Annales de l’institut Fourier (AIF)

Aiello, Valeriano

Nagnibeda, Tatiana

On the $3$-colorable subgroup $\protect \mathcal{F}$ and maximal subgroups of Thompson’s group $F$

Fang, Yanbo

Metrised ample line bundles in non-Archimedean geometry

Nguyen, Kieu Hieu

Un cas PEL de la conjecture de Kottwitz

Gelfert, Katrin

Ruggiero, Rafael O.

Geodesic flows modeled by expansive flows: Compact surfaces without conjugate points and continuous Green bundles

Agol, Ian

Stover, Matthew

Congruence RFRS towers

Lyczak, Julian

Order $5$ Brauer–Manin obstructions to the integral Hasse principle on log K3 surfaces

Balestrieri, Francesca

Johnson, Alexis

Newton, Rachel

Explicit uniform bounds for Brauer groups of singular K3 surfaces

Birkbeck, Christopher

Heuer, Ben

Williams, Chris

Overconvergent Hilbert modular forms via perfectoid modular varieties

Martínez, Ángel D.

Université Grenoble Alpes
ANNALES DE
L’INSTITUT FOURIER
Ángel D. Martínez
Córdoba’s differentiation theorem: revisited
Tome 73, n
o
2 (2023), p. 559-565.
https://doi.org/10.5802/aif.3535
Article mis à disposition par son auteur selon les termes de la licence
Creative Commons attribution – pas de modification 3.0 France
http://creativecommons.org/licenses/by-nd/3.0/fr/
C EN T R E
MER S ENN E
Les Annales de l’Institut Fourier sont membres du
Centre Mersenne pour l’édition scientifique ouverte
www.centre-mersenne.org e-ISSN : 1777-5310
Ann. Inst. Fourier, Grenoble
73, 2 (2023) 559-565
CÓRDOBA’S DIFFERENTIATION THEOREM:
REVISITED
by Ángel D. MARTÍNEZ (*)
Abstract. — In this paper we prove an exponential covering lemma implying
the three dimensional case of a well-known conjecture formulated by A. Zygmund
circa 1935 and solved by A. Córdoba in 1978. Our approach avoids a subtle argu-
ment involving the power series of the exponential function.
Résumé. — Dans cet article, on démontre un lemme de recouvrement exponen-
tiel impliquant le cas tridimensionnel d’une conjecture bien connue formulée par
A. Zygmund circa 1935 et prouvée par A. Córdoba en 1978. Notre approche évite
un argument subtil qui fati appel à la série entière de la fonction exponentielle.
1. Introduction
The concept of maximal function has been central in mathematical analy-
sis since its appeareance in Hardy and Littlewood’s seminal work [14]. It al-
lows to obtain the so-called Lebesgue’s differentiation theorem in a quanti-
tative way. In 1935 Jessen, Marcinkiewicz and Zygmund proved that instead
of balls or cubes one may also differentiate using the class Bn, of shrinking
n-dimensional rectangles (paralellepipeds) with sides parallel to the axes,
provided the function is locally in the class L1(1 + log+ L1)n−1(Rn). The
quantitative version of this result is known as the strong maximal theorem:
let us define
Mnf(x) = sup
x∈R∈Bn
1
m(R)
∫
R
|f(y)|dm(y)
where f is a locally integrable function and m denotes Lebesgues’s measure
in Rn. Then we have:
Keywords: Maximal functions, Differentiation theorem, Covering lemma.
2020 Mathematics Subject Classification: 42B25.
(*) The author was partially supported by MTM2014-56350-P project of the MCINN
(Spain). This material is based upon work supported by the National Science Foundation
under Grant No. DMS-1638352.
560 Ángel D. MARTÍNEZ
Theorem 1.1 (Jessen, Marcinkiewicz and Zygmund, [15]). — The class
Bn differentiates functions f which are locally in L1(1 + log+ L1)n−1(Rn),
that is, we have
lim
diam(R)→0
1
m(R)
∫
R
f(y)dm(y) = f(x) for a.e. x ∈ Rn
where the limit is taken over all rectangles such that x ∈ R ∈ Bn. Quanti-
tatively, the following inequality
m{x ∈ Rn : Mnf(x) > α} ⩽ Cn
∫
Rn
f(x)
α
(
