BOUNDS FOR A MAXIMAL DYADIC SUM OPERATOR

Transcription

1 L p BOUNDS FOR A MAXIMAL DYADIC SUM OPERATOR LOUKAS GRAFAKOS, TERENCE TAO, AND ERIN TERWILLEGER Abstract. The authors prove L p bounds in the range <p< for a maximal dyadic sum operator on R n. This maximal operator provides a discrete multidimensional model of Carleson s operator. Its boundedness is obtained by a simple twist of the proof of Carleson s theorem given by Lacey and Thiele [6] adapted in higher dimensions [8]. In dimension one, the L p boundedness of this maximal dyadic sum implies in particular an alternative proof of Hunt s extension [3] of Carleson s theorem on almost everywhere convergence of Fourier integrals.. The Carleson-Hunt theorem A celebrated theorem of Carleson [] states that the Fourier series of a squareintegrable function on the circle converges almost everywhere to the function. Hunt [3] extended this theorem to L p functions for <p<. Alternative proofs of Carleson s theorem were provided by C. Fefferman [] and by Lacey and Thiele [6]. The last authors proved the theorem on the line, i.e. they showed that for f in L R) the sequence of functions S N f)x) = fξ)e πixξ dξ ξ N converges to fx) for almost all x R as N. This result was obtained as a consequence of the boundedness of the maximal operator Cf) = sup S N f) N>0 from L R) intol, R). In view of the transference theorem of Kenig and Tomas [4] the above result is equivalent to the analogous theorem for Fourier series on the circle. Lacey and Thiele [5] have also obtained a proof of Hunt s theorem by adapting the techniques in [6] to the L p case but this proof is rather complicated compared with the relatively short and elegant proof they gave for p =. Investigating higher dimensional analogues, Pramanik and Terwilleger [8] recently adapted the proof of Carleson s theorem by Lacey and Thiele [6] to prove weak type, ) bounds for a discrete maximal operator on R n similar to the one which arises in the aforementioned proof. After a certain averaging procedure, this result provides an alternative proof of Sjölin s [0] theorem on the weak L boundedness of Date: December, Mathematics Subject Classification. Primary 4A0. Secondary 4A4. Key words and phrases. Fourier series, almost every convergence. Grafakos is supported by the NSF. Tao is a Clay Prize Fellow and is supported by a grant from the Packard Foundation.

2 LOUKAS GRAFAKOS, TERENCE TAO, AND ERIN TERWILLEGER maximally modulated Calderón-Zygmund operators on R n. The purpose of this note is to extend the result of Pramanik and Terwilleger [8] to the range <p< via a variation of the L L, case. Particularly in dimension, the theorem below yields a new proof of Hunt s theorem i.e. the L p boundedness of C for <p< ) using a variation of the proof of Lacey and Thiele [6].. Reduction to two estimates We use the notation introduced in [6] and expanded in [8]. A tile in R n R n is a product of dyadic cubes of the form n n I j = [m j k, m j + ) k ), j= j= where k and m j are integers for all j =,,,n. We denote a tile by s = I s ω s, where I s ω s =. The cube I s will be called the time projection of s and ω s the frequency projection of s. For a tile s with ω s = ωs ωs... ωs n, we can divide each dyadic interval ωs j into two intervals of the form ωs j =ωs j,cωs)) j ωs j [cωs), j )) for j =,,...,n. Then ω s can be decomposed into n subcubes formed from all combinations of cross products of these half intervals. We number these subcubes using the lexicographical order on the centers and denote the subcubes by ω si) for i =,,..., n. A tile s is then the union of n semi-tiles given by si) =I s ω si) for i =,,..., n. We let φ be a Schwartz function such that φ is real, nonnegative, and supported in the cube [ /0, /0] n. Define φ s x) = I s φ x cis ) I s n ) e πicω s)) x, where cj) is the center of a cube J. As in [6] and [8], we will consider the dyadic sum operator D r f) = s χ ωsr) N)φ s, s D f,φ where r n is a fixed integer, N : R n R n is a fixed measurable function, D is a set of tiles, and f,g is the complex inner product fx)gx) dx. R The following theorem is the main result of this article. Theorem. Let <p<. Then there is a constant C n,p independent of the measurable function N, of the set D, and of r such that for all f L p R n ) we have ) D r f) L p R n ) C n,p f L p R n ). In one-dimension, using the averaging procedure introduced in [6], it follows that the norm estimate ) implies Cf) L p C p f L p,

