Existence of extremizers for Fourier restriction to the moment curve
aa r X i v : . [ m a t h . C A ] J a n EXISTENCE OF EXTREMIZERS FOR FOURIER RESTRICTIONTO THE MOMENT CURVE
CHANDAN BISWAS AND BETSY STOVALL
Abstract.
We show that the restriction and extension operators associatedto the moment curve possess extremizers and that L p -normalized extremizingsequences of these operators are precompact modulo symmetries. Introduction
This article concerns the existence of extremizers and compactness (modulo sym-metries) for the restriction/extension inequalities associated to the moment curve.More precisely, we consider the operator E f ( x ) := Z R e ix · γ ( t ) f ( t ) dt, γ ( t ) := ( t, t , . . . , t d ) , which was shown by Drury [8] to extend as a bounded linear operator from L p ( R )to L q ( R d ) if and only if q > d + d +22 and q = d ( d +1)2 p ′ for d ≥
2. We prove thatfor all ( p, q ) in this range, there exist nonzero functions f such that kE f k q = kEk L p → L q k f k p . We prove, moreover, that whenever ( p, q ) = (1 , ∞ ), L p -normalizedextremizing sequences (i.e. those that saturate the operator norm) possess subse-quences that converge, modulo the application of symmetries of the operator, to anextremizing function.Our argument uses a modified version of the concentration-compactness frame-work of Lions [15] and the related Method of Missing Mass of Lieb [14]. Suchmethods have been well-studied for the L -based restriction/extension problemsassociated to various hypersurfaces (see [3], [4], [5], [7], [9], [11]), and the resultingtheory has been an important step towards breakthroughs in the study of long-timebehavior of various dispersive equations, including NLS, NLW, and other equations([2], see also [10], [12] and the references therein). Higher order equations are asso-ciated to surfaces whose curvature degenerate on some lower dimensional sets, andthus it may be of use to develop the concentration-compactness theory on lowerdimensional manifolds (some preliminary work on this connection is in [16]). Thuswe begin this study with a manifold at the opposite dimensional extreme. As wewill see later on, resolving our problem in the full exponent range makes inacces-sible certain tools that are available in previously considered cases. We encounteragain the lack of Hilbert space structure of L p , which had been previously addressedin [18], and now must also contend with the failure of an effective analogue of thebilinear approach introduced by B´egout–Vargas in [1] in certain exponent ranges. Date : January 12, 2021.
Key words and phrases.
Fourier extension, Fourier restriction, localization, profile decomposi-tion, extremizer.This work is supported in part by NSF grant DMS-1653264.
We now turn to the precise formulation of our results, for which we introducesome notation for the various symmetries of the operator E : L p → L q , by whichwe mean elements S of the automorphism group of L p ( R ) for which there existcorresponding elements T of the automorphism group of L q ( R d ) obeying E ◦ S = T ◦E . The key symmetries for our analysis are the dilations, the frequency translations,and the modulations. More precisely, the dilations are given by: f f λ := λ − p f ( λ − · ) , E f λ ( x ) = λ d ( d +1)2 q E f ( D λ ( x )) , D λ ( x ) = ( λx , . . . , λ d x d );The translations are given by: f τ t f := f ( · − t ) , E ( τ t f )( x ) = L t ( E f )( x ) , where L t is the boost L t g ( x ) := e ix · γ ( t ) g ( A Tt x ), and A t is the unique element in GL ( d ) for which γ ( t + t ) = γ ( t ) + A t γ ( t ) ( A t is lower triangular, with ones onthe diagonal); The modulations are given by:( m x f )( t ) := e − ix · γ ( t ) f ( t ) , E ( m x f ) = τ x E f. Let 1 ≤ p < d + d +22 , and set B p := kEk L p → L q . We say that f ∈ L p is anextremizer of (the L p ( R ) → L q ( R d ) inequality for) E if f kE f k q = B p k f k p .We say that { f n } ⊆ L p is an extremizing sequence for (the L p → L q inequality for) E if f n n and lim n →∞ kE f n k q k f n k p = B p . We are most interested in normalizedextremizing sequences, that is, extremizing sequences { f n } with k f n k p = 1, for all n . Our main result is the following. Theorem 1.1.
For d ≥ there exist extremizers of the L p ( R ) → L q ( R d ) inequalityfor E for every ( p, q ) satisfying ≤ p < d + d +22 and q = d ( d +1)2 p ′ . Moreover,when p > , given any extremizing sequence of E , there exists a subsequence thatconverges to an extremizer in L p ( R ) , after the application of a suitable sequence ofsymmetries. This immediately yields a related result for the corresponding restriction opera-tor.
Corollary 1.2.
The analogous result holds for the restriction operator R g ( t ) := b g ( γ ( t ))) ; namely, for d ≥ there exist extremizers of the L r ( R d ) → L s ( R ) inequalityfor R for every ( r, s ) satisfying ≤ r < d + d +2 d + d and r ′ = d ( d +1)2 s . Moreover, when r > , given any extremizing sequence of R , there exists a subsequence that, afterthe application of a suitable sequence of symmetries, converges in L r ( R ) to anextremizer. We give the short proof of the corollary now.
Proof.
In the case r = 1, the first conclusion is elementary, and so we assumethat r >
1. Let { g n } be an L r -normalized extremizing sequence of R . Let f n := B − ( s − s ′ |R g n | s − R g n . Then lim n k f n k s ′ = 1. On the other hand, by duality, B s ′ = B − ( s − s ′ lim n kR g n k ss = lim n h g n , E f n i ≤ k g n k r kE f n k r ′ ≤ B s ′ , so { f n } is an extremizing sequence for E , the norms of whose elements tend to 1.Applying Theorem 1.1 to {k f n k − s ′ f n } , there exist symmetries { S n } of E such thatalong a subsequence, { S n f n } converges in L s ′ to some extremizer f for E . Let T n denote the corresponding L r ′ automorphism, that is, T n ◦ E = E ◦ S n . Since S n f n = XISTENCE OF EXTREMIZERS FOR FOURIER RESTRICTION TO THE MOMENT CURVE3 B − ( s − s ′ |R T n g n | s − R T n g n , replacing { g n } by { T n g n } if necessary, we may assumethat S n equals the identity for all n and that { f n } converges to f . By Banach-Alaoglu, { g n } converges weakly to some g ∈ L r and so k g k r ≤ lim n k g n k r = 1. Onthe other hand B s ′ = lim n h g n , E f n i = h g, E f i ≤ B s ′ k g k r , so k g k r = 1. By Theorem 2 .
