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THE RESTRICTION THEOREM OF STEIN AND TOMAS MUSTAZEE RAHMAN Contents 1. Introduction 1 ∗ 2. Fourier transform of measures and the T T method 2 3. Oscillatory integrals and decay of the Fourier transform of the spherical measure 3 4. Proof of the Stein-Tomas restriction theorem 5 4.1. Optimality of the theorem: Knapp’s example 7 5. Connection of the restriction conjecture to problems involving Kakeya sets 8 6. Application: Controlling the size of solution to Schrödinger’s equation using size of initial data 14 7. Acknowledgements 15 References 15 1. Introduction If f : Rn → R is a L1 function then its Fourier transform fˆ is a L∞ function. However, not all L∞ functions can be represented as the Fourier transform of some L1 function, as is wellknown, say, from the Reimann-Lebesgue lemma. On the other hand the Fourier transform is an unitary isomorphism from L2 onto itself. If 1 < p < 2 then from the Hausdorff-Young 0 inequality it follows that the Fourier transform maps Lp into Lp where p0 is the Holder dual of p. Perhaps motivated by these observations Stein asked how small is the image of Lp 0 in Lp under the Fourier transform. One way to measure such ‘smallness’ is by restricting the Fourier transform to some submanifold of Rn and them studying the decay or regularity properties of the restricted functions. For example, restricting the Fourier transform of an L1 function to a point of Rn (evaluation) shows that the image of L1 under the Fourier transform is not all of L∞ since the former consists of continuous functions while evaluation is not welldefined for the latter. Stein’s restriction problem asks about the restriction property of the Fourier transform of an Lp function to the sphere S n−1 . More specifically, given f ∈ Lp (Rn ) for what q does it hold that ||fˆ |S n−1 ||Lq (S n−1 ) ≤ Cp,q,n ||f ||Lp (Rn ) (1.1) The best possible result for q = 2 is given by the Stein-Tomas theorem. Theorem 1.1 (Stein-Tomas). If f ∈ Lp (Rn ) with 1 ≤ p ≤ Cp,n ||f ||Lp (Rn ) . 2n+2 n+3 then ||fˆ |S n−1 ||L2 (S n−1 ) ≤ When p = 1 the theorem is trivial since ||fˆ |S n−1 ||L2 (S n−1 ) ≤ ||fˆ||L∞ |S n−1 |1/2 ≤ ||f ||L1 |S n−1 |1/2 . Also the theorem cannot hold for p = 2 since then fˆ is a generic L2 function, and such functions can easily blowup on the zero set S n−1 . These notes will give the proof of the theorem 1 2 MUSTAZEE RAHMAN for p < 2n+2 using Tomas’s argument (see [6]). The endpoint case p = 2n+2 was established by n+3 n+3 Stein using a complex interpolation method (see [5] for a proof). We will assume the reader is familiar with the Fourier transform, the Reisz-Thorin interpolation method and Young’s inequality. In the next section we will give an equivalent interpretation of the restriction 0 theorem using Banach space duality that turns the restriction estimate into an Lp → Lp estimate. In section 3 we shall consider oscillatory integrals and obtain point-wise decay estimates for the Fourier transform of the (n − 1)-dimensional Lebesgue measure on S n−1 . In section 4 we will outline the proof of Theorem 1.1 using Tomas’s interpolation argument. In section 5 we will discuss connections between the general restriction conjecture and related conjectures involving Kakeya sets, and finally in section 6 we will give an application of the Stein-Tomas theorem to Lp estimates of solutions to Schrödinger’s PDE. 2. Fourier transform of measures and the T T ∗ method Let µ be a complex valued regular Borel measure on Rn of bounded variation |µ|(Rn ). The Fourier transform of µ is Z e2πix·ξ dµ(x). µ̂(ξ) = Rn For a regular Borel measure µ of bounded variation and a Schwartz function f the convolution µ ∗ f is defined as Z f (x − y) dµ(y). µ ∗ f (x) = Rn One can easily verify that µ̂ is a bounded continuous function which satisfies the ReimannLebesgue lemma. The following lemma expresses some properties of the Fourier transform of measures that one is usually familiar with for functions. Their proofs are analogous to the proofs for functions. Lemma 2.1. Let µ and ν be measures of bounded variation and f a Schwartz function. Then (1) µ[ ∗ f = µ̂fˆ. (2) fcµ = fˆ ∗ µ̂ (Note that f µ is a new measure). (3) R µ̂dν = R ν̂dµ (the