3 L p BOUNDS FOR A MAXIMAL DYADIC SUM OPERATOR 3 which is the Carleson-Hunt theorem. Using the Marcinkiewicz interpolation theorem [] and the restricted weak type reduction of Stein and Weiss [], estimate ) will be a consequence of the restricted weak type estimate ) D r χ F ) L p, R n ) C n,p F p, <p< which is supposed to hold for all n-dimensional sets F of finite measure. But to show that a function g lies in L p,, it suffices to show that for every measurable set E of finite measure, there is a subset E of E which satisfies E E and also p gx) dx A E p ; E this implies that g L p, R n ) c p A, where c p is a constant that depends only on p. Let Cn, q) be the weak type q, q) operator norm for the Hardy-Littlewood maximal operator. Given a set E of finite measure we set { Ω= Mχ F ) > F ) } q Cn, q), E where we choose q so that p<q if F > E and q<pif F E. Note that in the first case the set Ω is empty. Using the L q to L q, boundedness of the Hardy-Littlewood maximal operator, we have Ω E and hence E E. Thus estimate ) will follow from D r χ F )x) dx C n,p E p 3) p F p, E where C n,p depends only on p and dimension n. The required estimate 3) will then be a consequence of the following two estimates: χ F,φ s χ ωsr) N)φ s x) dx C n,p,q E p 4) p F p, and 5) E s D I s*ω s D I s Ω χ F,φ s χ E N [ω sr) ],φ s C n,p,q E p p F p. 3. The proof of estimate 4) Following [7], we denote by ID) the dyadic grid which consists of all the time projections of tiles in D. For each dyadic cube J in ID) we define D J := {s D : I s = J} and a function ) γ ψ J x) := J x cj) +, J n where γ is a large integer to be chosen shortly. For each k =0,,,... we introduce families F k = { J ID) : k J Ω, k+ J * Ω }.

4 4 LOUKAS GRAFAKOS, TERENCE TAO, AND ERIN TERWILLEGER We may assume F E, otherwise the set Ω is empty and 4) is trivial. We begin by controlling the left hand side of 4) by χf φ s χωs) Nx))φ s x) dx E 6) J ID) J Ω k=0 J ID) J F k s DJ) E s DJ) χf φ s Nx))φ χωsr) sx) dx Using the fact that the function Mχ F ) is an A weight with A -constant bounded above by a quantity independent of F, it is easy to find a constant C 0 < such that for each k =0,,... and J F k we have 7) χf,ψ J J inf F ) J C k 0 inf Mχ F ) Cn, q) q C k 0 J F ) q J k+ J E since k+ J meets the complement of Ω. For J F k one also has that E k J = and hence 8) ψ J y) dy E ψ J y) dy J Cγ kγ. k J) c Next we note that for each J ID) and x R n there is at most one s = s x D J such that Nx) ω sxr). Using this observation along with 7) and 8) we can therefore estimate the expression on the right in 6) as follows 9) C C C k=0 J ID) J F k k=0 F E J ID) J F k ) q F ) q E χf φ sx χωsxr) Nx))φ sx x) dx E E χf,ψ J ψj xt) dx C0 k J k=0 J F k C 0 γ ) k J J F k k=0 E ψ J x) dx and at this point we pick γ so that C 0 γ <. It remains to control J F k J for each nonnegative integer k. In doing this we let Fk be all elements of F k which are maximal under inclusion. Then we observe that if J Fk and J F k satisfy J J then dist J,J c ) = 0 otherwise J would be contained in J and thus k+ J k J Ω.) But for any fixed J in Fk and any scale m, all the cubes J in J F k of sidelength m that touch J are concentrated near the boundary of J and have total measure at most m n J n ) n. Summing over all integers m with

