STRONG TYPE INEQUALITIES AND AN ALMOST-ORTHOGONALITY PRINCIPLE FOR FAMILIES OF MAXIMAL OPERATORS ALONG DIRECTIONS IN R 2

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1 STRONG TYPE INEQUALITIES AND AN ALMOST-ORTHOGONALITY PRINCIPLE FOR FAMILIES OF MAXIMAL OPERATORS ALONG DIRECTIONS IN R 2 ANGELES ALFONSECA Abstract In this aer we rove an almost-orthogonality rincile for maximal oerators over arbitrary sets of directions in R 2 Namely, we obtain L -bounds for an oerator of this tye from the corresonding L -bounds of the maximal functions associated to a certain artition of the set of directions, and from the articular structure of this artition We give alications to several tyes of maximal oerators 1 Introduction In this aer we continue the study of maximal functions along directions in R 2, initiated in [1] in connection with a certain notion of lanar almostorthogonality Let Ω 0 be an ordered subset of [0, π 4 ) We denote its elements by θ 1, θ 2,, with π 4 = θ 0 > θ 1 > θ 2 > > θ > We shall refer to Ω 0 as the searating set, and to its elements θ as the searators For each 1, we have a set Ω [θ, θ 1 ), with θ Ω The maximal oerators associated to these sets are defined as M Ω0 f(x, y) = su 1 h f(x t cos θ, y t sin θ ) dt 2h, and M Ω f(x, y) = h>0, su h>0,θ Ω 1 2h h h h f(x t cos θ, y t sin θ) dt, for 1 The maximal function over all these directions, that is, on Ω = Ω is M Ω f(x, y) = su M Ω f(x, y) We want to determine L -bounds of M Ω from the corresonding L - bounds of each M Ω and the articular structure of the searating set Ω 0 In [1], the author roved with F Soria and A Vargas the following relationshi 2000 Mathematics Subect Classification 42B25 Key words and hrases: Maximal oerators, strong-tye estimates Research artially suorted by TMR network Harmonic Analysis, and by Grant BFM2001/0189 1

2 2 A Alfonseca between the L 2 -norms of M Ω and M Ω : there exist constants C 1 and C 2, indeendent of the set Ω, such that (1) M Ω 2 L 2 L C 2, 1 su M Ω 2 L 2 L + C 2, 2 M Ω0 2 L 2 L 2, As a corollary, one gets a simle roof of a beautiful result by N Katz [7], solving a conecture that was oen for many years: if Ω has cardinality N > 1, then M Ω L 2 L 2, C(log N)α, for some constants C and α indeendent of Ω Actually, Katz roves it with the shar exonent α = 1 2 Another direct consequence of (1) is the following If Ω 0 is a lacunary sequence, then (2) su M Ω L 2 L 2, C su M Ω L 2 L 2,, ie we bound the norm of a suremum by the suremum of the norms Notice that we cannot directly get strong tye 2 inequalities by interolation between this and estimates on L 1, since Ω is an infinite set Inequalities (1) and (2) show that the oerators M Ω satisfy an almostorthogonality rincile when we look at their weak tye L 2 norms Our aim is to find a similar relation between the strong-tye norms of these maximal functions Instead of the geometrical arguments used in [1], we shall follow the aroach of Nagel, Stein, Wainger [8] (see also [9]) In order to do that, we need to introduce a Littlewood-Paley decomosition and the associated square function All this work is resented in Section 2 The last section contains several alications 2 The Main Result Let Ω 0 = {θ 1 > θ 2 > > θ > } be a subset of [0, π 4 ) For each 1, we consider sets Ω [θ, θ 1 ), where we take θ 0 = π 4 We shall also assume that θ Ω We define the maximal oerators associated to the sets Ω, 0, as M Ω0 f(x, y) = su 1 h f(x t cos θ, y t sin θ ) dt 2h, and, if 1 M Ω f(x, y) = h>0, su h>0,θ Ω 1 2h h h h for f S We also define the maximal function M Ω f(x, y) = su M Ω f(x, y) f(x t cos θ, y t sin θ) dt, In order to state our result, we need to introduce a certain square function associated with Ω 0 For each 1, set δ = θ 1 θ Let us consider the following angular sectors