multiplication formula). The next lemma is a version of Plancheral’s formula for measures. Its proof follows from Lemma 2.1 and the realization that ĝ¯ = ḡˇ for any Schwartz function. Lemma 2.2. Let µ be a measure of bounded variation and f, g be Schwartz functions. Then Z Z ˆ ¯ f ĝ dµ = (µ̂ ∗ ḡ)f dx. Now we recall an important fact concerning the duality of Lp spaces. For 1 ≤ p < ∞ the norm of an Lp function f can be expressed as Z p0 ||f ||Lp (µ) = sup | f g dµ| : g ∈ L (µ), ||g||Lp0 (µ) = 1 (2.1) p where p0 = p−1 is the Hölder dual conjugate to p. In fact, provided p > 1 we can and will 0 0 restrict g to all Schwartz functions with Lp norm 1 since such functions are dense in Lp . The THE RESTRICTION THEOREM OF STEIN AND TOMAS 3 density argument also implies that it suffices to prove Theorem 1.1 for Schwartz functions. We now give an equivalent formulation of Theorem 1.1 by using the duality of Lp spaces. Theorem 2.1. Let µ be a measure of bounded variation on Rn and let p0 denote the Hölder dual of p. The following are equivalent: (1) ||fˆ||L2 (µ) ≤ C||f ||Lp for all Schwartz functions f (restriction form). (2) ||fcµ||Lp0 ≤ C||f ||L2 (µ) for all f ∈ L2 (µ) (extension form). 0 (3) ||µ̂ ∗ f ||Lp0 ≤ C 2 ||f ||Lp for all Schwartz functions f (Lp → Lp form). These equivalent formulations are often called the T T ∗ method because it states that if we consider T ∗ to be the operation of restricting the Fourier transform of f ∈ Lp to S n−1 then 0 0 T ∗ is bounded if and only if its adjoint T : L2 (S n−1 ) → Lp or T T ∗ : Lp → Lp is bounded. Proof. Let’s first show the equivalence of (1) and (2). Supposing that (1) holds for any Schwartz function g we get from the multiplication formula in Lemma 2.1 that Z Z | g fcµdx| = | ĝf dµ| ≤ ||g||L2 (µ) ||f ||L2 (µ) ≤ C||g||Lp ||f ||L2 (µ) where the last inequality follows from using (1) on g. From the relation in 2.1 the claim of (2) follows after taking the sup over all g with Lp norm 1. To get (1) from (2) we work backwards; for any Schwartz function g and f ∈ L2 (µ) we have Z Z | ĝf dµ| = | g fcµdx| ≤ ||g||Lp ||fcµ||Lp0 ≤ C||g||Lp ||f ||L2 (µ) where the last inequality comes from assuming (2). Taking the sup over all f ∈ L2 (µ) gives (1) for all Schwartz functions g. Now we consider the equivalence of (1) and (3). Assuming (1) let f and g be any Schwartz functions. Then using Lemma 2.2 we get Z Z ¯ ¯ | g(µ̂ ∗ f )dx| = | ĝ fˆdµ| ≤ ||ĝ||L2 (µ) ||fˆ||L2 (µ) ≤ C 2 ||g||Lp ||f ||Lp Taking the sup over all Schwartz functions g with Lp norm 1 we get ||µ̂ ∗ f¯||Lp0 ≤ C 2 ||f ||Lp , which is the same as (3). Finally assuming (3) use Lemma 2.2 with both functions set to f to get Z Z ¯ˆ 2 ˆ ˆ ||f || 2 = f f dµ = f (µ̂ ∗ f¯)dx ≤ ||f ||Lp ||µ̂ ∗ f¯|| p0 ≤ C 2 ||f ||2 p L (µ) L L which is the square of (1). 3. Oscillatory integrals and decay of the Fourier transform of the spherical measure We begin our discussion of oscillatory integrals by studying the decay of the functional Z I(λ) = e−iλφ(x) ψ(x)dx Rn as |λ| → ∞. It is assumed that the phase function φ(x) is C ∞ and ψ(x) is in C0∞ . The behaviour of I(λ) depends on the critical points of φ. If φ has no critical points in the support of ψ then I(λ) decays faster than any negative power of |λ|. On the other hand if φ has non-degenerate critical points in the support of ψ then the decay rate is only of order 4 MUSTAZEE RAHMAN O(|λ|−n/2 ). The situation becomes more complicated if φ has degenerate critical points with only the 1-dimensional case being resolved completely. Theorem 3.1. Let I, φ, and ψ be as above. If φ has no critical points in the support of ψ then for every integer d > 0 there exists a constant Cd dependent on φ and ψ such that |I(λ)| ≤ Cd |λ|−d . Proof. Fix x0 ∈ supp(ψ) and note that since ∇φ(x0 ) 6= 0 we can assume w.l.o.g. that ∂xn φ(x0 ) 6= 0. Consider the equation y = φ(x) in a neighbourhood of (x0 , y0 ) where y0 = φ(x0 ). Let Qnr (p) denote the open cube in Rn of side length 2r centered about p with sides parallel to the coordinate axes and write x0 = (x10 , . . . , xn0 ). By the implicit function theorem there exists a δ > 0 and an unique C ∞ function u(x1 , . . . , xn−1 , y) mapping the cube Qnδ (x10 , . . . , xn−1 , y0 ) into a neighbourhood of xn0 such