IZABELLA LABA AND HONG WANG

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1 DECOUPLING AND NEAR-OPTIMAL RESTRICTION ESTIMATES FOR CANTOR SETS IZABELLA LABA AND HONG WANG Abstract. For any α (0, d), we construct Cantor sets n R d of Hausdorff dmenson α such that the assocated natural measure µ obeys the restrcton estmate fdµ p C p f L 2 (µ) for all p > 2d/α. Ths range s optmal except for the endpont. Ths extends the earler wor of Chen-Seeger and Shmern-Suomala, where a smlar result was obtaned by dfferent methods for α = d/ wth N. Our proof s based on the decouplng technques of Bourgan-Demeter and a theorem of Bourgan on the exstence of Λ(p) sets. 1. ntroducton We defne the Fourer transform f(ξ) = e 2πx ξ f(x)dx ξ R d. If µ s a measure on R d, we wll also wrte fdµ(ξ) = e 2πx ξ f(x)dµ(x) ξ R d. (1) We are nterested n estmates of the form ĝdµ p C g L q (µ) g L q (µ), where the constant may depend on the measure µ and on the exponents p, q, but not on f. If µ s a probablty measure, we trvally have ĝdµ g L 1 (dµ) g L q (dµ), so that (1) holds wth p = and all q [1, ]. In general, t s not possble to say more than that. However, the problem becomes more nterestng f we restrct attenton to specfc well-behaved classes of measures. There s a vast lterature on restrcton estmates for smooth manfolds (see e.g. [19], [22], [23] for an overvew and a selecton of references). It s well nown that (1) cannot hold wth p < (and any q) when µ s supported on a flat manfold such as a hyperplane. On the other hand, such estmates are possble f µ s the surface measure on a curved manfold M, wth the range of exponents p, q dependng on the geometry of M, n partcular on ts dmenson, smoothness and curvature. Date: July 27,

2 2 IZABELLA LABA AND HONG WANG In the model case when µ s the Lebesgue measure on the sphere S d 1 R d, the classc Tomas-Sten theorem states that (1) holds wth q = 2 and p 2d+2. It was d 1 furthermore conjectured by Sten that for q =, the range of p could be mproved to p > 2d ; ths has been proved for d = 2, but remans open n hgher dmenson, d 1 wth the current best results due to Guth [9], [10]. The conjectured range p > 2d for the sphere, f true, would be the best possble. d 1 Ths follows by lettng f 1 and usng the well nown statonary phase asymptotcs for dµ. The range of p n the Tomas-Sten theorem s also nown to be optmal. Here, the sharpness example s provded by the Knapp constructon where f s the characterstc functon of a small sphercal cap of dameter δ 0. We are nterested n the case when µ s a fractal measure on R d, sngular wth respect to Lebesgue. Here, agan, addtonal assumptons are necessary to mae nontrval estmates of the form (1) possble. For example, f µ s the natural selfsmlar measure on the Cantor ternary set, an easy calculaton shows that (1) cannot hold for any p <. However, f we assume that µ obeys an addtonal Fourer decay condton, then the followng result s nown. Here and below, we use B(x, r) to denote the closed ball of radus r centered at x. Theorem 1. Let µ be a Borel probablty measure on R d. Assume that there are α, β (0, d) and C 1, C 2 0 such that (2) µ(b(x, r)) C 1 r α x R d, r > 0, (3) µ(ξ) C 2 (1 + ξ ) β/2 ξ R d Then for all p (4d 4α + 2β)/β, the estmate (1) holds wth q = 2. Theorem 1 s due to Mocenhaupt [16] and Mtss [15] n the non-endpont range; the endpont was settled later by Ba and Seeger [1]. In the case α = β = d 1, ths recovers the Tomas-Sten theorem for the sphere. The range of exponents p n Theorem 1 s nown to be the best possble n dmenson 1, n the sense that for any 0 < α β < 1, there exsts a probablty measure µ on R, supported on a set of Hausdorff dmenson α and obeyng (2) and ((3), such that (1) fals for all p < (4 4α + 2β)/β, see [12], [3]. The examples are based on a constructon due to Hambroo and Laba [12]: the dea s to modfy a random Cantor-type constructon so as to embed a lower-dmensonal Cantor subset that has much more arthmetc structure than the rest of the set. Ths can be vewed as an analogue of the Knapp example for fractal sets. A hgher-dmensonal varant of the constructon wth d 1 < α β < d s gven n [13]. On the other hand, there exst specfc measures on R d for whch the range of exponents n (1) can be mproved further. If µ s supported on a set of Hausdorff dmenson α < d, t s easy to see usng energy ntegrals that (1) cannot hold outsde

