Carleson's theorem : proof, complements, variations

Publ. Mat. 48 (2004), 251–307 CARLESON’S THEOREM: PROOF, COMPLEMENTS, VARIATIONS Michael T. Lacey Abstract Carleson’s Theorem from 1965 states that the partial Fourier sums of a square integrable function converge pointwise. We prove an equivalent statement on the real line, following the method developed by the author and C. Thiele. This theorem, and the proof presented, is at the center of an emerging theory which complements the statement and proof of Carleson’s theorem. An outline of these variations is also given. Contents 1. Introduction 252 2. Decomposition 254 2.1. Complements 260 3. The Central Lemmas 261 3.1. Complements 264 4. The Density Lemma 265 4.1. Complements 266 5. The Size Lemma 266 5.1. Complements 270 6. The Tree Lemma 272 6.1. Complements 276 7. Carleson’s Theorem on Lp, 1 < p 6= 2 <∞276 7.1. The case of |F|<1 3277 7.2. The case of |F| ≥ 3 278 7.3. Complements 282 8. Remarks 282 2000 Mathematics Subject Classification. 42A20, 42B20, 42B25. Key words. Pointwise convergence, Fourier series, singular integrals, phase plane analysis. This work has been supported by an NSF grant. These notes are based on a series of lectures given at the Erwin Schr¨odinger Institute, in Vienna, Austria. I am indebted to the Institute for the opportunity to present these lectures. 252 M. T. Lacey 9. Complements and Extensions 284 9.1. Equivalent formulations of Carleson’s theorem 284 9.2. Fourier series near L1, Part 1 285 9.3. Fourier series near L1, Part 2 285 9.4. The Wiener-Wintner question 287 9.5. E. M. Stein’s maximal function 289 9.6. Fourier series in two dimensions 291 9.7. The bilinear Hilbert transforms 296 9.8. Multilinear oscillatory integrals 297 9.9. Hilbert transform on smooth families of lines 298 9.10. Schr¨odinger operators, scattering transform 299 References 301 1. Introduction L. Carleson’s celebrated theorem of 1965 [14] asserts the pointwise convergence of the partial Fourier sums of square integrable functions. We give a proof of this fact, in particular the proof of Lacey and Thiele [50], as it can be presented in brief self contained manner, and a number of related results can be seen by variants of the same argument. We survey some of these variants, complements to Carleson’s theorem, as well as open problems. We are concerned with the Fourier transform on the real line, given by b f(ξ) = Ze−ixξf(x)dx for Schwartz functions f. For such functions, it is an important elementary fact that one has Fourier inversion, (1.1) f(x) = lim N→∞ 1 2πZN −Nb f(ξ)eixξ dξ, x ∈R, the inversion holding for all Schwartz functions f. Indeed, 1 2πZN −Nb f(ξ)eixξ dξ =DN∗f(x), where DN(x) := sin Nx πx is the Dirchlet kernel. The convergence in (1.1) for Schwartz functions follows from the classical facts Z∞ −∞ DN(x)dx = 1, lim N→∞ Z|x|≥ DN(x)dx = 0,  > 0. Carleson’s Theorem on Fourier Series 253 L. Carleson’s theorem asserts that (1.1) holds almost everywhere, for f∈L2(R). The form of the Dirchlet kernel already points out the essential difficulties in establishing this theorem. That part of the kernel that is convolution with 1 xcorresponds to a singular integral. This can be done with the techniques associated to the Calder´on Zygmund theory. In addition, one must establish some uniform control for the oscillatory term sin Nx, which falls outside of what is commonly considered to be part of the Calder´on Zygmund theory. For technical reasons, we find it easier to consider the equivalent one sided inversion, (1.2) f(x) = lim N→∞ 1 2πZN −∞ b f(ξ)eixξ dξ. Schwartz functions being dense in L2, one need only show that the set of functions for which a.e. convergence holds is closed. The standard method for doing so is to consider the maximal function below, which we refer to as the Carleson operator (1.3) Cf(x) := sup NZN −∞ b f(ξ)eixξ dξ, x ∈R. There is a straight forward proposition. Proposition 1.4. Suppose that the Carleson operator satisfies (1.5) |{Cf(x)> λ}| .λ−2kfk2 2, f ∈L2(R), λ > 0. Then, the set of functions f∈L2(R)for which (1.2) holds is closed and hence all of L2(R). Proof: For f∈L2(R), we should see that Lf:= lim sup N→∞ f(x)−1 2πZN −∞ b f(ξ)eixξ dξ= 0 a.e. To do so, we show that for all  > 0, |{Lf> }| .. We take gto be a smooth compactly supported function so that kf−gk2≤3/2. Now Fourier inversion holds for g, whence Lf≤ C(f−g) + |f−g|. Then, by the weak type inequality, (1.5), we have |{C(f−g)> }| .−2kf−gk2 2.. This is a standard proposition, which holds in a general context, and serves as one of the prime motivations for considering maximal operators. Note in particular that we are not at this moment claiming that Cis a bounded operator on L2. Inequality (1.5) is the so called weak L2bound, 254 M. T. Lacey and we shall utilize the form of this bound in a very particular way in the proof below. It was one of L. Carleson’s great achievements to invent a method to prove this weak type estimate. Theorem 1.6. The estimate (1.5) holds. As a consequence, (1.2) holds for all f∈L2(R), for almost every x∈R. Carleson’s original proof [14] was extended to Lp, 1 <p<∞, by Hunt. Also see [67]. Fefferman [26] gave an alternate proof that was influential by the explicit nature of it’s “time frequency” analysis, of which we have more more to say below. We follow the proof of Lacey and Thiele [50]. More detailed comments on the history of the proof, and related results will come later. The proof will have three stages, the first being an appropriate decomposition of the Carleson operator. The second being an introduction of three lemmas, which can be efficiently combined to give the proof of our theorem. The third being a proof of the lemmas. We do not keep track of the value of generic absolute constants, instead using the notation A.Biff A≤KB for some constant K. And A≃Biff A.Band B.A. The notation 1Adenotes the indicator function of the set A. For an operator T,kTkpdenotes the norm of Tas an operator from Lpto itself. 2. Decomposition The Fourier transform is a constant times a unitary operator on L2(R). In particular, we shall take the Plancherel’s identity for granted. Proposition 2.1. For all f, g ∈L2(R), hf, gi=chb f, bgi for appropriate constant c=1 2π. The convolution of fand ψis given by f∗ψ(x) = Rf(x−y)ψ(y)dy. We shall also assume the following lemma. Lemma 2.2. If a bounded linear operator Ton L2(R)commutes with translations, then Tf =ψ∗f, where ψis a distribution, which is to say a linear functional on Schwartz functions. In addition, the Fourier transform of T f is given by c Tf =b ψb f. Carleson’s Theorem on Fourier Series 255 Let us introduce the operators associated to translation, modulation and dilation on the real line. Tryf(x) := f(x−y),(2.3) Modξf(x) := eiξxf(x),(2.4) Dilp λf(x) := λ−1/pf(x/λ),0< p ≤ ∞, λ > 0.(2.5) Note that the dilation operator preserves Lpnorm. These operators are related through the Fourier transform, by (2.6) d Try= Mod−y,\ Modξ= Trξ,d Dil2 λ= Dil2 1/λ . And we should also observe that the Carleson operator commutes with translation and dilation operators, while being invariant under modulation operators. For any y, ξ ∈R, and λ > 0, Try◦C =C ◦Tr,Dil2 λ◦C =C ◦Dil2 λ,C ◦Modξ=C. Thus, our mode of analysis should exhibit the same invariance properties. We have phrased the Carleson operator in terms of modulations of the operator P−f(x) = R0 −∞ b f(ξ)eixξ dξ, which is the Fourier projection on to negative frequencies. Specifically, since multiplication of fby an exponential is associated with a translation of b f, we have (2.7) Cf= sup N|P−(eiN·f)|. A characterization of the operator P−will be useful to us. Proposition 2.8. Up to a constant multiple, P−is the unique bounded operator on L2(R)which (a) commutes with translation (b) commutes with dilations (c) has as it’s kernel precisely those functions with frequency support on the positive axis. Proof: Let Tbe a bounded operator on L2(R) which satisfies these three properties. Condition (a) implies that Tis given by convolution with respect to a distribution. Such operators are equivalently characterized in frequency variables by c Tf =τb ffor some bounded function τ. Condition (b) then implies that τ(ξ) = τ(ξ/|ξ|) for all ξ6= 0. A function f is in the kernel of Tiff b fis supported on the zero set of τ. Thus (c) implies that τis identically 0 on the positive real axis, and non-zero on the negative axis. Thus, Tmust be a multiple of P−. 