11 in [13] { g n } converges in L r to g , from which weadditionally conclude that g is an extremizer of R . (cid:3) Notation.
We write A . B to denote A ≤ CB where C may depend on thedimension d and the exponent p , and whose value may change from one line to thenext but is independent of A and B . For the rest of the article we assume that d ≥
2. 2.
A refined extension estimate
The purpose of this section is to prove two refinements (Propositions 2.1 and 2.11)of Drury’s theorem, both of which will be used in the proof of Theorem 1.1.
Proposition 2.1.
Let < p < d + d +22 , and let q := d ( d +1)2 p ′ . There exist < θ = θ p < and c p > such that kE f k q . (cid:18) sup k ∈ Z sup I ∈D k sup n ≥ − c p n k f nI k p (cid:19) − θ k f k θp , f ∈ L p . (2.1) Here D k denotes the set of all dyadic intervals of length k , and f nI := f χ I χ {| f | < n k f k p | I | − p } . Proof of Proposition 2.1.
We will prove the proposition in two steps : First, in therange p < d + 2, we will prove the inequality (2.1) by using a multilinear extensionestimate and a variant of an argument of B´egout–Vargas [1]; then, we will adapt realinterpolation methods to deduce this bound for larger values of p . The significanceof p < d + 2 is that it ensures that ( qd ) ′ > pd , which allows for a d -linear-to-linearvariant of the bilinear-to-linear argument of Tao–Vargas–Vega [19]. We start withthe following lemma. Lemma 2.2.
Let I , . . . , I d be intervals of length one, and assume that there existssome k , ≤ k < d , such that for all j ≤ k and j ′ > k , dist( I j , I j ′ ) & . Then for f j supported on I j and q > d + d +22 , k d Y j =1 E f j k qd . d Y j =1 k f j k s , s := ( qd ) ′ . (2.2) Proof.
Changing variables, d Y j =1 E f j ( x ) = Z F ( ξ ) e iξ · x dξ where ξ = P dj =1 γ ( t j ), t < · · · < t d , F ( ξ ) = P σ ∈ S d Q dj =1 f σ ( j ) ( t j ) Q i We set a := ( qd ) ′ . Since a < (cid:0) k d Y j =1 E f j k qd (cid:1) a . k F k aa = Z d Y j =1 | f j | a ( t j ) Y i Lemma 2.3. Let q > d + d +22 and let I , . . . , I d be intervals of length r > , andassume that there exists some k , ≤ k < d , such that for j ≤ k < j ′ , dist( I j , I j ′ ) & r . Then for functions f j supported on I j , ≤ j ≤ d , k d Y j =1 E f j k qd . r d ( s ′ − d ( d +1)2 q ) d Y j =1 k f j k s , s := ( qd ) ′ . (2.4) (cid:3) Our next step will be to use a modified version of the Whitney decomposition,and almost orthogonality argument of [19].Consider the line ∆ := { ( t, . . . , t ) : t ∈ R } and the annular tubes T r := { ξ : r ≤ dist( ξ, ∆) ≤ r } . We cover T n with axis-parallel dyadic cubes of side length 2 n − K d ,with K d sufficiently large for later purposes. Let Q denote the (finitely overlapping)collection of all such dyadic cubes, and let Q n denote the subcollection consistingof those cubes having sidelength 2 n . For Q ∈ Q n and c > 0, we let cQ denote thecube with the same center as Q and sidelength c n , and c · Q denote the image of Q under the dilation ξ cξ .Define Γ( t ) := P dj =1 γ ( t j ). Then ∆ equals the preimage Γ − ( d · γ ) (Indeed Γ isknown to be one-to-one on each of the sets { t σ (1) ≤ · · · ≤ t σ ( d ) } , with σ ∈ S d apermutation). Lemma 2.4. Let Q ∈ Q n , Q ′ ∈ Q n ′ , and assume that Γ(5 Q ) ∩ Γ(5 Q ′ ) = ∅ . Then | n − n ′ | . and dist( Q, Q ′ ) . n .Proof. We may write Q = 2 n · ( Q l + ~k ) and Q ′ = 2 n ′ · ( Q l ′ + ~k ′ ), where k, k ′ , l, l ′ ∈ Z d , ~k = ( k, . . . , k ) , ~k ′ = ( k ′ , . . . , k ′ ), with l, l ′ ∈ [ − N, N ] d \ ([ − M, M ′ ] d + ∆), and Q l := [0 , d + l, Q l ′ = [0 , d + l ′ . XISTENCE OF EXTREMIZERS FOR FOURIER RESTRICTION TO THE MOMENT CURVE5 Here N, M are large constants that depend on K d . We define ρ ( ξ ) := d X i =1 | ξ i | i , δ ( ξ ) := min t ∈ R ρ ( A − t ( ξ − dγ ( t ))) , and let t ( ξ ) denote the minimum of all t with δ ( ξ ) = ρ ( A − t ( ξ − dγ ( t ))). (By basiccalculus, we can see that these minima are attained.) For ξ ∈ Γ(5 Q l ), | t ( ξ ) | . δ ( ξ ) ∼ 1. This is because Γ(5 Q l ) is compact and does not intersect d · γ , provided K d is sufficiently large. Applying our symmetries, for ξ ∈ Γ(5 Q ), | t ( ξ ) − k | . n and δ ( ξ ) ∼ n . Analogous estimates hold for ξ ∈ Γ(5 Q ′ ), so if Γ(5 Q ) and Γ(5 Q ′ )intersect, we must have that | n − n ′ | . | k − k ′ | . n , yielding the lemma. (cid:3) Lemma 2.5. There exist