3 DECOUPLING AND NEAR-OPTIMAL RESTRICTION ESTIMATES FOR CANTOR SETS 3 of the range p 2d/α, even f q =. (See e.g. [12, Secton 1]; the counterexample s provded by the functon f 1.) It turns out that there are measures for whch ths range s n fact realzed, wth examples provded by Chen [2], Shmern and Suomala [17], and Chen and Seeger [4]. In partcular, Chen and Seeger [4] proved that for d 1 and α = d/, where N, there are measures supported on a set of Hausdorff dmenson α, obeyng (2) and (3) wth β = α, for whch (1) holds for all p 2d/α. The proofs are based on regularty of convolutons: assumng that α = d/, the ey ntermedate step s to prove that the -fold self-convoluton µ µ s absolutely contnuous. Ths method, however, does not yeld optmal exponents when α d/ wth nteger. Our man result s as follows. Theorem 2. Let d N and 0 < α < d. Then there exsts a probablty measure supported on a subset of [0, 1] d of Hausdorff dmenson α such that: for every 0 < γ < α, there s a constant C 1 (γ) such that (4) µ(b(x, r)) C 1 (γ)r γ x R d, r > 0 for every β < mn(α/2, 1), there s a constant C 2 (β) > 0 such that (5) µ(ξ) C 2 (β)(1 + ξ ) β ξ R d, (6) for every p > 2d/α, we have the estmate ĝdµ p C 3 (p) g L 2 (µ) g L 2 (µ). Ths complements the results of [2], [4], [17], and provdes a matchng (except for the endpont) result for all dmensons 0 < α < d that are not of the form α = d/. The frst man ngredent of our constructon s Bourgan s theorem on Λ(p) sets [5] (see also Talagrand [20]). In ts full generalty, Bourgan s theorem apples to general bounded orthogonal systems of functons. We state t here n the specfc case of exponental functons on the unt cube n R d. Ths provdes an optmal restrcton estmate on each sngle scale n the Cantor constructon. Theorem 3. (Bourgan [5]) Let p > 2. For every N N suffcently large, there s a set S = S N {0, 1,..., N 1} d of sze t c 0 N 2d/p such that (7) a S c a e 2πa x L p [0,1] d C(p)( a S c a 2 ) 1/2 wth the constants c 0 and C(p) ndependent of N. (The set S s called a Λ(p)-set.) To pass from here to restrcton estmates for multscale Cantor sets, we use the decouplng technques of Bourgan and Demeter [6], [7]. Ths produces localzed restrcton estmates of the form (8) f L 2 (µ) C ɛ R ɛ f L p ([ R,R] d ),

4 4 IZABELLA LABA AND HONG WANG or equvalently by dualty, (9) ĝdµ L p ([ R,R] d ) C ɛ R ɛ g L 2 (µ) for all ɛ > 0, wth constants ndependent of R. The R ɛ factors account for the fact that we lose a constant factor at each step of the teraton. We wll try to mnmze these losses by applyng Bourgan s theorem to an ncreasng sequence of values of N, but we wll not be able to avod them completely. Fnally, we use a varant of Tao s epslon removal lemma [21] to deduce the global restrcton estmate (6) from (8). Ths removes the R ɛ factors, but at the cost of losng the endpont exponent p = 2d/α. It s not clear whether the endpont estmate can be obtaned wth our current methods. Our proof of the localzed restrcton estmate (8) s fully determnstc. However, the epslon removal lemma requres a pontwse Fourer decay estmate for µ. Randomzng our constructon enables us to prove the estmate (5) va an argument borrowed from [14], [17]. Ths proves the Fourer decay part of Theorem 2, and s also suffcent to complete the epslon removal argument. If d 2, or f d 2 and α d 2, the Cantor set supportng µ n Theorem 2 s a Salem set (.e. ts Fourer dmenson s equal to ts Hausdorff dmenson). The condton α d 2 s necessary for ths type of constructons to produce a Salem set, for the same reasons as n [17]. We note, however, that our proof of (6) wth p > 2d/α does not requre optmal Fourer decay and that the estmate (5) for any β > 0 would suffce. 2. The decouplng machnery We wll use the decouplng machnery developed by Bourgan and Demeter [6], [7]. In ths paper, we wll follow the conventons of [7], wth the surface measure on a parabolod replaced by the natural measure on a Cantor set. We use X Y to say that X CY for some constant C > 0, and X Y to say that X Y and X Y. The constants such as C, C, etc. and the mplct constants n may change from lne to lne, and may depend on d and p, but are ndependent of varables or parameters such as x, N, R, j, l. For quanttes that depend on parameters such as ɛ, we wll wrte X(ɛ) ɛ Y (ɛ) as shortcut for for every ɛ > 0 there s a constant C ɛ > 0 such that X(ɛ) C ɛ Y (ɛ). We wrte [N] = {0, 1,..., N 1} and B(x, r) = {y R d : x y r}. We use to use the Eucldean (l 2 ) norm of a vector n R d, the cardnalty of a fnte set, or the d-dmensonal Lebesgue measure of a subset of R d, dependng on the context. Occasonally, we wll also use the l norm on R d : f x = (x 1,..., x d ) R d, we wrte x = max( x 1,..., x d ). We wll also sometmes use F for the Fourer transform, so that Ff = f.