256 M. T. Lacey We move towards the tool that will permit us to decompose Carleson operator, and take advantage of some combinatorics of the timefrequency plane. We let Dbe a choice of dyadic grids on the line. Of the different choices we can make, we take the grid to be one that is preserved under dilations by powers of 2. That is (2.9) D={[j2k,(j+ 1)2k) : j, k ∈Z}. Thus, for each interval I∈ D and k∈Z, the interval 2kI={2kx:x∈I} is also in D. Atile is a rectangle s∈ D × D that has area one. We write a tile as s=I×ω, thinking of the first interval as a time interval and the second as frequency. The requirement of having area one is suggested by the uncertainty principle of the Fourier transform, or alternatively, our calculation of the Fourier transform of the dilation operators in (2.6). Let Tdenote the set of all tiles. While tiles all have area one, the ratio between the time and frequency coordinates is permitted to be arbitrary. See Figure 1 for a few possible choices of this ratio. Figure 1. Four different aspects ratios for tiles. Each fixed ratio gives rise to a tiling for the time frequency plane. Each dyadic interval is a union of its left and right halves, which are also dyadic. For an interval ωwe denote these as ω−and ω+respectively. We are in the habit of associating frequency intervals with the vertical axis. So ω−will lie below ω+. Associate to a tile s=Is×ωsthe rectangles s±=Is×ωs±. These two rectangles play complementary roles in our proof. Carleson’s Theorem on Fourier Series 257 Fix a Schwartz function ϕwith 1[−1/9,1/9] ≤bϕ≤1[−1/8,1/8]. Define a function associated to a tile sby (2.10) ϕs= Modc(ωs−)Trc(Is)Dil2 |Is|ϕ where c(J) is the center of the interval J. Notice that ϕshas Fourier transform supported on ωs−, and is highly localized in time variables around the interval Is. That is, ϕsis essentially supported in the timefrequency plane on the rectangle Is×ωs−. Notice that the set of functions {ϕs:s∈ T} has a set of invariances with respect to translation, modulation, and dilation that mimics those of the Carleson operator. It is our purpose to devise a decomposition of the projection P−in terms of the tiles just introduced. To this end, for a choice of ξ∈R, let (2.11) Qξf=X s∈T 1ωs+(ξ)hf, ϕsiϕs. We should consider general values of ξfor the reason that the dyadic grid distinguishes certain points as being interior, or a boundary point, to an infinite chain of dyadic intervals. And moreover, for a given ξ, only certain tiles can contribute to the sum above, those tiles being determined by the expansion of ξin a binary digits. See Figure 2. Let us list some relevant properties of these operators. ξ Figure 2. Some of the tiles that contribute to the sum for Qξ. The shaded areas are the tiles Is×ωs+. 258 M. T. Lacey Proposition 2.12. For any ξ, the operator Qξis a bounded operator on L2, with bound independent of ξ. Its kernel contains those functions with Fourier transform supported on [ξ, ∞), and it is positive semidefinite. Moreover, for each integer k, Qξ= Dil2 2−kQξ2−kDil2 2k (2.13) Qξ,k Tr2k= Tr2kQξ,k,(2.14) where Qξ,k =Ps∈T |Is|≤2k 1ωs+(ξ)hf, ϕsiϕs. Proof: Let {ω(n) : n∈Z}be the set of dyadic intervals for which ξ∈ω(n)+, listed in increasing order, thus ··· ⊂ ω(n)⊂ω(n+ 1) ⊂···. Let T(n) = {s∈ T :ωs=ω(n)}, and Q(n)f=X s∈T (n)hf, ϕsiϕs. The intervals ω(n)−are disjoint in n, and since ϕshas frequency support in ωs−, it follows that the operators Q(n)are orthogonal in n. The boundedness of Qξreduces therefore to the uniform boundedness of Q(n) in n. Two operators Q(n)and Q(n0)differ by composition with a modulation operator and a dilation operator that preserves L2norms. Thus, it suffices to consider the L2norm bound of a Q(n)with |Is|= 1 for all s∈ T(n). Using the fact that ϕsis a rapidly decreasing function, we see that (2.15) |hϕs, ϕs0i| .dist(Is, Is0)−4. Now that the spatial length of the tiles is one, the tiles are separated by integral distances. Since Pnn−4<∞, kQ(n)fk2 2=X s∈T (n)X s0∈T (n)hf, ϕsihϕs, ϕs0ihϕs0, fi .sup n∈ZX s∈T (n)|hf, ϕsihf, ϕ(Is+n)×ω(n)i| .X s∈T (n)|hf, ϕsi|2. Carleson’s Theorem on Fourier Series 259 The last inequality following by Cauchy-Schwarz. The last sum is easily controlled, by simply bringing in the absolute values. Since |hf, ϕsi|2. R|f|2|ϕs|dx X s∈T (n)|hf, ϕsi|2≤Z|f|2X s∈T (n)|ϕs|dx ≤ kfk2 2sup xX s∈T (n)|ϕs(x)| .kfk2 2. This completes the proof of the uniform boundedness of Qξ. Since all of the functions ϕsthat contribute to the definition of Qξ have frequency support below ξ, the conclusion abut the kernel of the operator is obvious. And that it is positive semidefinite, observe that (2.16) hQξf, fi=X s∈T ξ∈ωs+ |hf, ϕsi|2≥0. In particular, hQξϕs, ϕsi 6= 0 for s∈ T(n). To see (2.13) recall (2.6) and our specific choice of grids, (2.9). To see (2.14), observe that if I∈ D has length at most 2k, then I+ 2kis also in D. As the lemma makes clear, Mod−ξQξModξserves as an approximation to P−. A limiting procedure will recover P−exactly. Consider (2.17) Q= lim Y→∞ZB(Y) Dil2 2−λTr−yMod−ξQξModξTryDil2 2λµ(dλ, dy, dξ). Here, B(Y) is the set [1,2]×[0, Y ]×[0, Y ], and µis normalized Lebesgue measure. Notice that the dilations are given in terms of 2λ, so that in that parameter, we are performing an average with respect to the multiplicative Haar measure on R+. Apply the right hand side to a Schwartz function f. It is easy to see that as k→ −∞, the terms ModξTryDil2 2λQξ,k f tend to zero uniformly in the parameters ξ,y, and λ, with a rate that depends upon f. Here, Qξ,k is as in (2.14). Similarly, as k→ ∞, the terms ModξTryDil2 2λ(Q −Qξ,k)f 266 M. T. Lacey Fix k. Select from Ska subset S0 kof tiles satisfying {2kIs×ωs:s∈ S0 k} are pairwise disjoint, and if s∈ Skand s0∈S0 kare tiles such that 2kIs×ωs and 2kIs0×ωs0intersect, then |Is| ≤ |Is0|. It is clearly possible to select such a subset. And since the tiles in Skare incomparable with respect to ‘<’, we can use (4.2) to estimate X s∈Sk|Is| ≤ 2k+1 X s0∈S0 k |Is0| ≤2 c2−kδ−1. That is, we see that (4.3) holds, completing our proof. 4.1. Complements. 4.4. Let Sbe a set of tiles for which there is a constant Kso that for all dyadic intervals J,X s∈S Is⊂J |Is| ≤ K|J|. Then for all 1 ≤p < ∞, and intervals J, X s∈S Is⊂J 1Isp .Kp|J|1/p. In fact, Kp.p. 5. The Size Lemma Set σ= size(S). We will need to construct a collection of trees T∈ e Tlarge, with Slarge =ST∈e Tlarge T, and (5.1) X T∈e Tlarge |IT|.σ−2, as required by (3.10). The selection of trees T∈e Tlarge will be done in conjunction with the construction of +trees T+∈e Tlarge+. This collection will play a critical role in the verification of (5.1). The construction is recursive in nature. Initialize Sstock := S,e Tlarge := ∅,e Tlarge+ := ∅. Carleson’s Theorem on Fourier Series 267 While size(Sstock)> σ/2, select a +tree T+⊂ Sstock with (5.2) ∆(T+)>σ 2|IT+|. In addition, the top of the tree IT+×ωT+should be maximal with respect to the partial order ‘<’ among all trees that satisfy (5.2). And c(ωT+) should be minimal, in the order of R. Then, take Tto be the maximal tree (without reference to sign) in Sstock with top IT+×ωT+. After this tree is chosen, update Sstock := Sstock −T, e Tlarge := e Tlarge ∪T, e Tlarge+ := e Tlarge+ ∪T+. Once the while loop finishes, set Ssmall := Sstock and the recursive procedure stops. It remains to verify (5.1). This is a orthogonality statement, but one that is just weaker than true orthogonality. Note that a particular enemy is the is the situation in which hϕs, ϕs0i 6= 0. When ωs=ωs0, this may happen, but as we saw in the proof of Proposition 2.12, this case may be handled by direct methods. Thus we are primarily concerned with the case that e.g. ωs−⊂6=ωs0−. A central part of this argument is a bit of geometry of the timefrequency plane that is encoded in the construction of the +trees above. Suppose there are two trees T6=T0∈e Tlarge+, and tiles s∈Tand s0∈T0such that ωs−⊂6=ωs0−, then, it is the case that Is0∩IT=∅. We refer to this property as ‘strong disjointness’. It is a condition that is strictly stronger than just requiring that the sets in the time-frequency plane below are disjoint in T. [ s∈T Is×ωs−,T∈e Tlarge+. To see that strong disjointness holds, observe that ωT⊂ωs⊂6=ωs0−. Thus ωT0lies above ωT. That is, in our recursive procedure, Twas constructed first. If it were the case that Is0∩IT6=∅, observe that one interval would have to be contained in the other. But tiles have area one, thus, it must be the case that Is0⊂IT. That means that s0would have been in the tree (the one without sign) that was removed from Sstock before T0was constructed. This is a contradiction which proves strong disjointness. See Figure 3. 