On the existence of maximizers for a family of Restriction Theorems
aa r X i v : . [ m a t h . A P ] J u l ON THE EXISTENCE OF MAXIMIZERS FOR A FAMILY OFRESTRICTION THEOREMS
LUCA FANELLI, LUIS VEGA, AND NICOLA VISCIGLIA
Abstract.
We prove the existence of maximizers for a general family of re-strictions operators, up to the end-point. We also provide some counterxam-ples in the end-point case.
In the sequel we shall denote by dµ any positive measure on R dξ . For every fixed dµ we define T µ : C (supp( dµ )) ∋ ˆ h ( ξ ) → Z e ix · ξ ˆ h ( ξ ) dµ ∈ C ∞ ( R dx )Given two Banach spaces X, Y we denote by L ( X, Y ) the space of linear and con-tinuous operators between X and Y . Definition 0.1.
A measure dµ on R dξ satisfies the restriction condition w.r.t. p ∈ [1 , ∞ ] (shortly ( RC ) p ) provided that T µ ∈ L ( L ( dµ ) , L p ( R dx )). Definition 0.2.
Assume that dµ satisfies ( RC ) p then we say that there is a max-imizer for T µ w.r.t. p provided that there exists ˆ h ∈ L ( dµ ) such that: k ˆ h k L ( dµ ) = 1and k T µ ˆ h k L p ( R dx ) = k T µ k L ( L ( dµ ) ,L p ( R dx )) . Definition 0.3.
Assume that dµ satisfies ( RC ) p then we say that ˆ h n ∈ L ( dµ ) isa maximizing sequence for T µ w.r.t. p provided that: k ˆ h n k L ( dµ ) = 1and lim n →∞ k T µ ˆ h n k L p ( R dx ) = k T µ k L ( L ( dµ ) ,L p ( R dx )) . We have the following
Theorem 0.1.
Let dµ be a positive compactly supported measure on R dξ and let p ( µ ) = inf { ≤ p ≤ ∞| ( RC ) p holds for dµ } . Then for every max { , p ( µ ) } < p ≤ ∞ there exists a maximizer for T µ w.r.t. p . More precisely for every maximizingsequence ˆ h n ( ξ ) for T µ w.r.t. p, there exists x n ∈ R d such that e ix n · ξ ˆ h n ( ξ ) is compactin L ( dµ ) . Mathematics Subject Classification.
Key words and phrases.
Fourier restriction Theorems, Strichartz estimates.
In order to treat the case p = ∞ we shall use the following general fact whoseproof is inspired by ([1],[4]). Proposition 0.1.
Let H be a Hilbert space and T ∈ L ( H , L p ( R d )) for a suitable p ∈ (2 , ∞ ) . Let { h n } n ∈ N ∈ H such that: (1) k h n k H = 1 ; (2) lim n →∞ k T h n k L p ( R d ) = k T k L ( H ,L p ( R d )) ; (3) h n ⇀ ¯ h = 0;(4) T ( h n ) → T (¯ h ) a.e. in R d .Then h n → ¯ h in H , in particular k ¯ h k H = 1 and k T (¯ h ) k L p ( R d ) = k T k L ( H ,L p ( R d )) .Remark . The main difference between Proposition 0.1 and Lemma 2.7 in [4]is that we only need to assume weak convergence in the Hilbert space H for themaximizing sequence h n . On the other hand the argument in [4] works for operatorsdefined between general Lebesgue spaces and not necessarily in the Hilbert spacesframework. Remark . We shall use Proposition 0.1 by choosing H = L ( dµ ). The mainpoint is that in the assumptions of Proposition 0.1 we do not assume a-priori thealmost everywhere convergence of the maximizing sequence (which in our concretecontext cannot be easily checked).Next result shows that in general Theorem 0.1 cannot be extended to the end-point case p = p ( µ ).For every M > dµ M = δ P M , P M = { ( ξ, | ξ | ) , ξ ∈ R , | ξ | ≤ M } ; dµ M = δ P M , P M = { ( ξ, | ξ | ) , ξ ∈ R , | ξ | ≤ M } ; dσ M = 1 p | ξ | δ C M , C M = ∪ ± { ( ξ, ±| ξ | ) , ξ ∈ R , | ξ | ≤ M } where we have denoted in general by δ S the flat measure on S . Remark . Notice that the restriction operators associated to the measures dµ M , dµ M , dσ M are strictly related to the Strichartz estimates associated respectively tothe Schr¨odinger equation in 1-D, 2-D and to the wave equation in 3-D (providedthat the initial data are localized in frequencies).We have the following Theorem 0.2.