5 DECOUPLING AND NEAR-OPTIMAL RESTRICTION ESTIMATES FOR CANTOR SETS 5 Followng [7], we wll use cube-adjusted weghts. An R-cube wll be a d-dmensonal cube of sde length R, wth all sdes parallel to coordnate hyperplanes. Unless stated otherwse, we wll assume R-cubes to be closed. If I s an R-cube centered at c, we defne w I (x) = ( 1 + ) 100 x c R and ( ) 1/p 1 F L p (w I) = F p w I. I If η : R [0, ) s a functon (usually Schwartz), and I s as above, we wll wrte ( ) x c η I (x) = η. R If g : R C s a functon, I s an nterval, and σ s a measure (whch wll usually be clear from context), we wll wrte E I g = F 1 (1 I gdσ). We wll use the followng tools from Bourgan-Demeter, whch we restate here n a verson adapted to our settng. Lemma 1. (Reverse Hölder nequalty, [7, Corollary 4.2]). Let 1 p q. If I s a 1/R-cube and J s an R-cube, then (10) E I g L q (w J ) E I g L p (w J ) wth the mplct constant ndependent of R, I, J, g. Lemma 2. (L 2 decouplng, [7, Proposton 6.1]). Let I be a /R-cube for some N, and let I = I 1 I be a tlng of I by 1/R-cubes dsjont except for ther boundares. Then for any R-cube J we have (11) E I g 2 L 2 (w J ) j E Ij g 2 L 2 (w J ). Lemma 3. (Band-lmted functons are locally constant, cf. [8, 2.2]) There s a non-negatve functon η L 1 (R d ) such that the followng holds. For every R > 0, and every ntegrable functon h : R d C supported on a 1/R-cube I, there s a functon H : R [0, ) such that: H s constant on each sem-closed R-cube J ν := Rν + [0, R) d, ν Z d, ĥ(x) H(x) ( ĥ η R)(x) for all x R, where η R (y) = 1 η( y ). In R d R partcular, (12) H L 1 (R d ) η L 1 (R d ) ĥ L 1 (R d ).

6 6 IZABELLA LABA AND HONG WANG Proof. Replacng h by h( c) and ĥ(x) by e 2πc x ĥ(x) f necessary, we may assume that I = [0, 1 R ]d. Let χ be a non-negatve Schwartz functon such that χ 1 on [0, 1] d and that χ(x) s non-negatve, radally symmetrc and decreasng n x. Then χ(r ) 1 on I, and χ(r ) = 1 χ( ). Defne R d R and η(x) := sup y x 1 χ(y) H(x) := sup{ ĥ(y) : x, y belong to the same J ν}. Clearly, η s ntegrable and H s constant on each J ν. We have ĥ(x) H(x) by defnton. To prove the second nequalty, we note that h = hχ(r ), so that ĥ = ĥ χ(r ). Suppose that x J ν for some ν Z d, then for each y J ν we have ĥ(y) ĥ(z) 1 R χ(y z d R )dz Snce x y R, we have y z x z R R = y x R 1, so that by the defnton of η we have η( x z y z ) χ( ). Hence R R ĥ(y) ĥ(z) 1 R η(x z d R )dz = ( ĥ η R)(x), and the desred nequalty follows upon tang the supremum over y J ν. Fnally, by Fubn s theorem and rescalng we have H L 1 (R d ) ĥ L 1 (R d ) η R L 1 (R d ) = ĥ L 1 (R d ) η L 1 (R d ). Corollary 1. For every R > 0, M N, every ntegrable functon h : R d supported on an (MR) 1 -cube I, and every R-cube J, we have C (13) ĥ L 1 (w J ) 1 M d ĥ L 1 (R d ). Proof. Let L ν = MRν + [0, MR) d for ν Z d. Let H be the functon provded by Lemma 3 wth R replaced by MR, so that on each L ν we have H(x) H ν for some

7 DECOUPLING AND NEAR-OPTIMAL RESTRICTION ESTIMATES FOR CANTOR SETS 7 constant H ν 0. Then ĥ L 1 (w J ) H(x)w J (x)dx = H ν w J (x)dx ν L ν H ν w J (x)dx ν R = H ν R d w [0,1] d(x)dx. ν R Let C 1 := R d w [0,1] d(x)dx, then ĥ L 1 (w J ) C 1 H ν R d = C 1 H M d ν (MR) d ν ν = C 1 H(x)dx M d 1 M d ĥ L 1 (R d ), where at the last step we used (12). R 3. Sngle-scale decouplng We begn wth a sngle-scale decouplng nequalty for Cantor sets wth Λ(p) alphabets. We wll need the followng contnuous verson of Theorem 3. Lemma 4. Let p > 2, and let S [N] d be as n Theorem 3. Then for all h supported on E := S + [0, 1] d we have the nequalty (14) ĥ L p ([0,1] d ) C(p) h L 2 (R d ) Proof. We have a S By (7), t follows that c a e 2πa x L = sup f, c a e 2πa x p ([0,1] d ) f L p ([0,1] d =1 ) a S = sup f, c a δ a f L p ([0,1] d =1 ) a S f L p ([0,1] d =1 ) = sup sup sup c a l 2 (S) =1 f L p ([0,1] d =1 ) a S a S c a f(a) c a f(a) C(p),

