DECOUPLING INEQUALITIES IN HARMONIC ANALYSIS AND APPLICATIONS

Transcription

1 DECOUPLING INEQUALITIES IN HARMONIC ANALYSIS AND APPLICATIONS 1

2 NOTATION AND STATEMENT S compact smooth hypersurface in R n with positive definite second fundamental form ( Examples: sphere S d 1, truncated paraboloid P d 1 = {(ξ, ξ 2 ) R d 1 R; ξ 1} ) S δ = δ-neighborhood of S S δ τ τ where τ is a δ δ δ rectangular box in R d τ δ For f L p (R d ), f τ = ( ˆf τ)ˇis the Fourier restriction of f to τ

3 THEOREM (B-Demeter, 014) Assume supp ˆf S δ Then f p ε δ ε( τ f τ 2 p )1 2 for 2 p 2(d + 1) d 1 3

4 Range p 2d d 1 obtained earlier (B, 013) Exponent p = 2(d+1) d 1 is best possible EXAMPLE f = 1 Sδ Then f δ (1+ x ) d 1 2 and f p δ f τ δ d with τ the polar of τ f τ τ p δ d+1 2 ( τ f τ 2 p )1 2 ( 1 δ )d 1 4 δ d+3 4 = δ ( 1 δ )d+1 2p = δ d+3 4 4

5 Original motivations PDE Smoothing for the wave equation (TWolff, 2000) Schrödinger equation on tori and irrational tori (B, 92 ) Spectral Theory Eigenfunction bounds for spheres Diophantine applications, new approach to mean value theorems in Number Theory, bounds on exponential sums 5

6 MOMENT INEQUALITIES FOR TRIGONOMETRIC POLYNOMIALS WITH SPECTRUM IN CURVED HYPERSURFACES S R d smooth with positive curvature λ 1, E = Z d λs = lattice points on λs Let φ(x) = ξ E a ξ e 2πixξ φ is 1-periodic Apply decoupling theorem to rescaled f(y) = with δ = λ 2 φ L p ([0,1] d ) λε ( ξ E ξ E a ξ e 2πiy ξ λ η(λ 2 y) a ξ 2 )1 2 for p 2(d + 1) d 1

7 COROLLARY (S = S d 1 ) Eigenfunctions φ E of d-torus T d, ie φ E = Eφ E, satisfy φ p E ε φ 2 for p 2(d + 1) d 1 (Known for d = 3 for arithmetical reason, new for d > 3) 7

8 SCHRODINGER EQUATIONS AND STRICHARTZ INEQUALITIES Linear Schrödinger equation on R d iu t + u = 0, u(0) = φ L 2 (R d ) u(t) = e it φ = ˆφ(ξ)e i(xξ+t ξ 2) dξ STRICHARTZ INEQUALITY: e it φ p C φ L 2 with p = 2(d+2) x,t d Local wellposedness fo Cauchy problem for NLS on R d iu t + u + αu u p 2 = 0 u(0) = φ H s (R d ) s s 0 with s 0 defined by p 2 = 4 d 2s 0 (p 2 [s] if p 2Z) 8

9 TORI AND IRRATIONAL TORI φ L 2 (T d ) φ(x) = a ξ e 2πixξ ξ Z d e it φ = More generally (irrational tori) ξ Z d a ξ e 2πi(xξ+t ξ 2 ) e it φ = a ξ e 2πi(xξ+tQ(ξ)) where Q(ξ) = α 1 ξ α dξd 2, α j > 0 ξ Z d PROBLEM Which L p -inequalities are satisfied? (periodic Strichartz inequalities) 9

10 EXAMPLES (B, 92) d = 1 Assume φ(x) = ξ Z, ξ R a ξ e 2πixξ e it φ L 6 [ x, t 1] R ε φ 2 Moreover, taking a ξ = 1 R for ξ R, a ξ = 0 for ξ > R, e it φ 6 (log R) 1/6 (Classical Strichartz inequality fails in periodic case!) d = 2 e it φ 4 R ε φ 2 Proven using arithmetical techniques (# lattice points on ellipses) d 3 Only partial results Irrational tori No satisfactory results for d 2 10

