On the Boundedness of the Carleson Operator near L 1
aa r X i v : . [ m a t h . C A ] A ug ON THE BOUNDEDNESS OF THE CARLESONOPERATOR NEAR L VICTOR LIE
Abstract.
Based on the tile discretization elaborated in [14], we de-velop a Calderon-Zygmund type decomposition of the Carleson opera-tor. As a consequence, through a unitary method that makes no useof extrapolation techniques, we recover the previously known resultsregarding the largest rearrangement invariant space of functions withalmost everywhere convergent Fourier series. Introduction
In this paper we analyze some aspects concerning the behavior of the Car-leson operator near L . Elaborating on an idea introduced in [14], we designa Calderon-Zygmund type decomposition of the Carleson operator which,besides the direct interest in our problem, may prove useful in other relatedtopics. In particular, through this technique we are able to encompass thepreviously known results regarding the problem of the largest rearrangementinvariant space of (integrable) functions for which the a.e. convergence ofthe Fourier Series holds. The relevant point here though is not the ability ofreproving these results but rather the existence of a method that avoids thelimitations of the extrapolation techniques - the main ingredient on whichall the previous results rely on.As we will see, most of the difficulty and interest resides in the tile de-composition of the Carleson operator. Once this decomposition is achieved,everything else follows naturally. In the present paper we are focussing onthe method rather than on obtaining the best possible space on which theCarleson operator is finitely almost everywhere. With respect to the latterthe best current result belongs to Arias de Reyna ([2]).A significant improvement of his result seems to require an original idea,this paper setting just the first step in opening up a new direction of inves-tigation. Date : October 17, 2018.
Key words and phrases.
Time-frequency analysis, Carleson’s Theorem. This approach has as a consequence the removal of the exceptional sets in the tiledecomposition and thus gives direct strong L bounds for the Carleson operator. Also thisis one of the key ingredients in providing the full range for the L p bounds (1 < p < ∞ ) ofthe Polynomial Carleson operator. See also 3) in the Remarks section.
These being said, let us state the precise theme of interest regarding thebehavior of the Fourier Series near L : Open problem.
What is the largest Banach rearrangement invariantspace ( Y, k · k Y ) with Y ⊆ L ( T ) for which the Carleson operator defined by T : C ∞ ( T )
7→ D ′ ( T ) with (1) T f ( x ) := sup N ∈ N (cid:12)(cid:12)(cid:12)(cid:12)Z T e i N ( x − y ) cot( x − y ) f ( y ) dy (cid:12)(cid:12)(cid:12)(cid:12) , obeys the relation (2) k T f k , ∞ . k f k Y ∀ f ∈ Y ? Observation.
Notice that from Stein’s maximal principle ( [19] ) the aboveopen problem is equivalent to asking for the largest Banach rearrangementinvariant space ( Y, k · k Y ) with Y ⊆ L ( T ) for which the partial FourierSeries { S n f } n have the property (3) S n f ( x ) n →∞ −→ f ( x ) a.e. x ∈ T ∀ f ∈ Y. Significant literature has been written on this subject. The major break-through was made by Carleson ([6]), who showed that L ( T ) ⊂ Y . Later,Hunt ([10]) extended this result by showing that L p ( T ) ⊂ Y for any 1 < p < ∞ . A new influential proof of Carleson’s result was given by Fefferman in[9]. From this on, the problem evolved at a slower rate towards the limitingindex p = 1: Sj¨olin, proved in [16], that one may take in (2) Y = L (log L ) (ifone requires strong L bounds in (2)) or even L log L log log L for just L , ∞ bounds. Next, Soria ([17],[18]) constructed a larger space B ∗ ϕ ⊂ Y . In [1],Antonov showed that (2) holds for Y = L log L log log log L . Finally, com-bining elements from Antonov’s and Soria’s approaches with techniques onlogconvex quasi-Banach spaces, Arias de Reyna ([2]) proved that QA ⊂ Y ,where QA is a quasi-Banach space described in the Appendix.The following chain of inclusions holds:(4) L (log L ) ( L log L log log L ( B ∗ ϕ , L log L log log log L ( QA ( L log L .
The main result in this paper is given by We set D ′ ( T ) the class of distributions supported on the torus. More recently ([12]), Lacey and Thiele, combining ideas from both [6] and [9], provideda third approach to Carleson’s theorem on the pointwise convergence of the Fourier Series. See the Appendix for definition. Notice that there is no order relation between the spaces B ∗ ϕ and L log L log log log L (for more details see [2]). Theorem.
There exists a partition of the family of tiles P = [ n ∈ N P n , with T = X n T P n such that for each n ∈ N we have a) Given f ∈ L nonzero, there exists a further decomposition P n = [ α ∈ Z P αn , such that for any n ∈ N we have (5) supp T P αn ⊆ { M f > − α } , (6) k T P αn f k . − α |{ M f > − α }| , where here M stands for the dyadic Hardy-Littlewood maximal function.In particular, we deduce that (7) k T P n f k . k f k L log L . b) For f ∈ L the following holds: (8) k T P n f k , ∞ . k f k . c) If < p < ∞ and p ∗ = min { p, p ′ } with p ′ the H¨older conjugate of p ,then there exists an absolute constant δ > such that (9) k T P n f k p . p − δ n/p ∗ k f k p . d) If f ∈ L p with < p ≤ ∞ then (10) k T f k , ∞ . p k f k log e k f k p k f k . As a direct application of our Theorem we have the following
Corollary.