8 8 IZABELLA LABA AND HONG WANG so that f(a) l 2 (S) C(p) f L p ([0,1] d ) Smlarly, for any translate S + z of S we have Integratng n z [0, 1] d, we get f(a) l 2 (S+z) C(p) f L p ([0,1] d ) (15) f 2 L 2 (E) = [0,1] d f(a) l 2 (S+z)dz C(p) 2 f 2 L p ([0,1] d ). Argung agan by dualty, we have f L 2 (E) = sup g L 2 (E) =1 = sup g L 2 (E) =1 g1 E fdx F(g1 E ) fdx Usng (15), and tang the supremum over f wth f L p ([0,1] d ) 1, we get whch s (14) wth h = g1 E. F(g1 E ) L p ([0,1] d ) C(p) g L 2 (E), We note that the concluson of Lemma 4 remans true f we assume that h s supported on S + [ 1/2, 3/2] d nstead of E. Ths s proved by wrtng h as a sum of 2 d functons supported on translates of E and applyng Lemma 4 to each of them. We can now prove our frst decouplng nequalty. Lemma 5. Let S [N] d be a Λ(p)-set as n Theorem 3, and let E = S + [0, 1] d. Let f : R C be a functon such that g := f = s supported on E. For each a S, let g a = g1 a+[0,1] d, and defne f a va f a = g a. Then (16) f 2 L p (w I ) C(p) 2 a S for any 1-cube I. f a 2 L p (w I ) Proof. We frst rewrte the rght-hand sde of (16) usng Lemmas 2 and 1 wth σ equal to the Lebesgue measure. We have so that E a+[0,1] dg = ĝ a = f a, E [0,N] dg = ĝ = f, f 2 L 2 (w I ) a S f a 2 L 2 (w I ) a S f a 2 L p (w I ).

9 DECOUPLING AND NEAR-OPTIMAL RESTRICTION ESTIMATES FOR CANTOR SETS 9 Therefore to prove (16), t suffces to prove that (17) f 2 L p (w I ) C(p) 2 f 2 L 2 (w I ) Let η be a nonnegatve Schwartz functon such that η(x) = η( x), η 1 on [ 1, 1] d and supp η [ 1 2, 1 2 ]d. We wll prove that (18) f 2 L p (I) C(p) 2 f 2 L 2 (η I ) for every 1-cube I. By a coverng argument [7, Lemma 4.1], ths mples (17). We may assume that I = [0, 1] d. (If I = z +[0, 1] d ], we may replace f by f z = f( + z) and observe that f z (ξ) = e 2πz ξ f(ξ) s agan supported n E.) Let h = g ( η)ˇ, so that ĥ = f η and h s supported on S + [ 1/2, 3/2] d. Snce η 1 on [0, 1] d, we have f L p ([0,1] d ) f η L p ([0,1] d ). By Lemma 4 and the remar after ts proof appled to h, as clamed. f η L p ([0,1] d ) C(p) h L 2 (R) = f η L 2 (R) = f L 2 (η) 4. The Cantor set constructon Our proof of Theorem 2 s based on the constructon of a multscale Λ(p) Cantor set of dmenson α. Let α (0, d), p = 2d/α, and let {n j } j N be a sequence of postve ntegers. For the constructon of the measure µ n Theorem 2, we wll assume the followng condtons on n j : (19) n 1 n 2..., n, (20) ɛ > 0 C ɛ > 0 N n +1 C ɛ (n 1... n ) ɛ. However, large parts of our proof wor under weaer assumptons. In partcular, our localzed restrcton estmate n Lemma 8 contnues to hold f n j = n for all j. We also note here that n order for (20) to hold, t s enough to assume that (19) holds and that n j grow slowly enough, for example (21) wll suffce. For each j N, let n j+1 j + 1 n j j Σ j = Σ j (n j, t j, c 0, C(p)) = {S [n j ] d : S = t j and (7) holds wth N = n j } By Theorem 3, there are c 0, C(p) > 0 (ndependent of j) and t j wth t j c 0 n 2d/p j such that Σ j s non-empty for all j. Henceforth, we fx these values of c 0, C(p) and

