Comptes Rendus Mathématique

Ce Xu

Weiping Wang

Dirichlet type extensions of Euler sums

Vanderléa R. Bazao

César R. de Oliveira

Pablo A. Diaz

Comptes Rendus
Mathématique
Vanderléa R. Bazao, César R. de Oliveira and Pablo A. Diaz
On the Birman–Krein Theorem
Volume 361 (2023), p. 1081-1086
Published online: 24 October 2023
https://doi.org/10.5802/crmath.473
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. 1081-1086
https://doi.org/10.5802/crmath.473
Functional analysis / Analyse fonctionnelle
On the Birman–Krein Theorem
Sur le théorème de Birman–Krein
Vanderléa R. Bazaoa, César R. de Oliveira∗, b and Pablo A. Diazb
a Faculdade de Ciências Exatas e Tecnologias, UFGD, Dourados, MS, 79804-970 Brazil
b Departamento de Matemática, UFSCar, São Carlos, SP, 13560-970 Brazil
E-mails: vanderleabazao@ufgd.edu.br, oliveira@dm.ufscar.br, pdaraos@gmail.com
Abstract. It is shown that if X is a unitary operator so that a singular subspace of U is unitarily equivalent
to a singular subspace of U X (or XU ), for each unitary operator U , then X is the identity operator. In other
words, there is no nontrivial generalization of Birman–Krein Theorem that includes the preservation of a
singular spectral subspace in this context.
Résumé. On montre que si X est un opérateur unitaire tel qu’un sous-espace singulier de U est unitairement
équivalent à un sous-espace singulier de U X (ou XU ), pour chaque opérateur unitaire U , alors X est
l’opérateur d’identité. En d’autres termes, il n’y a pas de généralisation non triviale du théorème de Birman–
Krein qui inclut la préservation d’un sous-espace spectral singulier dans ce contexte.
2020 Mathematics Subject Classification. 47A55, 47A10, 81Q10.
Funding. CRdO thanks the partial support by CNPq (under contract number 303689/2021-8), and PAD
thanks the support by CAPES (Brazilian agencies).
Manuscript received 14 June 2022, accepted 31 January 2023.
1. Introduction
We are interested in preservations of spectral types of unitary operators U , on a Hilbert space H ,
under multiplicative (compositions, in fact) perturbations, that is, if X is another unitary opera-
tor, the perturbation has the form
U 7→U X . (1)
This is a right perturbation, and U 7→ XU is a left perturbation. Note that both XU and U X
are again unitary operators. Some notation: we shall denote the identity operator by 1 and
T1 ∼= T2 means that the linear operators T1 and T2 are unitarily equivalent; µ ⊥ ν means that
the measures µ and ν are mutually singular; T= {c ∈C | |c| = 1} is the one-dimensional torus and
the letter c will always denote an element of T.
The main question to be investigated is what kind of spectral types of (any) U are preserved
under such nontrivial perturbations X ̸= 1, and this study was motivated by the corresponding
∗Corresponding author.
ISSN (electronic) : 1778-3569 https://comptes-rendus.academie-sciences.fr/mathematique/
1082 Vanderléa R. Bazao, César R. de Oliveira and Pablo A. Diaz
question, addressed by Howland [7], in the context of additive perturbations of self-adjoint op-
erators. Howland has concluded that there is no nontrivial generalization of Kato–Rosenblum
Theorem, that is, only absolutely continuous spectra (with respect to Lebesgue measure) can be
(always) preserved under some nonzero perturbation (for instance, under trace class perturba-
tions). Our main conclusion is similar and appear in Theorem 6, that is, there is no generaliza-
tion of Birman–Krein Theorem [1], so that only the absolutely continuous spectral component of
(any) U can be preserved under certain multiplicative unitary perturbation X ̸=1 (either right or
left perturbations).