11 THEOREM Let Q be a positive definite quadratic form on R d Then ξ Z d, ξ R a ξ e 2πi(xξ+tQ(ξ)) L p [ x, t 1] R ε( aξ 2) 1 2 for p 2(d+2) d COROLLARY Local wellposedness of NLS on T d for s > s 0 Recent results of Killip-Visan 11

12 Proof Define f(x, t ) = ( ξ R, Q(ξ) R 2 ξ R ) a ξ e 2πi(x ξ R +t Q(ξ) R 2 ) η(r 2 x, R 2 t ) S = { ( y, Q(y) ) ; y R d, y 1} R d+1 From decoupling inequality f L p (B R 2) Rε R 2(d+1) ( ) 1/2 p aξ 2 and setting x = Rx, t = R 2 t f L p (B R 2) = Rd+2 p = R 2(d+1) p aξ e 2πi(xξ+tQ(ξ)) L p x <R, t <1 aξ e 2πi(xξ+tQ(ξ)) L p x <1, t <1 12

13 MOMENT INEQUALITIES FOR EIGENFUNCTIONS (M, g) compact, smooth Riemannian manifold of dimension d without boundary ϕ E = Eϕ E (E = λ 2 ) THEOREM (Hormander, Sogge) ϕ E p cλ δ(p) ϕ E 2 where δ(p) = d 1 ( 1 ) p ( 1 d 2 1 ) 1 p 2 if 2(d + 1) 2 p d 1 if 2(d + 1) d 1 p Estimates are sharp for M = S d 13

14 FLAT TORUS T d CONJECTURE ϕ E p c p ϕ E 2 if p < 2d d 2 and ϕ E p c p λ (d 2 2 d p ) ϕ E 2 if p > 2d d 2 14

15 THEOREM (B Demeter, 014) ϕ E p λ ε ϕ E 2 if d 3 and p 2(d + 1) d 1 ( ) and ϕ E p C p λ (d 2 2 d p ) ϕ E 2 if d 4 and p > 2(d 1) d 3 ( ) ( ) follows from ( ) combined with distributional inequality t 2d 1 d 3 [ϕ > t] λd 3 2 +ε for t > λ d 1 4 (B, 093) (Hardy-Littlewood + Kloosterman) 15

16 E = sum of 2 squares ESTIMATES ON T 2 multiplicity E = number N of representations of E as sum of two squares E = p e α α N = 4 (1 + e α ) On average, N (log E) 1/2 THEOREM (Zygmund Cook) ϕ E 4 C ϕ E 2 P 1 2 P 16

17 Estimating the L 6 -norm number of solutions of P 1 + P 2 + P 3 = P 4 + P 5 + P 6 with P i E = {P Z + iz; P 2 = E} COMBINATORIAL APPROACH Use of incidence geometry ANALYTICAL APPROACH: Use of the theory of elliptic curves (conditional) 17

18 THEOREM (Bombieri B, 012) ϕ E 6 N ε ϕ E 2 For all p <, ϕ E p C ϕ E 2 for most E Conditional to GRH and Birch, Swinnerton-Dyer conjecture ϕ E 6 N ε ϕ E 2 for most smooth numbers E 18

19 THE COMBINATORIAL APPROACH Consider equation P 1 + P 2 + P 3 = (A, B) with P j = (x j, y j ) E Set u = x 1 + x 2, v = y 1 + y 2 establishing correspondence between E E and P = E + E P 1 P 2 Introduce curves C A,B : (A u) 2 + (B v) 2 = E family C of circles of same radius E (pseudo-line system) Need to bound A,B C A,B P 2

20 USE OF SZEMEREDI-TROTTER THEOREM FOR PSEUDO-LINE SYSTEMS I(P, C) = number of incidences between point P and curves C C( P 2/3 C 2/3 + P + C ) bound N 7/2 E = 65, P = 112, C = 372 REMARK Erdös unit distance conjecture would imply N 3+ε

