ON PLATE DECOMPOSITIONS OF CONE MULTIPLIERS. 1. Introduction

Transcription

1 ON PLATE DECOMPOSITIONS OF CONE MULTIPLIERS GUSTAVO GARRIGÓS AND ANDREAS SEEGER Abstract. An imortant inequality due to Wolff on late decomositions of cone multiliers is nown to have consequences for shar L results on cone multiliers, local smoothing for the wave equation, convolutions with radial ernels, Bergman rojections in tubes over cones, averages over finite tye curves in R 3 and associated maximal functions. We observe that the range of in Wolff s inequality, for the conic and the sherical versions, can be imroved by using bilinear restriction results. We also use this inequality to give some imroved estimates on square functions associated to decomositions of cone multiliers in low dimensions. This gives a new L 4 bound for the cone multilier oerator in R Introduction Let Γ = {(τ, ξ) R R d : τ = ξ } denote the forward light-cone in R d+1, d. For fixed c > 0 and small δ > 0, we consider δ-neighborhoods of the truncated cone Γ δ (c) = {(τ, ξ) R d+1 : 1 τ and τ ξ cδ}, with the usual decomosition into lates subordinated to a δ-searated sequence in the shere {ω } S d 1 : (1.1) Let { Π (δ) = (τ, ξ) Γ δ (c) : ξ ξ ω dist(ω, ω ) δ if. c δ } ; (1.) α() := d( 1 1 ) 1, the standard Bochner-Riesz critical index in d dimensions. Then Wolff s inequality asserts that for all ε > 0 (1.3) f C ε δ α() ε( f Date: Aril 11, 007. The results of this aer were resented by G.G. at the Conference on Harmonic Analysis and its Alications at Saoro, 005, and a reliminary version has been included in the Hoaido University Technical Reort series in Mathematics (No. 103, December 005). G.G. thans Hoaido University and Prof. Tachizawa for the invitation and suort. Secial thans also to Aline Bonami for many suggestions and discussions related to this toic. G.G. was suorted in art by the Euroean Commission, within the IHP Networ HARP , contract number HPRN-CT HARP, and also by Programa Ramón y Cajal and grant MTM , MCyT (Sain). A.S. was suorted in art by NSF grant DMS ) 1/

2 G. GARRIGÓS AND A. SEEGER rovided that (1.4) su f Π (δ). The ower α() is otimal for each (excet erhas for ε > 0), and the inequality is conjectured to hold for all > + 4 d 1. In his fundamental wor [], Wolff develoed a method to rove such inequalities for large values of, and obtained a ositive answer for d = and > 74. Subsequently the method has been extended in the aer by Laba and Wolff [11] to higher dimensions. It is shown there that (1.3) holds for > d 7 when d 3 and > + d 3 when d 4. In this aer we modify the weaest art of their roof to obtain a better range of exonents in all dimensions (see Table 1 below). The imrovement relies on certain square-function bounds which follow from Wolff s bilinear Fourier extension theorem, [3]. Dimension [], [11] Imrovements Conjecture d = > 74 > := /3 > 6 d = 3 > 18 > 3 := 15 > 4 d = 4 > 8.4 > 4 := 7.8 > 10 3 d 5 > + 8 d 3 > d := + 8 d 3 (1 1 d+1 ) > + 4 d 1 Table 1. Range of exonents for the validity of (1.3) for light-cones. Theorem 1.1. Let d and d as in Table 1. Then, under the assumtion (1.4) the inequality (1.3) holds for all ε > 0 and all d. Remar: Various further more technical imrovements on the range of Theorem 1.1 (and by imlication on the results of Corollaries 1.3 and 1.4 below) have been obtained by the authors, and also by Wilhelm Schlag (ersonal communication). We intend to tae u these matters in a joint aer (cf. [9]). A similar result can be roved for decomositions of sheres in R d. We now let U δ (c) = {ξ R d : ξ 1 cδ}, and consider the decomosition into lates subordinated to a δ-searated sequence in the shere {ω } S d 1, { B (δ) = ξ U δ (c) : ξ/ ξ ω c } δ. Theorem 1.. The analogue of Wolff s inequality for the shere, (1.5) f C ε δ α() ε( ) 1/, f su f B (δ) holds for + 8 d 1 4 (d 1)d and all ε > 0.,