that y = φ(x1 , . . . , xn−1 , u) for all 0 (x1 , . . . , xn−1 , y) ∈ Qnδ (x10 , . . . , xn−1 , y0 ). The image of (x1 , . . . , xn−1 , y) → (x1 , . . . , xn−1 , u) 0 covers a small cubic neighbourhood of x0 of side length δ 0 , say. Cover supp(ψ) by these neighbourhoods and reduce to a finite subcover Q1 , . . . , Qk . Let h1 , . . . , hk be a smooth partition of unity subordinate to the subcover given by the Qi . Using this partition of unity we can write k Z X I(λ) = e−iλφ(x) ψ(x)hj (x) dx. j=1 It suffices to show that each of the summands decay as in the statement of the theorem. If h(x) is supported on Qnδ0 (x0 ) then we can make the change of variables x = (x1 , . . . , xn−1 , u) following the earlier setup. Then the integral becomes Z I(λ) = e−iλφ(x) ψ(x)h(x) dx Z = e−iλy α(x1 , . . . , xn−1 , y) dxdy Qn δ where α is a new smooth function supported on Qnδ (x10 , . . . , xn−1 , y0 ) which comes from the 0 change of variables and multiplication by the Jacobian factor |∂y u|. The new integration is of course carried out in the x1 , . . . , xn−1 , y coordinates. Since α is smooth and of compact support, integrating the last equation by parts d times in y and using the triangle inequality we deduce that |I(λ)| ≤ Cd |λ|−d with the constant Cd depending on the L1 norm of ∂yd α. When φ has non-degenerate critical points, i.e. points p such that ∇φ(p) = 0 but the Hessian Hφ (p) has non zero determinant, then it becomes technically harder to prove decay of I(λ) to the order of O(|λ|−n/2 ). What is needed is the Morse lemma which allows a change of variables that reduces a smooth function with a non-degenerate critical point to a quadratic form locally. We state the Morse lemma first. Lemma 3.1 (Morse lemma). Let φ : U → Rn be a smooth map where U ⊂ Rn is a neighbourhood of the origin. Suppose ∇φ(0) = 0 but det Hφ (0) 6= 0. Then there exists a smooth change of variables x = f (y) where f : Qnr (0) → Rn satisfies f (0) = 0, Im(f) ⊃ Qnρ (0), det Df(x) 6= 0 P for all x ∈ Qnr (0), and φ(f (y)) = ni=1 µi yi2 with y = (y1 , . . . , yn ) and all µi 6= 0. We will simply state the result about the decay of I(λ) in the presence of non-degenerate critical points. A proof can be found in [5]. Even though the technicalities are harder in this proof the recipe is localization, reduction to normal form using the Morse lemma, and integration by parts as in the previous theorem. THE RESTRICTION THEOREM OF STEIN AND TOMAS 5 Theorem 3.2. Let I, φ, and ψ be as before. If φ has only non-degenerate critical points in the support of ψ then there exists a constant C such that |I(λ)| ≤ C|λ|−n/2 . One of the important applications of Theorems 3.1 and 3.2 is to estimate the decay of the Fourier transform of a measure supported on a smooth compact hypersurface of non-vanishing Gaussian curvature. We will specifically work with the sphere S n−1 and the Fourier transform of its (n−1)-dimensional Lebesgue measure. Throughout the remaining discussion we denote by σ the (n − 1)-dimensioanal Lebesgue measure on S n−1 with n fixed but arbitrary. Cover S n−1 by charts as follows. The first two charts covering the ‘north’ and ‘south’ poles are given by the maps p φ1 : x → (x, 1 − |x|2 ) for x ∈ D(0, 1/2) ⊂ Rn−1 , p φ2 : x → (x, − 1 − |x|2 ) for x ∈ D(0, 1/2) ⊂ Rn−1 . The remaining maps φ3 , . . . , φk are chosen similarly to cover the remaining ‘poles’ of S n−1 so that their images along with the images of φ1 and φ2 cover S n−1 . Now notice that since σ is invariant under orthogonal transformations, σ̂(ξ) will also remain unchanged under such transformations. So to estimate |σ̂(ξ)| it suffices to estimate |σ̂(λen )| where λ = |ξ|. Let h1 , . . . , hk be a partition of unity subordinate to the images of φi . Z σ̂(λen ) = e−2πiλ x·en dσ(x) = k Z X e−2πiλ x·en hj (x) dσ(x) j=1 √ Z −2πiλ = e D(0,1/2) + k Z X j=3 1−|y|2 ψ (y) p 1 dy + 1 − |y|2 √ ψ2 (y) 2 e−2πiλ− 1−|y| p dy 1 − |y|2 D(0,1/2) Z e−2πiλ φj (y)·en ψj (y)Jj (y) dy. Dj The maps ψj Jj coming from the change of variables are smooth p and supported inside Dj which is the domain of φj . The first two phase functions ± 1 − |y|2 above have critical points at y = 0 but their Hessian there is ±2I(n−1)×(n−1) . The remaining phase functions φj · en have no critical points in Dj . So by combining Theorems 3.1 and 3.2 we deduce that |σ̂(λen )| ≤ Cn |λ|−(n−1)/2 . For arbitrary ξ ∈ Rn , after making the obvious adjustments we get ˆ ≤ |σ(ξ)| Cn (1 + |ξ|) n−1 2 . (3.1) The properties of the sphere used in this argument are that it is a smooth compact hypersurface of non-vanishing Gaussian curvature, and the decay estimate 3.1 holds for the Fourier transform of the induced (n − 1)-dimensional Lebesgue measure of any such manifold. 