1 + log+
f(x)
α
)n−1
dm(x)
holds for some constant Cn that depends only on the dimension.
Their proof relies on an iteration scheme based on the one dimensional
result and making use of the product structure of Rn. Quite surprisingly the
Orlicz space, L1(1 + log+ L1)n−1(Rn), is sharp if one assumes all its func-
tions to be differentiated by Bn (cf. [15], theorem 8). It should be pointed
out that the quantitative part of the statement appeared later (cf. [10]
or [12]).
Nowadays the standard proof of the weak (1, 1) inequality for the Hardy-
Littlewood maximal function (i.e. the case of cubes or balls in Rn) relies
on the Vitali covering lemma which allows to deduce this estimates as a
consequence of the geometry of balls. Standard references are the treatises
of de Guzmán [13] and Stein [18] where Cn = 5n.
It was not until 1975 when the exponential type covering needed to prove
geometrically the strong maximal theorem was finally discovered (cf. [8]).
The proof obtained by the A. Córdoba and R. Fefferman makes use of
certain sparseness properties. Let us introduce the one we shall need, using
their notation:
(P1) A sequence of rectangles {Rj} satisfies the P1 sparseness property
if for any k the following inequality holds:
m
Rk ∩ ⋃
j<k
Rj
 ⩽ 12m(Rk).
In 1978 A. Córdoba took advantage of the geometric approach to settle
Zygmund’s conjecture in the three-dimensional case:
Theorem 1.2 (A. Córdoba, [3, 4, 5]). — Given any function ϕ non
decreasing in each variable separately let Bϕ(R3) consist of rectangles
whose lengths are of the form (x, y, ϕ(x, y)). Then it differentiates L1(1 +
log+ L1)(R3).
ANNALES DE L’INSTITUT FOURIER
CÓRDOBA’S DIFFERENTATION THEOREM: REVISITED 561
In fact, Córdoba’s differentiation theorem is stronger but let us state it
in this way for the sake of the exposition’s clarity. His elegant proof relies
heavily on the three dimensional setting. No purely analytical proof has
been found to-date. For applications of this result we refer the reader to the
original works of A. Córdoba. Zygmund’s curiosity was not satisfied by this
and asked for further generalizations of this result in arbitrary dimension.
Some of them were dismissed in [17] by F. Soria. His counterexamples have
side lenghts of the form (s, ϕ(s, t), ψ(s, t)) where the functions ϕ and ψ are
monotone increasing. Recently more counterexamples have been found by
G. Rey in [16]. The rest remain an intriguing open question.
In the next section we provide a geometric covering result that implies
Theorem 1.2, we refer the reader to the literature for this implication (cf. [3,
5, 8]). Our approach neatly shows why and where the argument breaks
down in the higher dimensional case.
Acknowledgments
The author would like to express his gratitude to Antonio Córdoba for
introducing him to the subject in a graduate course delivered in Fall 2014
at UAM-ICMAT and Eric Latorre for his (unpublished) lecture notes [1].
The author is indebted to the anonymous referees for their careful reading
of the manuscript, suggestions and for pointing out references [10] and [12]
to his attention.
2. Proof of Theorem 1.2
Let us now state the main result of this paper:
Theorem 2.1 (A. Córdoba, [3]). — Given any function ϕ non decreas-
ing in each variable separately let Bϕ(R3) consist of rectangles whose lengths
are of the form (x, y, ϕ(x, y)). There exist dimensional constants C, c > 0,
but otherwise independent, such that for any {Rα} ⊆ Bϕ(R3) there exists
a subsequence {Rj} such that
(1) m
(⋃
α
Rα
)
⩽ Cm
⋃
j
Rj
 .
(2)
∫⋃
j
Rj
exp c
∑
j
χRj
 dm ⩽ Cm
⋃
j
Rj
 .
TOME 73 (2023), FASCICULE 2
562 Ángel D. MARTÍNEZ