3 PLATE DECOMPOSITIONS OF CONE MULTIPLIERS 3 Again (1.5) is conjectured to hold for the otimal range > + 4/(d 1). It has been nown to hold for > + 8/(d 1); this follows from a modification of the argument in [11], see also [10]. Note that in two dimensions the range is imroved from reviously > 10 to > 8. Remar: Theorem 1. may be extended to convex surfaces with nonvanishing Gaussian curvature and similarly Theorem 1.1 may be extended to cones with d 1 ositive rincial curvatures. This can be achieved by using scaling and induction on scales arguments such as in of [16], see also the article by Laba and Pramani [10] for related results. We roceed to list some of the nown imlications of Theorem 1.1. Corollary 1.3. Let d and d as in Table 1. Then (i) For all > d, α > d 1 d, we have ( (1.6) e it f ) 1/ L (R d ) dt f L α (R ). d (ii) For all ( d, ), α > d 1 d 1 the Fourier multilier (1.7) m α (τ, ξ) = (1 ξ /τ ) α + defines a bounded oerator in L (R d+1 ). (iii) Let K S (R d ) be radial, let ϕ C 0 (R d \ {0}) so that ϕ is radial and not identically zero, and let ε > 0. Let K t = F 1 [ϕ K(t )]. Then for all Schwartz functions f and 1 < < K f C ε su K t +ε f. t>0 d d 1 (iv) Let χ C0 (R) and s γ(s) R 3 a smooth curve satisfying n j=1 θ, γ(j) (s) = 0 for every unit vector θ and every s su χ. For t > 0 define the convolution oerator A t by A t f(x) = f(x tγ(s))χ(s)ds. Suose that max{n, 3 + /3} < <. Then A t mas L (R 3 ) into the L -Sobolev sace L 1/ (R3 ). Moreover the maximal function Mf = su t A t f defines a bounded oerator on L (R 3 ). Parts (i), (ii), (iii) are standard consequences of Theorem 1.1; see [] for (i) and the local version of (ii). The global version follows by results on dyadic decomositions of multiliers and L Calderón-Zygmund theory (see [6] or [17]). The roof of Theorem 1.6 in [15] together with these arguments can be used to deduce (iii) from Theorem 1.1. For (iv) see [16]. Besides the connection to cone multiliers a major motivation for this aer is the relevance of inequalities for late decomositions for the boundedness roerties of the Bergman rojection in tube domains over full light cones, see [1], []. Denote by (Y ) = y 0 y the Lorentz form and consider the forward light cone on which is ositive; Λ d+1 = {Y = (y 0, y ) R R d : y 0 y > 0, y 0 > 0}. Let T d+1 C d+1 be the tube domain over Λ d+1, i.e. T d+1 = R d+1 + iλ d+1.

4 4 G. GARRIGÓS AND A. SEEGER Let w γ (Y ) = (Y ) γ and consider the weighted sace L (T d+1, w γ ) with norm ( ) 1/. F,γ = F (X + iy ) γ (Y ) dy dx T d+1 Let P γ be the orthogonal rojection maing the weighted sace L (T d+1, w γ ) to its subsace A γ consisting of the holomorhic functions. Only the case γ > 1 is interesting since A γ = {0} for γ 1. We are interested in the L boundedness roerties of P γ. For γ > 1 the oerator P γ can only be bounded on L (T d+1, w γ ) in the range (1.8) 1 + d 1 (γ + d + 1) (γ + d + 1) < < 1 +, d 1 see e.g. Theorem 4.3 in [3], and (1.8) is indeed the conjectured range for L boundedness. Corollary 1.4. Let d and d as in Table 1. Then for all γ d 1 ( d (d+1) d 1 ), the Bergman rojection P γ is a bounded oerator in L (T d+1, w γ ) in the shar range (1.8). Beyond Corollary 1.4 both Theorem 1.1 and Theorem 4.4 below have imlications for the range of boundedness of the Bergman rojector P γ in natural weighted mixed norm saces. For the derivation of Corollary 1.4 and further discussion of mixed norm estimates we refer to [] (cf. in articular Proosition 5.5 and Corollaries 5.1 and 5.17). Our aroach to Theorem 1.1 is based on bilinear methods, for which we consider a closely related inequality: (1.9) ( 1/. f C α δ α f ) One can conjecture the validity of (1.9) for all α > 0 and all < < + 4 d 1, but for the moment no ositive result for any such seems to be nown. The limiting oint = (d+1) d 1 should be the hardest case, since by interolation and Hölder s inequality it imlies both (1.9) and (1.3) in all the conjectured ranges. This ind of inequalities arises naturally in the study of weighted mixed norm inequalities for the Bergman rojection oerator P γ, see []. We shall deduce Theorem 1.1 by using a stronger version of (1.9) for = (d + 3)/(d + 1), but with a ower of 1/δ which is (robably) not otimal. Namely under the assumtion (1.4) we have (1.10) f (d+3) C ε δ d 1 4(d+3) ε ( f ) 1/ (d+3), d+1 d+1 for all ε > 0. We rove this inequality in using the bilinear aroach of Tao and Vargas [0, 5] and the otimal bilinear cone extension inequality of T. Wolff [3], see Proosition.3 below. By Minowsi s inequality and interolation (1.10) trivially imlies non otimal estimates for the inequality (1.9) for all (, ) (see Corollary.4 below). In 3 we use these to refine a art of Wolff s roof of (1.3) and obtain the new shar estimates for large announced in Table 1. In 4 we imrove on some of the square-function results in low dimensions; these yield in articular the following estimate for the cone multilier in R +1. ( Theorem 1.5. Suose α > 5 4 ) 44. Then the cone Fourier multilier 41 mα defines a bounded 11 oerator on L 4 (R 3 ) and the local smoothing result (1.6) holds in two dimensions.