The condition ( RC ) holds for dµ M and ( RC ) holds for dµ M and dσ M for every < M ≤ ∞ . However there are not maximizers for T µ M , T µ M , T σ M w.r.t. to p=6, p=4, p=4 (respectively) provided that M = ∞ .Remark . In [2] it is proved the existence of maximizers for the restriction onthe sphere S w.r.t. to p = 4 (which turns out to be the end-point value for therestriction on S ). In the best of our knowledge this is the unique result concern-ing existence of maximizers for the end-point restriction problem on a compactmanifold. N THE EXISTENCE OF MAXIMIZERS FOR A FAMILY OF RESTRICTION THEOREMS 3 Proof of Theorem 0.2
We work with dµ M (the same argument works for dµ M and dσ M ). Notice thatvalidity of ( RC ) for dµ M follows from the usual Strichartz estimates k e it ∆ f k L ( R ) ≤ C k f k L ( R ) . Moreover the maximization problemsup k ˆ g k L dµ M )=1 k T µ M (ˆ g ) k L ( R ) is equivalent to(1.1) sup k h k L R )=1 ,supp ˆ h ( ξ ) ⊂ ( − M,M ) k e it ∆ h k L ( R ) . On the other hand by an elementary rescaling argument we get:(1.2) sup k h k L R )=1 ,supp ˆ h ( ξ ) ⊂ ( − M,M ) k e it ∆ h k L ( R ) = sup k h k L R )=1 k e it ∆ h k L ( R ) . By the previous identity it is easy to deduce that if a maximizer exists for (1.1)then it is necessarily a maximizer for(1.3) sup k h k L R )=1 k e it ∆ h k L ( R ) but this is absurd since by [3] there are no maximizers for (1.3) which are compactlysupported in the Fourier variables.2. Proof of Proposition 0.1 and Theorem 0.1
Proof of Prop 0.1
By using the Br´ezis and Lieb Lemma (see [1]) we get: k T ( h n ) − T (¯ h ) k pL p ( R d ) = k T ( h n ) k pL p ( R d ) − k T (¯ h ) k pL p ( R d ) + o (1)and by the hypothesis (3) in the Proposition we get k h n − ¯ h k H = k h n k H − k ¯ h k H + o (1) . In particular since h n is by hypothesis a maximizing sequence for T we get(2.1) k T k L ( H ,L p ( R d )) = ( k T ( h n ) − T (¯ h ) k pL p ( R d ) + k T (¯ h ) k pL p ( R d ) + o (1)) p k h n − ¯ h k H + k ¯ h k H + o (1) ≤ ( k T ( h n ) − T (¯ h ) k L p ( R d ) + k T (¯ h ) k L p ( R d ) + o (1)) k h n − ¯ h k H + k ¯ h k H + o (1)where we have used the inequality( a + b + c ) t ≤ a t + b t + c t ∀ a, b, c > t ≤
1. The estimate above implies(2.2) k T k L ( H ,L p ( R n )) ≤ ( k T k L ( H ,L p ( R n )) k h n − ¯ h k L p ( R n ) + k T (¯ h ) k L p ( R n ) + o (1)) k h n − ¯ h k H + k ¯ h k H + o (1)and hence k T k L ( H ,L p ( R d )) ( k h n − ¯ h k H + k ¯ h k H + o (1)) ≤ ( k T k L ( H ,L p ( R d )) k h n − ¯ h k L p ( R d ) + k T (¯ h ) k L p ( R d ) + o (1)) LUCA FANELLI, LUIS VEGA, AND NICOLA VISCIGLIA which is equivalent to k T k L ( H ,L p ( R d )) ( k ¯ h k H + o (1)) ≤ ( k T (¯ h ) k L p ( R d ) + o (1)) . In particular the previous estimate implies k T k L ( H ,L p ( R d )) ≤ (cid:13)(cid:13)(cid:13) T (cid:16) ¯ h k ¯ h k H (cid:17)(cid:13)(cid:13)(cid:13) L p ( R d ) anddue to the definition of k T k L ( H ,L p ( R d )) it implies easily the following(2.3) k T k L ( H ,L p ( R d )) k ¯ h k H = k T (¯ h ) k L p ( R d ) . On the other hand by (2.1) we can deduce(2.4) k T k L ( H ,L p ( R n )) ≤ ( k T k L ( H ,L p ( R n )) k ¯ h k H + k T ( h n − ¯ h ) k L p ( R n ) + o (1)) k h n − ¯ h k H + k ¯ h k H + o (1)and we easily get k T k L ( H ,L p ( R d )) ( k h n − ¯ h k H + o (1))(2.5) ≤ ( k T ( h n − ¯ h ) k L p ( R d ) + o (1)) . Notice that either k h n − ¯ h k H = o (1) (and in this case we can conclude) or (up tosubsequence) inf n ∈ N k h n − ¯ h k H ≥ ǫ > . In particular by (2) we get k T k L ( H ,L p ( R d )) ≤ (cid:13)(cid:13)(cid:13)(cid:13) T (cid:18) h n − ¯ h k h n − ¯ h k H (cid:19)(cid:13)(cid:13)(cid:13)(cid:13) L p ( R d ) + o (1))which by definition of k T k L ( H ,L p ( R d )) necessarily implies k T k L ( H ,L p ( R d )) = (cid:13)(cid:13)(cid:13)(cid:13) T (cid:18) h n − ¯ h k h n − ¯ h k H (cid:19)(cid:13)(cid:13)(cid:13)(cid:13) L p ( R d ) + o (1))and equivalently(2.6) k T k L ( H ,L p ( R d )) k h n − ¯ h k H = k T ( h n − ¯ h ) k L p ( R d ) + o (1)) . By combining the first identity in (2.1) with (2.3) and (2.6) we get(2.7) 1 = ( k h n − ¯ h k p H + k ¯ h k p H + o (1)) p k h n − ¯ h k H + k ¯ h k H + o (1)i.e. ( k h n − ¯ h k p H + k ¯ h k p H ) p = k h n − ¯ h k H + k ¯ h k H + o (1) . Since we are assuming p ∈ (2 , ∞ ) it is easy to deduce by a convexity argument thatthe previous inequality implies k ¯ h k H = 1 and k h n − ¯ h k H = o (1) (actually we haveexcluded the possibility k ¯ h k H = 0 and k h n − ¯ h k H = 1 + o (1) since by assumption¯ h = 0). (cid:3) Proof of Thm 0.1The case p = ∞ Let ˆ h n ∈ L ( dµ ) be a maximizing sequence for T µ w.r.t. p (where p is as inthe assumptions). First step : there is a sequence x n ∈ R d such that ˆ g n ( ξ ) = e ix n · ξ ˆ h n ( ξ ) has a weaklimit different from zero in L ( dµ ) N THE EXISTENCE OF MAXIMIZERS FOR A FAMILY OF RESTRICTION THEOREMS 5
In order to verify this property we prove that there is x n such that T µ (( e ix n · ξ ˆ h n ( ξ )) = τ x n T µ (ˆ h n ( ξ ))has a weak limit different from zero (here τ y denotes the translation of vector y ).Notice that by definition we have(2.8) k T µ ˆ h n k L p ( R nx ) → k T µ k L ( L ( dµ ) ,L p ( R nx )) > . By using the ( RC ) ¯ p condition for a suitable p ( µ ) < ¯ p < p we get k T µ ˆ h n k L ¯ p ( R nx ) ≤ k T k L ( L ( dµ ) ,L ¯ p ( R nx )) k ˆ h n ( ξ ) k L ( dµ ) and hence(2.9) sup n ∈ N k T µ ˆ h n k L ¯ p ( R dx ) ≡ S < ∞ . Next notice that we have the following inequality: k T µ ˆ h n k L p ( R dx ) ≤ k T µ ˆ h n k θL ¯ p ( R dx ) k T µ ˆ h n k − θL ∞ ( R dx ) where p = θ ¯ p . By combining this fact with (2.8) and (2.9) we deduce(2.10) k T µ ˆ h n k L ∞ ( R dx ) ≥ ǫ > . Notice also that we have (by compactness of the support of dµ ) k T µ ˆ h n k L ∞ ( R dx ) ≤ k ˆ h k L ( dµ ) p k dµ k and k∇ x T µ ˆ h n k L ∞ ( R dx ) = k T µ ( iξ ˆ h n ) k L ∞ ( R dx ) ≤ k ξ ˆ h n k L p k dµ k ≤ p k dµ k sup ξ ∈ supp ( µ ) | ξ | ! k ˆ h n k L ( dµ ) (where k dµ k = R dµ ). Hence(2.11) sup n ∈ N k T µ ˆ h n k W , ∞ ( R dx ) < ∞ . By (2.10) there exist x n such that | T µ ˆ h n ( x n ) | ≥ ǫ > | τ x n T µ ˆ h n (0) | ≥ ǫ > . On the other hand by (2.11) we get k τ x n T µ ˆ h n k W , ∞ ( B (0 , are uniformly bounded and hence by the Ascoli-Arzel´a Theorem τ x n ( T µ ˆ h n ( ξ ))has an uniform limit in B (0 , Second step : conclusion of the proof
Notice that k ˆ g n ( ξ ) k L = 1 LUCA FANELLI, LUIS VEGA, AND NICOLA VISCIGLIA and k T µ (ˆ g n ) k L p ( R nx ) = k T µ (ˆ h n ) k L p ( R nx ) . Hence ˆ g n is a maximizing sequence for T µ . On the other hand by the previousstep it is easy to check that all the hypothesis of Proposition 0.1 are satisfied if wechoose T = T µ , H = L ( dµ ) and we fix as a maximizing sequence ˆ g n . The case p = ∞ Following the computations done above we have thatlim n →∞ k T µ ˆ h n k L ∞ ( R nx ) = k T µ k L ( L ( dµ ) ,L ∞ ( R dx )) and moreover(2.13) sup n ∈ N k T µ ˆ h n k W , ∞ ( R dx ) < ∞ . In particular there is a sequence x n ∈ R d such thatlim n →∞ k T µ ˆ h n ( x n ) k = k T µ k L ( L ( dµ ) ,L ∞ ( R dx )) . As in the previous case we introduce ˆ g n = e ix n · ξ ˆ h ( ξ ) and it is easy to deduce thatˆ g n is still maximizing sequence with the extra property that(2.14) lim n →∞ | T µ (ˆ g n )(0) | = k T µ k L ( L ( dµ ) ,L ∞ ( R dx )) . By the Ascoli-Arzel´a theorem (that can be applied due to (2.13)) in conjunctionwith (2.14) we conclude that if ¯ g is the the weak limit of ˆ g n in L ( dµ ) then neces-sarily k T µ (¯ g ) k L ∞ ( R nx ) ≥ | T µ ¯ g (0) | = k T µ k L ( L ( dµ ) ,L ∞ ( R dx )) . On the other hand by semicontinuity of the norm L ( dµ ) we have that k ¯ g k L ( dµ ) ≤
1. By combining this fact with the definition of k T µ k L ( L ( dµ ) ,L ∞ ( R dx )) we easilydeduce that k ¯ g k L ( dµ ) = 1 and hence ˆ g n is compact in L ( dµ ). References [1]
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Luca Fanelli: Universidad del Pais Vasco, Departamento de Matem ´ aticas, Apartado644, 48080, Bilbao, Spain E-mail address : [email protected] Luis Vega: Universidad del Pais Vasco, Departamento de Matem ´ aticas, Apartado 644,48080, Bilbao, Spain E-mail address : [email protected] Nicola Visciglia: Universit´a di Pisa, Dipartimento di Matematica, Largo B. Pon-tecorvo 5, 56100 Pisa, Italy
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