10 10 IZABELLA LABA AND HONG WANG t j. By the well nown upper bounds on the sze of Λ(p) sets (see [5]), we must n fact have (22) c 0 n 2d/p j t j c 1 n 2d/p j for some constant c 1 ndependent of j. Let N = n 1... n and T = t 1... t. We construct a Cantor set E of Hausdorff dmenson α as follows. Defne A 1 = N 1 1 S 1, E 1 = A 1 + [0, N 1 1 ] d, for some S 1 Σ 1. For every a A 1, choose a Λ(p) set S 2,a Σ 2 wth S 2,a = t 2, and let A 2,a = a + N2 1 S 2,a, A 2 = A 2,a, E 2 = A 2 + [0, N2 1 ] d. a A 1 We contnue by nducton. Let 2, and suppose that we have defned the sets A j and E j, j = 1, 2,...,. For every a A, choose S +1,a Σ +1 wth S +1,a = t +1, and let A +1,a = a + N 1 +1 S +1,a, A +1 = a A A +1,a, E +1 = A +1 + [0, N 1 +1 ]d. Ths produces a sequence of sets [0, 1] d E 1 E 2 E 3..., where each E j conssts of T j cubes of sde length N 1 j. For each j, let µ j = 1 E j 1 E j. We wll dentfy the functons µ j wth the absolutely contnuous measures µ j dx. It s easy to see that µ j converge wealy as j to a probablty measure µ supported on the Cantor set E := j=1 E j. We note that for each N 1 j -cube τ of E j, and for all l > j, we have µ j (τ) = µ l (τ) = µ(τ) = T 1 j. For the tme beng, the specfc choce of the sets A,a does not matter, as long as they are Λ(p)-sets of the prescrbed cardnalty. Our multscale decouplng nequalty n Proposton 1 and the localzed restrcton estmate n Corollary 2 do not requre any addtonal condtons. However, addtonal randomzaton of these choces wll become mportant later n provng our global restrcton estmate. Lemma 6. Assume that (19) and (20) hold. Then the set E has Hausdorff dmenson α. Moreover, for every 0 γ < α there s a constant C 1 (γ) such that (23) µ(b(x, r)) C 1 (γ)r γ x R d, r > 0. Proof. We frst note that (19), (20) and (22) mply that (24) N α 2ɛ j+1 ɛ N α ɛ j ɛ T j ɛ N α+ɛ j ɛ N α+2ɛ j 1.

11 DECOUPLING AND NEAR-OPTIMAL RESTRICTION ESTIMATES FOR CANTOR SETS 11 Indeed, from (22) we have c j 0Nj α T j c j 1Nj α, whch mples N α ɛ j ɛ T j ɛ N α+ɛ j by (19). The remanng two nequaltes n (24) follow from (20). We frst prove (23). If r > N1 1, then (23) holds trvally wth C 1 = N γ 1. Suppose now that N 1 j+1 < r N 1 j for some j 1. Then any ball B(x, r) ntersects at most a bounded number of the N 1 j -cubes of E j, so that µ(b(x, R)) T 1 j γ N γ j+1 rγ for all γ < α (the second nequalty n the sequence follows from (24)). To prove the dmenson statement, we only need to show that E has Hausdorff dmenson at most α, snce the lower bound s provded by (23). To ths end, t suffces to chec that for every ɛ > 0, and for all r > 0, the set E can be covered by C ɛ r α ɛ balls of radus r. Agan, t suffces to consder r > N1 1. Suppose that N 1 j+1 < r N 1 j, then E E j+1 can be covered by T j+1 balls of radus N 1 j+1, hence also of radus r. Snce T j+1 ɛ N α+ɛ j r α ɛ, the desred bound follows. 5. Multscale decouplng Our goal n ths secton s to derve the followng multscale decouplng nequalty for fnte teratons of Cantor sets. For a A, let τ,a = a + [0, N 1 ]d. If f : R d C s a functon, we defne f,a va f,a = 1 τ,a f. Proposton 1. There s a constant C 0 (p) (ndependent of ) such that for any N -cube J, and for any functon f wth supp f E, we have ( ) 1/p ( ) 1/2 (25) f p L p (w I ) C 0 (p) f,a 2 L p (w J ), I I a A where J = I I I s a tlng of J by 1-cubes. Proof. The dea s to terate Lemma 5. Applyng t to the set N 1 E 1 and a rescalng of f by N 1, we see that there s a constant C 1 (p) such that for any functon f wth supp f E 1, and for any N 1 -cube J, we have (26) f 2 L p (w J ) C 1 (p) 2 a A 1 f 1,a 2 L p (w J ). Smlarly, applyng Lemma 5 to a rescalng of f j,a by N j+1 for each a A j, we see that for any N j+1 -cube J and for any functon f wth supp f E j+1 we have (27) f j,a 2 L p (w J ) C 1 (p) 2 b A j+1,a f j+1,b 2 L p (w J ) wth the same constant C 1 (p). To connect the steps of the teraton, we wll need a smple lemma on mxed norms.