5 PLATE DECOMPOSITIONS OF CONE MULTIPLIERS 5 This is a small imrovement over the nown range α > 5/44 which follows from a combination of [0] and [3]. Notation. We shall use the notation A B if there is a constant (which may deend on d) so that A CB. For families (A δ, B δ ), δ 1 we use A δ B δ if for every ε (0, 1) there is a constant C ε so that A δ C ε δ ε B δ for δ < 1.. The bilinear estimate Following the aroach by Tao and Vargas, we first establish an equivalence between linear and bilinear versions of (1.10), which is a higher dimensional analogue of Lemma 5. in [0]. Lemma.1. Let d, and suose that for some [, ) and α > max{0, (d 1)(1/4 1/)} ( ) ( ) ( (.1) f C δ / α f ) 1/ ( ) 1/ f, ω Ω ω Ω ω Ω f holds for all f S(R d+1 ) with su f Π (δ), all airs of 1-searated subsets Ω, Ω S d 1 and all δ 1. Then, it must also hold (.) f C δ α ( f ) 1/, su f Π (δ). ω Ω We remar that the restriction on α for > 4 is never severe. To see this we note that the condition (d 1)(1/4 1/) α()/ holds iff d and that (.) cannot hold with α < α()/; this can be roved using Kna examles. Proof of Lemma.1. Let Φ : Q [0, 1] d 1 S d 1 be a smooth arametrization of (a comact subset of) the shere and let D denote the set of all dyadic intervals I Q with I δ d 1. As in [1,. 971], we may consider a Whitney decomosition Q Q = I J I J, where I J means: (i) I, J D and I = J ; (ii) If I > δ d 1, then I and J are not adjacent but their arents are. (iii) If I = δ d 1, then I, J have adjacent or equal arents. For simlicity, we assume (by slitting the shere into finitely many ieces) that all ω Φ(Q) and let y = Φ 1 (ω ) Q. We also denote D j = {I D : ( ) f = f f = y,y Q δ j 1 I = j(d 1) }. Then I,J D j I J ( ) ( f f ). To establish (.) we tae L / -norms in the above exression and use Minowsi s inequality in j, so that we reduce the roblem to show, for each j (.3) ( ) ( / f f ) (j δ) α max{1, j(d 1)(1 4/) ( } f ) 1/ I,J D j I J y I y J y I y J.

6 6 G. GARRIGÓS AND A. SEEGER Inequality (.3) is trivial when j δ since by assumtion the number of y s in each I is aroximately constant. We consider the general case δ < j 1. By construction we must have (.4) χ I+J 1. Indeed, if c I denotes the center of I, then I D j J I I + J ( c I + B c j ) + ( cj + B c j ) ci + B c j. Since for each I there are at most O(1) cubes J with J I, and since the centers c I are j searated, (.4) follows easily. From (.4) it follows that the functions F I,J = ( y I f ) ( y J f ) have airwise (almost) disjoint sectra when I J D j. We may conclude by orthogonality and standard interolation arguments (.5) I J D j F I,J ( ) /. max{1, j(d 1)(1 4/) } F I,J / / / I J D j (the case / = follows by orthogonality and the cases / = 1 and / = are trivial; see e.g. Lemma 7.1 in [0]). Next, we wish to use the bilinear assumtion (.1) to estimate F I,J /. This can only be used directly when j 1, since dist(i, J) 1. For other j s we must use Lorentz transformations to rescale the roblem. To do this, let {η 1,..., η d } be an orthonormal basis of R d with η 1 being the center of Φ(I). Then we define L SO(1, d) acting on a basis of R d+1 by L(1, η 1 ) = (1, η 1 ), L( 1, η 1 ) = σ δ ( 1, η σ 1) and L(0, η l ) = δ (0, η l), l =,..., d, where we choose σ = j δ (so that δ < σ < 1). The functions f L have now sectrum in (erhas a multile) of the lates Π (σ) corresonding to the σ-searated centers {L(1, ω )}. Moreover, by the choice of σ, the lates corresonding to y I and y J are c-searated, and therefore after a change of variables we can aly (.1) at scale σ to obtain ( ) ( ) F I,J / = f f / y I y J ( ( j δ) α f ) 1/ ( f ) 1/, y I and then also ( ) [ / F I,J / / ( j δ) α I J D j I J D j ( y I y J f ) 1/ ( y J f ) 1/ [ ( ( j δ) α f ) /] / ( j δ) α ( f ) 1/ I y I where in the second inequality we have used ab a + b followed by the imbedding l 1 l. Combining this with (.5) we obtain (.6) ( j δ) α max{1, j(d 1)(1 4/) ( } f ) 1/ / I J D j F I,J This roves (.3). By our assumtion on α we may sum in j and the lemma follows.,, ]