Our arguments follow the general lines of [7], and do not go along with the Cayley transform
since it does not take sums of self-adjoint operators to the multiplicative form (4) of unitary ones,
and since there were some nonobvious choices in the construction here, we have found the result
interesting enough to justify publication.
As an additional motivation, it is worth mentioning that multiplicative perturbations (1) of
unitary operators naturally appear in quantum versions of time-periodic kicked systems [3, 5].
Let A be a self-adjoint operator describing certain quantum system (a free one) that, at instants
of time jτ, j ∈Z, undergoes kicked perturbations by a self-adjoint operator B ; the formal operator
describing such system is
A + B ∑
j∈Z
δ(t − jτ),
whose time evolution between two consecutive kicks (the Floquet operator) is given by [3]
e−i B e−i Aτ, (2)
that is, multiplicative perturbations (e−i B ) of unitary operators (e−i Aτ). A historically important
model [3] is the so-called kicked rotator with A = −d2/dθ2 and B = κcosθ, 0 ≤ θ < 2π, and a
real parameter κ > 0; it is expected that, for “most” periods τ > 0, the corresponding Floquet
operator (2) has pure point spectrum and a localized dynamics, specially for κ > 1 (it is a region
where classical and quantum physics would disagree), but currently, there is a mathematical
proof [2] of quantum localized dynamics only for small enough κ.
A basic general discussion of multiplicative perturbations is given in Section 2, including
a precise statement of preservation of spectral types and of our main result, whose proof is
concluded in Section 3.
2. Multiplicative perturbations
Sometimes it is convenient to write the perturbation in the form X = e i Y , with Y a bounded self-
adjoint operator, so that
X = e i Y =
∞∑
j=0
(i Y ) j
j !
=1+
∞∑
j=1
(i Y ) j
j !
=1+W, (3)
with W =∑∞j=1 (i Y ) j / j !. In this way, we can write
U X =U (1+W ) =U +UW (4)
and if W is a trace class operator, then UW is also trace class and, by Birman–Krein Theorem, the
Lebesgue absolutely continuous subspaces of UW and U are unitarily equivalent for all U (the
same for left perturbations XU ).
An example is the case of multiplicative perturbations of rank one: given φ ∈H with ||φ|| = 1,
let Pφ be the projector operator onto the subspace Lin{φ} generated by φ, i.e.,
Pφξ=
〈
φ,ξ
〉
φ , ∀ ξ ∈H ,
Vanderléa R. Bazao, César R. de Oliveira and Pablo A. Diaz 1083
which is self-adjoint and idempotent. The corresponding perturbing unitary operator, with
“intensity” λ ∈R, is
Xλ = Xλ,φ := e iλPφ , (5)
and by writing e iλPϕ =1+W , it follows by (3) that
W ξ= (e iλ−1)Pφξ,
and the (right) perturbed operator has the action
Uλ :=U Xλ =U1+U (e iλ−1)Pφ, (6)
with U0 = U . To simplify statements, denote by µλψ the spectral measure of the pair (Uλ,ψ),
ψ ∈H , and usually one supposes that φ is cyclic for U , that is, the closure
Lin
{
U jφ | j ∈Z}=H ,
with U 0 =1. Note that ifφ is cyclic for U , then U kφ is also is cyclic for U , for all k ∈Z; furthermore,
if φ is cyclic for U , then it is also cyclic for Uλ for all λ ∈R.
A particularly important result, that will be employed ahead, is the following version, for
unitary operators, of the Aronszajn–Donoghue Theorem for self-adjoint ones (for proofs see [6,
Proposition 8.3] and [4, Proposition 9.1.14]):
Theorem 1. Let φ be cyclic for U . If λ1 −λ2 ̸= 2πn, for all n ∈ Z, then the singular parts of Uλ1
and Uλ2 are mutually singular, i.e., for all 0 ̸=ψ ∈ H , the singular parts of the spectral measures
µ
λ1
ψ and µ
λ2
ψ are mutually singular.
In order to discuss more general spectral subspaces, for ψ ∈ H , denote by µUψ the spectral
measure of the pair (U ,ψ) and, given a (nonzero) finite Borel measure µ on T, let