268 M. T. Lacey IT×ωT ss0 Figure 3. The proof of strongly disjoint trees. Note that the gray tile could be in the tree that was removed after the selection of the +-tree with the top indicated above. We use this strong disjointness condition, and the selection criteria (5.2), to prove the bound (5.1). The method of proof is closely related to the so called T T∗method. Set S0=ST∈e Tlarge+ T, and F:= X s∈S0hf, ϕsiϕs. The operator f7→ hf, ϕsiϕsis self-adjoint, so that σ2X T∈e Tlarge+ |IT|=hf, Fi ≤ kfk2kFk2. And so, we should show that (5.3) kFk2 2.σ2X T∈e Tlarge+ |IT|. This will complete the proof. This last inequality is seen by expanding the square on the left hand side. In particular, the left hand side of (5.3) is at most the sum of the Carleson’s Theorem on Fourier Series 269 two terms X s,s0∈S0 ωs=ωs0 hf, ϕsihϕs, ϕs0ihϕs0, fi(5.4) 2X s,s0∈S0 ωs⊂6=ωs0 |hf, ϕsihϕs, ϕs0ihϕs0, fi|.(5.5) For the term (5.4), we have the obvious estimate on the inner product |hϕs, ϕs0i| .1 + dist(Is, Is0) |Is|−4 . (Compare to (2.15).) Thus, by Cauchy-Schwarz, (5.4) .X s∈S0|hf, ϕsi|2.σ2X T∈e Tlarge+ |IT|. For the term (5.5), we need only show that for each tree T, (5.6) X s∈TX s0∈S0 ωs⊂6=ωs0 |hf, ϕsihϕs, ϕs0ihϕs0, fi|2.σ2|IT|. Here, S(s) := {s0∈ S0−T:ωs−⊂6=ωs0−}. The implied constant should be independent of the tree T. Now, the strong disjointness condition enters in two ways. For s∈T, and s0∈ S(s), it is the case that Is0∩IT=∅. But furthermore, for s0, s00 ∈ S(s), we have e.g. ωs−⊂ωs0−⊂ωs00 −, so that Is0∩Is00 is also empty. At this point, rather clumsy estimates of (5.6) are in fact optimal. The definition of size gives us the bound |hϕs0, fi| .p|Is0|σ. And, since ωs⊂ωs0, we have |Is| ≥ |Is0|, and Is, and Is0are, in the typical situation, far apart. An estimation left to the reader gives (5.7) |hϕs, ϕs0i| .p|Is0||Is|χIs(c(Is0)). 270 M. T. Lacey Thus, we bound the left side of (5.6) by X s∈TX s0∈S(s)|hf, ϕsihϕs, ϕs0ihϕs0, fi| .σ2X s∈T|Is0||Is|χIs(c(Is0)) .σ2X s∈TZ(IT)c|Is|χIs(x)dx .σ2|IT| (5.8) as is easy to verify. This completes the proof of (5.6), and so finishes the proof of Lemma 3.9. 5.1. Complements. 5.9. Concerning the inequality (5.8), for any tree T, we have X s∈TZIc T|Is|χIs(x)dx .|IT|. 5.10. Let Tbe a +tree and set FT=X s∈Thf, ϕsiϕs. Then, the inequality below is true. kFTk2≃"X s∈T|hf, ϕsi|2#1/2 .size(T)|IT|1/2. 5.11. With the notation above, assume that 0 ∈ωT. Then, size(T)≃sup J  |J|−1X s∈T Is⊂J |hf, ϕsi|2   1/2 ≃sup J"|J|−1ZJFT−|J|−1ZJ FT 2 dx#1/2 , where the supremum is over all intervals J. The last quantity is the BMO norm of FT. Carleson’s Theorem on Fourier Series 271 5.12. It is an important heuristic that for a collection Sof pairwise incomparable tiles, the functions {ϕs:s∈ S} are nearly orthogonal. The heuristic permits a quantification in terms of the following weak type inequality. Let Sbe a collection of tiles that are pairwise incomparable with respect to ‘<’. Then for all f∈L2and all λ > 0, X s∈Sλ|Is|.λ−2kfk2 2, where Sλ={s∈ S :|hf, ϕsi| > λp|Is|}. Note that this in an inequality about the boundedness of a sublinear operator from L2(R) to L2,∞(R×S). In the latter space, one uses counting measure on S. 5.13. Another important heuristic is that the notion of “strong disjointness” for trees is as “pairwise incomparable” is for tiles. Let e Tbe a collection of strongly disjoint trees. Show that for all f∈L2and all λ > 0, X T∈e Tλ |IT|.λ−2kfk2 2, where e Tλ={T∈e T: ∆(T)≥λ|IT|}. 5.14. Let Sbe a collection of tiles that are pairwise incomparable with respect to ‘<’. Show that for all 2 <p<∞, "X s∈S|hf, ϕsi|p#1/p .kfkp. Notice that the form of this estimate at p=∞is obvious. 5.15. Let e Tbe a collection of strongly disjoint trees. Then for all 2 < p < ∞,X T∈e T ∆(T)p.kfkp. 5.16. The Lpestimates of the previous two complements can in some instances be improved. For each integer k,     X s∈T |Is|=2k |hf, ϕsi|2 |Is|1Is    1/2p .kfkp,2< p < ∞. 272 M. T. Lacey This can be seen by showing that X s∈T |Is|=2k |hf, ϕsi|2 |Is|1Is.(Dil1 2kχ)∗|f|2. 6. The Tree Lemma We begin with some remarks about the maximal function, and a particular form of the same that we shall use at a critical point of this proof. Consider the maximal function Mf= sup I∈D 1I|hf, χIi|. It is well known that this is bounded on L2. A proof follows. Consider a linearized version of the supremum. To each I∈ D, associate a set E(I)⊂I, and require that the sets {E(I) : I∈ D} be pairwise disjoint. (Thus, for fixed f,E(I) is that subset of Ion which the supremum above is equal to |hf, χIi|.) Define Af=X I∈D 1E(I)hf, χIi. We show that kAk2is bounded by a constant, independent of the choice of the sets E(I). The method is that of T T∗. Note that for positive f A A∗f≤2X I∈D X |J|≤|I| 1E(I)hχI, χJih1E(J), fi .X I∈D 1E(I)hf, χIi. It follows that kA∗k2 2= sup kfk2=1hA∗f, A∗fi = sup kfk2=1hf, A A∗fi .kAk2 and so kAk2.1, as claimed. Carleson’s Theorem on Fourier Series 273 We shall have recourse to not only this, but a particular refinement. Let Jbe a partition of Rinto dyadic intervals. To each J∈ J, associate a subset G(J)⊂J, with |G(J)| ≤ δ|J|, where 0 < δ < 1 is fixed. Consider (6.1) Mδf:= X J∈J 1G(J)sup I⊃J|hf, χIi|. Then kMδk2.√δ. The proof is Z|Mδf|2dx =X J∈J |G(J)|sup I⊃J|hf, χIi| ≤δX J∈J |J|sup I⊃J|hf, χIi| ≤δZ|Mf|2dx .δkfk2 2. We begin the main line of the argument. Let δ= dense(T), and σ= size(T). Make a choice of signs εs∈ {±1}such that X s∈T|hf, ϕsihφs,1Ei| =ZEX s∈T εshf, ϕsiφsdx. By the “Schwartz tails”, the integral above is supported on the whole real line. Let Jbe a partition of Rconsisting of the maximal dyadic intervals Jsuch that 3Jdoes not contain any Isfor s∈T. It is helpful to observe that for such Jif |J| ≤ |IT|, then J⊂3IT. And if |J| ≥ |IT|, then dist(J, IT)&|J|. The integral above is at most the sum over J∈ J of the two terms below. X s∈T |Is|≤|J| |hf, ϕsi|ZJ∩E|φs|dx(6.2) ZJ∩E X s∈T |Is|>|J| εshf, ϕsiφs dx.(6.3) Notice that for the second sum to be non-zero, we must have J⊂3IT. 274 M. T. Lacey The first term (6.2) is controlled by an appeal to the “Schwartz tails”. Fix an integer n≥0, and only consider those s∈Tfor which |Is|= 2−n|J|. Now, the distance of Isto Jis at least &|J|. And, |hf, ϕsi|ZJ∩E|φs|dx ≤σδ(|Is|−1dist(Is, J))−10|Is|. The Is⊂IT, so that summing this over |Is|= 2−n|J|will give us σδ2−nmin|J|,|IT|(dist(J, IT)|IT|−1)−10. This is summed over n≥0 and J∈ J to bound (6.2) by .σδ|IT|, as required. Critical to the control of (6.3) is the following observation. Let (6.4) G(J) = J∩[ s∈T |Is|≥|J| N−1(ωs+). Then |G(J)|.δ|J|. To see this, let J0be the next larger dyadic interval that contains J. Then 3J0must contain some Is0, for s0∈T. Let s00 be that tile with Is0⊂Is00 ,|Is00 |=|J|, and ωT⊂ωs00 . Then, s0< s00, and by the definition of density, ZE∩N−1(ωs00 ) χIs00 dx ≤δ. But, for each sas in (6.4), we have ωs⊂ωs00 , so that G(J)⊂N−1(ωs00 ). Our claim follows. Suppose that Tis a −tree. That means that the tiles {Is×ωs+:s∈T} are disjoint. We use an estimation absent of any cancellation effects. Then, the bound for (6.3) is no more than |G(J)|X s∈T |Is|≥|J| |hf, ϕsiφs|∞ .