4. Proof of the Stein-Tomas restriction theorem We begin by localizing once again. Let 0 ≤ ψ(x) ≤ 1 be a smooth radial function on Rn such that ψ(x) = 0 for |x| ≤ 1/2 and ψ(x) = 1 for |x| ≥ 1. Put φ(x) = ψ(2x) − ψ(x) and 6 MUSTAZEE RAHMAN φk (x) = φ(2−k x) for k ≥ 0. The function φk has support inside {x : 2k−2 ≤ |x| ≤ 2k } and for |x| ≥ 1/2 we have X φk (x) = 1. k≥0 P By putting φ−1 (x) = 1 − k≥0 φk (x) we see that it is smooth, supported inside |x| ≤ 1/2, P 0 and k≥−1 φk (x) = 1. To prove the Stein-Tomas theorem we will work with the Lp → Lp formulation from Theorem 2.1. Let Kj = σ̂φj for j ≥ −1. If f is a Schwartz function then P it is easy to see that the following point-wise equality holds: σ̂ ∗ f = K−1 ∗ f + j≥0 Kj ∗ f . So by Fatou’s lemma ||σ̂ ∗ f ||Lp0 ≤ ||K−1 ∗ f ||Lp0 + X ||Kj ∗ f ||Lp0 (4.1) j≥0 and it suffices to show that the right hand side is bounded by a constant factor of ||f ||Lp . For the first term we use Young’s inequality to get ||K−1 ∗ f ||Lp0 ≤ ||K−1 ||Lq ||f ||Lp where 1 2 = . Clearly q ≥ 1 since p0 > 2n+2 > 2 if p < 2n+2 . As K−1 is continuous and of compact q p0 n−1 n+3 support the quantity ||K−1 ||Lq is finite and independent of f . Now let’s handle the remaining terms. We will do so by first deriving a L1 → L∞ estimate and then a L2 → L2 estimate for the terms Kj ∗ f . We will then interpolate in between 0 to get a suitable Lp → Lp estimate that is summable in j. Notice that Kj is supported in {2j−2 ≤ |x| ≤ 2j } for all j ≥ 0. Combining this with the decay of σ̂ in 3.1 we get that ||Kj ||L∞ ≤ M 2−j(n−1)/2 where the constant M depends on φ and the constant Cn from the decay estimate 3.1. So by Young’s inequality we get the L1 → L∞ inequality ||Kj ∗ f ||L∞ ≤ ||Kj ||L∞ ||f ||L1 ≤ M 2−j(n−1)/2 ||f ||L1 . (4.2) Using the trivial but important inequality ||uv||L2 ≤ ||u||L∞ ||v||L2 we get the new estimate ˆ ˆ \ ||Kj ∗ f ||L2 = ||K j ∗ f ||L2 = ||K̂j f ||L2 ≤ ||K̂j ||L∞ ||f ||L2 = ||K̂j ||L∞ ||f ||L2 . (4.3) Since σ is symmetric we know that σ̌(ξ) = σ̂(−ξ) = σ̂(ξ). With g (a) (x) := g(ax) for a > 0 ˆ j ) ∗ σ. Since φ̂ is a Schwartz function nj (2 d ˆ ˆ ˆ we now get that K̂j = φ j σ̂ = φj ∗ σ̌ = φj ∗ σ = 2 φ there exists Mn such that |φ̂(ξ)| ≤ Mn (1 + |ξ|)−n . Also there is a constant An such that σ(D(p, r)) ≤ An rn−1 for all balls D(p, r) ⊂ Rn . In fact, this inequality only requires that our THE RESTRICTION THEOREM OF STEIN AND TOMAS 7 (n − 1)-dimensional manifold be compact. As a result, Z nj |K̂j (ξ)| ≤ 2 Mn (1 + 2j |ξ − y|)−n dσ(y) Rn Z nj = 2 Mn (1 + 2j |ξ − y|)−n dσ(y) D(ξ,2−j ) XZ + 2nj Mn (1 + 2j |ξ − y|)−n dσ(y) k≥0 D(ξ,2−j+k+1 )\D(ξ,2−j+k ) " # ≤ 2nj Mn σ(D(ξ, 2−j )) + X 2−nk σ(D(ξ, 2−j+k+1 )) k≥0 " ≤ 2nj Mn An 2−j(n−1) + # X 2−nk 2(−j+k+1)(n−1) k≥0 j = 3Mn An 2 . So from 4.3 we derive the L2 → L2 estimate ||Kj ∗ f ||L2 ≤ Bn 2j ||f ||L2 . (4.4) We can now interpolate between the estimates 4.2 and 4.4 using the Riesz-Thorin interpolation theorem. For 0 ≤ θ ≤ 1 we set 1−θ θ 1 = + , p 1 2 1 1−θ θ = + . q ∞ 2 This gives us that q = p0 , p0 = 2/θ, and 1 ≤ p ≤ 2. If we fix p to be in the range 1 ≤ p < 2n+2 < 2 then the interpolation theorem gives n+3 ||Kj ∗ f ||Lp0 ≤ M 1−θ B θ (2−j(n−1)/2 )1−θ 2jθ ||f ||Lp . (4.5) Upon simplifying we see that j ( p20 (n+1)−(n−1)) M 1−θ B θ (2−j(n−1)/2 )1−θ 2jθ = Q2 where Q depends on M, B and p0 . When 1 ≤ p < (2n + 2)/(n + 3) then p0 > (2n + 2)/(n − 1), which makes p20 (n + 1) − (n − 1) < 0. Thus ||Kj ∗ f ||Lp0 is summable in j from 4.5 and upon summing and using 4.1 we get ||σ̂ ∗ f ||Lp0 ≤ Cp,n ||f ||Lp . This finishes the proof of the Stein-Tomas theorem. 