ANALYSIS CLUB. Restriction Theory. George Kinnear, 7 February 2011

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1 ANALYSIS CLUB Restriction Theory George Kinnear, 7 February 011 The content of this note is base on [Tao10] an [Tao03]. 1 Restriction an extension Our setting is a smooth compact hypersurface S in R (e.g. the unit sphere S in R 3 ), with surface measure σ. Given f L 1 (R ), we have the Fourier transform ˆf (x) = e πix.ξ f (x) x R which by Riemann-Lebesgue is a boune, continuous function vanishing at infinity. Thus if we restrict ˆf to S, we get a meaningful function which has finite L q norm for every q. However, starting with f L (R ), we get ˆf L (R ) by Plancherel. There is no meaningful way to restrict an arbitrary L function to a set of measure zero such as the hypersurface S. The question arises: what happens for 1 < p <? QUESTION 1 RESTRICTION PROBLEM R S (p q) For which p an q o we have ˆf f L q (S,σ) L p (R ) (1) for all f ( S, say). If p an q are a pair for which (1) hols, we say R S (p q) is true. The earlier remarks show that R S (1 q) is true for all 1 q while R S ( q) is false for all 1 q. PROPOSITION If R S (p q) is true then R S ( p q) is true for all p p an q q. PROOF Höler an Sobolev inequalities. Now if (1) hols for a certain p an q, we have sup ˆf 1 L q (S,σ) f p =1 1

2 which by uality means sup f p =1 sup g L q (S,σ) =1 ˆf (ξ)g(ξ) σ(ξ) 1. Swapping the sup s an applying Parseval, followe by unoing the uality, we have sup sup f (ξ)ĝσ(ξ) x 1 g L q ((S,σ) =1 f p =1 sup ĝσ p 1 g L q ((S,σ) =1 ĝσ p g q. () In fact, these steps are reversible so () is equivalent to (1). This gives us an alternative question. QUESTION 3 EXTENSION PROBLEM R S (q p ) For which p an q o we have Fσ F L p (R ) L q (S,σ) (3) for all F ( S, say). If p an q are a pair for which (3) hols, we say R S (q p ) is true. Necessary conitions Taking F 1 in (3) shows that we nee σ L p. For the sphere, it is well-known that σ(x) = O ((1 + x ) (n 1)/) so we must have p > n n 1 (i.e. p < n n+1 ). There is another choice of F, known as the Knapp example, which is essentially the characterisitic function of a small cap on the sphere. Plugging this in to (3) shows that we must also have q n 1 n + 1 p. The restriction conjecture CONJECTURE 4 RESTRICTION CONJECTURE These necessary conitions are sufficient, i.e. if p < n n + 1 an q n 1 n + 1 p then R S (p q) is true. The following is a classic partial result in this irection.

3 3 THEOREM 5 TOMAS-STEIN (1975) ( ) R S (p q) is true for all 1 p n+ n+3, with q = n 1 n+1 p. Note that Proposition means that we can replace q = with q PROOF ( ) n 1 n+1 p. See [Ste93, p386] for the full proof. The following is a sketch proof of the restricte result (i.e. with f = χ E for some set E) in the enpoint case p = n+ n+3, where q =. We want to show ( ) 1/ f (ξ) σ(ξ) f S n 1 L n+ n+3 (R n ) We write the L norm as f f σ = f (σˇ f ) = f (σ 1ˇ f ) + f (σ ˇ f ) where we have split σˇ(ξ) = σˇ(ξ)φ( ξ λ ) + σˇ(ξ)(1 φ( ξ λ )) with φ a stanar bump. Thus σ 1ˇ is supporte on ξ λ an σ ˇ on ξ λ. The appropriate choice of λ will be mae later. Using the fact that σˇ(ξ) ξ (n 1)/, we have σ ˇ(ξ) λ n 1. On the other han, σ 1 = σ φ 1/λ, so σ 1 is σ sprea out on scale λ 1 maintaining mass 1. So σ 1 λ. So applying Höler an Plancherel to the first term, an Höler an Young on the secon, we get f σ = f (σ 1ˇ f ) + f (σ ˇ f ) Taking f = χ E we have f σ 1 f + f 1 σ ˇ f 1 C f λ + f n 1 1 λ. f σ E λ + E λ n 1 an choosing λ so that both terms are the same (i.e. taking λ = E /(n+1) ) we get f σ E n+3 n+1 = f (n+1) n+3 i.e. that the restriction estimate hols from p = (n+1) n+3 to L (S n 1 ). The most recent results on this problem have use bilinear or multilinear techniques, which we will now look at. 3 Bilinear estimates Note that when p is an even integer, Fσ expane out using Plancherel; for instance Fσ = Fσ Fσ 1/ 4 p in the extension problem (3) can be = Fσ Fσ 1/.

4 4 This means R S (q 4) is equivalent to Fσ Fσ F L q (S,σ) which involves no Fourier transforms, so coul be proven by more irect methos. The iea can be partially extene to p which are not even integers, since the squaring step can still be carrie out on R S (q p ) to give Fσ Fσ p / F L q (S,σ). This leas us to consier more general estimates of the form F p 1 σ 1 F σ F 1 / L q (S 1,σ 1 ) F L q (S,σ ) (4) for arbitrary pairs of smooth compact hypersurfaces S 1, S with respective surface measures σ 1, σ, an F 1, F supporte on S 1, S. DEFINITION If (4) hols, we say R S 1,S (q q p /) is true. Such bilinear estimates are more general than the linear ones, but if we restrict to S 1 = S = S, we have equivalence: PROPOSITION 6 PROOF This is because R S,S (q q p /) says R S,S (q q p /) R S (q p ) F 1 σ F σ F p 1 / L q (S,σ) F L q (S,σ) for all F 1, F supporte on S, which by polarisation is equivalent to Fσ Fσ p / F L q (S,σ), i.e. R S (q p ). Rather than putting S 1 = S = S, we might instea put some conitions on S 1, S an hope to prove bilinear estimates in that setting. We will see something along those lines later. Multilinear restriction There is no reason to limit ourselves to squaring out the norm to get a bilinear estimate; in imensions we can consier a -linear estimate F j σ j p / F L j q (S j,σ j ). For instance, in [BCT06] the following is obtaine:

5 5 THEOREM 7 NEAR-OPTIMAL MULTILINEAR RESTRICTION For ε > 0, p 1 1, q p, F j σ j L p / (B(0,R) R ε assuming the S j are smooth enough, an are transverse. F L j q (S j,σ j ) The assumption that the S j are transverse (i.e. their normals are never coplanar) is quite restrictive in particular, we certainly can t make them all S an use Proposition 6 to make progress on the restriction conjecture. In [Tao10] Tao iscusses a new result of Bourgain an Guth, which he says interpolates between the Theorem of [BCT06] an a result of Córoba to obtain progress on the restriction conjecture.

6 Bibliography [BCT06] Jonathan Bennett, Anthony Carbery, an Terence Tao. On the multilinear restriction an Kakeya conjectures. Acta Math., 196():61 30, 006. [Ste93] Elias M. Stein. Harmonic analysis: real-variable methos, orthogonality, an oscillatory integrals, volume 43 of Princeton Mathematical Series. Princeton University Press, Princeton, NJ, With the assistance of Timothy S. Murphy, Monographs in Harmonic Analysis, III. [Tao03] Terence Tao. Recent progress on the restriction conjecture, 003. [Tao10] Terence Tao. The Bourgain-Guth argument for proving restriction theorems. Blog post, December