δσ|J|. This is summed over J⊂3ITto get the desired bound. Carleson’s Theorem on Fourier Series 275 Suppose that Tis a +tree. (This is the interesting case.) Then, the tiles {Is×ωs−:s∈T}are pairwise disjoint, and we set F= Mod−c(ωT)X s∈T εshf, ϕsiϕs. Here it is useful to us that we only use the “smooth” functions ϕsin the definition of this function. Note that kFk2.σp|IT|, which is a consequence of the definition of size and Proposition 2.12. Set τ(x) = sup{|Is|:s∈T, N(x)∈ωs+}, and observe that for each J, and x∈J, X s∈T |Is|≥|J| εshf, ϕsiφs(x) = X s∈T τ(x)≥|Is|≥|J| εshf, ϕsiϕs(x). This is so since all of the intervals ωs+must contain ωT, and if N(x)∈ωs+, then it must also be in every other ωs0+that is larger. What is significant here is that on the right we have a truncation of the sum that defines F. This last sum can be dominated by a maximal function. For any τ > 0 and J∈ J, let Fτ,J = Mod−c(ωT)X s∈T τ≥|Is|≥|J| εshf, ϕsiϕs. This function has Fourier support in the interval −7 8|J|−1,−1 4τ−1. In particular, recalling how we defined ϕ, we can choose 1 16 < a, b < 1 4so that Fτ,J = (Dil1 a|J|ϕ−Dil1 bτ ϕ)∗F. We conclude that for x∈J, |Fτ(x),J(x)|.MδF(x), where Mδis defined as in (6.1). The conclusion of this proof is now at hand. We have X J∈J |J|≤3|IT|ZG(J)|Fτ(x),J |dx .ZS|J|≤3|IT|G(J) MδF dx .[ |J|≤3|IT| G(J) 1/2 kMδFk2 .δp|IT|kFk2 .σδ|IT|. 282 M. T. Lacey 7.3. Complements. 7.17. In the inequality (7.2), one can show that the constants Kpon the right hand side obey Kp.p2 p−1. 7.18. If it is the case that for some 0 <α<1, we have the inequality sup T0⊂T|sh(T0)|−1/αk∆T0kα<∞ then, the stronger estimate below holds. k∆Tk1.|sh(T)|. 7.19. In (7.2), we assert the restricted weak type inequality for 1<p<∞. The weak type estimate for 2 < p < ∞is in fact directly available. That is, for 2 < p < ∞, and f∈Lpof norm one, |{CNf > λ}| .λ−p, λ > 0. The key point is to take advantage of the fact that fis locally square integrable. A very brief sketch of the argument follows. (1) It suffices to prove the inequality above for λ= 1. (2) Define Ω = {M|f|2>1}, and show that |Ω|.1. (3) Define sums as in (7.8) and (7.9), and control each term separately. One will need to replace the Size Lemma as stated with 5.15. 8. Remarks 8.1. After Carleson [14] proved his theorem, Hunt [32] extended the argument to Lp, for 1 <p<∞. A similar extension, in the Walsh Paley case, was done by Billiard [9], in the case of L2, and Sj¨olin [67], for all 1 <p<∞. The Carleson theorem has equivalent formulations on the groups R,T, and Z. The last case, of the integers, was explicitly discussed by M´at´e [55]. This paper was overlooked until recently. 8.2. Fefferman [26] devised an alternate proof, which proved to be influential through it’s use of methods of analysis that used both time and frequency information in an operator theoretic fashion. The proof of Lacey and Thiele [50] presented here borrows several features of that proof. The notion of tiles, and the partial order on tiles is due to Fefferman [26]. Likewise, the Density Lemma and the Tree Lemma, and the proof of the same, have clear antecedents in this paper. Carleson’s Theorem on Fourier Series 283 8.3. Those familiar with the Littlewood Paley theory know that it is very useful in decoupling the scales of operators, like those of the Hilbert transform. The Carleson operator is, however, not one in which scales can be decoupled. This is another source of the interest in this theorem. 8.4. We present the proof of the Carleson theorem on the real line due to the presence of the dilation structure. 8.5. We choose to express the Carleson operator in terms of the projection P−. This operator is a linear combination of the identity operator and the Hilbert transform given by Hf(x) := lim →0Z<|y|<1/ f(x−y)dy y. Hence, an alternate form of the Carleson operator is (8.6) sup N|H◦ModNf|= sup NZeiNyf(x−y)dy y. This form is suggestive of other questions related to the Carleson Theorem, a point we rely upon below. 8.7. Despite the fact that Carleson’s operator maps L2into itself, all three known proofs of Carleson’s theorem establish the weak type bound on L2. The strong type bound must be deduced by interpolation. On the other hand, the weak type bound is a known consequence of the pointwise convergence of Fourier series. This was observed by Calder´on, as indicated by a footnote in [84], and is a corollary of a general observation of Stein [70]. 8.8. Hunt and Young [33] have established a weighted estimate for the Carleson operator. Namely for a weight win the class Ap, the Carleson operator maps Lp(w) into itself, for 1 < p < ∞. The method of proof utilizes the known Carleson bound, and distribution inequalities for the Hilbert transform. 8.9. The Proposition 2.8 has a well known antecedent in a characterization of (a constant times) the Hilbert transform as the unique operator A such that Ais bounded operator on L2that commutes with dilation, is invariant under dilations, A2is the identity, but is not itself the identity. See [71]. 8.10. The inequality (3.3) eschews all additional cancellations. It shows that all the necessary cancellation properties are already encoded in the decomposition of the operator. In addition the combinatorial model of the Carleson operator is in fact unconditionally convergent in s∈ T . 284 M. T. Lacey This turns out to be extremely useful fact in the course of the proof: one is free to group the tiles in anyway that one likes. 8.11. The Size Lemma should be compared to Rubio de Francia’s extension of the classical Littlewood Paley inequality [66]. Also see the author’s recent survey of this theorem [44]. 8.12. A +tree Tis a familiar object. Aside from a modulation by c(ωT), it shares most of the properties associated with sums of wavelets. In particular, if 0 ∈ωT, note that size(T)≃X s∈Thf, ϕsiϕsBMO where the last norm is the BMO norm. 8.13. The key instance of the Tree Lemma is that of a +tree. This case corresponds to a particular maximal function applied to a function associated to the tree. It is this point at which the supremum of Carleson’s theorem is controlled by a much tamer supremum: The one in the ordinary maximal function. 8.14. The statement and proof of the size lemma, Lemma 3.9, replaces the initial arguments of this type that are in [47]. This argument has proven to be very flexible in it’s application. And, in some instances it produces sharp estimates, as explained by Barrionuevo and Lacey [8]. 8.15. The set of functions Tk:= {s∈ T :|Is|= 2k}is an example of a Gabor basis. For appropriate choice of ϕ, the operator Akf:= X s∈T |Is|=2k hf, ϕsiϕs is in fact the identity operator. See the survey of Daubechies [25]. 9. Complements and Extensions 9.1. Equivalent formulations of Carleson’s theorem. The Fourier transform has a formulation on each of the Euclidean groups R,Zand T. Carleson’s original proof worked on T. Fefferman’s proof translates very easily to R. M´at´e [55] extended Carleson’s proof to Z. Each of the statements of the theorem can be stated in terms of a maximal Fourier multiplier theorem, and we have stated it as such in this paper. Inequalities for such operators can be transferred between these three Euclidean groups, and was done so by Auscher and Carro [7]. Carleson’s Theorem on Fourier Series 285 9.2. Fourier series near L1, Part 1. The point of issue here is the determination of that integrability class which guarantees the pointwise convergence of Fourier series. The natural setting for these questions is the unit circle T= [0,1], and the partial Fourier sums SNf(x) = X |n|≤Nb f(n)e2πinx,b f(n) = Z1 0 f(x)e−inx dx. In the positive direction, one seeks the “smallest” function ψsuch that if RTψ(f)dx < ∞, then the Fourier series of fconverge pointwise. Antonov [1] has found the best result to date, Theorem 9.1. For all functions f∈L(log L)(log log log L)(T), the partial Fourier series of fconverges pointwise to f. This extends the result of Sj¨olin [67], [68], who had the result above, but with a double log where there is a triple log above. Arias de Reyna [2], [3] has noted an extension of this theorem, in that one can define a rearrangement invariant Banach space B, so that pointwise convergence holds for all f∈B, and Bcontains L(log L)(log log log L). The method of proof takes as it’s starting point, the distributional estimate of (7.4). One seeks to “extrapolate” these inequalities to the setting of the theorem above and Antonov nicely exploits the explicit nature of the kernels involved in this maximal operator. Also see the work of Sj¨olin and Soria [69] who demonstrate that Antonov’s approach extends to other maximal operator questions. 