5 L p BOUNDS FOR A MAXIMAL DYADIC SUM OPERATOR 5 m J n, we obtain a bound which is at most a multiple of J. We conclude that J = J c n J c n Ω J F k J F k J F k J J since elements of Fk are disjoint and contained in Ω. Inserting this estimate in 9) and using that the Hardy-Littlewood maximal operator is of weak type, ), we obtain the required bound F ) q C Ω C F C E p p F p E for the expression on the right in 6) and hence for the expression on the left in 4). J F k 4. The proof of estimate 5) In proving estimate 5) we may assume that E by a simple scaling argument. The scaling changes the sets D, Ω, and the measurable function N but note that the final constants are independent of these quantities.) In addition all constants in the sequel are allowed to depend on n and p as described above. We may also assume that the set D is finite. Note that under the normalization of the set E, our choice of q is as follows: q<pif F c 0 and p<q when F >c 0 where c 0 is a fixed number in the interval, ), in fact c 0 = E ) We recall that a finite set of tiles T is called a tree if there exists a tile t T such that all s T satisfy s<twhich means I s I t and ω t ω s.) In this case we call t the top of T and we denote it by t = tt ). A tree T is called an r-tree if ω tt )r) ω sr) for all s T. For a finite set of tiles Q we define the energy of a nonzero function f with respect to Q by ) Ef; Q) = sup f,φ s, f L R n ) T I tt ) where the supremum is taken over all r-trees T contained in Q. We also define the mass of a set of tiles Q by I u sup MQ) = sup s Q u Q s<u E N [ω ur) ] + x ciu) I u /n ) γ dx. We now fix a set of tiles D and sets E and F with finite measure recall E ). We define P to be the set of all tiles in D with the property I s * Ω. Given a finite set of tiles P, find a very large integer m 0 one can construct a sequence of pairwise disjoint sets P m0, P m0, P m0, P m0 3,... such that P = m 0 j= and such that the following properties are satisfied P j

6 6 LOUKAS GRAFAKOS, TERENCE TAO, AND ERIN TERWILLEGER a) Eχ F ; P j ) j+)n for all j m 0. b) MP j ) j+)n for all j m 0. c) E χ F ; P \ P m0 P j ) ) jn for all j m 0. d) M P \ P m0 P j ) ) jn for all j m 0. e) P j is a union of trees T jk such that k I tt jk ) C 0 jn for all j m 0. This can be done by induction, see [], [6], and is based on an energy and a mass lemma shown in [8]. The following lemma is the main ingredient of the proof and will be proved in the next section. Lemma. There is a constant C such that for all measurable sets F and all finite set of tiles P which satisfy I s * Ω for all s P, we have Eχ F ; P ) C F q Note that this gives us decay no matter if F is large or small due to the choice of q the reader is reminded that if F c 0 then q [,p) while if F c 0 then q p, ].) We also recall the estimate below from [8]. Lemma. There is a finite constant C such that for all trees T,allf L R n ), and all measurable sets E with E we have 0) f,φs χ E N [ω sr) ],φ s C I tt ) Ef; T ) MT ) f L R n ). Given the sequence of sets P j as above, we use a), b), e), the observation that the mass is always bounded by, and Lemmata and to obtain χf,φ s χ E N [ω sr) ],φ s s P = j j s P j k χf,φ s χ E N [ω sr) ],φ s jk χf,φ s χ E N [ω sr) ],φ s C I ttjk ) ET jk ) MT jk ) F j k C F I ttjk ) min j+)n,c F q ) min, j+)n ) j k C F jn min jn, F q ) min, jn ) j C F q + log F q ) C min, F ) + log F ) C p F p