7 PLATE DECOMPOSITIONS OF CONE MULTIPLIERS 7 We turn to the roof of (a generalization of) the square function estimate (1.10). We shall use the following statement of Wolff s Fourier extension theorem. Wolff s bilinear estimate. [3,. 680]. Let d+3 d+1, ε > 0 and let E, E be 1-searated subsets of Γ 1/N. Then, for all smooth f and g suorted in E and E, and all N-cubes Q, we have (.7) f ĝl (Q) C ε N 1+ε f g. Denote by Q Q(δ 1/ ) a tiling of R d+1 with cubes Q of disjoint interior and sidelength δ 1/, with centers c Q in δ 1 Z d+1. Proosition.. Let d, and suose that su f Π (δ), su ĝ Π (δ) and let Ω, Ω S d 1 be 1-searated subsets. Suose (d+3) d+1 q and let (.8) µ() = d 4 d + 1. Then, for all ε > 0 (.9) ( Q Q(δ 1/ ) ( ω Ω f ) ( ω Ω g ) / L q/ (Q) ) / δ µ() ε ( ω Ω f ) 1/ ( ) 1/ g. ω Ω Proof. Let ψ S(R d+1 ) be so that su ψ B 1/10 and ψ(x) > 1 if x i, i = 1,..., d + 1; then n Z ψ( + d+1 n) 1. Let ψ Q = ψ( δ( c Q )), so that Q ψ Q 1. We write F Q = ( ) f ψq and G Q = ( ) ψq, ω Ω ω Ω g so that the suorts of F Q and ĜQ are 1-searated sets in Γ δ. Thus, we can use Wolff s estimate (.7) with N = δ 1/ to obtain ( (.10) ) ( ) f Lq/(Q) F Q G Q L q/(q) δ1/ F Q Ĝ Q. ω Ω ω Ω g Now, by almost orthogonality we can write F Q f ψ ( Q = f ) 1/ ψq and similarly for G Q. We write S Ω = ( ω Ω f ) 1/, S Ω = ( ω Ω g ) 1/, raise (.10) to the ower / and sum in Q. Thus ( ( ) ( f Q ω Ω ω Ω g ) / L q/ (Q) ) / ( δ Q S Ω ψ Q / and by the Cauchy-Schwarz and Hölder inequalities the right hand side is, SΩ ψ Q / ) /

8 8 G. GARRIGÓS AND A. SEEGER ( ) δ SΩ ψ Q 1/ ( SΩ ψ Q Q ( δ SΩ ψ Q Q Q Q 1+/) 1/( δ 1 (d+1)( 1 1 ) S Ω SΩ Q ) 1/ SΩ ψ Q Q 1+/) 1/ which yields the assertion. We combine Proosition. for q = and Lemma.1 to obtain Proosition.3. Let d, let µ() be as in (.8) and suose that > (d+3) d+1. Then, for all ε > 0 (.11) f C ε δ µ() ε ( f ) 1/ if su f Π (δ). We may aly Minowsi s inequality on the right hand side of (.11) and obtain (1.9) for the limiting case = (d + 3)/(d + 1). It turns out this is all what is needed to obtain the claimed imrovements in Theorem 1.1. This inequality can also be interolated with the trivial estimates for L and L to give: Corollary.4. The inequality (1.9) holds for all α > d 1 4 ( 1 1 (d+3) ), when α > d 1 4 (d+) (d+3) (1 (d+1) ) when d An imrovement of Wolff s estimate d+1 and for all We turn to Theorem 1.1. The roof in [, 11] for inequality (1.3) is based on a subtle localization rocedure, induction on scales and certain combinatorial arguments. Here we only discuss the modifications leading to the claimed imrovements based on Proosition.3. A more selfcontained exosition with further imrovements will be in [9]. For simlicity, when δ is fixed (and small) we use the notation A B to indicate the inequality A C ε δ ε B for all ε > 0. Recall that the number of lates Π (δ) covering Γ δ is aroximately δ d 1. Also, throughout this section we fix q(d) = (d + 3)/(d + 1). Due to various reductions (see [11, 3]), it is enough to show that, for all f with su ˆf Π (δ) and f 1, and for all λ > 0 we have (3.1) { f > λ} λ (d 1) d δ f where f = f. In [, 11] it is observed that, by Chebyshev s inequality, this roerty trivially holds for small enough λ; namely for all λ δ d We use (1.10) to enlarge this range of λ. Lemma 3.1. Let q = q(d) = (d + 3)/(d + 1). Then, inequality (3.1) holds for all (3.) λ δ d 1 + q 4( q).