9.3. Fourier series near L1, Part 2. In the negative direction, Kolmogorov’s fundamental example [36], [37] of an integrable function with pointwise divergent Fourier series admits a strengthening to the following statement, as obtained by K¨orner [40]. Theorem 9.2. For all ψ(x)=o(log log x), there is a function f: [0,2π]→ Rwith divergent Fourier series, and R|f|ψ(f)dx < ∞. The underlying method of proof was, in some essential way, unsurpassed until quite recently, when Konyagin [38], [39] proved Theorem 9.3. The previous theorem holds assuming only (9.4) ψ(x) = o slog x log log x!. 286 M. T. Lacey There is a related question, on the growth of partial sums of Fourier series of integrable functions. Hardy [31] showed that for integrable functions f, one has Snf=o(log n) a.e., and asked if this is the best possible estimate. This question is still open, with the best result from below following from Konyagin’s example. With ψas in (9.4), there is an f∈L1(T) with lim sup n→∞ Snf ψ(n)=∞for all x∈T. Let δtdenote the Dirac point mass at t∈T. The method of proof is to construct measures µ=K−1 K X k=1 δtk a set E⊂Twith measure at least 1/4, and choices of integers Nfor which sup n<N|Snµ(x)| ≥ ψ(N), x ∈E. Kolmogorov’s example consists of uniformly distributed point masses, whereas Konyagin’s example consists of point masses that have a distribution reminiscent of a Cantor set. 9.3.1. Probabilistic series. It is of interest from the point of view of probability and ergodic theory, to consider the version of the Hilbert transform and Carleson theorem that arises from the integers. Here, we consider the probabilistic versions. Let Xkbe independent and identically distributed copies of a mean zero random variable X. The question is if the sum ∞ X k=1 Xk k converges a.s. Without additional assumption on the distribution of X, a necessary and sufficient condition is that EXlog(2 + |X|)<∞. One direction of this is in [53]. If however Xis assumed to be symmetric, integrability is necessary and sufficient. This addresses the issue of the Hilbert transform. Carleson’s theorem, in this language, concerns the convergence of the series Y(t) := ∞ X k=1 Xk ke2πikt for all t∈T. Carleson’s Theorem on Fourier Series 287 The role of the quantifiers should be emphasized. Convergence holds for all t∈T, on a set of full probability. Given this, abstract results on 0–1 Laws assure us that if the series converges for all t, off of a single set of probability zero, then the limiting function is continuous with probability one. The paper of Talagrand [79] gives necessary and sufficient conditions for the convergence of this series. Theorem 9.5. Let Xkbe independent identically distributed copies of a mean zero symmetric random variable X.X∈L(log log L)iff the series Y(t)converges to a continuous function on Talmost surely. The assumption of symmetry should be added to the statement of the theorem in [79]. Cuzick and Lai [24] provide an example of a non symmetric mean zero X∈L(log log L) for which the series Y(t) is divergent. This series is a borderline series in that it just falls out of the scope of the powerful theory of Marcus and Pisier [54] on random Fourier series. My thanks to several people who provided me with some references for this section. They are James Campbell, Ciprian Demeter, Michael Lin, and Anthony Quas. 9.4. The Wiener-Wintner question. A formulation of Carleson’s operator on Zis CZf(j) := sup τ sup NX 0<|k|<N f(j−k)eiτk k. See [55]. Unaware of this work which followed soon after Carleson, Campbell and Petersen [12] considered this operator on `2, with equivalence in `pestablished by Assani and Petersen [5], [12]. Also see Assani, Petersen and White [6], for these and other equivalences. The latter authors had additional motivations from dynamical systems, which we turn to now. Calder´on [11] observed that inequalities for operators on Zwhich commute with translation can be transferred to discrete dynamical systems. Let (X, µ) be a probability space, and T:X→Xa map which preserves µmeasure. Thus, µ(T−1A) = µ(A) for all measurable A⊂X. A Carleson operator on (X, µ, T ) is Cmps f(x) := sup τ sup NX 0<|k|<N f(Tkx)eiτk k. And it is a consequence of Calder´on’s observation and Carleson’s theorem that this operator is bounded on L2(X). 288 M. T. Lacey There is however a curious point that distinguishes this case from the other settings of Euclidean groups. It is the case that one has pointwise convergence of lim N→∞ X 0<|k|<N f(Tkx)eiτk kexists for all τ holding for almost every x∈X? The boundedness of the maximal function Cmps shows that this would hold on a closed set in L2(X). The missing ingredient is the dense class for which the convergence above holds. Unlike the setting of Euclidean groups, there is no natural dense class. This conjecture was posed by Campbell and Petersen [12]. Conjecture 9.6. For all measure preserving systems (X, µ, T), and all f∈L2(X), we have the following: µ  lim N→∞ X 0<|k|<N f(Tkx)eiτk kexists for all τ  = 1. A theorem of Wiener and Wintner [83] provides a classical motivation of this question. This theorem concerns the same phenomena, but with the discrete Hilbert transform replaced by the averages. Theorem 9.7. For all measure preserving systems (X, µ, T ), and all f∈L2(X), we have the following: µ(lim N→∞ N−1 N−1 X k=0 f(Tkx)eiτk exists for all τ)= 1. This theorem admits a simple proof. And note that this theorem trivially supplies a dense class in all Lpspaces, 1 ≤p < ∞. The Wiener-Wintner theorem has several interesting variants, for which one can phrase related questions by replacing averages by Hilbert transforms. As far as is known to us, none of these questions is answered. An attractive theorem proved by Lesigne [51], [52] is Theorem 9.8. For any measure preserving system (X, µ, T )and all integrable functions f, there is a subset Xf⊂Xof full measure so that for all x∈Xf, all polynomials p, and all 1-periodic functions φ, the limit below exists: lim N→∞ N−1 N X n=1 φ(p(n))f(Tnx). Carleson’s Theorem on Fourier Series 289 Extending this theorem to the Hilbert transform would be an extraordinary accomplishment, whereas if one replaced the discrete dynamical system by flows, it could be that the corresponding result for the Hilbert transform might be within reach. In connection to this, Arkhipov and Oskolkov [4] have proved the following theorem. Theorem 9.9. For all integers d, sup deg(p)=dX n6=0 eip(n) n<∞, with the supremum formed over all polynomials of degree d. This is a far more subtle fact than the continuous analog stated in (9.10). Arkhipov and Oskolkov use the Hardy Littlewood Circle method of exponential sums, with the refinements of Vinogradov. See [4], [62], [63]. By Plancherel, this theorem shows that for a polynomial pwhich maps the integers to the integers, the operators on the integers given by Tpf(j) = X n6=0 f(x−p(n)) is a bounded operator on `2(Z). Stein and Wainger have established `2mapping properties for certain Radon transforms [76], [77]. 