7 L p BOUNDS FOR A MAXIMAL DYADIC SUM OPERATOR 7 for all <p<. We observe that the choice of q was made to deal with the logarithmic presence in the estimate above. Had we taken q = p throughout, we would have obtained the sought estimates with the extra factor of + log F. Looking at the penultimate inequality above, we note that we have actually obtained a stronger estimate than the one claimed in 3). Rescaling the set E and taking q to be either or, we have actually proved that for every measurable set E of finite measure, there is a subset E of E such that for all measurable sets F of finite measure we have D r χ F ) dx C E min, F ) + log F ). E E E This will be of use to us in section The proof of Lemma It remains to prove Lemma. Because of our normalization of the set E we may assume that Ω = {Mχ F ) >c F q } for some c>0. Fix an r-tree T contained in P and let I t = I tt ) be the time projection of its top. We write the function χ F as χ F 3It + χ F 3It) c. We begin by observing that for s in P one has χ F 3It) c,φ s C γ I s inf I s Mχ F ) + dist3i t) c,ci s ) I s n ) γ C γ I s F q )γ Is n I t since I s meets the complement of Ω for every s P. Square this inequality and sum over all s in T to obtain χ c,φ F 3It) s C I t F q, where the last estimate follows by placing the I s s into groups G m of cardinality at most mn so that each element of G m has size mn I t. We now turn to the corresponding estimate for the function χ F 3It. At this point it will be convenient to distinguish the case F >c 0 from the case F c 0. In the case F >c 0 the set Ω is empty and therefore χ F 3It,φ s C χ F 3It L C I t C I t F q, where the first estimate follows follows from the Bessel inequality 3) which holds on any r-tree T ; the reader may consult [8] or prove it directly. We therefore concentrate on the case F c 0. In proving Lemma we may assume that there exists a point x 0 I t such that Mχ F )x 0 ) c F q, otherwise there is nothing to prove. We may also assume that the center of ω tt ) is zero, i.e. cω tt ) ) = 0, otherwise we may work with a suitable modulation of the function χ F 3It in the Calderón-Zygmund decomposition below.

8 8 LOUKAS GRAFAKOS, TERENCE TAO, AND ERIN TERWILLEGER We write the set Ω = {Mχ F ) >c F q } as a disjoint union of dyadic cubes J l such that the dyadic parent J l of J l is not contained in Ω and therefore F J l F J l c F q J l. Now some of these dyadic cubes may have size larger than or equal to I t. Let J l be such a cube. Then we split J l in J l I t cubes J l,m each of size exactly I t. Since there is an x 0 I t with Mχ F )x 0 ) c F q, it follows that F J l,m c F q It + disti t,j l,m ) ) n ). I t n We now have a new collection of dyadic cubes {J k } k contained in Ω consisting of all the previous J l when J l < I t and the J l,m s when J l,m I t. In view of the construction we have c F q Jk when < I t F J k c F q Jk + disti ) ) n t,j k ) when = I t I t for all k. We now define the bad functions b k = χ Jk 3I t F J k 3I t F χ Jk which are supported in J k, have mean value zero, and they satisfy b k L R n ) c F q Jk + disti ) n t,j k ). I t We also set g = χ F 3It k b k the good function of the above Calderón-Zygmund decomposition. We check that that g L R n ) C F q. Indeed, for x in Jk we have F J k when < I t gx) = F 3I t J k F 3I t when = I t I t and both of the above are at most a multiple of F q ; the latter is because there is an x 0 I t with Mχ F )x 0 ) c F q. Also for x k J k ) c =Ω c, gx) =χ F 3It x) which is at most Mχ F )x) c F q. We conclude that g L R n ) C F q. Moreover g L R n ) k F 3I t J k J k dx + χ F 3It L R n ) C F 3I t C F q It