9 PLATE DECOMPOSITIONS OF CONE MULTIPLIERS 9 Proof. Let β = d 1 4(d+3). By Chebyshev s inequality and (1.10), we have { f > λ} λ q f q q δ qβ λ q ( f q) q/ and estimate ( f ) q/ q δ d 1 q/ (q/) f q q δ d 1 ( q 1) f su f q. Since by assumtion f 1 and by almost orthogonality f f, it suffices to show d 1 qβ that in the desired range of λ we have δ ( q 1) λ q δ (d 1) d λ which is equivalent to (3.). At this oint one can roceed exactly as in the roof of Proosition 3. of [11] (or. 177 in [], when d = ). The desired gain comes from using λ δ d 1 + q(d) 4( q(d)) (rather than λ δ d in ste (54) of [11] (or (68) of []). For comleteness, we shall briefly setch this rocedure here, referring always to the notation in [11]. Localizing with N-cubes as in Lemma 6.1 of [11], one can find a collection of functions {f } with sectrum in Γ δ and a number (3.3) λ (λδ d 1 4 +ε, cδ d 1 4 ) ) so that { f > λ} { f > λ } and (3.4) card ( P(f ) ) λ λ δ 3d 1 4. Here P(f ) refers to the set of lates in the wave-acet decomosition of f. When the cardinality of this set is small, a further localization argument and induction on scales allows to conclude the theorem (see Lemmas 6. and 6.3 in [11]). In [11, ], the size of card(p(f )) which ensures the validity of these arguments is controlled in three different ways, each deending on a different combinatorial estimate (3.5) card ( P(f ) ) c ε δ ε λ, or (3.6) card ( P(f ) ) c ε δ 3d 3 8 +ε λ 4, or, in three dimensions (i.e. d = ) only, (3.7) card ( P(f ) ) c ε δ ε λ 9. the last estimate being by far the most difficult (see Lemmas 5. and 5.3 in [11] and Lemma 3. in []). Given the lower bound for λ in (3.3) and (3.8) λ δ d 1 + q 4( q)

10 10 G. GARRIGÓS AND A. SEEGER and given (3.4) it remains to verify the estimates (3.5) in the claimed range > d, d 5, (3.6) for > d, d = 3, 4 and (3.7) for >. This is straightforward. By (3.4) and (3.8) we have card ( P(f ) ) δ ε λ δ d 1 q(d) ( q(d)) δ 3d 1 4 q(d) ( q(d)) 3d 1 which gives in the case d 4 the assertion (3.5) if d 1 4 > 0 or, after a short comutation > q(1 + d 3 ) = + 8 d d 3 d+1. This is the asserted range if d 5. Next we examine the validity of the inequality (3.6) under condition (3.8). We now have card ( P(f ) ) C ε λ 4 δ 3d 1 4 ε λ λ λ4 δ 3d 1 4 ε λ 4 δ d 1 +ε δ 5d 3 4 3ε δ (d 1)+ q(d) q(d) This quantity is δ ε δ 3d 3 8 λ 4 if and only if 5d 3 4 (d 1) + q(d) 3d 3 q(d) + 4ε < 8, which yields the range gives > q(d)( d 7 ). Notice that this inequality amounts to > 7.8 if d = 4 and > 15 if d = 3 which is the assertion in those cases. Finally we consider the case d = when q() = 10/3. By (3.4) we need to have λ λ δ 5/4 ε c ε δ 11/8 λ 9, i.e. λ δ 1/8 ε c ε λ 7 rovided that λ > λδ 1/4+ε. Thus taing the smallest ossible λ yields δ 35/8 10ε λ 9 and this has to hold for all λ satisfying (3.8), i.e. λ δ 1 + q() λ 4. 4( q()). Taing the minimal λ this is achieved if 35/8 10ε < 9/ 9q/(4 4q) with q = q() = 10/3. Solving in and letting ε 0 yields the range > 19q() = /3. On the roof of Theorem 1.5. The roof is similar to the roof of Theorem 1.1. Instead of (1.10) we use a square function inequality for the shere (3.9) q f C ε δ α(q)/ ε ( f ) 1, q su f B (δ), with α(q) = d(1/ 1/q) 1/, and q = (d + )/d. In two dimensions this is an old observation by C. Fefferman ([8]), and holds for q = 4 with ε = 0. In general the roof of (3.9) is rather analogous to the roof of Proosition.3; one uses Tao s bilinear Fourier extension inequality [19] (see also [1] for related results). Unlie (.11) in the conic case the inequality (3.9) is essentially otimal for the given range q (d + )/d. We omit further details. 4. More on square functions We shall now discuss some imrovements of the square function estimate in Proosition.3 in low dimensions; thus we see for estimates of the form (4.1) f C ε δ β ε ( f ) 1/, su f Π (δ). for some β < µ() = d/4 (d+1)/. We shall assume throughout this chater the Wolff assumtion, Hyothesis Π (δ), (4.) W(w, d). For all δ (0, 1) and all families {h } of functions satisfying su ĥ w h C ε δ α(w) ε( 1/w, h w) w

