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Université Grenoble AlpesANNALES DEL’INSTITUT FOURIERÁngel D. MartínezCórdoba’s differentiation theorem: revisitedTome 73, no2 (2023), p. 559-565.https://doi.org/10.5802/aif.3535Article mis à disposition par son auteur selon les termes de la licenceCreative Commons attribution – pas de modification 3.0 Francehttp://creativecommons.org/licenses/by-nd/3.0/fr/C EN T R EMER S ENN ELes Annales de l’Institut Fourier sont membres duCentre Mersenne pour l’édition scientifique ouvertewww.centre-mersenne.org e-ISSN : 1777-5310Ann. Inst. Fourier, Grenoble73, 2 (2023) 559-565CÓRDOBA’S DIFFERENTIATION THEOREM:REVISITEDby Ángel D. MARTÍNEZ (*)Abstract. — In this paper we prove an exponential covering lemma implyingthe three dimensional case of a well-known conjecture formulated by A. Zygmundcirca 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 parA. Zygmund circa 1935 et prouvée par A. Córdoba en 1978. Notre approche éviteun argument subtil qui fati appel à la série entière de la fonction exponentielle.1. IntroductionThe 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 insteadof balls or cubes one may also differentiate using the class Bn, of shrinkingn-dimensional rectangles (paralellepipeds) with sides parallel to the axes,provided the function is locally in the class L1(1 + log+ L1)n−1(Rn). Thequantitative version of this result is known as the strong maximal theorem:let us defineMnf(x) = supx∈R∈Bn1m(R)∫R|f(y)|dm(y)where f is a locally integrable function and m denotes Lebesgues’s measurein 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 Foundationunder Grant No. DMS-1638352.560 Ángel D. MARTÍNEZTheorem 1.1 (Jessen, Marcinkiewicz and Zygmund, [15]). — The classBn differentiates functions f which are locally in L1(1 + log+ L1)n−1(Rn),that is, we havelimdiam(R)→01m(R)∫Rf(y)dm(y) = f(x) for a.e. x ∈ Rnwhere the limit is taken over all rectangles such that x ∈ R ∈ Bn. Quanti-tatively, the following inequalitym{x ∈ Rn : Mnf(x) > α} ⩽ Cn∫Rnf(x)α(1 + log+f(x)α)n−1dm(x)holds for some constant Cn that depends only on the dimension.Their proof relies on an iteration scheme based on the one dimensionalresult and making use of the product structure of Rn. Quite surprisingly theOrlicz 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 pointedout 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) relieson the Vitali covering lemma which allows to deduce this estimates as aconsequence of the geometry of balls. Standard references are the treatisesof de Guzmán [13] and Stein [18] where Cn = 5n.It was not until 1975 when the exponential type covering needed to provegeometrically the strong maximal theorem was finally discovered (cf. [8]).The proof obtained by the A. Córdoba and R. Fefferman makes use ofcertain sparseness properties. Let us introduce the one we shall need, usingtheir notation:(P1) A sequence of rectangles {Rj} satisfies the P1 sparseness propertyif for any k the following inequality holds:mRk ∩ ⋃j<kRj ⩽ 12m(Rk).In 1978 A. Córdoba took advantage of the geometric approach to settleZygmund’s conjecture in the three-dimensional case:Theorem 1.2 (A. Córdoba, [3, 4, 5]). — Given any function ϕ nondecreasing in each variable separately let Bϕ(R3) consist of rectangleswhose lengths are of the form (x, y, ϕ(x, y)). Then it differentiates L1(1 +log+ L1)(R3).ANNALES DE L’INSTITUT FOURIERCÓRDOBA’S DIFFERENTATION THEOREM: REVISITED 561In fact, Córdoba’s differentiation theorem is stronger but let us state itin this way for the sake of the exposition’s clarity. His elegant proof reliesheavily on the three dimensional setting. No purely analytical proof hasbeen found to-date. For applications of this result we refer the reader to theoriginal works of A. Córdoba. Zygmund’s curiosity was not satisfied by thisand asked for further generalizations of this result in arbitrary dimension.Some of them were dismissed in [17] by F. Soria. His counterexamples haveside lenghts of the form (s, ϕ(s, t), ψ(s, t)) where the functions ϕ and ψ aremonotone increasing. Recently more counterexamples have been found byG. Rey in [16]. The rest remain an intriguing open question.In the next section we provide a geometric covering result that impliesTheorem 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 breaksdown in the higher dimensional case.AcknowledgmentsThe author would like to express his gratitude to Antonio Córdoba forintroducing him to the subject in a graduate course delivered in Fall 2014at UAM-ICMAT and Eric Latorre for his (unpublished) lecture notes [1].The author is indebted to the anonymous referees for their careful readingof the manuscript, suggestions and for pointing out references [10] and [12]to his attention.2. Proof of Theorem 1.2Let 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 lengthsare 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 existsa subsequence {Rj} such that(1) m(⋃αRα)⩽ Cm⋃jRj .