9.5. E. M. Stein’s maximal function. A prominent theme of the research of Stein and Wainger concerns oscillatory integrals, with polynomial phases. It turns out to be of interest to determine what characteristics of the polynomial govern allied analytic quantities. In many instances, this characteristic is just the the degree of the polynomial. For instance, the following is a corollary to a theorem of Stein and Wainger from 1970 [75]. Namely, that one has a bound (9.10) sup deg(P)=dZeip(y)dy y.1, d = 1,2,... A conjecture of Stein’s concerns an extension of Carleson’s maximal operator to one in which one forms a supremum over all polynomial choices of phase with a fixed degree. Thus, 290 M. T. Lacey Conjecture 9.11. For each integer d, the maximal function below maps Lpinto itself for 1< p < ∞. Cdf(x) = sup deg(P)=dZeip(y)f(x−y)dy y. Note that the case of d= 1 corresponds to Carleson’s theorem. Let us set C0 dto be the maximal operator above, but with the the restriction that the polynomials pdo not have a linear term. It is useful to make this distinction, as it is the linear terms that are intertwined with the Fourier transform. Stein [73] considered the purely quadratic terms, and showed that C0 2 maps Lpinto itself for all 1 <p<∞. The essence of the matter is the bound on L2, and there his argument is a variant on the method of T T∗, emphasizing a frequency decomposition of the operator. Stein and Wainger [78] have proved that C0 dis bounded on all Lp’s, for all d≥2. Again the L2case is decisive and the argument is an application of the T T∗method, but with a spatial decomposition of the operator. Let us comment in a little more detail about how these results are proved. If, for the moment, one consider a fixed polynomial P(y), and the oscillatory integral (9.12) TPf(x) := ZeiP (y)f(x−y)dy y. One may utilize the scale invariance of the the Hilbert transform kernel to change variables. With the correct change of variables, one may assume that the polynomial P(y) = Pd j=1 ajyjsatisfies P|aj|= 1. Then, it is evident that for |y|<1, say, that the integral above is well approximated by a truncation of the Hilbert transform. Thus, it is those scales of the operator larger than 1 that must be controlled. It is a consequence of the van der Corput estimates that some additional decay can be obtained from these terms. In particular, one has this estimate. To set notation, in the one dimensional case only, set P~a(x) = adxd+···+a1x, ~a = (ad,...,a1). Lemma 9.13. Let χbe a smooth bump function. Then we have the estimate \ eiP~a(y)χ(y)(ξ)∞.(1 + k~ak1)−1/d. In particular, by the Plancherel identity, we have the estimate (9.14) [eiP~a(y)χ(y)] ∗f(x)2.(1 + k~ak1)−1/dkfk2. Carleson’s Theorem on Fourier Series 291 Notice that these estimates are better than the trivial ones. And that the second estimate can be interpolated to obtain a range of Lpinequalities for which one has decay, with a rate that depends upon the degree and the Lpspace in question. In a discussion of the extensions of this principle in for example [73], [78], one establishes appropriate extensions of this last lemma, always seeking some additional decay that arises from the polynomial. For instance, in [78], Stein and Wainger prove a far reaching extension of this principle. Lemma 9.15. There is a constant δ > 0, depending only on the degree d, so that we have the estimate kSλk2.(1 + λ)−δ, for all λ > 0, where Sλf(x) := sup k~ak1≥λ a1=0 sup t>0|[Dil1 teiP~a(·)χ]∗f|. It is essential in this supremum be formed over polynomials P~a which do not have a linear term. Ionescu has pointed out that this lemma is not true with the linear term included, even in the case of second degree polynomials. The example, which we will see again below, begins by taking a function f(x), and replacing it by the function g(x) = eiλx2f(x). Then, in the supremum defining Sλabove, take the dilation parameter to be t= 1, and the polynomial to be P(y) = −y2+ 2xy. Note that as we are taking a supremum, we can in particular take a polynomial that depends upon x. In this example, the modulation of fby “chirp” is then canceled out by the choice of P. There is no decay in the estimate. This estimate is special to the case of the second power, so it is natural to guess that it plays a distinguished role in these considerations. This also points out an error in the author’s paper [43]. (The error enters in specifically at the equation (2.9). The phase plane analysis of that paper might yet find some use.) At this point, the resolution of Stein’s conjecture is not settled. And it appears that a positive bound of the operator C2will in particular require a novel phase plane analysis with quadratic phase. This should be compared to the notion of degeneracy in Section 9.8 below. 9.6. Fourier series in two dimensions. In this section we extend the Fourier transform to functions of the plane b f(ξ) = Zf(x)eix·ξdx 298 M. T. Lacey And they say that a polynomial Phas a power decay property if there is a δ > 0, so that for all fj∈L∞(Vj), we have the estimate |Λ(f1,...,fn)|.(1 + |λ|)−δ n Y j=1kfjk∞. From this estimate, a range of power decay estimates hold in all relevant products of Lpspaces. This should be compared to Lemma 9.13 and in particular (9.14) below. Clearly, there are obstructions to a power decay property, and this obstruction can be formalized in a definition. A polynomial Pis said to be degenerate (relative to {Vj})if there exist polynomials pj:Vj→ Rsuch that P=Pn j=1 pj◦πj. Otherwise Pis nondegenerate. In the case n= 0, where the collection of subspaces {Vj}is empty, Pis considered to be nondegenerate if and only if it is nonconstant. And in the example (9.25), we see that P(y) = 2x2+ 2y2= (x+y)2+ (x−y)2 is degenerate. It is natural to conjecture that non degeneracy is sufficient for a power decay property. This is verified in a wide range of special cases in the paper by Christ, Li, Tao, and Thiele [20], by a range of interesting techniques. It is of interest to determine if the natural conjecture here is indeed correct. 9.9. Hilbert transform on smooth families of lines. This question has its beginnings in the Besicovitch set, which we already mentioned in connection to spherical summation of Fourier series. One may construct Besicovitch sets with these properties. For choices of 0 < , α < 1, there is a Besicovitch set Kin the square [0,4]2say, for which Khas measure at most , and there is a function g:R2→T, so that for a set of x’s in [0,4]2of measure &1, K∩ {x+tv(x) : t∈R} contains a line segment of length one, and vis H¨older continuous of order α. One can ask if the H¨older continuity condition is sharp. A beautiful formulation of a conjecture in this direction is attributed to Zygmund. Conjecture 9.26. Let v:R2→Tbe H¨older continuous (of order 1). Then for all square integrable functions f, f(x) = lim t→0(2t)−1Zt −t f(x−uv(x)) du a.e (x). This is a differentiablity question, on a choice of lines specified by v. The only stipulation is that vis H¨older continuous. This is only known Carleson’s Theorem on Fourier Series 299 under more stringent conditions on v, such as analytic due to Stein [72], or real analytic due to Bourgain [10]. There is a partial result due to Katz [35] (also see [34]) that demonstrates at worst “log log” blowup assuming the H¨older continuity of v. The question is open, even if one assumes that v∈C1000. The difficulty in this problem arises from those points at which the gradient of vis degenerate; assumptions such as analyticity certainly control such degeneracies. Stein [72] posed the Hilbert transform variant, namely defining Hvf(x) := p.v. Z1 −1 f(x−yv(x))dy y, is it the case that there is a constant cso that if kvkH¨ol < c, then Hvmaps L2(R2) into itself. A curious fact about this question is that this inequality, if known, implies Carleson’s theorem for one dimensional Fourier series. To see this, observe that the symbol for the transform is ψ(ξ·v(x)), where ψis the Fourier transform of y−11{|y|<1}. Suppose the vector field is of the form v(x) = (1, ν(x1)) where we need only assume that ν is H¨older continuous of norm 1 say, and consider the trace of the symbol on the line ξ2=−N. Then, the symbol is ψ((ξ1, N)·(1, ν(x1)) = ψ(ξ1−Nν(x1)). We conclude that this symbol defines a bounded linear operator on L2(R), with bound that is independent of N. That is, for any Lipschitz function ν(x1), and any N > 1 the symbol ψ(ξ1−Nν(x1)) is the symbol of a bounded linear operator on L2(R). By varying Nand ν, we may replace Nν(x1) by an arbitrary measurable function. This is the substance of Carleson’s theorem. But the implication is entirely one way: A positive answer to the family of lines question seems to require techniques quite a bit more sophisticated than those that imply Carleson’s theorem. Recently Lacey and Li [45] have been able to obtain a partial answer, assuming only that the vector field has 1 + derivatives. Theorem 9.27. Assume that v∈C1+for some  > 0. Then the operator Hvis bounded on L2(R2). The norm of the operator is at most kHvk2.