11 PLATE DECOMPOSITIONS OF CONE MULTIPLIERS 11 where α(w) = d(1/ 1/w) 1/. Cf. Table 1. We note that in view of the embedding L (l ) L (l ) the inequality (4.) trivially imlies (4.1) with β = α(), for w <. Another trivial observation is that (4.1) holds with β (d 1)/4 in view of the Cauchy-Schwarz inequality, as f (x) δ (d 1)/4 ( f (x) ) 1/ for every x. The method for our imrovement over the exonent min{µ(), d 1 4 } will be limited to the case where (4.3) α() < min{µ(), d 1 4 } which holds if and only if < min{ (d 1) d, 4d (d+3) d 1 }. We have the additional restriction > d+1 in Proosition.3. Summarizing we shall get an imrovement which is limited to d =, 3, 4 and to the ranges (4.4) d =, 10/3 < < min{8, w}, d = 3, 3 < < 4, d = 4, 14/5 < < 3. We emhasize that square-function estimates such as (4.1) cannot a riori be interolated when subject to the Fourier suort condition (1.4). We shall however start with a reliminary result which is roved using an interolation. We let ϕ be a bum function adated to the late Π (δ) satisfying the natural estimates, so that ϕ equals 1 on the late, and is suorted on the double late. Define the oerator P by (4.5) P f = ϕ f. Each P is bounded on L (R d+1 ), 1, with uniform bounds. Lemma 4.1. Let d =, suose that hyothesis W(w, ) holds. Let (4.6) β = β (, w) = 3w 13 9w 40 6w 0 (6w 0) and let r = r(, w) be defined by 1 (4.7) r(w, ) = 1 w ( 3 10 ) 6w 0 Then, for 10/3 w, for all families {g } with g S(R d+1 ). ( P g C ε δ β ε g r) 1/r, Proof. By W(, w) and the embedding L (l ) l (L ) we have the inequality w P g C ε δ (α(w)+ε)( ) 1/w P g w w ( C ε δ (α(w)+ε) g w) 1/w w (4.8).

12 1 G. GARRIGÓS AND A. SEEGER We also observe that for 4 ( (4.9) P g ) 1/ ( C(1 + log δ 1 ) 1/ 1/ g ) 1/ Indeed the left hand side is estimated by ( su ω L (/) P g ωdx) 1/ su ω L (/) ( ) 1/ g M δ ωdx where M δ is a Besicovitch-tye maximal oerator associated to the light cone which is bounded on L with norm O( log( + δ 1 )) if δ < 1/, see [7], [14]. Thus Hölder s inequality imlies (4.9). Now we can combine Proosition.3 with resect to the double lates, and f = P g, and (4.9) to obtain (4.10) ( 10/3 P g C ε δ 1 0 ε g ) 1/ 10/3. After a little arithmetic the claimed bound follows by interolation between (4.8) and (4.10). Since r(, w) in Lemma 4.1 we immediately get Corollary 4.. Let d =, suose that hyothesis W(w, ) holds. Then for all family of functions {f } with su f Π (δ) the estimate (4.1) holds for 10/3 w with β = β (, w). In articular note that β (4, w) = 3w 1 4w 80 so that β (4, 6) = 3/3. If we use the exonent obtained in Theorem 1.1, i.e. w = = 190/3 we get only β (4, ) = 89/70 which is worse than 5/44 exonent that is already nown from [0], [3]. For large values of w one can imrove on the result of Corollary 4.. Our aroach will be similar to the one by Tao and Vargas [0] in + 1 dimensions. By using W(w, ) in that aroach one can slightly imrove on the reviously nown exonents. Theorem 4.3. Let d 4, and as in (4.4). If hyothesis W(w, d) holds then for all family of Schwartz functions {f } with su f Π (δ) the estimate (4.1) holds with ( d+1 (d+3) 1 (4.11) β = µ() d 1 to The roof will be given in 5. d+1 (d+3) + 1 ( 1) (w 1) )( 1 d ) (d 1) In + 1 dimensions Theorem 4.3 yields inequality (4.1) for the range 10/3 w with β equal (4.1) β (, w) = 1 (3 0)w ; (10 + 3)w in articular we have β (4, w) = 5w 0 44w 164 which (with w) occurs in Theorem 1.5. We comare with (4.6). Notice that 3/3 = β (4, 6) < β (4, 6) = 1/10. A straightforward comutation shows the inequality β (, w) < β (, w) holds if and only if (9 30)w +( )w+3(3 10) > 0 and after factoring we see that for 10/3 < < w we have β (, w) < β (, w) if and only if ( 10 3 )(w 3 3 )(w ) > 0. Thus for any (10/3, w) we have (4.13) β (, w) < β (, w) w > 3 3