The following are true:
1) (
Carleson-Hunt , [6], [10]) k T f k p . p k f k p for any < p < ∞ .
2) (
Sj¨olin , [16]) k T f k . k f k L (log L ) . For E ⊆ [0 , measurable k T χ E k , ∞ . | E | log e | E | .
4) (
Sj¨olin , [16]) k T f k , ∞ . k f k L log L log log L .
5) (
Arias de Reyna , [2]) k T f k , ∞ . k f k QA . In particular 5) also implies This second decomposition depends on f . Here, if J = ( c − | J | , c + | J || | ) is any given interval, we use the standard notation b J ( b >
0) to designate the interval ( c − b | J | , c + b | J || | ). Moreover, if I = S n ∈ N J n with each J n an interval, then we set b I := S n ∈ N b J n . VICTOR LIE
6) (
Soria , F., [17],[18]) k T f k , ∞ . k f k B ∗ ϕ .
7) (
Antonov , [1]) k T f k , ∞ . k f k L log L log log log L . Comment.
In fact, as a consequence of d) in our Theorem, we obtain thatfor < p ≤ ∞ one has k T f k , ∞ . p k f k QA p , where QA p is the quasi-Banach space defined by QA p := { f : T C | f measurable , k f k QA p < ∞} with (11) k f k QA p := inf ∞ X j =1 (1 + log j ) k f j k log e k f j k p k f j k | f = P ∞ j =1 f j , P ∞ j =1 | f j | < ∞ a.e. . But as it turns out, based on an observation of Louis Rodriguez-Piazzakindly provided to me by Arias de Reyna, the spaces QA p are equivalent inthe sense that k f k QA p ≈ p k f k QA ∞ . Notice that QA ∞ is the original space QA defined in [2] ; for an interestingstudy concerning the properties of the space QA one should consult [7] . Acknowledgments:
We thank Arias de Reyna for reading the manuscriptand supplying with useful comments.2.
Discretization of the operator
In this section we decompose the operator T in components { T P } P whichare “well” time-frequency localized. We follow the procedure in [9].Let T be the Carleson operator defined by T f ( x ) := sup N ∈ R | Z T x − y e i N ( x − y ) f ( y ) dy | , which after linearization becomes T f ( x ) = Z T x − y e i N ( x ) ( x − y ) f ( y ) dy , where N is some arbitrary measurable function (which from now will befixed).Choose now ψ an odd C ∞ function such thatsupp ψ ⊆ { y ∈ R | < | y | < } which has the property1 y = X k ≥ ψ k ( y ) ∀ < | y | < , where by definition ψ k ( y ) := 2 k ψ (2 k y ) (with k ∈ N ). Thus, we have that
T f ( x ) = X k ≥ T k f ( x ) := X k ≥ Z T e i N ( x ) y ψ k ( y ) f ( x − y ) dy . Take the canonical dyadic grid on T (time grid) - denoted by D T and thecorresponding canonical dyadic grid on R (frequency grid) - denoted with D F . A tile P will consist from a tuple [ ω, I ] ∈ D F × D T with the propertythat | ω | = | I | − . The collection of all such tiles will be denoted with P .Further, for each P = [ ω, I ] ∈ P we set E ( P ) := { x ∈ I | N ( x ) ∈ ω } .With this being said, for | I | = 2 − k ( k ≥
0) and P = [ ω, I ] ∈ P we definethe operators T P on L ( T ) by T P f ( x ) = (cid:26)Z T e i N ( x ) y ψ k ( y ) f ( x − y ) dy (cid:27) χ E ( P ) ( x ) . Notice that the Carleson operator obeys(12)
T f ( x ) = X P ∈ P T P f ( x ) . Finally, whenever
P ⊆ P is a family of tiles we set T P := X P ∈P T P . The proof of the Corollary.
The proof of 1).
We want to show that for 1 < p < ∞ (13) k T f k p . p k f k p . This is a trivial application of statement c) in our Theorem. Indeed, wehave k T f k p ≤ X n ∈ N k T P n f k p . p X n ∈ N − δn/p ∗ k f k p . k f k p . The proof of 2).
We want to show that(14) k T f k . k f k L (log L ) . We will use the following decomposition for each l ∈ Z define Q l := { x ∈ T | | f ( x ) | ∈ [2 l , l +1 ) } . Then we have that(15) k f k L (log L ) ∼ X l ∈ Z l | Q l | (log 1 | Q l | ) . Thus, using duality, for proving (14) will be enough to show (16) Z Q l | T ∗ g | . | Q l | (log 1 | Q l | ) k g k ∞ . Here we abuse the language and refer to D T also as the collection of all dyadic intervalsfor the specified grid. Same for D F . VICTOR LIE
Taking f = χ Q l = χ Q in the dual statements of a) and c) in our Theoremwe deduce Z Q | T P n ∗ g | . | Q | log 1 | Q | k g k ∞ and Z Q | T P n ∗ g | . | Q | / − n δ k g k ∞ . Thus Z Q | T P n ∗ g | . X n min {| Q | log 1 | Q | , | Q | / − n δ }k g k ∞ . | Q | (log 1 | Q | ) k g k ∞ . The proof of 3).
We are interested in(17) k T χ E k , ∞ . | E | log e | E | . Just apply d) with f = χ E .3.4. The proof of 4).