[1 + log+kvkC1+]2. 9.10. Schr¨odinger operators, scattering transform. There is a beautiful line of investigation relating Schr¨odinger equations in one dimension to aspects of the Fourier transform, and in particular, Carleson’s theorem. There is a further connection to scattering 300 M. T. Lacey transforms and nonlinear Fourier analysis. All in all, these topics are extremely broad, with several different sets of motivations, and a long list of contributors. We concentrate on a succinct way to see the connection to Carleson’s theorem, an observation made explicitly by Christ and Kiselev [15], [16], also see [65]. The basic object is a time independent Schr¨odinger operator on the real line, H=−d2 dx2+V where Vis an appropriate potential on the real line. The idea is that if V is small, in some specific senses, then the spectrum of Hshould resemble that of −d2 dx2. In particular eigenfunctions should be perturbations of the exponentials. Standard examples show that one should seek to show that for almost all λ, the eigenfunctions of energy λ, that is the solutions to (H−λ2I) are bounded perturbations of e±iλx. Seeking such an eigenfunction, one can formally write u(x) = eiλx +1 iλ Z∞ x sin(λ(x−y))V(y)u(y)dy. Iterating this formula, again formally, one has u(x)=eiλx +1 iλ Z∞ x sin(λ(x−y))V(y)eiλy dy(9.28) +1 (iλ)2ZZx≤y1≤y2 sin(x−y1)sin(y1−y2)V(y1)V(y2)u(y2)dy1dy2.(9.29) Observe that (9.28) no longer contains u, and is a linear combination of eiλx Z∞ x e2iλyV(y)dy(9.30) eiλx Z∞ x V(y)dy.(9.31) One seeks estimates of these in the mixed norm space of say, L2 λL∞ x. From such estimates, one deduces that for almost all λ, there is an eigenfunction with is a perturbation of eiλx. Concerning (9.30), notice that if V∈L2, we can, by Plancherel, regard Vas b f, for some f∈L2. The desired estimate is then a consequence Carleson’s Theorem on Fourier Series 301 of Carleson’s theorem. This is indicative of the distinguished role that L2plays in this subject. Also of the intertwining of the roles of frequency and time that occur in the subject. Concerning (9.31), unless V∈L1, there is no reasonable interpretation that can be placed on this term. In practice, a different approach than the one given here must be adopted. If one continues the expansion in (9.29), one gets a bilinear operator with features that resemble both the Carleson operator, and the bilinear Hilbert transform. See the papers by Muscalu, Tao, and Thiele [57], [59]–[61]. We refer the reader to these papers by Christ and Kiselev [15]–[18]. For a survey of this subject, see [19]. The reader should also consult the ongoing investigations of Muscalu, Tao, and Thiele [58]. This paper begins with an interesting summary of the perspective of the nonlinear Fourier transform. References [1] N. Yu. Antonov, Convergence of Fourier series, in: “Proceedings of the XX Workshop on Function Theory” (Moscow, 1995), East J. Approx. 2(2) (1996), 187–196. [2] J. Arias de Reyna,“Pointwise convergence of Fourier series”, Lecture Notes in Mathematics 1785, Springer-Verlag, Berlin, 2002. [3] J. Arias de Reyna, Pointwise convergence of Fourier series, J. London Math. Soc. (2) 65(1) (2002), 139–153. [4] G. I. Arkhipov and K. I. Oskolkov, A special trigonometric series and its applications, (Russian), Mat. Sb. (N.S.) 134(176), no. 2 (1987), 147–157, 287; translation in Math. USSR-Sb. 62(1) (1989), 145–155. [5] I. Assani and K. Petersen, The helical transform as a connection between ergodic theory and harmonic analysis, Trans. Amer. Math. Soc. 331(1) (1992), 131–142. [6] I. Assani, K. Petersen and H. White, Some connections between ergodic theory and harmonic analysis, in: “Almost everywhere convergence, II” (Evanston, IL, 1989), Academic Press, Boston, MA, 1991, pp. 17–40. [7] P. Auscher and M. J. Carro, On relations between operators on RN,TNand ZN,Studia Math. 101(2) (1992), 165–182. [8] J. Barrionuevo and M. T. Lacey, A weak-type orthogonality principle, Proc. Amer. Math. Soc. 131(6) (2003), 1763–1769 (electronic). 302 M. T. Lacey [9] P. Billard, Sur la convergence presque partout des s´eries de Fourier-Walsh des fonctions de l’espace L2(0,1), Studia Math. 28 (1966/1967), 363–388. [10] J. Bourgain, A remark on the maximal function associated to an analytic vector field, in: “Analysis at Urbana, Vol. I” (Urbana, IL, 1986–1987), London Math. Soc. Lecture Note Ser. 137, Cambridge Univ. Press, Cambridge, 1989, pp. 111–132. [11] A.-P. Calder´ on, Ergodic theory and translation-invariant operators, Proc. Nat. Acad. Sci. U.S.A. 59 (1968), 349–353. [12] J. Campbell and K. Petersen, The spectral measure and Hilbert transform of a measure-preserving transformation, Trans. Amer. Math. Soc. 313(1) (1989), 121–129. [13] A. Carbery, A. Seeger, S. Wainger and J. Wright, Classes of singular integral operators along variable lines, J. Geom. Anal. 9(4) (1999), 583–605. [14] L. Carleson, On convergence and growth of partial sumas of Fourier series, Acta Math. 116 (1966), 135–157. [15] M. Christ and A. Kiselev, WKB asymptotic behavior of almost all generalized eigenfunctions for one-dimensional Schr¨odinger operators with slowly decaying potentials, J. Funct. Anal. 179(2) (2001), 426–447. [16] M. Christ and A. Kiselev, Scattering and wave operators for one-dimensional Schr¨odinger operators with slowly decaying nonsmooth potentials, Geom. Funct. Anal. 12(6) (2002), 1174–1234. [17] M. Christ and A. Kiselev, WKB and spectral analysis of onedimensional Schr¨odinger operators with slowly varying potentials, Comm. Math. Phys. 218(2) (2001), 245–262. [18] M. Christ and A. Kiselev, Maximal functions associated to filtrations, J. Funct. Anal. 179(2) (2001), 409–425. [19] M. Christ and A. Kiselev, One-dimensional Schr¨odinger operators with slowly decaying potentials: spectra and asymptotics (or: Baby Fourier analysis meets toy quantum mechanics), Lecture Notes, University of Arkansas Spring Lecture Series (2002). [20] M. Christ, X. Li, T. Tao and C. M. Thiele, On multilinear oscillatory integrals, nonsingular and singular, Arxiv, math.CA/0311039 (2003). [21] G. Cohen and M. Lin, Laws of large numbers with rates and the one-sided ergodic Hilbert transform, Illinois J. Math. 47(4) (2003), 997–1031. Carleson’s Theorem on Fourier Series 303 [22] R. R. Coifman, Remarks on weak type inequalities for operators commuting with translations, Bull. Amer. Math. Soc. 74 (1968), 710–714. [23] A. C´ ordoba and R. Fefferman, On the equivalence between the boundedness of certain classes of maximal and multiplier operators in Fourier analysis, Proc. Nat. Acad. Sci. U.S.A. 74(2) (1977), 423–425. [24] J. Cuzick and Tze Leung Lai, On random Fourier series, Trans. Amer. Math. Soc. 261(1) (1980), 53–80. [25] I. Daubechies,“Ten lectures on wavelets”, CBMS-NSF Regional Conference Series in Applied Mathematics 61, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 1992. [26] C. Fefferman, Pointwise convergence of Fourier series, Ann. of Math. (2) 98 (1973), 551–571. [27] C. Fefferman, On the divergence of multiple Fourier series, Bull. Amer. Math. Soc. 77 (1971), 191–195. [28] C. Fefferman, On the convergence of multiple Fourier series, Bull. Amer. Math. Soc. 77 (1971), 744–745. [29] C. Fefferman, The multiplier problem for the ball, Ann. of Math. (2) 94 (1971), 330–336. [30] L. Grafakos, T. Tao and E. Terwilleger,Lpbounds for a maximal dyadic sum operator, Math. Z. 246(1–2) (2004), 321–337. [31] G. H. Hardy, On the summability of Fourier’s series, J. London Math. Soc. 12 (1913), 365–372. [32] R. A. Hunt, On the convergence of Fourier series, in: “1968 Orthogonal Expansions and their Continuous Analogues” (Proc. Conf., Edwardsville, Ill., 1967), Southern Illinois Univ. Press, Carbondale, Ill., pp. 235–255. [33] R. A. Hunt and Wo Sang Young, A weighted norm inequality for Fourier series, Bull. Amer. Math. Soc. 80 (1974), 274–277. [34] N. H. Katz, Maximal operators over arbitrary sets of directions, Duke Math. J. 97(1) (1999), 67–79. [35] N. H. Katz, A partial result on Lipschitz differentiation, in: “Harmonic analysis at Mount Holyoke”, (South Hadley, MA, 2001), Contemp. Math. 320, Amer. Math. Soc., Providence, RI, 2003, pp. 217–224. [36] A. N. Kolmogorov, Une s´erie de Fourier-Lebesque divergente presque partout, Fund. Math. 4(1923), 324–328. [37] A. N. Kolmogorov, Une s´erie de Fourier-Lebesque divergente presque partout, C. R. Acad. Sci. Paris S´er. I Math. 183 (1926), 1327–1329. 