9 L p BOUNDS FOR A MAXIMAL DYADIC SUM OPERATOR 9 since the J k are disjoint. It follows that g L R n ) C F q F q It = C F q It. Using the simple Bessel inequality 3) g, φ s C g L R n ) we obtain the required conclusion for the function g. For a fixed s P and J k we will denote by dk, s) = dist J k,i s ). Then we have the following estimate for all s and k: dk, t) ) n I s 3 4) b k,φ s C γ F q Jk since + dk,t) + dk,s). I t n I s n We also have the estimate + I t n + dk,s) I s n q Jk I s 3 ) C γ F γ+n + dk,s) ) γ I s n 5) b k,φ s C γ F q Is + dk,s) ). γ I s n To prove 4) we use the fact that the center of ω tt ) = 0 which implies that φ s obeys size estimates similar to I s φ s ) and the mean value property of b k to obtain bk,φ s = b k y) J φ s y) φ s cj k )) ) C γ I s 3 dy b k L sup k ξ J k + ξ cis) ). γ I s n To prove estimate 5) we note that b k,φ s C γ I s inf Mb k ) ) I s + dk,s) ) γ and that Mb k ) Mχ F )+ F 3I t J k Mχ Jk ) and since I s * Ωwehaveinf Is Mχ F ) c F q while the second term in the sum above was observed earlier to be at most C F q. Finally we have the estimate b k,φ s C γ F q Jk I s 6) + dk,s) ) γ I s n which follows by taking the geometric mean of 4) and 5). Now for a fixed s P we may have either J k I s or J k I s = since I s is not contained in Ω.) Therefore for fixed s P there are only three possibilities for J k : a) J k 3I s I s n

10 0 LOUKAS GRAFAKOS, TERENCE TAO, AND ERIN TERWILLEGER b) J k 3I s = c) J k I s =, J k 3I s, and J k * 3I s. Observe that case c) is equivalent to the following statement: c) J k I s =, dk, s) =0,and n I s. Let us start with case c). Note that for each I s there exists at most n choices of J k with the above properties. Thus for each s in the sum below we can pick one J ks) at a cost of n, which is harmless. Also note that since dk, s) =0and n I s, we must have that I s J k. But I s I t and I t implies that J k 3I t. Now for a given J k and a fixed scale m, there are at most m # of sides) + n possibilities of I s such that mn = I s and dk, s) =0. Using 5) we obtain b k,φ s n ) b k s),φ s k: J k as in c) C n F q I s C n F q for which J k as in c) m mn J ks) = I s mn J ks) C n F q m m # of sides) + n ) mn k C n F q It, where we have used the disjointness of the J k s. This finishes case c). We now consider case a). Using 4) we can write k: J k as in a) b k,φ s ) Cγ F q 3 ) k: J k 3I s I s 3 and we control the expression inside the parenthesis above by ) ) 3 J I s 3 k 3 k: J k 3I s k: J k 3I s J k 3I s I s in view of the Cauchy-Schwarz inequality and of the fact that the dyadic cubes J k are disjoint and contained in 3I s. Finally note that the last sum above adds up to at most C n since for every dyadic cube J k there exist at most n ++# of sides) dyadic cubes of a given size whose triples contain it. The required estimate C n,γ F q It now follows. Finally we deal with case b) which is the most difficult case. We split the set of k into two subsets, those for which J k 3I t and those for which J k * 3I t, recall I t.) Whenever J k * 3I t we have dk, s) dk, t). In this case we use

11 L p BOUNDS FOR A MAXIMAL DYADIC SUM OPERATOR Minkowski s inequality below and estimate 6) with γ>nto obtain the estimate k: J k *3I t b k,φ s ) k: J k *3I t b k,φ s ) C γ F q k: J k *3I t C γ F J q k dk, t) γ k: J k *3I t C γ F q It γ n C γ F q It γ n I s γ n ) dk, s) γ dk, t) γ k: J k *3I t l= k:dk,t) l I t n I s γ n ) l I t n ) γ. But note that all the J k with dk, t) l I t n are contained in l+ I t and since they are disjoint we can estimate the last sum above by C lm I t l I t n ) γ. The required estimate C γ F q It now follows. Next we consider the sum below in which we use estimate 4) 7) I s C γ F q C γ F q b k,φ s I s { [ I s ) I s 3 Is n dk, s) ) γ ) ) 3 Is γ ][ n I s dk, s) I s I s dk, s) I s n ) γ ]}. The second sum above can be estimated by I s J k x cis ) I s n ) γ dx I s 3I s) c x cis ) I s n ) γ dx I s C γ.