(2)∫⋃jRjexp c∑jχRj dm ⩽ Cm⋃jRj .TOME 73 (2023), FASCICULE 2562 Ángel D. MARTÍNEZProof. — it is well known that without loss of generality we may assumethat 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. Thisreduction respects (1) from the statement (cf. [8]). We will denote by R∗ arectangle centered in the same point and dilated by a factor of three. Theintroduction of this will be apparent later when we require intersections tobe clean.We select the rectangles following the following algorithm. Having chosenk − 1 of them, we will choose Rk to be the next enlisted rectangle R thatsatisfies(2.1) 1m(R)∫Rexpk−1∑j=1χR∗j dm ⩽ 3.This can be shown to imply property (2) from the statement. DenotingIk =∫⋃kj=1Rjexp k∑j=1χR∗j dmlet us observe thatIk =∫⋃k−1j=1Rj\Rkexpk−1∑j=1χR∗j dm+ ∫Rkexp k∑j=1χR∗jdm.The choice of of Rk shows that Ik ⩽ Ik−1 + 3em(Rk) which by inductionimplies Ik ⩽ 3e∑kj=1 m(Rj). From this and the P1 property our claimfollows easily, namely(2.2)∫⋃kj=1Rjexp k∑j=1χR∗jdm ⩽ 6em⋃jRj .This shows that, following this sieve, part (2) will hold with c = 1 and itonly rests to show that it also respects (1). To do so we observe that for arejected rectangle R the negation of the inequality (2.1) implies3 <∞∑k=0Akekholds, where we are using the notation A0 = 1 andAk =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 toexactly k rectangles. Let us denote by R1j those rectangles previous to R inANNALES DE L’INSTITUT FOURIERCÓRDOBA’S DIFFERENTATION THEOREM: REVISITED 563the selection process whose first and third side lengths are greater to thoseof R. Analogously, from the remaining rectangles, we denote by R2j thosewith second and third lengths exceeding those of R. Decomposing Ak indifferent cases (namely, the k = r + s intersections being the result of theintersection of r rectangles from the first class and s from the second) theabove inequality is equivalent to3 <∞∑k=0∑r+s=kAr,sekwhereAr,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 withthree 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 2564 Ángel D. MARTÍNEZLet us introduce a related piece of notation now. Let a0 = 1 and ingeneralar =m({x ∈ R :∑j χR1∗j (x) = r})m(R)for r > 0. Similarly, let b0 = 1 andbs =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 that3 <( ∞∑r=0arer)( ∞∑s=0bses).(At this point the proof breaks down in higher dimensions.) Reading backwhat this means it impliesmaxℓ=1,2{1mxℓ(R)∫Rexp(k∑i=1χR∗i)dmxℓ}>√3 − 1where mxℓ is the length in the ℓth direction. The integral mean is a maximalfunction of the exponential restrected to any line intersecting R and parallelto the xℓ axis. We will denote such a (one dimensional) maximal operator byMℓ. The proof concludes estimating the size of the set covered by rejectedrectangles. For this purpose one observes that it is included in{x : M1 exp(k∑i=1χR∗i)(x) >√3 − 1 or M2 exp(k∑i=1χR∗i)(x) >√3 − 1}whose measure can be bounded appropiately by the weak L1 boundednessof the Hardy-Littlewood maximal function. Indeed, the maximal functionsappearing above are one dimensional means in the first and second coordi-nates alone to which the weak L1 bound applies linewise. Integrating thisL1 bounds on the rest of variables and using Theorem 1.1 for n = 1 onegetsm⋃βRβ ⩽ 5√3 − 1∫⋃Riexp(k∑i=1χR∗i)dmwhere β extends over the indices of rejected rectangles. This can be boundedusing equation (2.2) to conclude (1). □Remark. — It is easy to show that this argument also works for a familysatisfying that in some order any given R has two sidelengths smaller thanany previous rectangle.ANNALES DE L’INSTITUT FOURIERCÓRDOBA’S DIFFERENTATION THEOREM: REVISITED 565BIBLIOGRAPHY[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 harmonicanalysis (Proc. Sem., Univ. Maryland, College Park, Md., 1979), Lecture Notes inMath., 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 MatemáticaIberoamericana, 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 withrespect 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-theoreticapplications”, Acta Math. 54 (1930), no. 1, p. 81-116.[15] B. Jessen, J. Marcinkiewicz & A. Zygmund, “Note on the differentiability ofmultiple 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. MARTÍNEZInstitute for Advanced Study Fuld Hall 412, 1Einstein Drive,Princeton, NJ 08540 (USA)amartinez@ias.eduTOME 73 (2023), FASCICULE 2

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