4.1. Optimality of the theorem: Knapp’s example. Earlier we mentioned that when considering possible inequalities of the from in 1.1 the Stein-Tomas restriction theorem is optimal for q = 2. We’ll now see why this is the case. We will work with the extension form of the theorem as given in (2) of 2.1. So we want to show that if for some p0 we have ||fˆσ||Lp0 ≤ C||f ||L2 (S n−1 ) then we must have p0 ≥ 2n+1 . For 0 < δ ≤ 1 consider the spherical n−1 cap Cδ = {x ∈ S n−1 : 1 − x · en ≤ δ 2 } . 8 MUSTAZEE RAHMAN √ Note that for any x ∈ S n−1 , |x − en |2 = 2(1 − x · en ). So x ∈ Cδ ⇔ |x − en | ≤ 2δ. In particular we have that |Cδ | ∼ δ n−1 in S n−1 . Now take f to be the characteristic function of Cδ . Then ||f ||L2 (S n−1 ) ∼ δ (n−1)/2 . Also consider all ξ such that |ξn | ≤ C1 δ −2 and |ξj | ≤ C1 δ −1 for all j < n where C1 will be very small. The set of all such ξ has volume about δ −(n+1) . So now Z −2πix·ξ c |f σ(ξ)| = e dσ(x) ZCδ −2πi(x−e )·ξ n e dσ(x) = Z Cδ ≥ cos(2π(x − en ) · ξ) dσ(x) . Cδ If C1 is chosen sufficiently small (and will be free of δ) then |(x − en ) · ξ| ≤ 1/6 for all x ∈ Cδ and ξ as stipulated above. Thus for all such ξ we will have 1 |fc σ(ξ)| ≥ |Cδ | ∼ δ n−1 . 2 −(n+1) As the set of such ξ has volume about δ we conclude that ||f ||Lp0 ≥ (const) δ n−1− n+1 p0 . So if the extension estimate in 2.1 holds then we must have δ (n−1)/2 ≥ (const) δ for all 0 < δ ≤ 1. This requires that n − 1 − n+1 p0 ≥ n−1− n+1 p0 n−1 , 2 or equivalently that p0 ≥ 2n+2 . n−1 The restriction conjecture. The restriction conjecture is a long standing open problem in analysis. Conjectured by Stein, it states that if f ∈ L∞ (S n−1 ) then 2n ||fc σ||Lp ≤ Cn,p ||f ||L∞ (S n−1 ) for all p > . (4.6) n−1 Taking f ≡ 1 and using the decay estimate 3.1 for σ̂ we see that σ̂ ∈ Lq if q(n − 1)/2 > n. Thus the choice for the range of p is reasonable. The conjecture was shown to be true for n = 2 by Fefferman and Stein in the 1970’s (see [5]). In fact for n ≥ 2, the slightly stronger estimate 2n ||fc σ||Lp (Rn ) ≤ Cn,p ||f ||Lp (S n−1 ) for all p > (4.7) n−1 can be shown to be equivalent (see [2]). The conjecture has strong connections to many other areas in analysis including the Kakeya conjecture. In the next section we will explore this connection. 5. Connection of the restriction conjecture to problems involving Kakeya sets Consider Knapp’s example from section 4.1 once again. We had a spherical cap Cδ centered at en or radius about δ and we took the Fourier transform of its characteristic function. Notice that this cap can be thought of at the intersection of the sphere with a rectangle centered at en of length about δ 2 in the en direction and lengths about δ in all other orthogonal directions. THE RESTRICTION THEOREM OF STEIN AND TOMAS 9 As such the Fourier transform of its characteristic function will be mostly constant in the dual rectangle (centered at the origin in frequency space) of length about δ −2 in the en direction and lengths about δ −1 in all other orthogonal directions due to the uncertainty principle. In the example we observed that the size of the Fourier transform in the dual rectangle was to the order of δ n−1 which is the volume of the cap itself. To discuss the Kakeya problem it will be convenient to replace dual rectangles with tubes. For δ > 0, e ∈ S n−1 and a ∈ Rn let Tδe (a) denote the tube in Rn that is centered at a, has axis in the direction e with length 1 in that direction, and with cross-sectional radius δ (see Fig. 1). e a 1 Figure 1. The tube Tδe (a). Let Cδ (e) denote the spherical cap of radius δ centered at e ∈ S n−1 . If fe is the characteristic n−1 function of Cδ (e) then Knapp’s example illustrates that fd on e σ has size to the order of δ −2 e the dilated tube δ Tδ (0). We can clearly translate these tubes anywhere is space with the effect of having to premultiply fe by some complex exponential e2πi a·x . Suppose we take lots of disjoint spherical caps of radius δ with collection of centers {ej } and then consider a function of the from X f= e2πi aj ·x fej . (5.1) j e Then fc σ should be essentially constant on the union of dilated tubes A = ∪j (δ −2 Tδ j (δ 2 aj ). Up to the dilation factor of δ −2 the set A contains an unit line segment in each of the directions e ej , namely the central segment of the tubes Tδ j (aj ). Since a maximal disjoint collection of caps of radius δ in S n−1 will have cardinality