12 LOUKAS GRAFAKOS, TERENCE TAO, AND ERIN TERWILLEGER Putting this estimate into 7), we have C γ F q C γ F q C γ F q C γ F q { { I s I s { I s { C γ F q It. I s ) 3 I s Is γ } n dk, s) 3 m log n 3 m log n 3 } mn mn } I s = mn dk, s) m ) γ } There is also the subcase of case b) in which I s. Here we have the two special subcases: I s 3J k = and I s 3J k =. We begin with the first of these special subcases in which we use estimate 5). We have 8) 9) > I s I s 3J k = C γ F q C γ F q b k,φ s > I s I s 3J k = [ > I s I s 3J k = ) I s I s I s γ ) n dk, s) γ I s γ n dk, s) γ ][ > I s I s 3J k = I s I s γ n dk, s) γ ]).

13 L p BOUNDS FOR A MAXIMAL DYADIC SUM OPERATOR 3 Since I s 3J k = we have that dk, s) x ci s ) for every x J k. Therefore the second term inside square brackets above satisfies I s γ n I s dk, s) x cis ) ) γ dx γ k J k I s n I s C γ. > I s I s 3J k = Putting this estimate into 9), we obtain C γ F q C γ F q I s C γ F q > I s I s 3J k = k:j k 3I t I s > I s I s 3J k I s γ ) n dk, s) γ I s γ ) n dk, s) γ mn m=0 s: I s 3J k I s = mn I s γ ) n. dk, s) γ Since the last sum above is at most a constant 8) satisfies the estimate C γ F q It. Finally there is the subcase of case b) in which I s and I s 3J k =. Here again we use estimate 5). We have 0) > I s I s 3J k b k,φ s ) Cγ F q I s > I s I s 3J k I s γ ) n. dk, s) γ Let us make some observations. For a fixed s there exists at most finitely many J k s contained in 3I t with size at least I s. Consider the following sets for α {0,,,...}, J α := {J k as in the sum above : α I s n dk, s) < α+ I s n }. We would like to know that for all α the cardinality of J α is bounded by a fixed constant depending only on dimension. This would allow us to work with a single cube J α s) from each set at the cost of a constant in the sum below. Fix α {0,,,...} and note that I s 3J k and dk, s) > α I s n implies that > αn I s. It is clear that the cardinality of J α would be largest if we had = α+ I s for all J k J α. Then the cube of size 7 n αn I s centered at I s would contain all elements of J k. This bounds the number of such elements by 7 n. )

14 4 LOUKAS GRAFAKOS, TERENCE TAO, AND ERIN TERWILLEGER Then using the Cauchy-Schwarz inequality we obtain ) n > I s I s 3J k I s γ n dk, s) γ 7 C n C n α= α= I s γ n distj α s),i s ) γ I s γ n distj α s),i s ) γ > I s I s 3J k I s γ n dk, s) γ αγ Putting this estimate into the right hand side of 0), the estimate C n,γ F q It follows as in the previous case. This concludes the proof of Lemma. now 6. applications We conclude by discussing some applications. We show how one can strengthen the results of the previous sections to obtain distributional estimates for the function D r χ F ) similar to those in the paper of Sjölin [0]. We showed in section 4 that for any measurable set E there is a set E of at least half the measure of E such that D r χ F ) dx C min E, F ) + log F ) ) E E for some constant C depending only on the dimension. For λ>0 we define E λ = { D r χ F ) >λ } and also Eλ = { Re D r χ F ) >λ } Eλ = { Re D r χ F ) < λ } Eλ 3 = { Im D r χ F ) >λ } Eλ 4 = { Im D r χ F ) < λ }. We apply ) to each set E j λ to obtain λ E j λ min Ej λ, F ) + log F E j λ ). Using this fact in combination with the easy observation that for a> a log a 0 = a λ λ log λ ), to obtain that E j λ C F { λ λ when λ< e cλ when λ. Since E λ 4 j= Ej λ we conclude a similar estimate for E λ.

