
tions of n = p +ai (p ∈P ,1 ≤ i ≤ `) is greater than 18 log`−1.6. That is, fA (n) ≥ 18 log`−1.6. Now
Corollary 4 follows immediately. 
3. Remarks
It is known that there is a positive proportion of positive odd numbers that can be represented as
p+2k with k ∈N and p ∈P (See Romanoff [9]) and there is an arithmetical progression of positive
odd numbers none of which can be represented as p +2k with k ∈N and p ∈P (see Erdős [5]).
Since one can take k ≤ c exp((4 − 28157 )m) for some positive constant c in Lemma 5 (see [8,
Theorem 16]), it follows that 18 in Theorem 2 and Corollary 3 can be improved to any constant
less than (4− 28157 )−1.
974 Yong-Gao Chen and Yuchen Ding
References
[1] Y.-G. Chen, “Romanoff theorem in a sparse set”, Sci. China, Math. 53 (2010), no. 9, p. 2195-2202.
[2] Y.-G. Chen, X.-G. Sun, “On Romanoff’s constant”, J. Number Theory 106 (2004), no. 2, p. 275-284.
[3] Y. Ding, “Extending an Erdős result on a Romanov type problem”, Arch. Math. 118 (2022), p. 587-592.
[4] Y. Ding, G.-L. Zhou, “Some application of the admissible sets”, preprint.
[5] P. Erdős, “On integers of the form 2k +p and some related problems”, Summa Brasil. Math. 2 (1950), p. 113-123.
[6] A. Granville, “Primes in intervals of bounded length”, Bull. Am. Math. Soc. 52 (2015), no. 2, p. 171-222.
[7] J. Maynard, “Small gaps between primes”, Ann. Math. 181 (2015), no. 1, p. 383-413.
[8] D. H. J. Polymath, “Variants of the Selberg sieve, and bounded intervals containing many primes”, Res. Math. Sci. 1
(2014), article no. 12.
[9] N. P. Romanoff, “Über einige Sätze der additiven Zahlentheorie”, Math. Ann. 109 (1934), p. 668-678.

On a conjecture of Erdős

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