to the order of δ −(n−1) we conclude that the set A will contain an unit line segment (up to dilation) in about δ −(n−1) directions. A set in Rn that contains a unit line segment in every direction is called a Kakeya set. Since the tube Tδe (a) is essentially a δ-neighbourhood of the line segment {a + te : −1/2 ≤ t ≤ 1/2} we can think of A as essentially the δ-neighbourhood of an almost Kakeya set. The function fc σ is intimately related to the geometry of Kakeya sets, and as we have seen from Knapp’s example, it is also at the boundary of satisfying the restriction theorem. One can make quantitative sense of this observation to the point that the restriction conjecture an stated in 4.7 implies the Kakeya set conjecture which states that the Hausdorff dimension of a Kakeya set in Rn is n. Let f ∈ L1loc (Rn ) and define the Kakeya maximal function of f as Z 1 ∗ fδ (e) = sup |f | . e a∈Rn |Tδ (a)| Tδe (a) The goal is to prove an inequality of the form ∀ > 0 ∃ Cn,p, such that ||fδ∗ ||Lp (S n−1 ) ≤ Cn,p, δ − ||f ||Lp (Rn ) for some p < ∞ . (5.2) 10 MUSTAZEE RAHMAN The following inequalities are trivial: ||fδ∗ ||L∞ ≤ ||f ||L∞ (5.3) ||fδ∗ ||L∞ ≤ δ −(n−1) ||f ||L1 (5.4) Due to the fact that there exists Kakeya sets in Rn of Lebesgue measure zero, one can not get rid of the δ − factor from 5.2. Indeed suppose the estimate 5.2 holds without the δ − factor. Let E be a Kakeya set in Rn and denote by Eδ its δ-neighbourhood. Then with f = χEδ we have fδ∗ (e) ∼ 1 for all e ∈ S n−1 so that ||fδ∗ ||pL ∼ 1 for all δ. But ||f ||Lp = |Eδ | → 0 as δ → 0 and estimate 5.2 fails without the δ − factor. Now take f = χD(0,δ) and note that D(0, δ) ⊂ Tδe (0) for all e ∈ S n−1 . Thus fδ∗ (e) = |D(0,δ)| ∼ δ so that ||fδ∗ ||Lp ∼ δ while |T e (0)| δ ||f ||Lp ∼ δ n/p . So for 5.2 to hold we must have p ≥ n. This sets up the Kakeya maximal function conjecture: prove that ∀ > 0 ∃ Cn, such that ||fδ∗ ||Ln (S n−1 ) ≤ Cn, δ − ||f ||Ln (Rn ) . (5.5) In the rest of this section we will prove that the restriction conjecture 4.7 implies the Kakeya maximal conjecture 5.5. It is also true that the Kakeya maximal conjecture implies the Kakeya set conjecture. A proof of this fact can be found in [7]. The proof of the former fact is due to Bourgain and appeared in [2] although similar arguments were used earlier by Fefferman [3] when dealing with the ball multiplier problem and by other authors in [1]. Lemma 5.1. Fix 1 < p < ∞ and let {ej } be any maximal collection of δ-separated points in S n−1 . Suppose the following condition holds: If a non-negative sequence yj satisfies P 0 δ n−1 j yjp ≤ 1 then for any centers aj the inequality ||yj χT ej (aj ) ||Lp0 ≤ Aδ δ holds. Then it also holds that ||fδ∗ ||Lp ≤ Cn,p Aδ ||f ||Lp for all f ∈ Lp (Rn ) . Proof. The relevant geometric fact is that if |e − e0 | < δ then some bounded number of tubes 0 with varying centres Tδe (a) cover the tube Tδe (a0 ) with a0 arbitrary. This directly leads to the inequality fδ∗ (e) ≤ Cn fδ∗ (e0 ). During the course of the proof the constant Cn will change from line to line but will depend n and p only (we shall write it as Cn,p ). We have " ||fδ∗ ||Lp ≤ #1/p XZ |fδ∗ (e)|p dσ(e) Cδ (ej ) j " ≤ Cn,p #1/p XZ |fδ∗ (ej )|p dσ(e) Cδ (ej ) j #1/p " ≤ Cn,p X = Cn,p δ n−1 δ n−1 |fδ∗ (ej )|p j X j yj |fδ∗ (ej )| THE RESTRICTION THEOREM OF STEIN AND TOMAS 11 P 0 with the non-negative sequence yj satisfying δ n−1 j yjp = 1. The last line comes from the R 0 e duality of `p and `p . There exists some collection of {aj } such that |fδ∗ ej | ≤ ej2 |f | T j (aj ) |Tδ (aj )| e and since |Tδ j (aj )| ∼ δ n−1 we get ||fδ∗ ||Lp ≤ Cn,p X Z yj j ≤ Cn,p || δ X |f | e Tδ j (aj ) yj χT ej (aj ) ||Lp0 ||f ||Lp (Hölder) δ j ≤ Cn,p Aδ ||f ||Lp . Theorem 5.1 (Kinchin’s inequality). Fix 1 < p < ∞ and let fj for 1 ≤ j ≤ N be functions in Lp (Rn ). Let j be independent random variables taking values {±1} with equal probability. Then " N # N X X p E || j fj ||Lp ∼ ||[ |fj |2 ]1/2 ||pLp . j=1 j=1 The comparative constants depend on p. Proof. By Fubini it suffices to prove that given aj with 1 ≤ j ≤ N one has " N # N X X E | j aj |p ∼ [ |aj |2 ]p/2 . j=1 j=1 For p = 2 we have equality since the j are independent, have mean zero and variance 1. To x −x 2 ≤ ex /2 we prove the ≤ we use the exponential moment method. Using the ineqality e +e 2 bound for λ > 0, ! N PN X t j=1 j aj tλ P j aj > λ = P e >e (t > 0) j=1 " ≤ E # Y etaj j e−tλ j Y etaj + e−taj ) e−tλ = ( 2 j t2 ≤ e2 Choose t = Pλ j a2j to conclude P ( PN j=1 j P j a2j −tλ e . − aj > λ) ≤ e 2 λ2 P 2 j aj . By symmetry of the distribu- tion of the j we get the same inequality with greater than replaced by less than inside P () and λ > 0 replaced by −λ < 0. Combining them we get P (| N X j=1 − j aj | > λ) ≤ 2e 2 λ2 P 2 j aj . 