13 PLATE DECOMPOSITIONS OF CONE MULTIPLIERS 13 so that the L result in Theorem 4.3 is better than the result of Corollary 4. in the range w > 3/3. We obtain the following corollary which yields Theorem 1.5. Corollary 4.4. Let d = and suose that W(w, ) holds for some w > 6. Let 10/3 < 4 and let α > min{β (, w), β (, w)} (i.e. α > β (, w)} if w > 3/3). Then (i) the smoothing inequality (1.6) holds true and (ii) the Fourier multilier m α in (1.7) defines a bounded oerator on L (R 3 ). We also observe that by interolation we obtain the analogous boundedness results for the range 4 w under the assumtion that α > 1 + 4(w ) (w 4) min{β (4, w)), β (4, w)}. If we use the result of Theorem 1.1 in + 1 dimensions (i.e. hyothesis W(w, ) with w = = 190/3) we obtain this result for α > β (, ) = ( ). which equals 445/3934 if = 4. This reresents a slight imrovement over the Tao-Vargas result [0] which yields the L 4 boundedness for α > = ; note that We also see from Corollary 4. that the validity of (1.3) for the otimal (conjectured) range 6 imlies the L 4 boundedness for α > 3/3 = ; however it has been conjectured that it should hold for all α > 0. Proof of Corollary 4.4. It remains to estimate the L norm of the square-function. But this is done exactly as in [13], [14] using the otimal L (/) (R 3 ) bounds of the aroriated Besicovitch maximal oerators (note that (/) > by the assumtion 4). 5. Proof of Theorem 4.3 We wor with the oerators P in (4.5) which localize in Fourier sace to the doubles of the lates Π (δ). We shall use the following consequence of W(w, d). Lemma 5.1. Suose that W(w, d) holds. Let w r and let = Then, for every ε > 0, (5.1) ( r P h C ε δ w α(w) ε 1/. h r) r r w+1 (i.e. r = (w 1)). Proof. This follows by interolation between the Wolff inequality (i.e. (4.8) in d dimensions) and the trivial bound P h h. To establish Theorem 4.3 we shall wor with the following Hyothesis H(γ,). For all δ < 1, ε > 0 (5.) ( h C ε δ γ ε h ) 1/, rovided that su ĥ Π (δ). By Proosition.3 we now already that for > (d + 3)/(d + 1) this inequality holds true with the exonent γ = µ() = d/4 (d + 1)/ and we see an imrovement in the ranges (4.4).

14 14 G. GARRIGÓS AND A. SEEGER We use Lemma 5.1 to rove the following roosition which amounts to an imroved version of Proosition 5.4 in [0] (who considered the case w = in the + 1 dimensional situation). As in we wor with a covering Q(δ 1/ ) of 1/δ cubes. Proosition 5.. Suose that < < w and suose that hyotheses H(γ, ) and W(w, d) hold. Let r = (w 1). Then (5.3) ( Q Q(δ 1/ ) rovided that su ĝ Π (δ). g L r (Q) ) 1/ Cε δ γ+α() ε ( g ) 1/. Proof. We grou the indices (and therefore the corresonding lates Π (δ) ) into O(δ (d 1)/4 ) disjoint families S l so that dist(ω, ω ) δ 1/4 for, S l. Define G l = S l g. As in the roof of Proosition. we also wor with the functions ψ Q adated to the cubes Q Q(δ 1/ ). By the suort roery of ψ Q the Fourier transform of ψ Q G l is suorted in a C δ late and these lates form an essentially disjoint late family. Therefore Lr G l ψq r G l (Q) (5.4) l l ( δ w α(w) ψ Q G l l r) 1/, by Lemma 5.1 with δ relaced by δ. By the suort roerty of ψ Q G l and Young s inequality (5.5) ψ Q G l r δ d+1 4 ( 1 1 r ) ψ Q G l and therefore ( Q l ) G l 1/ L r (Q) δ w α(w) + d+1 4 ( 1 1 )( r ψq G l Q,l ) 1/. A little algebra shows w α(w) + d (1 1 r ) = α() with r = (w 1). Using some straightforward estimation using the decay of the ψ Q we also get (5.6) ( Q l ) ( G l 1/ L r (Q) δ α()/ G l As Ĝl is suorted in a C δ late we may use rescaling arguments as in the roof of Lemma.1 to deduce from the hyothesis H(γ, ) alied with arameter δ that ( G l δ γ/ ) 1/ g S l l ) 1/