Of course, this result is implied by the claims 5)-7).Still, we think it is worth providing a different approach to this problem,one that isolates a relevant idea in Sj¨olin’s original proof and nicely adaptsit in the context of our Theorem.Our task is to prove that(18) k T f k , ∞ . k f k L log L log log L . The definition that we take here for L log L log log L is as in the originalpaper of Sj¨olin ([16]) given by the space of functions f ∈ L ( T ) for whichwe have k f k L log L log log L = R | f | log + | f | log + log + | f | < ∞ . Thus, setting as before Q l := { x ∈ T | | f ( x ) | ≈ l } , we can always assume Q l = ∅ for l ≤ k f k L log L log log L ≈ X l> l l log l | Q l | . Now fix l >
2. For χ l = χ Q l we run the tile partition described at pointa) of our Theorem. Then for each n ∈ N there exists a decomposition of P n = S α ∈ Z P αn such that(19) supp T P αn ⊆ { M χ l > − α } and k T P αn χ l k . − α |{ M χ l > − α }| . Split the set Z = A l ∪ B l ∪ C l where A l := { r ∈ N | − r < γ − l l − } ,B l := { r ∈ N | γ − l l − ≤ − r < γ − l } ,C l := { r ∈ N | − r ≥ γ − l } , with γ > T P α := X n ∈ N T P αn . Notice that unlike k f k L log L log log L , k f k L log L log log L is not a norm. We assume here that | Q l | 6 = 0. Then, based on (19), we have(20) supp T P α ⊆ { M χ l ≥ − α } , which implies that(21) X α ∈ C l | supp T P α | . X α ∈ C l |{ M χ l ≥ − α }| . X α ∈ C l α | Q l | . γ − l | Q l | . Thus the set S l = S α ∈ C l supp T P α can be excised since we have a goodcontrol on X l ∈ Z | S l | . γ − X l ∈ Z l | Q l | . γ − k f k L log L log log L . Next, it is useful to notice that based on (19) we have k T P αn χ l k . − α |{ M χ l ≥ − α }| . min { − α , | Q l |} , while based on d) in our Theorem we infer that k T P αn χ l k . − n δ | Q l | . From these we deduce(22) k T P α χ l k . α min { − α , | Q l |} . Then, from (22), we have that(23) X α ∈ B l k T P α χ l k . ( l + log 1 γ ) log l | Q l | , and respectively(24) X α ∈ A l k T P α χ l k . X α ∈ A l α − α . γ (1 + log 1 γ ) 2 − l l − . Putting together (21), (23) and (24) and choosing γ = c k f k L log L log log L (with c > Proposition 1.
Let f ∈ L log L log log L with k f k L log L log log L < .Then there exists A ⊆ [0 , with | A | ≤ k f k L log L log log L and C > anabsolute constant such that (25) k T f k L ( A c ) ≤ C k f k L log L log log L . Now, by a canonical density argument, we obtain that the sequenceof the partial Fourier sums { S n f ( x ) } n converges almost everywhere for f ∈ L log L log log L . Relation (18) follows from an application of Stein’smaximal principle ([19]). VICTOR LIE
The proof of 5)-8).
We will show that(26) k T f k , ∞ . p k f k QA p . We choose here to use the log-convexity result due to Kalton ([11]), asfurther described in [2]: Theorem. If { f j } j is a sequence of functions in L , ∞ ( T ) , then we have (27) k X j f j k , ∞ . X j (1 + log j ) k f j k ∞ . Take f ∈ W and set f = P ∞ j =1 f j with f j as in (11). Apply now (27) andpoint d) in our Theorem to conclude k T f k , ∞ . X j (1 + log j ) k T f j k ∞ . p ∞ X j =1 (1 + log j ) k f j k log e k f j k p k f j k . Finding structures in our family of tiles; main definitions
In this section we isolate the main concepts needed for further discretizingand organizing the family of tiles P . Our presentation here is based on thedefinitions and notations introduced in [14]. Definition 1. ( weighting the tiles )Let A be a (finite) union of dyadic intervals in [0 , and P be a finitefamily of tiles. For P = [ ω, I ] ∈ P with I ⊆ A we define the mass of P relative to the set of tiles P and the set A as being (28) A P , A ( P ) := sup P ′ =[ ω ′ ,I ′ ] ∈ P I ⊆ I ′⊆A | E ( P ′ ) || I ′ | (cid:6) ∆(10 P, P ′ ) (cid:7) N where N is a fixed large natural number. Definition 2. ( ordering the tiles )Let P j = [ ω j , I j ] ∈ P with j ∈ { , } . We say that P ≤ P iff I ⊆ I and ω ⊇ ω . We write P < P if P ≤ P and | I | < | I | . Notice that ≤ defines an order relation on the set P . Definition 3. ( modulated/scaled (maximal) Hilbert transform - “tree” )We say that a set of tiles P ⊂ P is a tree with top P if the followingconditions hold: ∀ P ∈ P ⇒ P ≤ P if P = [ ω , I P ] ∈ P and P ′ = [ ω , I P ] such that P ′ ≤ P then P ′ ∈ P if P , P ∈ P and P ≤ P ≤ P then P ∈ P . One can use a different approach to (26) that avoids the use of Kalton’s theorem onthe log convexity of k · k , ∞ . For this, one can follow the proof of 4). Definition 4. ( Carleson measure relative to a tree )We say that a set of tiles