21 PROBLEM Establish uniform L p -bound for some p > 2 for higher dimensional toral eigenfunctions Important to PDE s Control for Schrodinger operators on tori THEOREM (B-Burq-Zworski, 012) Let d 3, V L (T d ), Ω T d, Ω φ, an open set Let T > 0 There is a constant C = C(Ω, T, V ) such that for any u 0 L 2 (T d ) u 0 L 2 (T d ) C eit( +V ) u 0 L 2 (Ω [0,T ]) (uses Zygmund s inequality) More regular V : Anantharaman-Macia (011)

22 INGREDIENTS IN THE PROOF OF DECOUPLING THEOREM Wave packet decomposition Parabolic rescaling Multilinear harmonic analysis d = 2 bilinear square function inequalities (Cordoba 70 s) d 3 Bennett, Carbery, Tao (06) Multi-scale bootstrap 22

23 BENNETT - CARBERRY - TAO THEOREM S R d compact smooth hypersurface S 1,, S d S disjoint patches in transversal position ν(ξ 1 ) ν(ξ d ) > c for all ξ 1 S 1,, ξ d S d ν 2 ν 1 S 2 S S 1 Let µ 1,, µ d be measures on S, µ j σ = surface measure and suppµ j S j dµ j = ϕj dσ, ϕj L2 (σ) Take large ball B R R d Then d j=1 ˆµ j (x) 1 d L p (B R ) R ε d j=1 ϕ j L 2 (σ) with p = 2d d 1

24 MEAN VALUE THEOREMS AND DIOPHANTINE RESULTS (FIRST APPLICATIONS) (1) Around Hua s inequality n N e(nx+n 3 y) 6 dxdy = I6 (N) < CN 3 (log N) c (Hua, 1947) Arithmetically, I 6 (N) is the number of integral points n 1,, n 6 N on the Segre cubic x 1 + x 2 + x 3 = x 4 + x 5 + x 6 x x3 2 + x3 3 = x3 4 + x3 5 + x3 6 Vaughan-Wooley (95) I 6 (N) = 6N 3 +U(N), U(N) = O ( N 2 (log N) 5) De La Bretèche (07) Precise asymptotic for U(N)

25 Applications of decoupling inequality to S = {(t, t 3 ), t 1} R 2 B(N 2 ) n N ( n e N x + ( n N ) 3 y ) 6 dxdy N 3+ε By change of variables and periodicity n N e (nx+ n3 ) N y 6 1 = n N e (nnx+ n3 ) N y 6 N 3+ε which means that n 1 + n 2 + n 3 n 4 n 5 n 6 = 0 n n3 2 + n3 3 n3 4 n3 5 n3 6 < N has at most O(N 3+ε ) solutions with n i N 25

26 n N e (nx + n3 ) N y 6 dxdy N 3+ε is optimal CONJECTURE I 8 (N) = n N ( ) e nx + n 3 8 y dxdy N 4+ε KNOWN Hua (1947) I 10 (N) N 6+ε (optimal) Wooley (014) I 9 (N) N 5+ε (optimal) I 8 (n) N ε Consequence of (optimal) Vinogradov mean value theorem e(nx + n 2 y + n 3 12 z) dxdydz N 6+ε

27 (2) A MEAN VALUE THEOREM OF ROBERT AND SARGOS THEOREM (Robert-Sargos, 2000) n N e (n 2 x + n4 ) 6 N 3y dxdy N 3+ε Proof based on Poisson summation (Bombieri-Iwaniec method) Applications to exponential sums and Weyl s inequality THEOREM (B-D, 014) n N e (n 2 x + nk N k 1y ) 6 dxdy N 3+ε