3 Strong tye inequalities 3 = { (x, y) R 2 θ 1 ( ) x 20 δ arctan < θ } y 20 δ, and the wider sectors { = (x, y) R 2 θ 1 ( ) x 10 δ arctan < θ } y 10 δ Given 1, we ick a function ω, homogeneous of degree zero, C on S 1, identically equal to 1 in and vanishing outside The multilier oerator S associated to ω is defined by and the square function S by (S f) = ω f, Sf(x) = S f(x) 2 The roerties of S deend in a direct way on the geometry of Ω 0 We can now state the following two results 1/2 Theorem 1 With the above notation, given 2 <, there exists a finite constant C such that ( ) ] (3) M Ω f C [ M Ω0 f + su M Ω L L Sf For 1 < < 2, we get the following Theorem 2 If 1 < < 2, there exists a finite constant C such that [ ) ] (4) M Ω f C M Ω0 L L (su + M Ω L L S 2/ L L f Theorem 1 will follow from the two lemmas below Let us define the directional Hardy-Littlewood maximal function, in the direction of θ by M θ f(x) = su 1 h f(x t cos θ, y t sin θ)dt 2h Then we have h>0 h Lemma 3 For each 1 and for all θ Ω, (5) M θ f(x) C[M θ f(x) + MM θ (S f)(x)], where M is the ordinary Hardy-Littlewood maximal function

4 4 A Alfonseca Proof Our argument reresents a slight modification of the roof of Lemma 3 in [5] Choose a ositive function ψ C0 (R) with ψ = 1 on [ 1, 1] Fix 1 and θ Ω Then if f 0, M θ f(x) C su h>0 1 ψ 2h (6) = C su N h,,θ f(x) h>0 ( ) t f(x t cos θ, y t sin θ) dt = h Set m = ψ Take a function φ C 0 (R2 ), with φ(ξ) = 1 if ξ 1 We decomose (N h,,θ f) (ξ) as (N h,,θ f) (ξ) = m(hξ 1 cos θ + hξ 2 sin θ) f(ξ) = = m(hξ 1 cos θ + hξ 2 sin θ) φ(hδ ξ) f(ξ) + + m(hξ 1 cos θ + hξ 2 sin θ) (1 φ(hδ ξ)) (1 ω (hξ)) f(ξ) + + m(hξ 1 cos θ + hξ 2 sin θ) (1 φ(hδ ξ)) ω (hξ) f(ξ) = (7) = I h,,θ (f)(ξ) + II h,,θ (f)(ξ) + III h,,θ (f)(ξ) Consider the multilier in the first term, m(hξ 1 cos θ + hξ 2 sin θ)φ(hδ ξ) If we comose it with an aroriate rotation, we can write it in the form m(hη 1 )φ(hδ η) Then, differentiating with resect to η 1 and η 2, we see that the Fourier transform K 1 of m(hη 1 )φ(hδ η) satisfies z 1 α z 2 β K 1 (z) Cδ 1+β, so that the oerator K 1 f is bounded by the maximal function over rectangles of eccentricity δ and sides arallel to the coordinate axes Therefore su h I h,,θ (f) is bounded by M θ,δ f, the maximal function over rectangles of eccentricity δ and sides arallel to the direction θ Now, observe that θ θ δ, so the rectangles in the definition of M θ,δ f can be included in rectangles of comarable area and sides arallel to θ Hence, su I h,,θ f(x) CM θ f(x) h>0 The second term is treated in the same way The third term is clearly controlled by MM θ (S f) This finishes the roof of Lemma 3 For the roof of Theorem 1, we also need the following lemma Lemma 4 Assume that for some > 1, q 2 and for any sequence of functions {f }, one has 1/q 1/q (8) M Ω f q B f q =1 =1

5 Then Strong tye inequalities 5 (9) M Ω f C [ M Ω0 f + B Sf ], for some constant C Proof From the ointwise estimate (5) in Lemma 3, we get [ M Ω f(x) = su,θ Ω M θ f(x) C M Ω0 f(x) + su MM θ (S f)(x),θ Ω Using the Hardy-Littlewood maximal theorem, we have su MM θ (S f) C,θ Ω su M θ (S f),θ Ω Notice that su M θ (S f),θ Ω su (M θ (S f)) q θ Ω =1 ( = MΩ (S f) ) q =1 The hyothesis of the lemma imlies now 1/q su M θ (S f) B S f q,θ Ω =1 1/2 B S f 2 = B Sf =1 In the second inequality we have used that q 2 therefore, Lemma 4 1/q 1/q = ] This gives (9) and, Proof of Theorem 1 It is a simle consequence of Lemma 4, from the trivial observation that (8) holds with B = su M Ω L L and = q Proof of Theorem 2 For technical reasons, we shall rove Theorem 2 for a slightly different square function, S, adated to the sectors We take a function ω identically equal to 1 in and vanishing outside a slightly wider sector Then we define the oerator S by ( S f) = ω f, and consider the associated square function S