P ⊂ P is a sparse tree if P is a tree and forany P ∈ P we have (29) X P ′∈P IP ′ ⊆ IP | I P ′ | ≤ C | I P | , where here C > is an absolute constant. Definition 5. ( L ∞ control over union of trees )Fix n ∈ N . We say that P ⊆ P n is an L ∞ - forest (of n th -generation) ifi) P is a collection of separated trees, i.e. P = [ j ∈ N P j with each P j a tree with top P j = [ ω j , I j ] and such that (30) ∀ k = j & ∀ P ∈ P j P (cid:2) P k . ii) the counting function (31) N P ( x ) := X j χ I j ( x ) obeys the estimate kN P k L ∞ . n .Further, if P ⊆ P n only consists of sparse separated trees then we refer at P as a sparse L ∞ -forest . Definition 6. ( BM O control over union of trees )A set
P ⊆ P n is called a BM O - forest (of n th -generation) or simply a forest ifi) P may be written as P = [ j ∈ N P j with each P j an L ∞ - forest (of n th -generation);ii) for any P ∈ P j and P ′ ∈ P k with j, k ∈ N , j < k we either have I P ∩ I P ′ = ∅ or | I P ′ | ≤ j − k | I P | . As before, if
P ⊆ P n only consists of sparse L ∞ -forest, then we refer itas a sparse forest . Notice that if
P ⊆ P n is a forest then, due to ii) above, the countingfunction N P := P j N P j obeys the estimate kN P k BMO C . n . Discretization of the family of tiles
The mass decomposition - n discretization. In this section we partition the set P into S n ∈ N P n , with each P n a BM O − forest.The procedure described below is an adaptation of the one introducedby the author in [14] for proving the L p boundedness (1 < p < ∞ ) of thePolynomial Carleson operator.We start by constructing the family P according to the following algo-rithm: • Let P ,max be the collection of maximal tiles P ∈ P with | E ( P ) || I | ≥ and let I := { I | P = [ ω, I ] ∈ P ,max } . • Define the counting function N := P I ∈I χ I and verify that(32) kN k BMO C := sup J dyadic J ⊆ [0 , P I ⊆ JI ∈I | I || J | ≤ . and hence N ∈ BM O D ( R ). • Apply the John-Nirenberg inequality(33) |{ x ∈ J | |N ( x ) − R J N | J | | > γ }| . | J | e − c γ kN k BMOD ( R ) . for γ > c kN k BMO C (here c > |{ x ∈ J | X I ⊆ JI ∈I χ I ( x ) > γ }| . | J | e − c . • Based on (34), conclude that the set A := { x ∈ [0 , | X I ⊆ [0 , I ∈I χ I ( x ) > c kN k BMO C } obeys the relation | A | ≤ e − c . • Remove from P all the tiles which have the time interval not includedin the set A . Run again the above algorithm for the new collection P . This process ends in a finite number of steps since wlog we mayassume that the initial family P is finite.This way, after the k th repetition of our algorithm, we have constructedthe sets A k , P k,max , I k and the counting function N k .We now define the 1 − maximal set of tiles P max := S k P k,max , the col-lection of the time-intervals I := S k I k and finally the counting function N := P I ∈I χ I . Notice that kN k BMO D ( R ) ≤ kN k BMO C . Notice that from the above construction we have that • kN k BMO C . max k kN k k BMO C ; • for any l < k we have A k ⊂ A l and | A k | ≤ e − ( k − l ) c | A l | . Next, define P := { P = [ ω, I ] ∈ P | I * A & A P , [0 , ( P ) ∈ [2 − , ) } and further, by induction, construct P k := (cid:26) P = [ ω, I ] ∈ P | I * A k +11 , I ⊆ A k A P ,A k ( P ) ∈ [2 − , ) (cid:27) . Finally, setting P := [ k P k we end the construction of the family of tiles having the mass of order 1.Now suppose that we have constructed the sets { P k } k
Observations.
1) Remark that this partition of P n conserves the convexityproperty on which the boundedness of the trees is heavily relying. Moreprecisely we have that(44) if P < P < P such that P , P ∈ P αn then P ∈ P αn .
2) Notice that for any P ∈ P αn = ∅ we have that • either ∀ J ∈ J α − I P ∩ J = ∅ , • or if J ∈ J α − s.t. I P ∩ J = ∅ then | I P | > | J | .3) For P = [ ω P , I P ] ∈ P αn = ∅ let c ( I P ) be the center of the interval I P anddefine I P ∗ = [ c ( I P ) − | I P | , c ( I P ) − | I P | ] ∪ [ c ( I P ) + | I P | , c ( I P ) + | I P | ].Then we have that(45) supp T P ⊆ I P and supp T ∗ P ⊆ I P ∗ . Moreover writing I P ∗ = [ r =1 I rP ∗ with each I rP ∗ a dyadic interval of length | I P | we have the following property(46) R I rP ∗ | f || I P | < − α +10 .
4) Let
P ∈ P αn be a tree. Define P min the collection of minimal tiles in P .Further set J P ∗ := { I rP ∗ | P ∈ P min and r ∈ { , . . . , }} , and CZ ( J P ∗ ) the Calderon-Zygmund decomposition of the interval [0 , J P ∗ .Then, from (40) and (46) we deduce the following key property :(47) R I | f || I | < − α +10 ∀ I ∈ CZ ( J P ∗ ) . The proof of the Main Theorem.
Proof of a).