15 PLATE DECOMPOSITIONS OF CONE MULTIPLIERS 15 and hence ( Q l which is the assertion. ) ( G l 1/ L r (Q) Cε δ α()+γ ε ( ) 1/ g l S l C ε δ α()+γ ε ( g ) 1/ ) 1/ Proof of Theorem 4.3, cont. We first note that Hyothesis H(, µ()) holds by Proosition.3. Assuming that H(, γ) holds for some γ µ() the following estimate for bilinear exressions is an immediate consequence of Proosition 5.; here r = (w 1). ( (5.7) ( ) ( ) / ) / f δ α() γ ( f ) 1 Q Q ω Ω ω Ω g L r/ (Q) ω Ω ( ω Ω g ) 1 We now assume that Ω and Ω are searated as in Proosition. and interolate the inequalities (5.7) and (.9) with q = (d + 3)/(d + 1). As a result we obtain ( ( ) ( f Q Q ω Ω ω Ω g ) / L / (Q) ) / δ Γ(,γ) ( ω Ω f ) 1 ( g ) 1. ω Ω where Γ(, γ) = (1 ϑ)µ() + ϑ α() + γ ( 1 with ϑ = q 1 )/( 1 q 1 ). r By Lemma.1 we also obtain (5.8) f δ Γ(,γ) ( f ) 1/. ω Ω Notice that α() < Γ(, γ) µ() rovided that α() < γ µ(). Moreover γ = Γ(, γ) if and only if γ equals γ = 1 ( α() ) ϑ ( ) (1 ϑ)µ() + ϑ = µ() µ() α(). 1 ϑ/ ϑ The fixed oint is contained in the interval (α(), µ()) and one observes that Γ(, γ) < γ for γ < γ < µ(). Thus, if we define a sequence γ n by setting γ 0 = µ() and γ n+1 = Γ(, γ n ) for n 0, then γ n is decreasing and bounded below and converges to γ. We comute that ϑ/( ϑ) = (1/q 1/) / (1/q + 1/ /r) and α() µ() = (d )/4 (d 1)/ and see that γ is equal to the right hand side of (4.11). Thus (5.8) and an iteration yields the assertion of the theorem. References [1] D. Béollé, A. Bonami, M. Peloso and F. Ricci. Boundedness of weighted Bergman rojections on tube domains over light cones. Math. Z. 37 (001), [] D. Béollé, A. Bonami, G. Garrigós and F. Ricci. Littlewood-Paley decomositions related to symmetric cones and Bergman rojections in tube domains. Proc. London Math. Soc. (3) 89 (004), [3] D. Béollé, A. Bonami, G. Garrigós, C. Nana, M. Peloso and F. Ricci. Lecture notes on Bergman rojectors in tube domains over cones: An analytic and geometric viewoint. IMHOTEP J. Afr. Math. Pures Al. 5 (004)..

16 16 G. GARRIGÓS AND A. SEEGER [4] J. Bourgain. Besicovitch tye maximal oerators and alications to Fourier analysis, Geom. Funct. Anal. 1 (1991), [5] J. Bourgain. Estimates for cone multiliers, Geometric Asects of Functional Analysis, Oerator theory, Advances and Alications, vol. 77, ed. by J. Lindenstrauss and V. Milman, Birhäuser Verlag, [6] A. Carbery. Variants of the Calderón-Zygmund theory for L -saces. Rev. Mat. Iberoamericana (1986), [7] A. Córdoba. Geometric Fourier analysis. Ann. Inst. Fourier 3 (198), [8] C. Fefferman. A note on sherical summation multiliers. Israel J. Math. 15 (1973), [9] G. Garrigós, W. Schlag and A. Seeger, in rearation. [10] I. Laba and M. Pramani. Wolff s inequality for hyersurfaces. Proceedings of the 7th International Conference on Harmonic Analysis and Partial Differential Equations, El Escorial, 004. Collect. Math. 006, Vol. Extra, [11] I. Laba and T. Wolff. A local smoothing estimate in higher dimensions. J. Anal. Math. 88 (00), [1] S. Lee. Imroved bounds for Bochner-Riesz and maximal Bochner-Riesz oerators. Due Math. J. 1 (004), [13] G. Mocenhaut. A note on the cone multilier. Proc. Amer. Math. Soc. 117 (1993), [14] G. Mocenhaut, A. Seeger and C.D. Sogge. Wave front sets, local smoothing and Bourgain s circular maximal theorem. Annals of Math. 136, (199), [15] D. Müller and A. Seeger. Regularity roerties of wave roagation on conic manifolds and alications to sectral multiliers. Adv. Math. 161 (001), [16] M. Pramani and A. Seeger. L regularity of averages over curves and bounds for associated maximal oerators. Amer. J. Math., 19 (007), [17] A. Seeger. Some inequalities for singular convolution oerators in L -saces. Trans. Amer. Math. Soc. 308 (1988), [18] T. Tao. Endoint bilinear restriction theorems for the cone, and some shar null form estimates. Math. Z. 38 (001), [19] T. Tao. A shar bilinear restrictions estimate for araboloids. Geom. Funct. Anal. 13 (003), [0] T. Tao and A. Vargas. A bilinear aroach to cone multiliers II. Alications. Geom. Funct. Anal. 10 (000), [1] T. Tao, A. Vargas and L. Vega. A bilinear aroach to the restriction and Kaeya conjectures. J. Amer. Math. Soc. 11 (1998), [] T. Wolff. Local smoothing tye estimates on L for large. Geom. Funct. Anal. 10 (000), [3] T. Wolff. A shar bilinear cone restriction estimate. Ann. of Math. () 153 (001), G. Garrigós, De. Matemáticas C-XV, Universidad Autónoma de Madrid, 8049 Madrid, Sain address: gustavo.garrigos@uam.es A. Seeger, Deartment of Mathematics, University of Wisconsin-Madison, Madison, WI 53706, USA address: seeger@math.wisc.edu