6 6 A Alfonseca The roof follows the main circle of ideas that one can find in [3] We start by considering the oerator N h,,θ f(x) defined in the roof of Lemma 3 by (6) As we did there, we slit it into three terms, by (7) We already know that (10) su h,,θ Ω (I h,,θ (f) + II h,,θ (f)) M Ω0 f Without loss of generality, we can assume that Ω is a finite set Then there exists a minimal constant C(Ω) such that su III h,,θ (f) C(Ω) f h,,θ Ω Let us take a sequence of functions {g } By (7) and (10), su III h,,θ (g ) h,,θ Ω su N h,,θ (g ) + su I h,,θ (g ) + II h,,θ (g ) h,,θ Ω h,,θ Ω ( ) su h,,θ Ω N h,,θ(su g ) + M Ω 0 su g ( ) su h,,θ Ω III h,,θ(su g ) + 2 M Ω 0 su g Therefore, su III h,,θ (g ) (C(Ω) + 2 M h,,θ Ω Ω0 L L ) su g On the other hand, we have 1/ su III h,,θ (g ) h,θ Ω ( su su III h,,θ h,θ Ω L L ) 1/ g By interolation (taking θ = 2 ), 1/2 (11) su III h,,θ (g ) 2 h

7 (C(Ω) + 2 M Ω0 L L ) 1 θ (su Now, Strong tye inequalities 7 su III h,,θ h,θ Ω su III h,,θ (f) h,,θ Ω L L ) θ su III h,,θ (I S )(f) + su III h,,θ Ω h,,θ ( S f) 2 h,θ Ω 1/2 g 2 1/2 The first term in the last exression is 0, because of the definition of III h,,θ and S By (11), the second term is bounded by (C(Ω) + 2 M Ω0 L L ) 1 θ (su By the minimality of C(Ω), (12) (C(Ω) + 2 M Ω0 L L )1 θ (su C(Ω) su III h,,θ h,θ Ω su III h,,θ h,θ Ω L L L L ) θ Sf ) θ S L L From equation (12), we derive the following exression for C(Ω) ( ) ] C(Ω) 2 [ M Ω0 + su L L su 2/ III h,,θ S L L h,θ Ω Now, we notice that III h,,θ f M(N h,,θ ( ω f)), L L where M is the Hardy-Littlewood maximal function The Hardy-Littlewood maximal theorem and the uniform boundedness of the L 1 -norm of the ω imly that su su III h,,θ C h,θ Ω su M Ω L L, L L and thus Theorem 2 is roved

8 8 A Alfonseca 3 Some alications Our first alication of Theorem 1 will be a generalization of the result by Nagel, Stein and Wainger [8] Let Ω 0 = {θ } be a lacunary sequence of numbers in [0, π 4 ) tending to zero, that is, assume that there exists 0 < λ < 1 such that 0 θ +1 λθ for all N The above authors roved that M Ω0 is bounded in L for all 1 < We shall show that we can introduce new directions Ω between the lacunary searators of Ω 0 and still get a bounded oerator Theorem 5 Let Ω 0 be a lacunary sequence Then, under the hyothesis of Theorem 1, ) (13) M Ω f C (su M Ω L L f, 1 < < Theorem 5 also generalizes the result by Sögren and Sölin [9] In that aer they roved the same estimate but in the secial case in which Ω is given by a lacunary sequence, for all Our result does not deend on the articular structure of the Ω s, and it shows, in a neat way, that the orthogonality roerty mentioned in the introduction holds when the searating set is lacunary Proof Without loss of generality, we can assume that 1 2 < λ < 1 We shall also assume that there exists 0 < λ 0 < λ such that θ +1 λ 0 θ for all Actually we can take λ 0 = λ 2, and add some terms to the initial sequence in order to get this Obviously, the maximal oerator associated to the new sequence is greater than the maximal oerator of the former one We need this lower bound λ 0 in order to ensure that, for every 1, the sectors and +2 do not overla To get (13), we aly Theorems 1 and 2 In our case, because of Nagel, Stein and Wainger s result, M Ω0 L L C, and so we get and ) M Ω f C f + C (su M Ω L L Sf, if 2 <, ) M Ω f C f +C (su M Ω L L Finally, a standard argument, as in [8], gives and this roves (13) Sf C f, 1 < <, S 2/ L L f, if 1 < < 2