collections { ψ Q } Q ∈Q , { φ Q } Q ∈Q of smooth functions with supp φ Q ⊆ Γ(5 Q ) , such that φ Q ≡ on the support of ψ Q , P Q ψ Q ≡ on Γ( R d \ ∆) ,and { b ψ Q } Q ∈Q , { b φ Q } Q ∈Q are bounded in L .Proof. As in the proof of the previous lemma, we write Q ∈ Q as Q = 2 n · ( Q l + ~k ) with ~k = ( k, . . . , k ) and l in [ − N, N ] d \ ([ − M, M ′ ] d + ∆). Then Γ( cQ ) = D n ( A k Γ( cQ l ) + dγ ( k )). Let S Q ( ζ ) := D n ( A k Γ( ζ ) + dγ ( k )). Let φ Q l denote aSchwartz function that is supported on Γ(5 Q l ) and is identically 1 on Γ(3 Q l ) and˜ φ Q l a Schwartz function that is supported on Γ(3 Q l ) and identically 1 on Γ( Q l ).Define φ Q ( ξ ) := φ Q l ( S − Q ( ζ )), and ˜ φ Q analogously and set ψ Q := ˜ φ Q P Q ′∈Q ˜ φ Q ′ . Bythe previous lemma only a bounded number of terms (independent of the size of Q ) are non-zero in the sum in the denominator of ψ Q , and all these are smooth onthe scale of Q . Thus { φ Q ◦ S Q } and { ψ Q ◦ S Q } are bounded in the Schwartz class,hence { b φ Q } and { b ψ Q } are bounded in L . (cid:3) We will use the above lemma in the proof of the following lemma. By Lemma 2.3and the almost orthogonality lemma [19, Lemma 6.1], we obtain the following. Lemma 2.6. If q > d + d +22 , then kE f k dq . (cid:0)X n X I ∈D n ntd ( s ′ − d ( d +1)2 q ) k f I k dts (cid:1) t , where s := ( qd ) ′ , t := ( qd ) ′ , D n is the collection of all dyadic intervals of length n ,and f I := f χ I .Proof. Write 5 Q , Q ∈ Q as Q dj =1 I j . Then kE f k dq = k X Q ∈Q ( E f ) d ∗ b ψ Q k qd . (cid:0) X Q ∈Q k ( E f ) d ∗ b φ Q k t qd (cid:1) t (2.5) . (cid:0)X n X Q ∈Q n k d Y j =1 E f I j k t qd (cid:1) t . (cid:0)X n X Q ∈Q n ntd ( s ′ − d ( d +1)2 q ) d Y j =1 k f I j k ts (cid:1) t , where the first inequality uses almost orthogonality [19, Lemma 6.1], second in-equality uses the support condition on φ Q and then Young’s inequality, and thelast inequality follows from Lemma 2.3. The lemma follows because if Q ∈ Q n , S I j is covered by a bounded number of intervals in D n , and each dyadic intervalarises in only a bounded number of such coverings. (cid:3) CHANDAN BISWAS AND BETSY STOVALL Definition 2.7. We define a family of Banach spaces X p,q,r,s with norms k f k X p,q,r,s := (cid:0)X n (cid:0) X I ∈D n nr ( p − s ) k f k rL s ( I ) (cid:1) qr (cid:1) q . Then Lemma 2.6 states that kE f k q . k f k X p,dt,dt,s , for q > d + d +22 , p = ( qd + d ) ′ , t = ( qd ) ′ , s = ( qd ) ′ . Lemma 2.8. Assume that < s < p < r ≤ q < ∞ . Then L p ⊆ X p,q,r,s . Moreover,there exist c > , θ > such that if f ∈ L p with k f k p = 1 , then k f k X p,q,r,s . sup k ≥ sup I − c k k f kI k − θp k f k θp . Here the supremum is taken over all dyadic intervals I and f kI := f χ I χ {| f |≤ k | I | − p } . We note that this immediately implies Proposition 2.1 in the range p < d ( qd ) ′ ,i.e. when p < d + 2. Proof of Lemma 2.8. We will prove the superficially stronger estimate wherein wedenote f I := f χ I χ {| f |≤| I | − p } , f kI := f χ I χ { k − | I | − p < | f |≤ k | I | − p } . Thus f I := f χ I = P k ≥ f kI . By H¨older’s inequality, k f k X p,q,r,s = (cid:0)X n (cid:0) X I ∈D n nr ( p − s ) ( X k ≥ k f kI k ss (cid:1) rs (cid:1) qr (cid:1) q ≤ sup n sup I ∈D n sup k ≥ − c k (2 n ( p − s ) k f kI k s (cid:1) − θ × (cid:0)X n (cid:0) X I ∈D n nr ( p − s ) θ (cid:0)X k ≥ c ks k f kI k sθs (cid:1) rs (cid:1) qr (cid:1) q . By H¨older’s inequality, 2 n ( p − s ) k f kI k s ≤ k f kI k p , so it remains to bound the secondterm in the product on the right hand side.We begin with the k = 0 term. Since r > p , we may choose θ < rθ > p > s . Then using H¨older’s inequality repeatedly and finallysumming a geometric series, (cid:0)X n (cid:0) X I ∈D n nrθ ( p − s ) k f I k θrs (cid:1) qr ) q ≤ (cid:0)X n (cid:0) X I ∈D n nrθ ( p − rθ ) k f I k rθrθ (cid:1) qr (cid:1) q ≤ (cid:0)X n nrθ ( p − rθ ) k f χ {| f |≤ − np } k rθrθ (cid:1) r = (cid:0)Z | f | rθ X n < | f | − p nrθ ( p − rθ ) (cid:1) r = k f k pr p . Now we turn to the k ≥ c < c < c < ( p − s ) θs . Then severalapplications of H¨older’s inequality and the triangle inequality give (cid:0)X n (cid:0) X I ∈D n nr ( p − s ) θ (cid:0)X k ≥ c ks k f kI k sθs (cid:1) rs (cid:1) qr (cid:1) q XISTENCE OF EXTREMIZERS FOR FOURIER RESTRICTION TO THE MOMENT CURVE7 . (cid:0)X n (cid:0) X I ∈D n nrθ ( p − s ) X k ≥ c kr k f kI k rθs (cid:1) qr (cid:1) q . (cid:0)X k ≥ c kq X n nqθ ( p − s ) (cid:0) X I ∈D n k f kI k rθs (cid:1) qr (cid:1) q ≤ (cid:0)X k ≥ c kq (cid:0)X n nrθ ( p − s ) X I ∈D n k f kI k rθs (cid:1) qr (cid:1) q ≤ (cid:0)X k ≥ c kq (cid:0)X n ns ( p − s ) X I ∈D n k f kI k ss (cid:1) qθs (cid:1) q ≤ (cid:0)X k ≥ c kq (cid:0)X n ns ( p − s ) Z {| f |∼ − np k } | f | s (cid:1) qθs (cid:1) q ∼ (cid:0)X k ≥ c kq (cid:0) − k ( p − s ) k f k pp (cid:1) qθs (cid:1) q ∼ k f k pθs p . (cid:3) In the case of larger p (i.e., d + 2 ≤ p < d + d +22 ), we will (roughly speaking)interpolate the bound in Proposition 2.1, now established for sufficiently small p ,with Drury’s estimate kE f k q . k f k p . The details of this deduction are given in thenext two lemmas. Lemma 2.9. Let < p < d + d +22 and q = d ( d +1)2 p ′ . Let f ∈ L p and write f = P n n f n , where f n := 2 − n f χ E n , and E n := { n ≤ | f | < n +1 } . Then kE f k q . sup n kE n f n k νq k f k − νp , for some < ν < , depending only on p .Proof. We will prove the lemma by slightly adapting the proof of the Stein–Weissinterpolation theorem [17]. Write p = − θp + θp , for some 1 < p < p < p < d + d +22 ; set q i := d ( d +1)2 p ′ i , i = 0 , 1. Set ν := min i =0 , /p i − /q i /p i − /q i . Note that 0 < ν < g ∈ L q ′ and decompose g analogously to f : g = P n n g n , where g n =2 − n gχ F n , F n := { n ≤ | g | < n +1 } . We may assume that k f k p = k g k q ′ = 1 andthus P n np | E n | ∼ P m mq ′ | F m | ∼ f n , g m , hE f, g i . X n,m h n E f n , m g m i≤ X n,m k n E f n k νq k m g m k νq ′ min i =0 , k n E f n k − νq i k m g m k − νq ′ i . sup n k n E f n k νq X n,m min i =0 , k n f n k − νp i k m g m k − νq ′ i CHANDAN BISWAS AND BETSY STOVALL . sup n k n E f n k νq X n,m (2 n + m min i =0 , | E n | pi | F m | q ′ i ) − ν . It remains to bound the sum on the right side of this inequality.Were it the case that 2 n | E n | p = 2 m | F m | q ′ = 1, | E n | p | F m | q ′ ≤ | E n | p | F m | q ′ would hold if and only if nA ≤ − mB , where A := p ( p − p ) , B := q ′ ( q ′ − q ′ ) . In any case, X n,m (2 n + m min i =0 , | E n | pi | F m | q ′ i ) − ν ≤ X nA + mB ≤ (2 n + m | E n | p | F m | q ′ ) − ν + X nA + mB> (2 n + m | E n | p | F m | q ′ ) − ν (2.6)We begin with the first summand on the right of (2.6). Simple arithmetic,followed by H¨older’s inequality (since − νp + − νq ′ ≥ 1) gives X nA + mB ≤ (2 n + m | E n | p | F m | q ′ ) − ν = X nA + mB ≤ θ ( nA + mB )(1 − ν ) (2 np | E n | ) − νp (2 mq ′ | F m | ) − νq ′ ≤ X k ≤ θk (1 − ν ) X n (2 np | E n | ) − νp X ⌈ nA + mB ⌉ = k (2 mq ′ | F m | ) − νq ′ = X k ≤ θk (1 − ν ) X n (2 np | E n | ) − νp X k − nA − B Lemma 2.10. Proposition 2.1 holds for functions | f | ∼ λχ E , for λ > and E ameasurable subset of R , with bounds independent of λ, E .Proof. We may assume that λ = 1. Choose p , p , q , q , θ as in the proof ofLemma 2.9, with the additional assumption that p < d +2. By H¨older’s inequality,then the remark following Lemma 2.8 and Drury’s theorem, kE f k q ≤ kE f k − θq kE f k θq . (sup k ∈ Z sup I ∈D k sup n ≥ − c p n k ( χ E ) nI k p ) θ ′ k χ E k − θ − θ ′ p k χ E k θp , where 0 < θ ′ < − θ arises by applying Proposition 2.1 with exponents p , q (whereit has already been established). XISTENCE OF EXTREMIZERS FOR FOURIER RESTRICTION TO THE MOMENT CURVE9 The proof is now just a matter of unwinding the definition of ( χ E ) nI and per-forming some arithmetic. Observe that k ( χ E ) nI k p = ( | E ∩ I | p , if 1 < n | E | p | I | − p , , otherwise , and analogously with p in place of p . Thus we may rewritesup k ∈ Z sup I ∈D k sup n ≥ − c p n k ( χ E ) nI k p = sup k ∈ Z sup I ∈D k min { , (cid:0) | E | k (cid:1) cp p }| E ∩ I | p ∼ (cid:0) sup k ∈ Z sup I ∈D k sup n ≥ − c p n k ( χ E ) nI k p (cid:1) pp . Finally, we obtain kE f k q . (sup k ∈ Z sup I ∈D k sup n ≥ − c p n k ( χ E ) nI k p ) ϑ k χ E k − ϑp ∼ (sup k ∈ Z sup I ∈D k sup n ≥ − c p n k f nI k p ) ϑ k f k − ϑp , where ϑ = θ ′ pp . Note that 0 < ϑ < (1 − θ ) pp < (cid:3) Proposition 2.1 in the cases p ≥ d + 2 follows by first applying Lemma 2.9, thenapplying Lemma 2.10 to the supremum term in the conclusion of Lemma 2.9, andfinally observing that, in the decomposition in Lemma 2.9, | (2 m f m ) nI | ≤ | ( f ) nI | , for all integers m, n and intervals I . (cid:3) The following proposition is