We show that if $\varphi$ is a lacunary Fourier series and the equation $\psi (x) -\psi (x + \alpha) = \varphi(x), x \bmod 1$ has a measurable solution $\varphi$, then in fact the equation has a solution in L2.



This work of Michel Herman (1942-2000) appeared only as a preprint of the Mathematics Institute, University of Warwick, dated May 1976. It was turned into TEX format by Claire Desescures. Minor editorial work was done by Albert Fathi

Herman, Michael R.

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+  University of Warwick institutional repository: http://go.warwick.ac.uk/wrapThis paper is made available online in accordance with publisher policies. Please scroll down to view the document itself. Please refer to the repository record for this item and our policy information available from the repository home page for further information.  To see the final version of this paper please visit the publisher’s website. Access to the published version may require a subscription.  Author(s): MICHAEL ROBERT HERMAN Article Title: L2 regularity of measurable solutions of a finite-difference equation of the circle  Year of publication: 2004 Link to published version: http://dx.doi.org/10.1017/S0143385704000409 Publisher statement: None    http://journals.cambridge.org Downloaded: 02 Jun 2009 IP address: 137.205.202.8Ergod. Th. & Dynam. Sys. (2004), 24, 1277–1281 c© 2004 Cambridge University PressDOI: 10.1017/S0143385704000409 Printed in the United KingdomL2 regularity of measurable solutions of afinite-difference equation of the circle†MICHAEL ROBERT HERMANMathematics Institute, Warwick University, UKandCentre de Mathematiques, Ecole Polytechnique, Plateau de Palaiseau,91120 Palaiseau, FranceWe show that if ϕ is a lacunary Fourier series and the equation ψ(x) − ψ(x + α) =ϕ(x), x mod 1 has a measurable solution ϕ, then in fact the equation has a solution in L2.(1) We consider the circle T = R/Z and the translations (or rotations) Rα = x →x + α(α ∈ T).For 1 ≤ p ≤ +∞, let Lp = Lp(T, dx,C) with the norm ‖·‖p . The only measureconsidered is the Haar measure of T, dx = m. All equalities are to be considered m-almosteverywhere.(2) Let ϕ ∈ L1 and α ∈ T; we try to solveψ − ψ ◦ Rα = ϕ (*)with ψ measurable and the equality almost everywhere.If one supposes that ψ is in L1, then by identification of Fourier coefficients ifϕ(x) =∑k∈Zϕˆ(k)e2xikx ,then one hasψ(x) =∑k∈Zϕˆ(k)1 − e2π ikα e2πikx,(with the convention that 0/0 = 0). (Of course one has 0 = ∫Tϕ(x) dx).(3) The case when a = p/q (mod 1), (p, q) = 1. Then a necessary and sufficientcondition for measurable solutions to (*) isq−1∑i=0ϕ ◦ Riα = 0. (1)If (1) is satisfied then the equation (*) has solutions just as regular as is ϕ.† This work of Michel Herman appeared only as a preprint of the Mathematics Institute, University of Warwick,dated May 1976. It was turned into TEX format by Claire Desescures. Minor editorial work was done byAlbert Fathi.http://journals.cambridge.org Downloaded: 02 Jun 2009 IP address: 137.205.202.81278 M. R. Herman(4) The case when α is irrational. It is easy (by Fourier series) to construct ϕ ∈ L1 with∫Tϕ(x) dx = 0 and an irrational α such that the equation (*) has no solution in L1. By theergodicity of Rα, measurable solutions of (*) differ by a constant.If one looks for solutions of (*) which are only measurable then Anosov has shown thatone has necessarily ∫Tϕ(x) dx = 0 (for ϕ ∈ L1).Furthermore, Anosov has constructed ϕ ∈ Cω(T) with ∫Tϕ(x) dx = 0 and an irrational αsuch thatsupk =0∣∣∣∣ ϕˆ(k)1 − e2πikα∣∣∣∣ = +∞,but nevertheless the equation (*) has a measurable solution ψ (of course not in L1)(see [1]).We will show that the examples of Anosov cannot happen when ϕ is a lacunary Fourierseries.It is then easy to construct a ϕ with∫Tϕ(x) dx = 0 and an irrational α such that theequation (*) has no measurable solution ψ (since there is no L2 solution).For other examples see [6].