12 MUSTAZEE RAHMAN Now use that E[|X|p ] = p R∞ " E | 0 λp−1 P (|X| > λ)dλ to get N X # Z p ≤ 2p j aj | ∞ p−1 λ − e 2 λ2 P 2 j aj dλ 0 j=1 = Cp N X !p/2 a2j . j=1 To get the other inequality we use duality once again. X X a2j = E[| j aj |2 ] j j ≤ E[| X j aj |p ]1/p E[| X j 0 0 0 0 j aj |p ]1/p j !1/2 X ≤ Cp a2j E[| X j j aj |p ]1/p . j The reverse inequality follows for all p0 . Kinchin’s inequality is useful when one needs to find a function with desirable properties that are not dependent on intricate cancellation. These kind of properties stand in contrast to the characteristic of our earlier work where oscillation and cancellation were often at the heart of the matter. We are going to use Kinchin’s inequality to make quantitative sense of the statement that functions resembling 5.1 are mostly constant on the union of tubes. We are now ready to proceed with the proof that the restriction conjecture implies the Kakeya maximal function conjecture. Theorem 5.2. If the restriction conjecture 4.7 holds then the Kakeya maximal function conjecture 5.5 holds as well. Proof. We will be making use of Lemma 5.1. Let {ej } be a maximal collection of δ-separated e points in S n−1 and let aj be any collection of points in Rn . Let τj = δ −2 Tδ j (aj ) be the dilation ej −2 n−1 of the tube : 1 − ej · e ≤ Cδ 2 } √ Tδ (aj ) by δ . We can find spherical caps Cj = {e ∈ S of radius Cδ each and centered about ej such that they are disjoint provided C is chosen sufficiently small. Knapp’s example as discussed earlier then provides us with functions fj n−1 supported on Cj such that |fj | = 1 on Cj and |fc on τj . We now fix a non-negative j σ| ≥ B δ sequence yj , take independent random variables ωj as in Kinchin’s inequality and consider the function X fω = ωj yj fj . j Note that due to the fj having disjoint supports X q X q ||fω ||qLq = yj ||fj ||qLq ∼ yj δ n−1 . j j (5.6) THE RESTRICTION THEOREM OF STEIN AND TOMAS 13 On the other hand by a direct application of Kinchin’s inequality we have Z X q 2 q/2 d E[||fω σ||Lq ] ≥ C ( |yj fc j σ| ) Rn ≥ Cδ j q(n−1) Z X Rn = C δ q(n−1) Z j X Rn |yj2 χ2τj |q/2 |yj2 χτj |q/2 j In particular there exists some choice of ωj for which Z X q q(n−1) d ||fω σ||lq ≥ C δ | yj2 χτj |q/2 . Rn (5.7) j The constant C above changed from line to line but is dependent only on q and n. If the 2n restriction conjecture 4.7 holds then fix any q > n−1 and note that the conjecture along with 5.6 and 5.7 imply that Z X X q q(n−1) δ | yj2 χτj |q/2 ≤ C yj δ n−1 . Rn 0 Let p = q/2 and zj = X j j Then the above is equivalent to the following: if p0 > X 0 zjp δ n−1 ≤ 1 implies || zj χτj ||Lp0 ≤ Cδ −2(n−1) . yj2 . j n n−1 then j Rescaling this back by δ 2 we obtain X p0 X 2( n −(n−1)) zj δ n−1 ≤ 1 implies || zj χTj ||Lp0 ≤ Cδ p0 j 0 for all p > n . n−1 j Thus Lemma 5.1 implies that ||fδ∗ ||Lp ≤ C δ 2( pn0 −(n−1)) ||f ||Lp for all f ∈ Lp , p < n . (5.8) n As p0 & n−1 the term pn0 − (n − 1) % 0 and p % n. So for any > 0 there exists a p < n dependent on such that 5.8 translates to ||fδ∗ ||Lp ≤ C δ − ||f ||Lp . Interpolating this with the trivial L∞ bound in 5.3 we conclude that ||fδ∗ ||Ln ≤ C,n δ ||f ||Ln . To summarize, the restriction conjecture implies the Kakeya maximal function conjecture which in turn implies the Kakeya set conjecture. The key link between these implications is that the Fourier transform of very many disjoint spherical caps gives rise to δ-tubes oriented in many different directions in frequency space. These tubes are essentially the δ-thickened version of an almost Kakeya set. There has been some partial results regarding the converse of these implications, notably due to Bourgain [2] who used estimates about Kakeya sets to prove a restriction theorem beyond that of Stein-Tomas. It is not clear, however, whether either of the Kakeya conjectures imply the full restriction conjecture. 