Proof. — it is well known that without loss of generality we may assume
that a priori the rectangles {Rj} to be a finite sequence ordered by decreas-
ing length of their third side length and satisfying the P1 property. This
reduction respects (1) from the statement (cf. [8]). We will denote by R∗ a
rectangle centered in the same point and dilated by a factor of three. The
introduction of this will be apparent later when we require intersections to
be clean.
We select the rectangles following the following algorithm. Having chosen
k − 1 of them, we will choose Rk to be the next enlisted rectangle R that
satisfies
(2.1) 1
m(R)
∫
R
exp
k−1∑
j=1
χR∗
j
 dm ⩽ 3.
This can be shown to imply property (2) from the statement. Denoting
Ik =
∫⋃k
j=1
Rj
exp
 k∑
j=1
χR∗
j
 dm
let us observe that
Ik =
∫⋃k−1
j=1
Rj\Rk
exp
k−1∑
j=1
χR∗
j
 dm+ ∫
Rk
exp
 k∑
j=1
χR∗
j
dm.
The choice of of Rk shows that Ik ⩽ Ik−1 + 3em(Rk) which by induction
implies Ik ⩽ 3e
∑k
j=1 m(Rj). From this and the P1 property our claim
follows easily, namely
(2.2)
∫⋃k
j=1
Rj
exp
 k∑
j=1
χR∗
j
dm ⩽ 6em
⋃
j
Rj
 .
This shows that, following this sieve, part (2) will hold with c = 1 and it
only rests to show that it also respects (1). To do so we observe that for a
rejected rectangle R the negation of the inequality (2.1) implies
3 <
∞∑
k=0
Ake
k
holds, where we are using the notation A0 = 1 and
Ak =
m({x ∈ R :
∑
j χR∗j (x) = k})
m(R)
for k > 0. Notice that Ak is the proportion of points in R that belong to
exactly k rectangles. Let us denote by R1j those rectangles previous to R in
ANNALES DE L’INSTITUT FOURIER
CÓRDOBA’S DIFFERENTATION THEOREM: REVISITED 563
the selection process whose first and third side lengths are greater to those
of R. Analogously, from the remaining rectangles, we denote by R2j those
with second and third lengths exceeding those of R. Decomposing Ak in
different cases (namely, the k = r + s intersections being the result of the
intersection of r rectangles from the first class and s from the second) the
above inequality is equivalent to
3 <
∞∑
k=0
∑
r+s=k
Ar,se
k
where
Ar,s =
m({x ∈ R :
∑
j χR1∗j (x) = r and
∑
j χR2∗j (x) = s})
m(R) .
Figure 2.1. Schematic (x1, x2)-section of the intersection of R with
three vertical and two horizontal rectangles (respectivelyR1∗j andR2∗j ).
In different shades of gray A1,0, A0,1 (lighter) and A1,1 (darker).
TOME 73 (2023), FASCICULE 2
564 Ángel D. MARTÍNEZ
Let us introduce a related piece of notation now. Let a0 = 1 and in
general
ar =
m({x ∈ R :
∑
j χR1∗j (x) = r})
m(R)
for r > 0. Similarly, let b0 = 1 and
bs =
m({x ∈ R :
∑
j χR2∗j (x) = s})
m(R)
for s > 0. It is now obvious from the clean intersections of R and Rℓ∗j ,
ℓ = 1, 2, (see Figure 2.1) that Ar,s ⩽ arbs holds from which it follows that
3 <
( ∞∑
r=0
are
r
)( ∞∑
s=0
bse
s
)
.
(At this point the proof breaks down in higher dimensions.) Reading back
what this means it implies
max
ℓ=1,2
{
1
mxℓ(R)
∫
R
exp
(
k∑
i=1
χR∗
i
)
dmxℓ
}
>
√
3 − 1
where mxℓ is the length in the ℓth direction. The integral mean is a maximal
function of the exponential restrected to any line intersecting R and parallel
to the xℓ axis. We will denote such a (one dimensional) maximal operator by
Mℓ. The proof concludes estimating the size of the set covered by rejected
rectangles. For this purpose one observes that it is included in{
x : M1 exp
(
k∑
i=1
χR∗
i
)
(x) >
√
3 − 1 or M2 exp
(
k∑
i=1
χR∗
i
)
(x) >
√
3 − 1
}
whose measure can be bounded appropiately by the weak L1 boundedness
of the Hardy-Littlewood maximal function. Indeed, the maximal functions
appearing above are one dimensional means in the first and second coordi-