12 12 IZABELLA LABA AND HONG WANG Lemma 7. Let {c j } be a double-ndexed sequence (fnte or nfnte) wth c j 0. Then for p > 2, ( ) p/2 ( ( 2/p ) p/2. (28) cj) p c 2 j j j Proof. Let F j () = c 2 j, and G() = j F j() = j c2 j, so that ( ( p/2 ) 2/p. G p/2 = cj) 2 j On the other hand, by Mnows s nequalty G p/2 j F j () p/2 = j ( c p j) 2/p, and the lemma follows. We wll prove (25) by nducton n. For an m-cube J wth m N, let J = I I(J) I be a tlng of J by 1-cubes. Let C 2 be a constant such that (29) w I C 2 w J. I I(J) It s easy to see that such a constant exsts and can be chosen ndependently of J. We wll prove that (25) holds wth C 0 (p) = C 1 (p)c 1/p 2. To start the nducton, let J be an N 1 -cube, then by (29) and (26), f p L p (w I ) C 2 f p L p (w J ) I I(J) C 1 (p) p C 2 ( a A 1 f 1,a 2 L p (w J ) Ths s (25) for = 1. Suppose now that we have proved (25) for = j. Let J be an N j+1 -cube, and let J = L L I be a tlng of J by N j-cubes. Then f p L p (w I ) I I(J) f p L p (w I ) = L L I I(L) C 0 (p) jp L L ( a A j f j,a 2 L p (w L ) ) p/2 ) p/2

13 DECOUPLING AND NEAR-OPTIMAL RESTRICTION ESTIMATES FOR CANTOR SETS 13 by our nductve assumpton. Usng Lemma 7, a rescalng of (29), and (27), we see that f p L p (w I ) C 0(p) jp ( ) 2/p p/2 f j,a p L p (w L ) I I(J) a Aj L L C 0 (p) jp C 2 f j,a 2 L p (w J ) a Aj p/2 C 0 (p) jp C 2 C 1 (p) f j+1,a 2 L p (w J ) a A j+1 Ths ends the nductve step and the proof of the proposton. p/2. 6. From decouplng to localzed restrcton Lemma 8. Let E and µ be as n Secton 5. Let J be an N -cube. Then for all g L 2 (dµ), we have ĝdµ L p (J) C 0 (p) N d/p wth the mplct constants ndependent of. T 1/2 g L 2 (dµ) Proof. It suffces to prove that for all l >, and for all g L 2 (R) supported on E l, we have gdµ l L p (J) C 0 (p) N d/p T 1/2 g L 2 (dµ l ) wth the mplct constants ndependent of and l. The clam then follows by tang the lmt l. We contnue to use the Cantor set notaton from Sectons 4 and 5. For a A j, let g j,a = 1 a+[0,+n 1 j ] g. By Proposton 1 and Lemma 1, we have d gdµ ( ) 1/2 l L p (J) C 0 (p) g,a dµ l 2 L p (w J ) a A ( C 0 (p) N d p d ) 1/2 2 g,a dµ l 2 L 2 (w J ) a A For each a A, let B l,a be the set of l-th level descendants of a (more precsely, B l,a = {b A l : b + [0, N 1 l ] d a + [0, N 1 ]d }. Note that B l,a = T l /T. By

14 14 IZABELLA LABA AND HONG WANG Cauchy-Schwartz, so that g,a dµ l L 2 (w J ) ĝ l,b dµ l L 2 (w J ) b B l,a gdµ l L p (J) C 0 (p) N d p d 2 C 0 (p) N d p d 2 ( Tl T ( Tl T ( Tl T ) 1/2 ( ) 1/2 ( ĝ l,b dµ l 2 L 2 (w J ) b A l ) 1/2 ( ) d/2 N ( N l b B l,a ĝ l,b dµ l 2 L 2 (w J ) ) 1/2 b A l ĝ l,b dµ l 2 L 2 (R) ) 1/2. At the last step, we appled Corollary 1 to the functons (g l,b dµ l )( ) (g l,b dµ l )( ) supported on 2N 1 l -cubes. Snce ĝ l,b dµ l 2 L 2 (R) = g l,b dµ l 2 L 2 (R) b A l b A l we fnally have = gdµ l 2 L 2 (R) = N d l T 1 l g 2 L 2 (µ l ) ) 1/2 gdµ l L p (J) C 0 (p) N d p 1 2 ( Tl T ) 1/2 ( N N l = C 0 (p) N d/p T 1/2 g L 2 (dµ l ) ) d/2 ( ) N d 1/2 l g L2(µl) T l as clamed. Corollary 2. (Localzed restrcton estmate) Assume that (19) holds. Then for any ɛ > 0 we have the estmate (30) ĝdµ L p (J) C ɛ R ɛ g L 2 (dµ). for all R n 1 and for all R-cubes J. The constant C ɛ depends on ɛ, but not on g, R or J. Equvalently, for any f supported n J, we have (31) f L 2 (dµ) C ɛ R ɛ f L p (J).