14 MUSTAZEE RAHMAN 6. Application: Controlling the size of solution to Schrödinger’s equation using size of initial data ) of the Stein-Tomas restriction In the final section we will use the endpoint case (p0 = 2n+2 n−1 p0 theorem to get an upper bound on the L norm of the solution of a homogeneous Schrödinger equation in terms on the L2 norm of the initial data. This estimate is due to Robert Strichartz and these type of estimates are named in his honour. More examples of such estimates and their relation to the restriction theorem can be found in [5] and [4]. Consider the PDE 1 i ∂t u − ∆u = 0 2π u(x, 0) = f (x) . To solve this PDE take the Fourier transform in the x-variable on both sides of the equation. The PDE then gives rise to an ODE is the t-variable with initial condition fˆ. Solving the ODE using standard methods and using the Fourier inversion formula gives Z 2 u(x, t) = e2πi (x·ξ+t|ξ| ) fˆ(ξ) dξ . (6.1) Rn This solution can equivalently be represented as u(x, t) = (fˆµ)∨ where µ is the measure supported on the parabola {τ = |ξ|2 } ⊂ Rn+1 and given by Z Z g(ξ, |ξ|2 ) dξ g(ξ, τ ) dµ(ξ, τ ) = Rn Rn+1 for all continuos functions g on Rn+1 . The parabola is a smooth hypersurface of non vanishing Gaussian curvature but it is not compact. To deal with this difficulty take ψ ∈ C0∞ (Rn+1 ) such that ψ ≡ 1 on |τ |+|ξ| ≤ 1, and consider the measure ψµ which is now supported on a compact subset of the parabola. If f is any Schwartz function on Rn such that supp(f̂) ⊂ D(0, 1) ⊂ Rn then fˆψ = fˆ. Consequently ||(fˆψµ)∨ ||Lq (Rn+1 ) = ||(fˆµ)∨ ||Lq (Rn+1 ) and ||fˆ||L2 (ψµ) = ||fˆ||L2 (Rn ) = ||f ||L2 (Rn ) . For such f we apply the Stein-Tomas restriction theorem in dimension n + 1 with p0 = to get ||(fˆµ)∨ ||Lp0 (Rn+1 ) ≤ C||fˆ||L2 (ψµ) = C||f ||L2 (Rn ) . 2n+4 n In view of 6.1 this is equivalent to the inequality ||u||L2+ n4 (Rn+1 ) ≤ C ||f ||L2 (Rn ) . (6.2) Now suppose that f is Schwartz with supp(f̂) ⊂ D(0, λ). Put fλ (x) = f ( λx ) and uλ (x, t) = u( λx , λt2 ). Then supp(fˆλ ) ⊂ D(0, 1) and we have i ∂t uλ − 1 ∆uλ = 0 2π uλ (x, 0) = fλ (x) . Also 0 ||fλ ||L2 (Rn ) = λn/2 ||f ||L2 (Rn ) and ||uλ ||Lp0 (Rn+1 ) = λ(n+2)/p ||u||Lp0 (Rn+1 ) = λn/2 ||u||Lp0 (Rn+1 ) . THE RESTRICTION THEOREM OF STEIN AND TOMAS 15 So applying the inequality 6.2 to uλ and fλ we deduce that 6.2 holds for any Schwartz function f with fˆ having compact support. Since such functions are dense in L2 it follows that the estimate 6.2 holds for any solution u(x, t) of the Schrödinger equation with initial data f in L2 . This is the Strichartz estimate for the Schrödinger equation. 7. Acknowledgements I want to thank professor Jim Colliander for tweaking my interest in the subject matter, for directing me towards relevant literature, and for valuable discussions. I learned all of the material in Section 3 from professor Michael Goldstein; the example indicating the optimality of the Stein-Tomas restriction theorem and the discussion of the Kakeya problem is taken from the lecture notes of Tom Wolff [7], and the application to Strichartz estimates is derived from lecture notes of Wilhelm Schlag [4]. References [1] Beckner W., Carbery A., Semmes S., Soria F. A note on restriction of the Fourier transform to spheres. Bull. London Math. Soc. 21 (1989), 394-398. [2] Bourgain, J. Besicovitch type maximal operators and applications to Fourier analysis. Geometric and Functional Analysis 1 (1991), 147-187. [3] Fefferman, C. The multiplier problem for the ball. Ann. Math. 94(1971), 330-336. [4] Schlag, W. Harmonic Analysis notes. http://www.math.uchicago.edu/~schlag/book.pdf. [5] Stein, E. M. Harmonic Analysis. Princeton University Press, Princeton, N.J., 1993. [6] Tomas, P. A. A restriction theorem for the Fourier transform. Bull. Amer. Math. Soc. 81 (1975), 477-478. [7] Wolff, T. H. Lectures on Harmonic Analysis. American Mathematical Society University Lecture Series. 2002. http://www.math.ubc.ca/~ilaba/wolff/notes_march2002.pdf Department of Mathematics, University of Toronto, Toronto, Ontario M5S 2E4, Canada E-mail address: [email protected]