(5) Let + = ni be a lacunary sequence of positive integers: n0 = 1 and nn+1/ni ≥ q > 1for all i.Let  = + ∪ {0} ∪ (−+) be the symmetric sequence of integers.One denotesLp = {ϕ ∈ Lp | ϕˆ(n) = 0 if n /∈ }.One says that ϕ ∈ L1 is a lacunary Fourier series if there exists a lacunary sequence  asabove such that ϕ ∈ L1. Then one has, for all 1 ≤ p < +∞, ϕ ∈ Lp; and all the norms‖ · ‖p are equivalent on L2(see [5]).(6) We propose to prove the following.THEOREM. Let ϕ ∈ L2 and α ∈ T. If the equation(∗)ψ − ψ ◦ Rα = ϕhas a measurable solution ψ , then the equation has a solution in L2 and if α ∈ T−Q/Zthen in fact, by the ergodicity of Rα,ψ ∈ L2.To prove the theorem one needs the following lemmas.(7)LEMMA. Let f : T → T be a bijection preserving the Haar measure m.Let K be a measurable set of T. Let  > 0 and the set of integersA = {n ∈ Z | m(K ∩ f n(K)) ≥ m(K)2 − }.The set of integers A is relatively dense: there exists a positive integer k, such that{j, . . . , j + k} ∩ A = φ, for all j ∈ Z.For a proof see [3, p. 31].http://journals.cambridge.org Downloaded: 02 Jun 2009 IP address: 137.205.202.8L2 regularity of measurable solutions 1279(8)LEMMA†. Let L2 be given. There exist constants C > 0 and b (0 < b < 1) such that ifB ⊂ T is measurable with m(B) ≥ b, then for all ϕ ∈ L2 one hasC(∫B|ϕ(x)|2 dx)1/2≥ ‖ϕ‖2.Proof. Let 0 < a < 1 and ϕ ∈ L2 with ‖ϕ‖2 = 1. LetA(ϕ) ≡ A = {x ∈ T | |ϕ(x)| ≥ a}.We have ‖ϕ‖22 = 1 =∫T−A|ϕ(x)|2 dx +∫A|ϕ(x)|2 dx ≤ a2 +∫A|ϕ(x)|2 dx.One has by the Ho¨lder inequality1 ≤ ‖ϕ‖4(m(A))1/4 + a.Since the norms ‖ · ‖2 and ‖ · ‖4 are equivalent on L2, one has ‖·‖4 ≤ k‖·‖2, k being aconstant greater than 1.It follows thatm(A) ≥(1 − ak)4; (2)chooseb = 1 − 12(1 − ak)4.If B ⊂ T with m(B) ≥ b and if ϕ ∈ L2 with ‖ϕ‖2 = 1, we havem(A(ϕ) ∩ B) ≥ 12(1 − ak)4by (2), so ∫B|ϕ(x)|2 dx ≥ 12a2(1 − ak)4=(1C)2.The result follows byC(∫B|ϕ(x)|2 dx)1/2≥ ‖ϕ‖2. (9)LEMMA. Let ϕ ∈ L2. A necessary and sufficient condition for a ψ ∈ L2 that verifiesψ − ψ ◦ Rα = ϕ to exist is that supn∈N ‖ϕn‖2 < +∞ with ϕn =∑n−1i=0 ϕ ◦ Riα .For the proof see [4]. In fact it results from the more general lemma, which uses the factthat the unit ball of a reflexive Banach space is weakly compact, and the Markov–Kakutanifixed point theorem (affine version).† I thank Y. Meyer who brought to my attention the fact that Carleson has proved a stronger lemma (unfortunatelyunpublished): For every B with m(B) > 0 there exists C(m(B), q) > 0 such that one has the conclusion of thelemma. I thank B. Maurey for the proof proposed.http://journals.cambridge.org Downloaded: 02 Jun 2009 IP address: 137.205.202.81280 M. R. HermanLEMMA. Let L be a reflexive Banach space of norm ‖·‖ and u : L → L a continuouslinear operator. Given x ∈ L, a sufficient condition for the existence of a y ∈ L satisfyingy − u(y) = x to exist is thatsupn∈N∥∥∥∥n−1∑i=0ui(x)∥∥∥∥ < +∞;the condition is necessary if supn∈N ‖un‖ < +∞.(10) Proof of the theorem. Let L2 be given and be determined by item (8) (and that dependson ).Let  > 0 with (1 − )2 −  ≥ b.One starts with a measurable solution ofψ − ψ ◦ Rα = ϕ, (*)with ϕ ∈ L2. There exists a compact set K ⊂ T of measure ≥ 1 − , such that ψ|K iscontinuous. By (*) one hasψ − ψ ◦ Rnα =n−1∑i=0ϕ ◦ Riα ≡ ϕn.It follows that(∫K∩Rnα(K)|ϕn(x)|2 dx)1/2≤ 2 supx∈K|ψ(x)| < +∞.Let A = {n ∈ Z | m(K ∩ Rnα(K)) ≥ (1 − )2 −  ≥ b}. By item (7), thesubset A is a relatively dense sequence of integers, and let k be the integer of (7).Let B = {−k,−k + 1, . . . , k}. Since ϕn ∈ L2 by (8) one hassupn∈A‖ϕn‖2 = C1 < +∞.Let C2 = supn∈B‖ϕn‖2 < +∞. Since every n ∈ Z can be written as n = n1 + n2 withn1 ∈ A and n2 ∈ B and if n1, and n2 are positive integers, we haveϕn1+n2 = ϕn1 ◦ Rn2α + ϕn2 ;finally we deduce thatsupn∈Z‖ϕn‖2 ≤ C1 + C2and the theorem results from (9).