9 Strong tye inequalities 9 Theorem 5, in turn, gives us the following alication: let {θ } be a lacunary sequence as above and take, for each, a family of N directions {α,k } k=1,,n [θ, θ 1 ) Define, as usual, the maximal oerators M Ω by M Ω f(x, y) = su su 1 h f(x t cos α,k, y t sin α,k ) dt 2h, and h>0 k=1,,n Using Nets Katz s result [7], we have We also have the trivial bound This give us, by interolation, h M Ω f(x, y) = su M Ω f(x, y) su M Ω L 2 L2 C log N su M Ω L 1 L1, CN su M Ω L L C N 2 1 (log N) 2 2, 1 < 2 Then, using (13), we get M Ω f C N 2 1 (log N) 2 2 f, 1 < 2 Let us now see some other consequences of Theorem 1 Theorem 6 Let Ω 0 be a set of N uniformly distributed directions, and assume that ( ) su M Ω L 2 L 2 < Then ( [C 1 log N + C 2 M Ω f 2 for some universal constants C 1, C 2 )] su M Ω L 2 L 2 f 2, Proof It is ust a consequence of Theorem 1 Notice that M Ω0 L 2 L2 C log N, as Strömberg roved in [11] We also use the fact that, in this case, Sf 2 f 2 by Plancherel Remark A similar result can be obtained for other values of 2 In articular, for 2 4, we can use Cordoba s bound for the square function associated to these directions (see [4]) We have restricted ourselves to the case = 2 for the sake of simlicity

10 10 A Alfonseca We shall now consider more general sets Ω 0 In order to do this, we need some definitions Given an oen set U R, we say that a set of intervals {I β } is a Whitney decomosition for U (with constant C) if (i) I β U, for all β (ii) Iβ = U (iii) 1 C I β d(i β, U) C I β We shall say that a set S of finite cardinality N is of Whitney tye with constants (C 1, C 2 ) if for each s S there is a Whitney decomosition {I s β } of {s} c (with constant C 1 ) such that at most C 2 (log N) 2 of the I s β have nonemty intersection with S This tye of set was introduced by Katz in [6] Theorem 7 Assume Ω 0 is a set of Whitney tye of cardinality N > 1 Then ( M Ω f 2 C log N ) su M Ω L 2 L 2 f 2 Proof Since Ω 0 has cardinality N, M Ω0 L 2 L2 C log N, by Katz s result [7] In order to get the conclusion from Theorem 1, we must find an aroriate bound for the square function term in (3) We shall show that, for all ξ, ω (ξ) C(log N) 2 This estimate, together with Plancherel s theorem, will rove Theorem 7 Given θ k Ω 0 θ 0, let ξ k = (cos θ k, sin θ k ) Note that if θ k 1 > θ θ k and ξ = (cos θ, sin θ), then ω (ξ) ω (ξ k 1 ) + ω (ξ k ), so we need only rove ω (ξ k ) C(log N) 2 for all k = 1,, N Fix k and consider the Whitney decomosition {Iβ k} of {θ k} c given by the definition of a set of Whitney tye We shall show that for any fixed β, (14) ω (ξ k ) C, :θ I k β and this will imly that ω (ξ k ) = 1 + β :θ I β ω (ξ k ) = 1 + C(log N) 2 Let us rove (14) Observe that the width of the suort of ω is less than 3 2 (θ 1 θ ) So if (θ 1 θ ) 2 3 min( θ k θ, θ k θ 1 ),