28 PROOF Apply DCT with δ = N 1 2 to x <N 2 y <N This gives a bound N ε { I n N e ( n I (( n N ) 2x ( ) n ky ) 6 + dxdy ( ) N e(n 2 x + N k+1 n k y) where {I} is a partition of [ N 2, N ] in intervals of size N If I = [l, l + N], n = l + m, m < N n I e(n 2 x+n k+1 n k y) m< N and 6th moment bounded by N 3 2 +ε ( ) N ε( N ( N 3 2 )1 3 ) 3 = N 3+ε 6 dxdy )1 3 } 3 e ( m(2lx+kn k+1 l k 1 y)+m 2 x )

29 DECOUPLING FOR CURVES IN HIGHER DIMENSION Γ R d non-degenerate smooth curve, ie φ (t) φ (d) (t) 0 with φ : [0, 1] Γ a parametrization Let Γ 1,, Γ d 1 Γ be separated arcs and (Γ j ) δ a δ-neighborhood of Γ j Assume supp ˆf j (Γ j ) δ Apply a version of the hypersurface DCT to S = Γ Γ d 1 (Note that second fundamental form may be indefinite)

30 THEOREM ( ) (B-D) d 1 j=1 f j 1 d 1 2(d+1) L # (B N ) N 1 +ε d 1 2(d+1) j=1 [ fj,τ p L p # (B N) ] 1 2(d+1) where p = 2(d+1) d 1 N = δ 1 and {τ} a partition of Γ δ in size δ 1 2-tubes, THEOREM ( ) (B) Let d be even d/2 j=1 f j 2/d L 3d # (B N) N 1 6 +ε d/2 j=1 [ fj,τ 6 L 6 # (B N) ] 1 3d

31 MEAN VALUE THEOREMS Let d 3 and ϕ 3,, ϕ d : [ 0, 1 ] R be real analytic such that W ( ϕ,, ) 3 ϕ d 0 I 1,, I d 1 [ N 2, N ] Z intervals that are N separated THEOREM d 1 ( ( k ) ( k ) ( k ) 1 e kx 1 + k 2 x 2 + Nϕ 3 x 3 + Nϕ 4 x Nϕ d )x d N N N j=1 k I j N 1 2 +ε d 1 L 2(d+1) ([0,1] d ) 31

32 COROLLARY (Bombieri-Iwaniec, 86) Assume ϕ : [0, 1] R real analytic, ϕ > 0 Then k N ( ) ) e(kx 1 + k 2 k 8 x 2 + Nϕ x 3 dx1 dx 2 dx 3 N 4+ε N Essential ingredient in their work on ζ ( it ) THEOREM (Bombieri-Iwaniec, 86) ζ ( it ) t 9 56 for t (later improvement by Huxley, Kolesnik, Watt using refinements of the method) 32

33 EXPONENTIAL SUMS (Bombieri-Iwaniec method) (Huxley Area, lattice points and Exponential Sums, LMSM 1996) GOAL Obtaining estimates on exponential sum m M ( ( )) m e T F M EXAMPLE ζ( 1 2 By approximate functional equation, bounding + it ) reduces to sums m M ( e T log m ) M with M < T 1/2 33

34 STEP 1 Subdivide [ M 2, M ] in shorter intervals I of size N on which T F ( M m ) can be replaced by cubic polynomial cubic exponential sums of the form n N e(a 1 n + a 2 n 2 + µn 3 ) with a 1, a 2, µ depending on I and µ is small STEP 2 Conversion to new exponential sum by Poisson summation (stationary phase) of the form h H with b 1, b 2, b 3, b 4 depending on I e ( b 1 h + b 2 h 2 + b 3 h 3/2 + b 4 h 1/2) 34

35 STEP 3 Understanding the distribution of b 1 (I), b 2 (I), b 3 (I), b 4 (I) when I varies (the second spacing problem ) STEP 4 Use of the large sieve to reduce to mean value problems of the form A k (H) = k = 4, 5, h H e ( x 1 h+x 2 h 2 +x 3 H 1 2h 3/2 +x 4 H 1 2h 1/2) 2k dx THEOREM (B) Assume W (ϕ, ψ ) 0 and 0 < δ < < 1 Then h H ( e hx 1 + h 2 x ( h ) δ ϕ x ( h ) ) H ψ 10 x 4 dx1 dx 2 dx 3 dx 4 H [δ 3/4 H 7 + (δ + )H 6 + H 5 ]H ε COROLLARY (Huxley-Kolesnik, 1991) A 5 (H) H 5+ε THEOREM (Huxley, 05) ζ( it) t , = 0, 1561