(11) From the theorem we deduce the following: if ϕ ∈ L2α , α is irrational, and ψ ismeasurable and satisfies ψ − ψ ◦ Rα = ϕ, then ψ ∈ Lp for every 1 ≤ p < +∞ since ψis a lacunary Fourier series. In general, ψ /∈ L∞ even if ϕ is of class Cω as we will showby a classical example.http://journals.cambridge.org Downloaded: 02 Jun 2009 IP address: 137.205.202.8L2 regularity of measurable solutions 1281Construction of an irrational α. Let α = 1/(a1 + (1/(a2 +· · · ))) be the continued fractionof an irrational α (ai ≥ 1, ai ∈ N).If pn/qn are the convergents of α, one has q0 = 1, q1 = a1 and qn = anqn−1 + qn−2,if n ≥ 2. If x ∈ R and |||x||| is the distance of x to the nearest integer, one has|||qnα||| < 1qn+1≤ 1an+1qn.If one chooses the sequence (ai) so that it increases sufficiently rapidly, one easilyconstructs an irrational α such that, for every n ≥ 2, one has|||qnα||| ≤ e−qn. (+)Let us remark that, for every irrational α, (q2n)n∈N is a lacunary sequence of positiveinteger (in fact we have q2n+2/q2n ≥ 2 and also q2n+1/q2n−1 ≥ 2).Construction of ϕ. Let n ≥ 1 be a sequence of complex numbers satisfying∞∑n=1|c2n|2 < +∞ but∞∑n=1|c2n| = +∞.Let ϕ(x) = ∑∞n=1 c2n(1 − e2πiq2nα)e2πiq2nx .If α satisfies (+), then ϕ ∈ Cω(T,C) (and one has 0 = ∫Tϕ(x) dx).Let ψ(x) = ∑∞n=1 c2ne2πiq2nx ; one has ψ ∈ L2 (and ψ is a lacunary Fourier series).Furthermore, one hasψ − ψ ◦ Rα = ϕ.But ψ /∈ L∞, for if this was the case then, since ψ is a lacunary Fourier series, we wouldhave∑∞n=1|c2n| < +∞, which is contrary to the choice of the sequence (c2n) (see [5]).(12) We have shown a proposition in [2] that implies the following remark.Remark. Let ϕ : T → R be continuous (but not necessarily lacunary) and α irrational.We suppose that there exists ψ ∈ L∞ with ψ −ψ ◦Rα = ϕ; then ψ is almost everywhereequal to a continuous function.REFERENCES[1] D. V. Anosov. On an additive functional homology equation connected with an ergodic rotation of thecircle. Translation Math. U.S.S.R. Izvestija 74 (1973), 1257–1271.[2] M. R. Herman. Sur la conjugaison diffe´rentiable des diffe´omorphismes du cercle a` des rotations. Thesis,Orsay 1976.[3] K. E. Petersen. Introductory ergodic theory. Lecture Notes, University of North Carolina at Chapel Hill.[4] K. E. Petersen. On a series of cosecants related to a problem in ergodic theory. Compositio Math. 26 (1973),313–317.[5] A. Zygmund. Trigonometric Series. Cambridge University Press, 1959.[6] A. Y. Gordon. Funktsional’ny Analiz Evo Prilozheniya 9(4) (1975), 71–72 (in Russian).

Funktsional’ny Analiz Evo Prilozheniya

Introductory ergodic theory. Lecture Notes,

On an additive functional homology equation connected with an ergodic rotation of the circle.

Ona series ofcosecants related to a problem in ergodic theory.

Sur la conjugaison diff´ erentiable des diff´ eomorphismes du cercle ` a des rotations. Thesis,

Trigonometric Series.

L2 regularity of measurable solutions of a finite-difference equation of the circle

MICHAEL ROBERT HERMAN

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                  2 regularity of measurable solutions of a finite-difference equation of the circle
                  
               

L-2 regularity of measurable solutions of a finite-difference equation of the circle

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L2 regularity of measurable solutions of a finite-difference equation of the circle

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