Our aim is to show that with the notions previouslydefined we have(48) supp T P αn ⊆ { M f > − α } and k T P αn f k . − α |{ M f > − α }| . The first of the above conditions is an immediate consequence of the con-struction of the tile families { P αn } .For the second condition we need to analyze the structure of each P αn .Let P αn = S j ∈ N P α,jn be the decomposition of P αn in L ∞ − forests. Further,for j ∈ N , we decompose each L ∞ − forest P α,jn = S k P α,n,jk in maximaltrees. Set I P α,n,jk the time interval of the top of the tree P α,n,jk and define I P α,jn := S k I P α,n,jk and I P αn := S j I P α,jn respectively.Now, as a consequence of the construction in Section 5, for N α,nj := P k χ I Pα,n,jk and N α,n = P j N α,nj , we have(49) kN α,nj k ∞ . n ∀ j ∈ N and kN α,n k . n | I P αn | . Based on (49), it is thus enough to prove that for
P ⊂ P αn tree(50) Z | T P f | . − n − α | I P | . Indeed, from (50) and (49) we deduce(51) R | T P αn f | . − n − α kN α,n k . − α | I P α, n | . − α |{ M f > − α }| . Now for showing (50) we proceed as follows:Without loss of generality we may assume that all the tiles P ∈ P are ata constant frequency ω (the frequency of the tree). Define(52) L P ( f ) := X J ∈ CZ ( J P∗ ) R J f ( s ) e i ω s ds | J | χ J . Observe that as a consequence of (47) we have(53) |L P ( f ) | . − α χ I P . Setting now P the shift of P to the origin, we have Z | T P f | ≤ Z | T P ( e i ω · f ( · ) − L P ( f )( · )) | + Z | T P ( L P ( f )( · )) | . For the first term we use the mean zero condition Z I P ∗ { f ( · ) e i ω · − L P ( f )( · ) } = 0 ∀ P ∈ P and deduce that(54) Z | T P ( e i ω · f ( · ) − L P ( f )( · )) | . − n k f k L ( I P ) . For the second term, we use relation (53) and the L boundedness of theHilbert transform(55) Z | T P ( L P ( f )) | . | E ( P ) | − n/ kL P ( f ) k L ( I P ) . − n − α | I P | , where we set E ( P ) := S P ∈P E ( P ).Here we have used the key Carleson measure estimate(56) | E ( P ) | . − n | I P | . Indeed, for proving (56), we follow the reasoning from [13] (p. 481) andfor J P := { I P | P ∈ P min } , we set ˇ J ( P ) := (cid:8) I ⊂ I P | Exactly one of the left or right halvesof I contains an element of J P (cid:9) ∪ J P . and ˘ P = (cid:8) P = [ ω, I ] ∈ P | I ∈ ˇ J ( P ) (cid:9) . Then we have | E ( P ) | ≤ X P ∈ ˘ P | E ( P ) | . − n X I ∈ ˇ J ( P ) | I | . − n | I P | . This ends our proof.Notice that (48) implies k T P n k . X α k T P αn f k . k M f k . k f k L log L . Proof of b).
In this section we will show a slightly stronger statementthan the one claimed in our theorem. More precisely, we prove that theoperators { T P n } n are uniformly weak (1 ,
1) bounded:We claim that ∀ G ⊆ [0 , ∃ G ′ ⊆ G, | G ′ | > | G | such that (57) Z G ′ | T P n f | . k f k ∀ n ∈ N . Fix G ⊂ T and define G ′ as G ′ = { x ∈ G | M f ( x ) ≤ C k f k | G | } , where here C > G ′ ⊂ G with | G ′ | & | G | .Set now λ = C k f k | G | . It will be enough to prove that Lemma 1.
Let k ∈ N and suppose − α ≈ λ − k . Then the following relationholds: (58) Z G ′ | T P αn f | . − k k f k ∀ n ∈ N . This result relies on the tree estimate provided below
Lemma 2.
Let
P ⊂ P αn be a tree with top I P . Then we have (59) Z G ′ | T P f | . − α | G ′ ∩ E ( P ) | − n/ | I P | . Proof.