15 DECOUPLING AND NEAR-OPTIMAL RESTRICTION ESTIMATES FOR CANTOR SETS 15 Proof. Suppose that N < R N +1, and let J be an N +1 -cube contanng J. By Lemma 8 and (19), we have ĝdµ L p (J) ĝdµ L p (J ) C 0 (p) +1 N d/p +1 T 1/2 +1 g L 2 (dµ) C 0 (p) +1 N d/p +1 (c+1 0 N 2d/p +1 ) 1/2 g L 2 (dµ) (C 0 (p)c 1/2 0 ) +1 g L 2 (dµ) ɛ R ɛ g L 2 (dµ) as clamed. The second part (31) follows by dualty. 7. Global restrcton estmate Proposton 2. Assume that n obey (19) and (20). Suppose furthermore that µ obeys a pontwse decay estmate (32) µ(x) (1 + x ) β for some β > 0. Then for any q > p we have the estmate (33) ĝdµ L q (R) g L 2 (dµ). The mplct constant depends on the measure µ and on q, but not on g. Equvalently, (34) f L 2 (dµ) f L q (R). To prove ths, we adapt Tao s epslon-removal argument, see [21, Theorem 1.2]. It suffces to prove Lemma 9 below; once ths s done, the proof of the proposton s completed exactly as n [21], wth Lemma 9 replacng Tao s Lemma 3.2. Lemma 9. Assume that n, t, µ are as n Theorem 2, and let R > 0 be large enough. Suppose that {I 1,..., I M } s a sparse collecton of R-cubes, n the sense that ther centers x 1,..., x M are R B M B -separated for some large enough constant B (dependng on β). Then for any f supported on M j=1 I j, we have f L 2 (dµ) ɛ R Cɛ f L p (R). Here and below, the constant C n the exponent may depend on B, and may change from lne to lne, but s ndependent of R, M, ɛ, or f. Proof. We follow the outlne of Tao s argument, wth modfcatons necessary to adapt t to our settng. We frst note the followng estmate: f f s supported n an R-cube J wth R N l, then (35) f L 2 (dµ l ) ɛ R ɛ f L p (J).

16 16 IZABELLA LABA AND HONG WANG The mplct constant depends on ɛ, but not on f, R or J. Ths s proved as n Lemma 8 and Corollary 2, except that we do not tae the lmt l n the proof of Lemma 8. Let N be such that N R < N +1. We have E = T N d ; by (24), ths mples that (36) R 2d p d ɛ ɛ E ɛ R 2d p d+ɛ. Let f = f φ, where supp f I and φ = φ I for a fxed Schwartz functon φ such that φ 0, φ 1 on [ 1, 1] d, and supp φ [ 1, 1] d. Note that φ (x) = R d φ(rx), and n partcular φ s supported n [ R 1, R 1 ] d. Then f = φ f = φ f. By the support propertes of φ, for x E we actually have f(x) = φ ( f 1 E )(x), where we abuse the notaton slghtly and use E to denote both the -th stage set from the Cantor teraton and a CN 1 -neghbourhood of E. Ths s harmless snce ether set can be covered by a bounded number of translates of the other. We clam that the followng holds: for all r [1, 2], and for any collecton of functons F 1,..., F M L 2 (R d ), we have (37) F φ r ɛ R Cɛ E r/2 F r L 2 L (µ) 2 (R d ). Assumng the clam (37), we complete the proof of the lemma as follows. F = f 1 E, and observe that E r/2 F r L 2 (R d ) = f r L 2 (µ ). Applyng (37) to F wth r = p, and then usng (35), we get p f L 2 (dµ) = F φ p L 2 (µ) ɛ R Cɛ E p /2 ɛ R Cɛ f p L 2 (µ ) F p L 2 (R d ) Let ɛ R Cɛ f p L p (R d ) as requred. R Cɛ f p L p (R d )

17 DECOUPLING AND NEAR-OPTIMAL RESTRICTION ESTIMATES FOR CANTOR SETS 17 It remans to prove (37). We wll do so by nterpolatng between r = 1 and r = 2. For r = 1, t suffces to prove that for each, (38) F φ L 2 (µ) ɛ R Cɛ E 1/2 F L 2 (R d ), snce ths mples (37) by trangle nequalty. To prove (38), we nterpolate between L 1 and L estmates. Frst, we have by Fubn s theorem F φ L 1 (µ) F (x y) φ (y) dy dµ(x) ( ) = F (u) φ (v u) dµ(v) du R d sup µ(x + [ R 1, R 1 ] d ) F L 1 (R d ) x ɛ R d R 2d p +ɛ F L 1 (R d ) ɛ R Cɛ E 1 F L 1 (R d ) where at the last step we used (36). Interpolatng ths wth the pontwse bound sup F φ (x) F L (R d ) φ L 1 (R d ) F L (R d ) x we get (38). To complete the argument, we need to prove (37) wth r = 2. Defne the functons g va ĝ = F, so that g L 2 (R) = F L 2 (R) and F φ = ĝφ. We thus need to prove that (39) 2 ĝ φ ɛ R Cɛ E 1 g 2 L 2 L (µ) 2 (R d ). By translatonal nvarance and the rapd decay of φ, t suffces to prove (39) wth φ replaced by 1 B. Let R be the operator R(h) = ĥ E. By Corollary 2, R s a bounded operator from L p (J) to L 2 (µ) for any bounded cube J (wth norm dependng on J). We have to prove that R( ( ) 1/2 g 1 I ) ɛ R Cɛ E 1/2 g 2 L 2 L (µ) 2 (R d ) By the T T argument, t suffces to prove that ( 1 ) 2 1/2 ( I R R g j 1 Ij ɛ R Cɛ E 1 g 2 L 2 (R d ) j L 2 (R d ) ) 1/2