304 M. T. Lacey [38] S. V. Konyagin, On the divergence everywhere of trigonometric Fourier series, (Russian), Mat. Sb. 191(1) (2000), 103–126; translation in Sb. Math. 191(1–2) (2000), 97–120. [39] S. V. Konyagin, On divergence of trigonometric Fourier series everywhere, C. R. Acad. Sci. Paris S´er. I Math. 329(8) (1999), 693–697. [40] T. W. K¨ orner, Everywhere divergent Fourier series, Colloq. Math. 45(1) (1981), 103–118 (1982). [41] M. T. Lacey, Transferring the Carleson-Hunt theorem in the setting of Orlicz spaces, in: “Interaction between functional analysis, harmonic analysis, and probability” (Columbia, MO, 1994), Lecture Notes in Pure and Appl. Math. 175, Dekker, New York, 1996, pp. 307–314. [42] M. T. Lacey, On the bilinear Hilbert transform, in: “Proceedings of the International Congress of Mathematicians, Vol. II” (Berlin, 1998), Doc. Math. Extra Vol. II (1998), 647–656 (electronic). [43] M. T. Lacey, Carleson’s theorem with quadratic phase functions, Studia Math. 153(3) (2002), 249–267. [44] M. T. Lacey, Issues related to Rubio de Francia’s LittlewoodPaley Inequailty: A Survey, math.CA/0306417 (2003). [45] M. T. Lacey and X. Li, Maximal theorems for directional Hilbert transform on the plane, Preprint. [46] M. T. Lacey and C. M. Thiele,Lpestimates for the bilinear Hilbert transform, Proc. Nat. Acad. Sci. U.S.A. 94(1) (1997), 33–35. [47] M. T. Lacey and C. M. Thiele,Lpestimates on the bilinear Hilbert transform for 2 < p < ∞,Ann. of Math. (2) 146(3) (1997), 693–724. [48] M. T. Lacey and C. M. Thiele, On Calder´on’s conjecture for the bilinear Hilbert transform, Proc. Natl. Acad. Sci. USA 95(9) (1998), 4828–4830 (electronic). [49] M. T. Lacey and C. M. Thiele, On Calder´on’s conjecture, Ann. of Math. (2) 149(2) (1999), 475–496. [50] M. T. Lacey and C. M. Thiele, A proof of boundedness of the Carleson operator, Math. Res. Lett. 7(4) (2000), 361–370. [51] E. Lesigne, Spectre quasi-discret et th´eor`eme ergodique de Wiener-Wintner pour les polynˆomes, Ergodic Theory Dynam. Systems 13(4) (1993), 767–784. [52] E. Lesigne, Un th´eor`eme de disjonction de syst`emes dynamiques et une g´en´eralisation du th´eor`eme ergodique de Wiener-Wintner, Ergodic Theory Dynam. Systems 10(3) (1990), 513–521. Carleson’s Theorem on Fourier Series 305 [53] J. Marcinkiewicz and A. Zygmund, Sur les fonctions ind´ependentes, Fund. Math. 29 (1937), 60–90. [54] M. B. Marcus and G. Pisier, Characterizations of almost surely continuous p-stable random Fourier series and strongly stationary processes, Acta Math. 152(3–4) (1984), 245–301. [55] A. M´ at´ e, Convergence of Fourier series of square integrable functions, (Hungarian, with English summary), Mat. Lapok 18 (1967), 195–242. [56] C. Muscalu, T. Tao and C. M. Thiele, Multi-linear operators given by singular multipliers, J. Amer. Math. Soc. 15(2) (2002), 469–496 (electronic). [57] C. Muscalu, T. Tao and C. M. Thiele, A discrete model for the bi-Carleson operator, Geom. Funct. Anal. 12(6) (2002), 1324–1364. [58] C. Muscalu, T. Tao and C. M. Thiele, A Carleson theorem for a Cantor group model of the scattering transform, Nonlinearity 16(1) (2003), 219–246. [59] C. Muscalu, T. Tao and C. M. Thiele,Lpestimates for the “biest” I. The Walsh model, math.CA/0102083. [60] C. Muscalu, T. Tao and C. M. Thiele,Lpestimates for the “biest” II. The Fourier model, math.CA/0102084. [61] C. Muscalu, T. Tao and C. M. Thiele, A counterexample to a multilinear endpoint question of Christ and Kiselev, Math. Res. Lett. 10(2–3) (2003), 237–246. [62] K. I. Oskolkov, Schr¨odinger equation and oscillatory Hilbert transforms of second degree, J. Fourier Anal. Appl. 4(3) (1998), 341–356. [63] K. I. Oskolkov, A class of I. M. Vinogradov’s series and its applications in harmonic analysis, in: “Progress in approximation theory” (Tampa, FL, 1990), Springer Ser. Comput. Math. 19, Springer, New York, 1992, pp. 353–402. [64] E. Prestini and P. Sj¨ olin, A Littlewood-Paley inequality for the Carleson operator, J. Fourier Anal. Appl. 6(5) (2000), 457–466. [65] C. Remling, The absolutely continuous spectrum of one-dimensional Schr¨odinger operators with decaying potentials, Comm. Math. Phys. 193(1) (1998), 151–170. [66] J. L. Rubio de Francia, A Littlewood-Paley inequality for arbitrary intervals, Rev. Mat. Iberoamericana 1(2) (1985), 1–14. [67] P. Sj¨ olin, An inequality of Paley and convergence a.e. of WalshFourier series, Ark. Mat. 7(1969), 551–570 (1969). [68] P. Sj¨ olin, Convergence almost everywhere of certain singular integrals and multiple Fourier series, Ark. Mat. 9(1971), 65–90. 306 M. T. Lacey [69] P. Sj¨ olin and F. Soria, Remarks on a theorem by N. Yu. Antonov, Studia Math. 158(1) (2003), 79–97. [70] E. M. Stein, On limits of seqences of operators, Ann. of Math. (2) 74 (1961), 140–170. [71] E. M. Stein,“Singular integrals and differentiability properties of functions”, Princeton Mathematical Series 30, Princeton University Press, Princeton, N.J., 1970. [72] E. M. Stein, Problems in harmonic analysis related to curvature and oscillatory integrals, in: “Proceedings of the International Congress of Mathematicians, Vol. 1, 2” (Berkeley, Calif., 1986), Amer. Math. Soc., Providence, RI, 1987, pp. 196–221. [73] E. M. Stein, Oscillatory integrals related to Radon-like transforms, in: “Proceedings of the Conference in Honor of Jean-Pierre Kahane” (Orsay, 1993), J. Fourier Anal. Appl. Special Issue (1995), 535–551. [74] E. M. Stein,“Harmonic analysis: real-variable methods, orthogonality, and oscillatory integrals”, Princeton Mathematical Series 43, Monographs in Harmonic Analysis III, Princeton University Press, Princeton, NJ, 1993. [75] E. M. Stein and S. Wainger, The estimation of an integral arising in multiplier transformations, Studia Math. 35 (1970), 101–104. [76] E. M. Stein and S. Wainger, Discrete analogues in harmonic analysis. I. l2estimates for singular Radon transforms, Amer. J. Math. 121(6) (1999), 1291–1336. [77] E. M. Stein and S. Wainger, Discrete analogues in harmonic analysis. II. Fractional integration, J. Anal. Math. 80 (2000), 335–355. [78] E. M. Stein and S. Wainger, Oscillatory integrals related to Carleson’s theorem, Math. Res. Lett. 8(5–6) (2001), 789–800. [79] M. Talagrand, A borderline random Fourier series, Ann. Probab. 23(2) (1995), 776–785. [80] C. M. Thiele, Time-frequency analysis in the discrete phase plane, in: “Topics in analysis and its applications”, World Sci. Publishing, River Edge, NJ, 2000, pp. 99–152. [81] C. M. Thiele, The quartile operator and pointwise convergence of Walsh series, Trans. Amer. Math. Soc. 352(12) (2000), 5745–5766. [82] C. M. Thiele, Singular integrals meet modulation invariance, in: “Proceedings of the International Congress of Mathematicians, Vol. II” (Beijing, 2002), Higher Ed. Press, Beijing, 2002, pp. 721–732. Carleson’s Theorem on Fourier Series 307 [83] N. Wiener and A. Wintner, On the ergodic dynamics of almost periodic systems, Amer. J. Math. 63 (1941), 794–824. [84] A. Zygmund,“Trigonometric series. Vol. I, II”, Third edition, Cambridge Mathematical Library, Cambridge University Press, Cambridge, 2002. School of Mathematics Georgia Institute of Technology Atlanta GA 30332 USA E-mail address:[email protected] http://www.math.gatech.edu/~lacey Primera versi´o rebuda el 30 d’octubre de 2003, darrera versi´o rebuda el 18 de juny de 2004.