Let CZ ( J P ∗ ) be the Calderon-Zygmund decomposition described inthe Observations, Section 5.2.. As before, we can assume without loss ofgenerality that all the tiles P ∈ P are at a constant frequency ω = 0.Now following the description from a) we have Z G ′ | T P f | ≤ Z G ′ | T P ( f − L P ( f )) | + Z G ′ | T P ( L P ( f )) | . The second term is trivially bounded by the Cauchy-Schwarz inequalityand the L boundedness of the Hilbert transform(60) R G ′ | T P ( L P ( f )) | . | G ′ ∩ E ( P ) | − n/ kL P ( f ) k L ( I P ) . − α | G ′ ∩ E ( P ) | − n/ | I P | . A similar statement for a rougher mass-discretized family P n is proved in [8]. For this,the authors are using the mass-size decomposition technique presented in [12]. Then, theyinterpolate the resulting L -estimate with a “modified” L estimate to get good controlnear L - this reasoning is also used in our approach. For two positive quantities
A, B we write A ≈ B iff 2 − B < A < B . To set our problem in the context of this assumption we take advantage of thetranslation invariance of our statement and use a standard estimate (see [9]) that gives anerror of order 2 − α | G ′ ∩ E ( P ) | . For the first term we need to be more careful; we will show that for any g ∈ L ∞ ( T ) with supp g ⊆ G ′ we have(61) (cid:12)(cid:12)(cid:12)(cid:12)Z ( T P ∗ g ) ( f − L P ( f )) (cid:12)(cid:12)(cid:12)(cid:12) . − α | G ′ ∩ E ( P ) | − n/ | I P | k g k ∞ . At this point we make essential use of the mean zero property Z J ( f − L P ( f )) = 0 ∀ J ∈ CZ ( J P ∗ ) . Thus, for proving (61), it is enough to show that for g = gχ G ′ ∈ L ∞ (62) Z (cid:12)(cid:12)(cid:12) ( T P ∗ g − L P ( T P ∗ g )) ( f − L P ( f )) (cid:12)(cid:12)(cid:12) . − α | G ′ ∩ E ( P ) | − n/ | I P | k g k ∞ . For fixed J ∈ CZ ( J P ∗ ) and x ∈ J we have (cid:12)(cid:12)(cid:12)(cid:12) T P ∗ g ( x ) − | J | Z J T P ∗ g ( s ) ds (cid:12)(cid:12)(cid:12)(cid:12) =(63) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) | J | Z J X P ∈P| IP |≥| J | Z T [ ϕ k ( x − y ) − ϕ k ( s − y )] g ( y ) χ E ( P ) ( y ) dy ds (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) . | J | Z J X I P ∗ ⊇ J | I P | − | J | R E ( P ) ∩ G ′ | g || I P | ds . k g k ∞ χ J | J | X I P ∗ ⊇ J | J | | I P | | G ′ ∩ E ( P ) | . This last relation gives us
Z (cid:12)(cid:12)(cid:12) ( T P ∗ g − L P ( T P ∗ g )) ( f − L P ( f )) (cid:12)(cid:12)(cid:12) . − α − n k g k ∞ X J ∈ CZ ( J P∗ ) X I P ∗ ⊇ J | J | ( | J || I P | ) | G ′ ∩ E ( P ) | . − α | G ′ ∩ E ( P ) | − n/ | I P | k g k ∞ , thus proving (59).Here we have relayed on the following key relation:(64) X P ∈P X J ∈ CZ ( JP∗ ) J ⊆ IP ∗ ( | J || I P | ) | J | | G ′ ∩ E ( P ) | . | G ′ ∩ E ( P ) | | I P | . For proving this, take l ∈ N and set CZ ∗ l ( I P ) := { J ∈ CZ ( J P ∗ ) | J ⊆ I P ∗ , | J | ≈ − l | I P | } . Now (64) will be a consequence of(65) S l := X P ∈P X J ∈ CZ ∗ l ( I P ) | J | | G ′ ∩ E ( P ) | . ǫ l ( + ǫ ) | G ′ ∩ E ( P ) | | I P | , where here ǫ ∈ (0 ,
1) is some absolute constant.
For proving (65), we start by refining the set P as follows:For u ∈ { . . . , l } we set P u := { P ∈ P | u − < CZ ∗ l ( I P ) ≤ u } . Then, fix u and define S l,u := X P ∈P u X J ∈ CZ ∗ l ( I P ) | J | | G ′ ∩ E ( P ) | . Next, we decompose inductively the set P u as follows: set P the collectionof maximal (with respect to “ ≤ ”) tiles in P u ; repeat this procedure for thecollection P \ P and thus construct the set of maximal tiles P . Continuethis process until exhausting P u . Thus we end up with partitioning P u = m [ r =1 P r , into collections of (successively) maximal tiles.Applying now Cauchy-Schwarz we deduce that(66) S l,u ≤ | G ′ ∩ E ( P ) | m X r =1 X P ∈P r ( X J ∈ CZ ∗ l ( I P ) | J | ) . Finally, we observe that(67) m X r =1 X P ∈P r ( X J ∈ CZ ∗ l ( I P ) | J | ) . l | I P | . This last fact is a consequence of the construction of the sets {P r } r , thedefinition of CZ ∗ l ( I P ) and of the fact that for any J ∈ CZ ( J P ∗ ) one has { P ∈ P | J ∈ CZ ∗ l ( I P ) } . . Indeed, based on these above mentioned facts, one has that the maincontribution in the left hand side of (67) comes from the first O (2 l − u ) termsof the sum. We leave further details for the interested reader. (cid:3) We pass now to the proof of Lemma 1. With the notations from point a)and based on (59), for each L ∞ -forest P α,jn we decompose P α,jn = S k P α,n,jk into maximal trees and deduce Z G ′ | T P α,jn f | . X k ∈ N Z G ′ | T P α,n,jk f | . − α X k ∈ N | G ′ ∩ E ( P α,n,jk ) | − n/ | I P α,n,jk | ≤ − α − n/ { X k ∈ N | G ′ ∩ E ( P α,n,jk ) |} { X k ∈ N | I P α,n,jk |} . − α | G ′ ∩ E ( P α,jn ) | | I P α,jn | . From the Carleson measure condition imposed in Definition 6 ii), we have Z G ′ | T P αn f | . X j Z G ′ | T P α,jn f | . − α | G ′ | X j | I P α,jn | . − α | G ′ | | I P α, n | . Finally, using that(68) | I P α, n | . | ¯ J α | ≈ α Z ¯ J α | f | , and setting 2 − α ≈ λ − k we conclude that (58) holds.6.3. Proof of c).