18 18 IZABELLA LABA AND HONG WANG By Schur s test, ths follows from (40) sup 1 I R R1 Ij h L j 2 (R d ) ɛ R Cɛ E 1 h L 2 (R ). d We clam that (41) 1 I R R1 I h L 2 (R d ) ɛ R Cɛ E 1 h L 2 (R d ) and (42) 1I R R1 Ij h L 2 (R d ) M 2 h L 2 (R d ), j, wth constants ndependent of, j. Together, these two mply (40). We frst prove (41), By Lemma 2, Hölder s nequalty, and by (36), we have R1 I h L 2 (dµ) ɛ R ɛ 1 I h L p (R d ) ɛ R Cɛ R d 2 d p h L 2 (R d ) ɛ R Cɛ E 1/2 h L 2 (R d ). Ths mples (41) by the T T argument wth a fxed. For j, we note that R Rh = h µ, so that 1 I R R1 Ij s an ntegral operator wth the ernel K j (x, y) = 1 I (x)1 Ij (y) µ(x y). By (32), K(x, y) dy I j M Bβ R Bβ M 2 f B was chosen large enough dependng on β. The clamed estmate (42) now follows from Schur s test. 8. Fourer decay To complete the proof of Theorem 2, t now suffces to prove that the Cantor set n Secton 4 can be constructed so that (19), (20), and (5) all hold. Snce (5) mples (32), the restrcton estmate (6) wll follow from Proposton 2. In all our ntermedate results so far, t dd not matter how the Λ(p) alphabet sets S,a were chosen, as long as they had the prescrbed cardnaltes. Here, however, t s crucal to randomze the choce of S,a. Theorem 4. Let {n } N and {t } N be two determnstc sets of ntegers such that (19), (20), (22) all hold, and that Σ s non-empty for each (as provded by Bourgan s theorem). Let {µ } N {0} be a sequence of random measures on [0, 1] d such that: µ 0 = 1 [0,1] d, µ 1, µ 2,... are constructed nductvely va the teratve process descrbed n Secton 4,

19 DECOUPLING AND NEAR-OPTIMAL RESTRICTION ESTIMATES FOR CANTOR SETS 19 for each N, the sets S,a are chosen randomly and ndependently from Σ, wth probablty dstrbuton such that (43) E(µ (x) E n ) = µ 1 (x) x [0, 1] d. Then the lmtng Cantor measure µ almost surely obeys all conclusons of Theorem 2. An example of a random constructon of µ that meets the condton (43) s as follows. Choose n and t as ndcated n the theorem (recall that for (20) to hold, t suffces to assume (21)). For each N, choose a Λ(p) set B [n ] d such that B = t c 0 n 2d/p and (7) holds wth n = n, for some c 0, C(p) ndependent of. Let B = {B,v : v [n ] d }, B,v [n ] d, B,v = v + B mod (n Z) d Then B Σ (n, t, c 0, 2 d C(p)), snce any functon supported on B,v s a sum of at most 2 d functons supported on translates of B. Set A 0 = {0}. Let now 1, and assume that A 1 has been constructed. For each a A 1, choose a random v(, a) [n ] d so that P(v(, a) = v) = n d for each v [n ] d and the choces are ndependent for dfferent a A 1. Let S,a = B,v(a), a random translate of B, and contnue the constructon as n Secton 4. Then (43) holds by translatonal averagng, and all other assumptons of the theorem hold wth C(p) replaced by 2 d C(p). Instead of usng random translates of a sngle set B for each, we could choose a set B,a Σ for each a A 1, then let S,a = B(, a) + v(, a) mod (n Z) d, where v(, a) s a random translaton vector n [n ] d as above, chosen ndependently of B,a and ndependently of the choces made for all other a. Bourgan s theorem [5] shows that a generc subset of [N] d of sze about N 2d/p s a Λ(p) set, so that Σ (wth an approprate choce of c 0 and C(p)) should be large for most values of t j n the ndcated range, provdng many sets avalable for the constructon. Other varants are possble. We now turn to the proof of the theorem. Proof. By Lemma 6, E = supp µ has Hausdorff dmenson α, and µ obeys (4) for all 0 < γ < α. The Fourer decay estmate (5) s proved by a calculaton almost dentcal to that n [14, Secton 6] for a specal case n dmenson 1, and n [17, Theorem 14.1] (see also [18, Theorem 4.2]) for more general measures n hgher dmensons. The proof n [17, Theorem 14.1] can be followed here almost word for word, except for the trval changes n parameters to allow a varable sequence {n j } nstead of a constant one (see e.g. [3]). It s easy to chec that the proof goes through as long as log N +1 ɛ N ɛ, N. Snce log N +1 = log N +log n +1 ɛ N ɛ +nɛ +1, ths s a weaer condton than (20).

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