15 L p BOUNDS FOR A MAXIMAL DYADIC SUM OPERATOR 5 Next we obtain similar distributional estimates for maximally modulated singular integrals M such as the maximally modulated Hilbert transform i.e. Carleson s operator) or the maximally modulated Riesz transforms Mf)x) = sup x j y j eπiξ y fy) dy ξ R n R x y n n+. To achieve this in the one dimensional setting, one applies an averaging argument similar to that in [6] to both terms of estimate ) to recover a similar estimate with the Carleson operator. For more general homogeneous singular integrals with sufficiently smooth kernels, one applies the averaging argument to suitable modifications of the operators D r as in [8]. Then one obtains a version of estimate ) in which D r χ F ) is replaced by Mχ F ). The same procedure as above then yields the distributional estimate { Mχ F ) >λ } { C n F log ) when λ< λ λ e cλ when λ. which recovers Lemma. in [0]. It should be noted that the corresponding estimate { D r χ F ) >λ } { Cn F log ) when λ< ) λ λ e cλ when λ. obtained here for D r is stronger as it concerns an unaveraged version of all the aforementioned maximally modulated singular integrals M. Using the idea employed in Sjölin [9] we can obtain the following result as a consequence of ). Let B be a ball in R n. Proposition. i) If B fx) log+ fx) log + log + fx) dx <, then D r f) is finite a.e. on B. ii) If B fx) log+ fx) ) dx <, then D r f) is integrable over B. iii) For all λ>0 we have {x R n : D r f)x) >λ} Ce cλ/ f L where C, c only depend on the dimension in particular they are independent of the measurable function N : R n R n.) References [] L. Carleson, On convergence and growth of partial sums of Fourier series, Acta Math ), [] C. Fefferman, Pointwise convergence of Fourier series, Ann. of Math ), [3] R. Hunt, On the convergence of Fourier series, Orthogonal Expansions and their continuous analogues, Edwardsville, IL 967), D. T. Haimo ed), Southern Illinois Univ. Press, Carbondale IL, 968, [4] C. Kenig and P. Tomas, Maximal operators defined by Fourier multipliers, Studia Math ), [5] M. Lacey and C. Thiele, Convergence of Fourier series, preprint. [6] M. Lacey and C. Thiele, A proof of boundedness of the Carleson operator, Math. Res. Lett ), [7] M. Lacey and C. Thiele, On Calderón s conjecture, Ann. of Math ),

16 6 LOUKAS GRAFAKOS, TERENCE TAO, AND ERIN TERWILLEGER [8] M. Pramanik and E. Terwilleger, AweakL estimate for a maximal dyadic sum operator on R n, Ill. J. Math., to appear. [9] P. Sjölin, An inequality of Paley and convergence a.e. of Walsh-Fourier series, Ark. Matematik 7 968), [0] P. Sjölin, Convergence almost everywhere of certain singular integrals and multiple Fourier series, Ark. Matematik 9 97), [] E. M. Stein, Singular integrals and differentiability properties of functions, Princeton Univ. Press, Princeton NJ, 970. [] E. M. Stein and G. Weiss, An extension of theorem of Marcinkiewicz and some of its applications, J. Math. Mech ), Loukas Grafakos, Department of Mathematics, University of Missouri, Columbia, MO 65, USA address: loukas@math.missouri.edu Terence Tao, Department of Mathematics, University of California, Los Angeles, Los Angeles, CA 9004, USA address: tao@math.ucla.edu Erin Terwilleger, School of Mathematics, Georgia Institute of Technology, Atlanta, GA 3033 USA address: erin@math.gatech.edu