The central estimate for proving (9) is given by(69) k T P n f k . n − n/ k f k . We limit ourselves to just providing the main ideas. For the details of theproof the reader should consult [14].One first decomposes P n = S P jn with each P jn an L ∞ − forest, and thenproves that • k T P jn f k . n − n/ k f k . • for each k ∈ { , . . . , n − } the sequence { T P k + j nn } j ∈ N consists fromalmost orthogonal operators and thus one can apply Cotlar-Steinlemma.This ensures that (69) holds. Now, for 1 < p ≤
2, relation (9) is just aconsequence of (8), (69) and classical interpolation theory. For p > L case: one first proves the desired estimate for an L ∞ forest and then using the structure of the BM O -forest one is able toextend the initial result to the entire family P n (for more details see [14]).6.4. Proof of d).
We will show that for 1 < p < ∞ (70) k T f k , ∞ . p k f k log e k f k p k f k . As before, we start by reformulating (70) in the equivalent form ∀ G ⊂ T ∃ G ′ ⊂ G with | G ′ | & | G | such that(71) Z G ′ | T f | . p k f k log e k f k p k f k . Repeat the same construction as in a). Then, with the same notationsfrom c), we have(72) Z G ′ | T P n f | . p − δ n/p ∗ k f k p . In [4], one can found (without a proof) the following statement: For 1 ≤ p ≤ k T f k p, ∞ . k f k p log e k f k k f k p . The unpublished proof ([3]) of this result handled tome by Arias de Reyna, relies on Carleson’s original approach to the pointwise convergenceof the Fourier Series. Finally, one may notice that the case p = 1 in his result is equivalentwith the case p = 2 in (70). Combining (8) and (72) we conclude that Z G ′ | T f | ≤ X n ∈ N Z G ′ | T P n f | . p X n ∈ N min {k f k , − δ n/p ∗ k f k p } . p k f k log e k f k p k f k , proving the desired result. 7. Remarks
1) As mentioned in the introduction, the central part of our paper resideson the tile decomposition described in Section 5.The mass discretization of
T f is independent of f and results in thegeometric organization of P = S n P n which gives us both the n -decay foreach {k T P n f k p } n with 1 < p < ∞ ( f ∈ L p ) and the ability to sum withineach “scale” n the lengths of the time support of the maximal trees in P n .The Calderon-Zygmund discretization P n = S α P αn realizes the decompo-sition of the function T P n f depending on f and is designed to get a goodcontrol near L . This discretization accounts for the multi-frequency natureof our problem. A similar instance appeared in [15], where the authors areelaborating a “Calderon-Zygmund decomposition for multiple frequencies”for a given function f . Our approach though is quite different: instead of de-composing the input object (the function f ) at multiple frequencies imposinga mean zero condition of the initial function for each frequency, we rely onthe properties of the initial discretization P = S n P n and first decomposethe tile family P n in subfamilies { P αn } α followed by a further decompositionof the corresponding output object T P αn f in multiple pieces with each piecehaving the mean zero condition strictly relative to the frequency at whichit lives.In the setting of the present paper our procedure is more effective andunlike the one in [15] gives an explicit construction. On the other hand,the decomposition in [15] is more general and hence can be used in otherproblems which are not necessarily related to the Carleson operator.2) The treatment of the “forest” operators T P n in our Theorem offersa substitute to the classical theory of the Calderon-Zygmund operators:indeed, following the tile decomposition and the proofs of a) and b) onenotices that the weak (1 ,
1) bound is obtained by using a Vitali type coveringargument - in which we can sum the lengths of the intervals of the trees in thestructure of T P n f depending on the size of the maximal function associated The mass parameter depends on the function N which may be taken as just anarbitrary measurable function as long as the final estimates on the operator T do notdepend on it. This helps us in both summing the operators { T P n } n (and thus obtaining L p boundsfor 1 < p < ∞ ) and in interpolating with norm estimates near L . Here is the point where our technique overcomes the difficulty of treating the excep-tional sets. Here the algorithm described for the mass decomposition plays a fundamental role. to f . Also like in the classical theory, through our decomposition the entireweight moves on the L behavior of T P n where one needs to use orthogonalitymethods. To complete our parallelism, it would be of interest if our methodscould provide a satisfactory theory for the adjoint operator T P n ∗ near L .We think this topic deserves further investigation.3) Finally, in view of our approach, the following question appears asnatural: Open question.
Fix f ∈ L ( T ) . With the previous notations and defini-tions, set P α := [ n P αn , with α ∈ N . Is it then true that ∃ C > absolute constant such that for any α ∈ N (73) k T P α f k ≤ C k f k ?A positive answer to this question would imply that(74) k T f k L . k f k L log L , which is the best one can hope for if we require strong L bounds for theCarleson operator T .Of course it would be still very interesting if (73) holds with the L normreplaced by the L , ∞ norm in the left hand side.With the current technology we can only prove that k T P α f k ≤ C k f k L log L . Appendix - spaces near L In this section we briefly introduce the definitions of the relevant re-arrangement invariant Banach spaces which appeared in the previous litera-ture when studying the problem of the pointwise convergence of the FourierSeries near L . Definition 7.
Let ϕ : [0 , ∞ ) [0 , ∞ ) be an absolutely continuous functionwith the following properties: • ∃ C > such that ϕ ( t ) ≤ C ϕ ( t ) for all t ≥ ; • ϕ ′ ( t ) ≥ almost everywhere; • ϕ (0) = 0 ; • lim t →∞ ϕ ( t ) = ∞ . Then we define the space
L ϕ ( L ) of all (measurable) functions f for which (75) k f k L ϕ ( L ) := Z T | f | | ϕ ( f ) | < ∞ . A classical result in Banach space theory (see e.g. [5]) asserts
Proposition.