36 PROBLEM (Huxley) Obtain good bound on A 6 (H) THEOREM (B, 014) A 6 (H) H 6+ε This bound follows from THEOREM ( ) and is optimal Combined with the work of Huxley on second spacing problem COROLLARY ζ ( it ) t ε, = 0,

37 PROGRESS TOWARDS THE LINDELÖF HYPOTHESIS µ(σ) = inf (β > 0; ζ(σ + it) = 0 ( t β)) Bounds on µ( 1 2 ) 1 4 (Lindelöf, 1908) = 0, (Kolesnik, 1973) 1 6 (Hardy-Littlewood) = 0, (Kolesnik, 1982) = 1, 1650 (Walfisz, 1924) = 0, (Kolesnik, 1985) = 0, 1647 (Titchmarsh, 1932) 9 56 = 0, 1607 (Bombieri-Iwaniec, 1986) = 0, (Phillips, 1933) = 0, 1638 (Titchmarsh, 1942) = 0, 1574 (Huxley-Kolesnik, 1990) = 0, (Huxley, 2002) = 0, 1631 (Min, 1949) = 0, (Huxley, 2005)

38 SKETCH OF PROOF OF DECOUPLING IN 2D ( ) K p (N) C ε N γ+ε best bound in inequality for supp ˆf Γ 1N, τ : f L p #(B N ) ( )1 K p (N) f τ 2 2 L p τ # (B N) 1 N 1 N tube There is bilinear reduction (B-G argument) ( f 1 f 2 ) 1 2 L p # (B N) K p(n) 2 i=1 ( τ f i τ 2 L p # (B N) if supp f 1 (Γ 1 ) 1N, supp f 2 (Γ 2 ) 1N with Γ 1, Γ 2 Γ separated )1 4 By parabolic rescaling, if M N and τ : 1 M 1 M tube f τ L p # (B N) K p ( N M )( τ τ f τ 2 L p # (B N) )1 2

39 ( ) From bilinear L 4 -inequality, for M N, τ = 1 M 1 M tubes 2 ( i=1 τ f τ 2 )1 4 L 4 # (B M) f 1 L 2 # (B M ) f 2 L 2 # (B M ) = ( fτ i 2 i τ L 2 # (B M) Interpolation with trivial L L bound (using ware packet decomposition) ( i M ε i τ f i τ 2 ( )1 4 L p # (B M) M ε i τ f i τ 2 L p # (B M) with κ = p 4 p 2 and τ : 1 M 1 M 2 tubes M ε i ( τ f i τ 2 L 2 # (B M) )κ 4 i ( ( τ )1 4 τ f i τ 2 L p/2 # (B M) f i τ 2 L 2 # (B M) )κ 4 ( )1 fτ i 4 1 κ 2 i τ )1 κ 4 )1 4 L p # (B M)

40 Writing B N as a union of M -balls ( i τ f i τ 2 )1 4 L P # (B N) N ε i ( )κ 4 ( τ f i τ 2 L p # (B N) i τ f i τ 2 )1 4 1 κ L p # (B N) ( ) Iterate, considering coverings {τ s } 1 s s, τ s : N 2 s N 2 s+1 tubes τ s τ s 1 K p (N) N ε N 2 s 1 K p (N 1 2 s ) κ K p (N 1 2 s+1 ) κ(1 κ) K p (1) (1 κ) s { γ 2 s 1 1 (1 κ) s +κγ κ 2 s 1 (2(1 κ)) s 2κ 1 } (2(1 κ) = 4 p 2 < 1 for p > 6) κ 1 κ ( ) 1 (2(1 κ)) s γ Taking s large enough γ 2κ 1 κ γ = 0 for p = 6