Hµ(U ) :=
{
ψ ∈H |µUψ ≪µ
}
,
which is a closed subspace of the Hilbert space, whose orthogonal complement is{
ψ ∈H |µUψ ⊥µ
}
.
Finally, denote by [U ]µ the restriction of U to Hµ(U ), i.e., [U ]µ :=U |Hµ(U ).
Definition 2. A unitary operator X preserves µ on the right if [U X ]µ ∼= [U ]µ, for all unitary
operators U . And X preserves µ on the left if [XU ]µ ∼= [U ]µ, for all unitary operators U .
Remark 3.
(a) The simple case X = c1 translates the spectra in T and so does not preserve measures
if c ̸= 1.
(b) As already mentioned, by Birman–Krein Theorem, if X = 1+W with trace class W , then
X preserves the Lebesgue measure ℓ on T.
(c) If ν≪µ, then
[[U ]µ]ν = [U ]ν. (7)
(d) For real t , define the translated (in T) measure
µt (·) :=µ
(
e i t ·
)
. (8)
Then
[U ]µt = e i t [U ]µ. (9)
It is enough to discuss right or left preservation of a measure, as Proposition 4 ensures. After
this proposition we will just say “X preserves µ” (as already employed above).
1084 Vanderléa R. Bazao, César R. de Oliveira and Pablo A. Diaz
Proposition 4. X preserves µ on the right if and only if X preserves µ on the left.
Proof. Let X ∗ denotes the adjoint of X . If X preservesµ on the right, then pick X ∗U (which is also
a unitary operator); by hypothesis [X ∗U X ]µ ∼= [X ∗U ]µ, and since X is unitary, [X ∗U X ]µ ∼= [U ]µ
and one obtains that [X ∗U ] ∼= [U ]µ, and so X ∗ preserves µ on the left. Now one has [U ]µ =
[X ∗XU ]µ ∼= [XU ]µ, and so X preserves µ on the left.
Similarly, one concludes the reciprocal. □
Corollary 5. X preserves µ if and only if X ∗ preserves µ.
Theorems 6 states the main result of this note.
Theorem 6. If the unitary operator X ̸=1 preserves µ, then µ≪ ℓ.
The completion of the proof of this theorem is the subject of Section 3; in the following we
present some basic and useful properties.
Proposition 7. Suppose that X preserves µ. Then:
(a) If ν≪µ, then X preserves ν.
(b) X preserves µt , for all t ∈R.
(c) If X ∼= Y , then Y preserves µ.
(d) If PE is an orthogonal reducing projection for X (projection onto the closed subspace
E ⊂H ), then PE X preserves µ on PE (H ) = E. Recall that E⊥ will also be reducing for X .
(e) If Y also preserves µ, then X Y preserves µ.
Proof. (a). [U X ]ν ∼= [[U X ]µ]ν ∼= [Uµ]ν ∼= [U ]ν.
(b). [U X ]µt = e i t [U X ]µ = [e i tU X ]µ and since e i tU is a unitary operator, we have [e i tU X ]µ ∼=
[e i tU ]µ =Uµt ; hence [U X ]µt = [U ]µt .
(c). If X ∼= Y , that is, Y =V X V ∗ for some unitary operator V , then
[U Y ]µ = [UV X V ∗]µ = [V (V ∗UV X V ∗V )V ∗]µ
∼= [V ∗UV X V ∗V ]µ = [V ∗UV X ]µ ,
and since V ∗UV is unitary and X preserves µ, one obtains
[V ∗UV X ]µ ∼= [V ∗UV ]µ ∼= [U ]µ,
that is, [U Y ]µ ∼= [U ]µ.
(d). Write H = E ⊕E⊥; then X = X |E ⊕X |E⊥ , which can be written as
X =
(
X |E 0
0 X |E⊥
)
.
Now, if Ũ and Ṽ are unitary operators acting on E and E⊥, respectively, then
U =
(
Ũ 0
0 Ṽ
)
is unitary on H . Thereby,
[U ]µ ∼= [U X ]µ =
[(
Ũ 0
0 Ṽ
)(
X |E 0
0 X |E⊥
)]
µ
=
(
[Ũ X |E ]µ 0
0 [Ṽ X |E⊥ ]µ
)
,
and since [U ]µ =
(
[Ũ ]µ 0
0 [Ṽ ]µ
)
, by equating the first components, one gets [Ũ X |E ]µ ∼= Ũµ, and so X |E
preserves µ on E = PE (H ).
(e). Indeed, if U is a unitary operator, then U X is also unitary and
[U X Y ]µ ∼= [U X ]µ ∼= [U ]µ. □
Vanderléa R. Bazao, César R. de Oliveira and Pablo A. Diaz 1085
3. Proof of Theorem 6
It is supposed that X preserves µ; the goal is to show that µ≪ ℓ. By Remark 3(a), we may suppose
that X ̸= c1. By item (c) of Proposition 7, R∗X R also preserves µ, for any unitary operator R. Pick
the operator R = Xλ from (5), with λ = π and some normalized vector φ (that will be selected
ahead), that is,
Rξ= Xπξ= ξ+ (e iπ−1)
〈
φ,ξ
〉
φ= ξ−2〈φ,ξ〉φ,
for all ξ ∈H .