somewhat easier to use, though it only applies in amore limited range of exponents. The proof requires only a small modification inthe argument leading to Proposition 2.1. Proposition 2.11. Let < p < d + 2 , and let q := d ( d +1)2 p ′ . There exists < θ = θ p < such that for f ∈ L p , kE f k q . (cid:0) sup I | I | − p ′ kE f I k ∞ (cid:1) − θ k f k θp . (2.7) Here, the supremum is taken over dyadic intervals I , and f I := f χ I .Proof. Let q < q with θ := q q sufficiently close to 1 for later purposes. By (2.5),H¨older, Lemma 2.3, arithmetic, and another application of H¨older, kE f k q . (cid:0)X n X Q = Q I j ∈Q n k d Y j =1 E f I j k t qd (cid:1) td . (cid:0)X n X Q = Q I j ∈Q n k d Y j =1 E f I j k t (1 − θ ) ∞ k d Y j =1 E f I j k tθ q d (cid:1) td . (cid:0)X n X Q = Q I j ∈Q n k d Y j =1 E f I j k t (1 − θ ) ∞ ndtθ ( p − s ) d Y j =1 k f I j k tθs (cid:1) td ≤ (cid:0)X n X Q = Q I j ∈Q n (2 nd ( p − max j =1 ,...,d kE f I j k d ∞ ) t (1 − θ ) ndtθ ( p − s ) d Y j =1 k f I j k tθs (cid:1) td ≤ (cid:0) sup I | I | p − kE f I k ∞ (cid:1) − θ k f k θX p,dtθ,dtθ,s . Here s := ( q d ) ′ , t := ( qd ) ′ . For p < d + 2 and θ sufficiently close to 1, 1 < s
In this section we prove that any near extremizer of E is uniformly bounded andis supported on a compact set around the origin possibly after applying symmetry,if we allow ourselves to lose a small amount of L p -mass. Below is the precisestatement. Proposition 3.1. Let < p < d + d +22 , and let q := d ( d +1)2 p ′ . For each ǫ > ,there exist δ > and R < ∞ such that for each nonzero function f satisfying kE f k q ≥ B p (1 − δ ) k f k p , there exists a symmetry S such that the following holds. k Sf k L p (cid:0) {| t | >R }∪{| Sf | >R k f k p } (cid:1) < ǫ k f k p . We start with the following lemma. Lemma 3.2. Let < p < d + d +22 , and q := d ( d +1)2 p ′ . There exists a sequence ρ k → such that for every f ∈ L p ( R ) , there exists a sequence { I k } of dyadicintervals such that if { f >k } is inductively defined by f > := f, f k := f >k − χ {| f | < k | I k | − p k f k p } χ I k , f >k := f >k − − f k , (3.1) then for any measurable function | h >k | ≤ | f >k | , kE h >k k q ≤ ρ k k f k p . Proof. Let 0 f ∈ L p . Multiplying by a constant if needed, we may assume that k f k p = 1. By the Dominated Convergence Theorem, given f >k − , we choose adyadic I k to maximize k f k k p . With the sequence { f k } and { f >k } as defined in 3.1,let K ∈ N and set A K := sup I dyadic k ( f > K ) KI k p . By the maximality property of the I k , A K ≤ k f K + j k p , for each 0 ≤ j ≤ K . Henceby the disjointness of the supports of the f k ’s, KA pK ≤ K X j =1 k f K + j k pp ≤ . By Proposition 2.1, for any measurable function | h > K | ≤ | f > K | , kE h > K k q . max { − c p θ p K , K θp } . K − θpp . This completes the proof with ρ k := C ( k ) − θpp with θ p as in Proposition 2.1 and C a sufficiently large constant. (cid:3) Proof of Proposition 3.1. Since we can always take δ < , it suffices to considerthose f for which k f k p = 1 and kE f k q > B p k f k p . Let I k,f denote the dyadicintervals from Lemma 3.2 and set f ≤ k := f − f >k , and so f k = f >k − − f >k . XISTENCE OF EXTREMIZERS FOR FOURIER RESTRICTION TO THE MOMENT CURVE11 By Lemma 3.2, k f k f k p & k f . 1. Applying a symmetry if needed, wemay assume that I k f ,f is the unit interval. We will prove (under these assumptionson f ) that the conclusion of the proposition holds with S equal to the identity.If the conclusion were to fail, there would exist some ε > { f n } ⊆ L p with k f n k p = 1, k n := k f n . k f k n n k p & I k n ,f n = [0 , kE f n k q ≥ B p (1 − n − ), and k f n k L p (cid:0) {| t | >n }∪{| f n | >n } (cid:1) > ε. By pigeonholing and passing to a subsequence, we may assume that k n = k . n . Write I kn := I k,f n =: ξ kn + [0 , ℓ kn ] , k, n ∈ N . Passing to a subsequence, we may assume that { ξ kn } and { ℓ kn } converge in [ −∞ , ∞ ]and [0 , ∞ ], respectively, for each k and that {k f kn k p } converges for all k . For each k , say that the k is negligible if k f kn k p → 0, that the k is good if it is negligible or if { ξ kn } converges in R and { ℓ kn } converges in (0 , ∞ ). Say that the k is bad if it is notgood. For bad k : say k is long if ℓ kn → ∞ as n → ∞ , short if ℓ kn → n → ∞ ;otherwise it must be far , i.e. | ξ kn | → ∞ .We will prove that every k is good. Assuming this for now, we complete theproof of the proposition. Since every k is good, for each fixed K and sufficientlylarge n (depending on K ), | f n | χ ( {| t | >n }∪{| f n | >n } ) ≤ | f >Kn | . Therefore lim inf n →∞ k f >Kn k p > ε, for every K . Since the supports of the f ≤ Kn and f >Kn intersect on a set of measurezero, lim sup n →∞ k f ≤ Kn k p = (cid:0) k f n k pp − k f >Kn k pp ) p < (1 − ε p ) p . We conclude that kE f ≤ Kn k q < B p (1 − ε p ) p . On the other hand, by Lemma 3.2, kE f >Kn k q ≤ ρ K . Therefore kE f ≤ Kn k q ≥ B p − ρ K . For K sufficiently large, dependingon ε , B p (1 − ε p ) p > B p − ρ K , leading to a contradiction.It remains to prove that every k is good. Define g ≤ Kn := X k ≤ K, good f kn , b ≤ Kn := X k ≤ K, bad f kn . Then f ≤ Kn = g ≤ Kn + b ≤ Kn , and the supports of g ≤ Kn and b ≤ Kn have measure zerointersection and thus obey the L p orthogonality condition k f ≤ Kn k pp = k g ≤ Kn k pp + k b ≤ Kn k pp . Our next lemma shows that we have L q -orthogonality in limit. Lemma 3.3. For every ≤ K < ∞ , lim n →∞ kE f ≤ Kn k qq − (cid:0) kE g ≤ Kn k qq + kE b ≤ Kn k qq (cid:1) = 0 . Assuming Lemma 3.3, we complete the proof of the proposition by showing thatevery k is good. Suppose, by way of contradiction, that k is bad. By definition,lim sup n →∞ k f k n k p > ε , for some ε > 0. Passing to a subsequence, we may assumethat k f k n k p > ε for all n . Therefore k b ≤ Kn k p > ε for all K ≥ k and all n . Takinga smaller ε if needed, k g ≤ Kn k p ≥ k f k n k p > ε , for all K ≥ k and all n . We mayfurther assume that ε p < . We can use these L p estimates to bound the extensionfor sufficiently large K :lim sup n →∞ kE f ≤ Kn k qq ≤ lim sup n →∞ kE g ≤ Kn k qq + kE b ≤ Kn k qq ≤ lim sup n →∞ B qp (cid:0) k g ≤ Kn k qp + k b ≤ Kn k qp (cid:1) ≤ (cid:2) (1 − ε p ) qp + ε q (cid:3) B qp . For the last inequality, we have used that for g, b ≥ g p + b p ≤ b, g > ε , and ε p < , g q + b q ≤ (1 − ε p ) qp + ε q , which in turn follows from basic calculus.Crucially, (cid:2) (1 − ε p ) qp + ε q (cid:3) =: c ε < 1. On the other hand, by our hypothesis on { f n } and Lemma 3.2, B p = lim n →∞ kE f n k q = lim K →∞ lim n →∞ kE f ≤ Kn k q . (cid:3) It remains to prove Lemma 3.3. Proof of Lemma 3.3. By elementary calculus, for all q > a, b ≥ | ( a + b ) q − a q − b q | . q (cid:0) ab q − + ba q − (cid:1) . Since q > d + d +22 > 2, we may apply this inequality to our good and bad part ofthe function to see that kE f ≤ Kn k qq − kE g ≤ Kn k qq − kE b ≤ Kn k qq . q,K X k ≤ K good X k ′ ≤ K bad Z |E f kn ||E f k ′ n | q − + |E f kn | q − |E f k ′ n | . Moreover, by H¨older’s inequality, and boundedness of the E f jn , Z |E f kn ||E f k ′ n | q − + |E f kn | q − |E f k ′ n | ≤ (cid:0) kE f kn k q − q + kE f k ′ n k q − q (cid:1) kE f kn E f k ′ n k q . kE f kn E f k ′ n k q , so it suffices to prove that kE f kn E f k ′ n k q → k is good and k ′ is bad.To this end, choose q + < q < q − with q ± = d ( d +1)2 p ′± and q = q + + q − . Therefore p − < p < p + . Because k is good, f kn remains bounded in L p ± (in fact, in everyLebesgue space) as n → ∞ . When the k ′ is short, k f k ′ n k p − → 0. Therefore kE f kn E f k ′ n k q ≤ kE f kn k q + kE f k ′ n k q − . k f k ′ n k p − → . Similarly when the k ′ is long, k f k ′ n k p + → kE f kn E f k ′ n k q . k f k ′ n k p + → . XISTENCE OF EXTREMIZERS FOR FOURIER RESTRICTION TO THE MOMENT CURVE13 Finally, suppose the k ′ is far (and neither short nor long). We set L := lim l kn +lim l k ′ n , and assume that n is sufficiently large so that l kn + l k ′ n ≤ L and | ξ kn − ξ k ′ n | ≥ L . By the arithmetic-geometric mean inequality, Lemma 2.3, and H¨older’sinequality with s := ( qd ) ′ < p , kE f kn E f k ′ n k q ≤ k ( E f kn ) d − E f k ′ n k qd + kE f kn ( E f k ′ n ) d − k qd . | ξ kn − ξ k ′ n | − d ( s − p ) ( k f kn k d − s k f k ′ n k s + k f kn k s k f k ′ n k d − s ) . (cid:0) L | ξ kn − ξ k ′ n | ) d ( s − p ) → . (cid:3) A profile decomposition for frequency localized sequences Proposition 4.1. Let q = d ( d +1)2 p ′ > p > . Let { f n } be a sequence of measurablefunctions with supp f n ⊆ [ − R, R ] and | f n | ≤ R for all n . Then, after passing toa subsequence, there exist { x jn } n,j ≥ ⊆ R d and { φ j } ⊆ L p such that the followinghold with w Jn := f n − J X j =1 e − ix jn · γ φ j . (i) lim n →∞ | x jn − x j ′ n | = ∞ , for all j = j ′ ; (ii) e ix jn · γ f n ⇀ φ j , weakly in L p , for all j ; (iii) lim n →∞ kE f n k qq − P Jj =1 kE φ j k qq − kE w Jn k qq = 0 for all J ; (iv) lim J →∞ lim n →∞ kE w Jn k q = 0 ; (v) (cid:0)P ∞ j =1 k φ j k ˜ pp (cid:1) p ≤ lim inf n →∞ k f n k p , where ˜ p := max { p, p ′ } .Remark . Here we allow for the possibility that the φ j with j sufficiently largemight all be identically zero.The essential step is finding a nonzero weak limit. Lemma 4.3. Let q = d ( d +1)2 p ′ > p > with p < d + 2 . There exists C > suchthat for any sequence { f n } ⊆ L p with | f n | ≤ Rχ [ − R,R ] , k f n k p ≤ A, and kE f n k q ≥ ε, there exists a sequence { x n } ⊆ R d such that, after passing to a subsequence, e ix n · γ f n ⇀ φ , weakly in L p , for some φ ∈ L p with k φ k p & ε ( εA ) C . Here theimplicit constant is independent of R .Proof. By (2.7), for each n , there exists a dyadic interval I n such that ε ≤ (cid:0) | I n | − p ′ kE ( f n ) I n k ∞ (cid:1) θ A − θ . (4.1)By H¨older’s inequality, | I n | − p ′ kE ( f n ) I n k ∞ ≤ C R min {| I n | − p ′ , | I n ∩ [ − R, R ] | p } . Hence, in the terminology of the previous section, { I n } cannot be long, short,nor far, and so, after passing to a subsequence, we may assume that I n = I isindependent of n .By (4.1), for each n , there exists x n ∈ R d such that ε (cid:0) εA ) − θθ . | I | − p ′ |E ( f n ) I ( x n ) | = | I | − p ′ |E ( e ix n · γ f n ) I (0) | . By boundedness of the sequence { f n } , after passing to a subsequence, e ix n γ f n ⇀φ , weakly in L p , for some φ ∈ L p . Along this same subsequence, we then have e ix n γ ( f n ) I ⇀ φ I . By compactness of their support, E φ I (0) = lim n →∞ E ( e ix n · γ f n ) I (0) . Therefore by H¨older’s inequality, ε (cid:0) εA ) − θθ . | I | − p ′ |E φ I (0) | ≤ | I | − p ′ k φ I k ≤ k φ I k p . (cid:3) The remainder of the section is devoted to the proof of Proposition 4.1. Weprove the proposition first in the case p = 2, and then in the general case. Proof of Proposition 4.1 when p = 2 . In the case p = 2 , q = q := d ( d + 1), we mayreplace (v) with the stronger condition that for all J ,(v’) lim n →∞ k f n k − J X j =1 k φ j k − k w Jn k = 0 . Suppose that we are given 1 ≤ J < ∞ and sequences { x jn } n ∈ N ,j 0. By (ii), | w J − n | ≤ J Rχ [ − R,R ] , and by (v’),lim sup k w J − n k ≤ lim sup k f n k =: A. Therefore, by Lemma 4.3, there exists { x J n } ⊆ R d and a subsequence along which e ix J n · γ w J − n ⇀ φ J weakly in L , with k φ J k & ε C . This immediately implies thatlim n →∞ k e ix J n · γ w J − n k − k φ J k − k e ix J n · γ w J − n − φ J k = 0 , so (v’) holds with J = J . By the compact support condition, E ( e ix J n · γ w J − n ) →E φ J , a.e., so by the Brezis–Lieb lemma,lim n →∞ kE ( e ix J n · γ w J − n ) k qq − kE φ J k qq − kE ( e ix J n · γ w J − n − φ J ) k qq = 0 . Therefore (iii) holds with J = J .Suppose that | x J n − x j n | 6→ ∞ , for some j < J . Passing to a subsequence, wemay assume that x J n − x j n → x . Then multiplication by e i ( x J n − x j n ) · γ convergesto multiplication by e ix · γ in the strong operator topology. Moreover, by (i), multi-plication by e i ( x J n − x jn ) · γ , j = j , converges to zero in the weak operator topology.Thereforewk-lim e ix J n · γ w J − n = wk-lim e i ( x J n − x j n ) · γ ( e ix j n · γ f n − φ j ) + X j = j e i ( x J n − x jn ) · γ φ j = 0 . On the other hand, the left hand side of the preceding equals φ J , which is nonzero,a contradiction. Tracing back, (i) holds for indices bounded by J . The proof of XISTENCE OF EXTREMIZERS FOR FOURIER RESTRICTION TO THE MOMENT CURVE15 (ii) is similar: Along our subsequence,wk-lim e ix J n · γ f n = wk-lim J − X j =1 e i ( x J n − x jn ) · γ φ j + e ix J n · γ w J − n = J − X j =1 φ J . It remains to verify (iv). Let ε J := lim sup n →∞ kE w Jn k q . If ε J > ε > k φ J k & ε C infinitely often, which is impossible by(v’). (cid:3) Having proved Proposition 4.1 in the case p = p := 2, q = q := d ( d + 1), weturn to the general case. Proof of Proposition 4.1 for d + d = q = d + d p ′ > p > . Fix an exponent ∞ >q > d + d +22 so that q lies strictly between q and q . As { f n } is bounded in L , we may apply the L -based profile decomposition to determine { x jn } , { φ j } .That (i) and (ii) hold are immediate. That (iii) holds follows from the Brezis–Lieblemma and the above argument in the case q = q . The Brezis–Lieb lemma alsoimplies that (iii) holds in the case q = q , so kE w Jn k q is bounded, uniformly in J (albeit with a constant that depends on R ). Choosing θ so that q = θq + − θq ,lim J →∞ lim n →∞ kE w Jn k q . R lim J →∞ lim n →∞ kE w Jn k − θq = 0 , so (iv) holds as well.It remains to prove (v). Let us fix J and let ǫ > 0. We choose compactlysupported smooth nonnegative functions a, b satisfying sup b = R a = 1 and k a ∗ ( bφ j ) − φ j k p < ǫ for all 1 ≤ j ≤ J . We define for each j ≤ Jπ jn f ( t ) := a ∗ s ( e ix jn · γ ( s ) b ( s ) f ( s ))( t ) . The weak limit condition (ii), compactness of the supports of a and b , and anapplication of the Dominated Convergence Theorem imply thatlim n k π jn f n − a ∗ ( bφ j ) k p = 0 . Letting ǫ to 0, it suffices to prove thatlim n k P n k L p → l ˜ p ( L p ) ≤ ≤ p ≤ ∞ (4.2)where ˜ p = max( p, p ′ ) and P n := ( π jn ) Jj =1 . Validity of (4.2) is elementary for p = 1 , ∞ . By complex interpolation it suffices to prove (4.2) for p = 2, whichis equivalent to proving lim n k P ∗ n k l ( L ) → L ≤ 1. Note that P ∗ n F = P Jj =1 ( π jn ) ∗ F j for F = { F j } ∈ l ( L ) and ( π jn ) ∗ F j ( t ) = b ( t ) e ix jn · γ ( t ) ( a ∗ F j )( t ). Thus it suffices toshow that k J X j =1 ( π jn ) ∗ F j k L ≤ (1 + o Jn (1)) J X j =1 k F j k L (where, of course, o Jn (1) is independent of F and lim n →∞ o Jn (1) = 0). Now k J X j =1 ( π jn ) ∗ F j k L ≤ J X j =1 k F j k L + 2 X j = j ′ |h π j ′ n ( π jn ) ∗ F j , F j ′ i| , and thus it is enough to prove that π j ′ n ( π jn ) ∗ → π j ′ n ( π jn ) ∗ g ( u ) = R K jj ′ n ( u, s ) g ( s ) ds where K jj ′ n ( u, s ) = Z a ( u − t ) e i ( x jn − x j ′ n ) · γ ( t ) [ b ( t )] a ( t − s ) dt. Since both a and b have compact support and supp u K jj ′ n ( u, s ) ⊂ supp a + supp b ,by stationary phase, lim n sup s k K jj ′ n ( u, s ) k L u = 0 . By identical reasoning lim n sup u k K jj ′ n ( u, s ) k L s = 0 . Therefore π j ′ n ( π jn ) ∗ : g R K jj ′ n ( u, s ) g ( s ) ds goes to 0 in the operator norm topol-ogy. This completes the proof. (cid:3) L p convergence We are finally ready to prove Theorem 1.1. Let { f n } be an extremizing se-quence and ε > 0. By Proposition 3.1 after applying an appropriate sequence ofsymmetries, there exists R = R ε such that for all sufficiently large n kE f Rn k q ≥ B p − ε. Restricting R to lie in the positive integers, we may apply Proposition 4.1 along asubsequence (which is independent of R ) to decompose f Rn = P Jj =1 e ix Rn · γ ( t ) φ jR + r JRn , J < ∞ . Furthermore since { f Rn } is nearly extremizing, for each R , there existssome large profile φ jR . Indeed, for all large nB qp − ε ≤ kE f Rn k qq − ε ≤ ∞ X j =1 kE φ jR k qq ≤ B ˜ pp (cid:0) ∞ X j =1 k φ jR k ˜ pp (cid:1) max j kE φ jR k q − ˜ pq ≤ B ˜ pp max j kE φ jR k q − ˜ pq ≤ B qp max j k φ jR k q − ˜ pp . (5.1)Denoting this large φ jR by φ R , by Proposition 4.1 there exists { x Rn } ⊆ R d such that { e ix Rn · γ f Rn } converges weakly in L p to some function φ R . Since lim R k φ R k p = 1 =lim n k f n k p , by strict convexity (see Theorem 2 . . R lim n k f Rn − e − ix Rn · γ φ R k p = 0 . By the triangle inequality and Proposition 3.1lim R lim n k f n − e − ix Rn · γ φ R k p = 0 . (5.2)Therefore for all sufficiently large R , R lim sup n k e − ix R n · γ φ R − e − ix R n · γ φ R k p = o min( R ,R ) (1) . If {| x R n − x R n |} was unbounded for some R , R , then after passing through asubsequence, multiplication by e i ( x R n − x R n ) · γ tends to zero in the weak operatortopology. Thus, by H¨older’s inequality,1 . k φ R k p = lim n | R ( e i ( xR n − xR n ) · γ φ R − φ R ) φ R | φ R | p − dt |k φ R k p − p XISTENCE OF EXTREMIZERS FOR FOURIER RESTRICTION TO THE MOMENT CURVE17 . lim sup n k e − ix R n · γ φ R − e − ix R n · γ φ R k p , which contradicts (5.2). Thus, for all sufficiently large R , R , {| x R n − x R n |} remainsbounded as n goes to infinity. Applying an appropriate sequence of modulationsto the { f n } , we may assume that { x Rn } is bounded for all R . After passing toa subsequence, each { x Rn } converges to some x R ∈ R for every sufficiently largeinteger R . 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