The space
L ϕ ( L ) endowed with the norm (76) k f k L ϕ ( L ) := Z T f ∗ ( t ) ϕ ( f ∗ )( t ) dt < ∞ , becomes a Banach rearrangement space. In the topic treated in this paper three special choices for the function ϕ are of interest: • ϕ ( t ) = log(1 + t ) - defines the Zygmund space L log L ; • ϕ ( t ) = log(1 + t ) log log(1 + t ) - defines the space L log L log log L considered by Sj¨olin in [16]. • ϕ ( t ) = log(1+ t ) log log log(10+ t ) - defines the space L log L log log log L considered by Antonov in [1].To complete the picture and thus present the evolution until nowadays,we need to consider two more spaces:The first space was considered by F. Soria in [17]:Let B ϕ be the set of the measurable functions for which k f k ϕ := Z ∞ ϕ ( λ f ( t )) dt < ∞ where here λ f is the distribution function of f given by λ f ( t ) = |{ x ∈ T | | f ( x ) | > t }| . Take now the subspace B ∗ ϕ ⊂ B ϕ defined by B ∗ ϕ := { f | k f k ∗ ϕ < ∞} where k f k ∗ ϕ = R ∞ ϕ ( λ f ( t ))(1 + log( k f k ϕ ϕ ( λ f ( t )) )) dt. The pointwise convergence theory developed by Soria was addressing thespace B ∗ ϕ with ϕ ( s ) = s (1 + log + 1 s ).The second space was introduced by Arias de Reyna in [2]:Let QA be the quasi-Banach space defined as follows: QA := { f : T C | f measurable , k f k QA < ∞} where k f k QA := inf ∞ X j =1 (1 + log j ) k f j k log e k f j k ∞ k f j k | f = P ∞ j =1 f j , P ∞ j =1 | f j | < ∞ a.e. . With the exception of Zygmund’s L log L space, it is known ([16], [1], [17]and [2]) that all the other spaces considered here, namely L log L log log L , L log L log log log L , B ∗ ϕ and QA are rearrangement invariant spaces offunctions with almost everywhere convergent Fourier series. Here f ∗ stands for the decreasing rearrangement of f . References [1] N. Yu. Antonov. Convergence of Fourier series. In
Proceedings of the XX Workshopon Function Theory (Moscow, 1995) , volume 2, pages 187–196, 1996.[2] J. Arias-de Reyna. Pointwise convergence of Fourier series.
J. London Math. Soc. (2) ,65(1):139–153, 2002.[3] Juan Arias de Reyna.
About Carleson’s Theorem . 2000. unpublished.[4] Juan Arias de Reyna.
Pointwise convergence of Fourier series , volume 1785 of
LectureNotes in Mathematics . Springer-Verlag, Berlin, 2002.[5] Colin Bennett and Robert Sharpley.
Interpolation of operators , volume 129 of
Pureand Applied Mathematics . Academic Press Inc., Boston, MA, 1988.[6] Lennart Carleson. On convergence and growth of partial sumas of Fourier series.
ActaMath. , 116:135–157, 1966.[7] M. Carro, M. Mastylo, and L. Rodriguez-Piazza. Almost everywhere convergent ofFourier Series.
J. Fourier Anal. Appl. , (18):266–286, 2012.[8] Yen Do and Michael Lacey. On the Convergence of Lacunary Walsh-Fourier Series.Preprint, arXiv:1101.2461v2, 2011.[9] Charles Fefferman. Pointwise convergence of Fourier series.
Ann. of Math. (2) , 98:551–571, 1973.[10] Richard A. Hunt. On the convergence of Fourier series. In
Orthogonal Expansions andtheir Continuous Analogues (Proc. Conf., Edwardsville, Ill., 1967) , pages 235–255.Southern Illinois Univ. Press, Carbondale, Ill., 1968.[11] N. J. Kalton. Convexity, type and the three space problem.
Studia Math. , 69(3):247–287, 1980/81.[12] Michael Lacey and Christoph Thiele. A proof of boundedness of the Carleson opera-tor.
Math. Res. Lett. , 7(4):361–370, 2000.[13] Victor Lie. The (weak- L ) boundedness of the quadratic Carleson operator. Geom.Funct. Anal. , 19(2):457–497, 2009.[14] Victor Lie. The polynomial Carleson operator. Submitted, arXiv:1105.4504v1, 2011.[15] Fedor Nazarov, Richard Oberlin, and Christoph Thiele. A Calder´on-Zygmund de-composition for multiple frequencies and an application to an extension of a lemmaof Bourgain.
Math. Res. Lett. , 17(3):529–545, 2010.[16] Per Sj¨olin. An inequality of Paley and convergence a.e. of Walsh-Fourier series.
Ark.Mat. , 7:551–570 (1969), 1969.[17] Fernando Soria. Note on differentiation of integrals and the halo conjecture.
StudiaMath. , 81(1):29–36, 1985.[18] Fernando Soria. On an extrapolation theorem of Carleson-Sj¨olin with applications toa.e. convergence of Fourier series.
Studia Math. , 94(3):235–244, 1989.[19] E. M. Stein. On limits of seqences of operators.
Ann. of Math. (2) , 74:140–170, 1961.
Department of Mathematics, Princeton, NJ
E-mail address : [email protected]@math.princeton.edu