By Proposition 7, items (c) and (e), and Corollary 5, the operators R∗X R and
Z := X ∗R∗X R
preserve µ as well. Explicitly, one has
R∗X Rξ=R∗X (ξ−2〈φ,ξ〉φ)
=R∗ (X ξ−2〈φ,ξ〉 Xφ)
=R∗(X ξ)−R∗ (−2〈φ,ξ〉 Xφ)
=R∗(X ξ)−2〈φ,ξ〉R∗ (Xφ)
=X ξ−2〈φ, X ξ〉φ−2〈φ,ξ〉(Xφ−2〈φ, Xφ〉φ)
=X ξ+ (4〈φ,ξ〉〈φ, Xφ〉−2〈φ, X ξ〉)φ−2〈φ,ξ〉 Xφ,
and if ψ= X ∗φ,
Zξ= ξ+ [4〈φ,ξ〉〈φ, Xφ〉−2〈φ, X ξ〉]ψ−2〈φ,ξ〉φ =: (1+W )ξ,
with
W ξ= [4〈φ,ξ〉〈φ, Xφ〉−2〈φ, X ξ〉]ψ−2〈φ,ξ〉φ .
Since X ̸= c1, one may choose φ so that the set {φ,ψ} is linearly independent and so the
operator W has rank precisely 2 (note that if X = c1, then W = 0, as expected). Let e1 and e2 be
two normalized and independent eigenvectors of W and note that the corresponding eigenvalues
do not vanish, otherwise W would have rank smaller than 2. Observe that e1 and e2 are also
eigenvectors of the unitary operator Z , hence (one may suppose that) e1 ⊥ e2. If E is the
orthogonal complement of e1, which reduces Z , it follows, by Proposition 7(d), that the operator
Z2 := Z |E also preserves µ on E , and since we now have a nonzero rank one perturbation, Z2 has
the form (5), that is,
Z2 =1+ (e iλ−1)Pe2 ,
for some λ ̸= 2πk,k ∈Z.
Finally, pick a unitary operator U̇ on E with e2 one of its cyclic vectors; hence
U̇ Z2 = U̇ +U̇ (e iλ−1)Pe2
has the form (6). Since Z2 preserves µ,
[U̇ Z2]µ ∼= [U̇ ]µ
and, by Theorem 1, if µ has a (nonzero) singular component, Z2 could not preserve µ, and one
concludes that µ is absolutely continuous with respect to Lebesgue measure.
1086 Vanderléa R. Bazao, César R. de Oliveira and Pablo A. Diaz
References
[1] M. S. Birman, M. G. Kreı̆n, “On the theory of wave operators and scattering operators”, Dokl. Akad. Nauk SSSR 144
(1962), p. 475-478, English translation: Soviet Math. 3 (1962), p. 740–744.
[2] J. Bourgain, “Estimates on Green’s functions localization and the quantum kicked rotor model”, Ann. Math. 156
(2002), no. 1, p. 249-294.
[3] G. Casati, L. Molinari, “Quantum Chaos with time-periodic Hamiltonians”, Prog. Theor. Phys., Suppl. 98 (1989), p. 287-
322.
[4] J. A. Cima, A. L. Matheson, W. T. Ross, The Cauchy Transform, Mathematical Surveys and Monographs, vol. 125,
American Mathematical Society, 2006.
[5] M. Combescure, “Spectral properties of a periodically kicked quantum Hamiltonian”, J. Stat. Phys. 59 (1990), no. 3-4,
p. 679-690.
[6] S. R. Garcia, W. T. Ross, “Model spaces: a survey”, in Invariant subspaces of the shift operator. CRM workshop (Montréal
2013), Contemporary Mathematics, vol. 638, American Mathematical Society, 2015, p. 197-245.
[7] J. S. Howland, “On the Kato-Rosenblum Theorem”, Pac. J. Math. 123 (1986), p. 329-335.


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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
eigenspaces”, Proc. Lond. Math. Soc. 125 (2022), no. 4, p. 759-777.
[6] L. Fefferman, Charles, “The Multiplier Problem for the Ball”, Ann. Math. 94 (1971), no. 2, p. 330-336.
[7] L. Grafakos, Classical Fourier Analysis, Graduate Texts in Mathematics, Springer, 2014.
[8] ——— , Modern Fourier Analysis, Graduate Texts in Mathematics, Springer, 2014.
[9] Y. Katznelson, An Introduction to Harmonic Analysis, 3 ed., Cambridge Mathematical Library, Cambridge University
Press, 2004.
[10] G. N. Watson, A Treatise on the Theory of Bessel Functions, Cambridge Mathematical Library, Cambridge University
Press, 1995.
[11] G. M. Wing, “The Mean Convergence of Orthogonal Series”, Am. J. Math. 72 (1950), no. 4, p. 792-808.
[12] A. Zygmund, R. Fefferman, Trigonometric Series, 3 ed., Cambridge Mathematical Library, Cambridge University
Press, 2003.


Remarks on the $L^p$ convergence of Bessel–Fourier series on the disc

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