11 Strong tye inequalities 11 then ω (θ k ) = 0 On the other hand, if the reverse inequality is true, then (θ 1 θ ) > 2 3 min( θ k θ, θ k θ 1 2 Iβ k 3C 1 Now, since the intervals [θ, θ 1 ) are airwise disoint, there are at most [ 3C 1 2 ] + 1 of those θ Iβ k Hence (14) is roved This finishes the roof of Theorem 7 Finally, let us see a result in the sirit of the work of Barrionuevo [2] Theorem 8 There is a constant C 0 > 0 such that for every Ω 0 with (Ω 0 ) = N, we have ( ) (15) M Ω L 2 L 2 C 0 su M Ω L 2 L 2 N C 0 log N Proof The roof will be by induction on N If C 0 is big enough, (15) is true for small N Now, given a small ɛ to be determined later, we shall say θ l Ω 0 is bad if ω (θ l ) N ɛ, ie the overlaing of the ω s on θ l is high For a fixed bad θ l, we consider the indices (1) < (2) < < (k) < < l, such that ω (θ l ) 0 This means that [ θ l θ (k) δ (k) 20, θ (k) 1 + δ ] (k) 20 Then θ (k) form a lacunary sequence tending to θ l Indeed, θ l θ (k) < 1 20 θ (k) θ (k) 1 < 1 20 θ (k) θ (k 1), and, as (k) < l for all k, this imlies We define the sets θ (k) θ (k 1) θ l θ (k 1) 1 20 θ (k 1) θ (k 2) G (θ l ) = { < l : ω (θ l ) 0}, G + (θ l ) = { > l : ω (θ l ) 0} Then, for a bad θ l, (G (θ l ) G + (θ l )) N ɛ Set η1 = θ l(1), where l(1) = min{k : (G (θ k )) N ɛ } Next set η 2 = θ l(2), where l(2) = min{k : θ k / G (η 1 ) and [G (θ k ) \ G(η 1 )] N ɛ } and we roceed by induction For each k, the set G (η k ) is a lacunary sequence We similarly define η + k, associated to the sets G + The set

12 12 A Alfonseca {η + k, η k : k 1} has at most 2N 1 ɛ elements Then, alying Theorem 5, we have that for each k, (16) su M Ω L 2 L 2 C su M Ω L 2 L 2 G ± (η ± k ) G ± (η ± k ) Now we use the induction hyothesis on the cardinality of the searating set, taking instead of Ω 0 the set {η ± k } k which, as we ointed out, has at most 2N 1 ɛ elements Thus, (17) su k su M Ω L 2 L 2 C 0C su M Ω L 2 L 2(N 1 ɛ ) ɛ G ± (η ± k ) Set G = ( k G (η k )) ( k G + (η + k )) Then / G ω (θ) < N ɛ for all θ Hence Lemma 3 and an argument as in the roof of Theorem 7 give (18) su M Ωl L 2 L 2 C N ɛ su M Ωl L 2 L 2 l / G Combining (17) and (18), we get su M Ω L 2 L 2 ( C 0 C (N 1 ɛ ) ɛ + C N ɛ) su M Ω L 2 L 2 If we now choose ɛ = C 0 log N, we have C 0 C N ɛ2 + C C 0, for big C 0 deending on C, C, and (15) is roved Acknowledgements: I would like to exress my gratitude to Anthony Carbery and Jim Wright for the interesting conversations we had, and for their hositality when I visited Edinburgh University I would also like to thank the referee for his useful suggestions References [1] A Alfonseca, F Soria, A Vargas, A remark on maximal oerators along directions in R 2, Prerint [2] J A Barrionuevo, A note on the Kakeya maximal oerator, Mathematical Research Letters 3 (1995) [3] A Carbery, Differentiation in lacunary directions and an extension of the Marcinkiewicz multilier theorem, Ann Inst Fourier, Grenoble (1) 37 (1988) [4] A Córdoba, Geometric Fourier Analysis, Ann Inst Fourier, Grenoble (3) 32 (1982) [5] J Duoandikoetxea, A Moyua, Weighted inequalities for square and maximal functions in the lane, Studia Mathematica (1) 102 (1992), [6] N H Katz, Remarks on maximal oerators over arbitrary sets of directions, Bull London Math Soc (6) 31 (1999) [7] N H Katz, Maximal oerators over arbitrary sets of directions, Duke Mathematical Journal (1) 97 (1999), [8] A Nagel, E M Stein, S Wainger, Differentiation in lacunary directions, Proc Natl Acad Sci USA (3) 75 (1978),

13 Strong tye inequalities 13 [9] P Sögren, and P Sölin, Littlewood-Paley decomositions and Fourier multiliers with singularities on certain sets, Ann Inst Fourier, Grenoble (1) 31 (1981), [10] E M Stein, Singular integrals and differentiability roerties of functions, (Princeton University Press 1970) [11] J O Strömberg, Maximal functions associated to rectangles with uniformly distributed directions, Annals of Mathematics 107 (1978), Deartamento de Matemáticas, Universidad Autónoma de Madrid Madrid, Sain angelesalfonseca@uames