nates alone to which the weak L1 bound applies linewise. Integrating this
L1 bounds on the rest of variables and using Theorem 1.1 for n = 1 one
gets
m
⋃
β
Rβ
 ⩽ 5√
3 − 1
∫⋃
Ri
exp
(
k∑
i=1
χR∗
i
)
dm
where β extends over the indices of rejected rectangles. This can be bounded
using equation (2.2) to conclude (1). □
Remark. — It is easy to show that this argument also works for a family
satisfying that in some order any given R has two sidelengths smaller than
any previous rectangle.
ANNALES DE L’INSTITUT FOURIER
CÓRDOBA’S DIFFERENTATION THEOREM: REVISITED 565
BIBLIOGRAPHY
[1] A. Córdoba, “Calderón–Zygmund Theory and Beyond: Fourier Analysis, Uncer-
tainty Principles and Modeling (Lecture notes written by E. Latorre, Fall 2014)”,
unpublished.
[2] ——— , “On the Vitali covering properties of a differentiation basis”, Studia Math.
57 (1976), no. 1, p. 91-95.
[3] ——— , “s × t × Φ(s, t)”, Mittag-Leffler Institute Report, 9, 1978.
[4] ——— , “Maximal functions: a proof of a conjecture of A. Zygmund”, Bull. Amer.
Math. Soc. (N.S.) 1 (1979), no. 1, p. 255-257.
[5] ——— , “Maximal functions: a problem of A. Zygmund”, in Euclidean harmonic
analysis (Proc. Sem., Univ. Maryland, College Park, Md., 1979), Lecture Notes in
Math., vol. 779, Springer, Berlin, 1980, p. 154-161.
[6] ——— , “La razón geométrica del Teorema Fundamental del Cálculo”, Gac. R. Soc.
Mat. Esp. 3 (2000), no. 3, p. 435-446.
[7] ——— , All that Math: portraits of mathematicians as young researchers. Celebrat-
ing the Centennial of Real Sociedad Matemática Española, Revista Matemática
Iberoamericana, 2011, xiv+346 pages.
[8] A. Córdoba & R. Fefferman, “A geometric proof of the strong maximal theorem”,
Ann. of Math. (2) 102 (1975), no. 1, p. 95-100.
[9] ——— , “A geometric proof of the strong maximal theorem”, Bull. Amer. Math.
Soc. 81 (1975), no. 5, p. 941.
[10] N. A. Fava, “Weak type inequalities for product operators”, Studia Math. 42
(1972), p. 271-288.
[11] R. Fefferman & J. Pipher, “A covering lemma for rectangles in Rn”, Proc. Amer.
Math. Soc. 133 (2005), no. 11, p. 3235-3241.
[12] M. de Guzmán, “An inequality for the Hardy-Littlewood maximal operator with
respect to a product of differentiation bases”, Studia Math. 49 (1973/74), p. 185-
194.
[13] ——— , Differentiation of integrals in Rn, Lecture Notes in Mathematics, vol. 481,
Springer-Verlag, Berlin-New York, 1975, xii+266 pages.
[14] G. H. Hardy & J. E. Littlewood, “A maximal theorem with function-theoretic
applications”, Acta Math. 54 (1930), no. 1, p. 81-116.
[15] B. Jessen, J. Marcinkiewicz & A. Zygmund, “Note on the differentiability of
multiple integrals”, Fundamenta Mathematicae 25 (1935), no. 1 (eng), p. 217-234.
[16] G. Rey, “Another counterexample to Zygmund’s conjecture”, Proc. Amer. Math.
Soc. 148 (2020), no. 12, p. 5269-5275.
[17] F. Soria, “Examples and counterexamples to a conjecture in the theory of differ-
entiation of integrals”, Ann. of Math. (2) 123 (1986), no. 1, p. 1-9.
[18] E. M. Stein, Singular integrals and differentiability properties of functions, Prince-
ton Mathematical Series, vol. 30, Princeton University Press, Princeton, N.J., 1970,
xiv+290 pages.
Manuscrit reçu le 18 avril 2021,
révisé le 6 novembre 2021,
accepté le 1er décembre 2021.
Ángel D. MARTÍNEZ
Institute for Advanced Study Fuld Hall 412, 1
Einstein Drive,
Princeton, NJ 08540 (USA)
amartinez@ias.edu
TOME 73 (2023), FASCICULE 2


Córdoba’s differentiation theorem: revisited

Ota, Kazuto

On the rank-part of the Mazur–Tate refined conjecture for higher weight modular forms

Not a member yet

On the $3$ -colorable subgroup \protect \mathcal{F} and maximal subgroups of Thompson’s group $F$