Endpoint Strichartz estimates

Transcription

1 Endpoin Sricharz esimaes Markus Keel and Terence Tao (Amer. J. Mah. 10 (1998) ) Presener : Nobu Kishimoo (Kyoo Universiy) 013 Paricipaing School in Analysis of PDE 013/8/6 30, Jeju

2 1 Absrac of he paper We prove an absrac Sricharz esimae, which implies previously unknown endpoin Sricharz esimaes for he wave equaion (in dimension n 4) and he Schrödinger equaion (in dimension n 3). Three oher applicaions are discussed: local exisence for a nonlinear wave equaion; and Sricharz-ype esimaes for more general dispersive equaions and for he kineic ranspor equaion.

3 e: (X, dx) measure space, H Hilber space, Suppose: R, we have an operaor U() : H (X) which obeys Energy esimae: U()f C f H,, f H, (1) Decay esimae (one of he following): σ > 0 s.. U()U(s) g C s σ g 1 s, g 1 (X) () (unruncaed decay), or U()U(s) g C(1 + s ) σ g 1, s, g 1 (X) (3) (runcaed decay), where U(s) : (X) H is he adjoin of U(s).

4 In paricular, we consider he following cases: 3 X = R n, H = (R n ), and (i) Schrödinger case: [U()f](x) = [e i f](x) = 1 (4πi) n/ R n e x y 4i f(y) dy. (ii) Wave case: [U()f](x) = [e i P N f](x) = F 1[ e i φ N Ff ] (x), where N Z and P N is a ilewood-paley projecion o { ξ N}. U() saisfies (1), and [Schrödinger case] for n 1, U() saisfies () wih σ = n. [wave case] for n, U() saisfies (3) wih σ = n 1.

5 Definiion 1.1. We say ha he exponen pair (q, r) is σ-admissible if 4 q, r, (q, r, σ) (,, 1), and 1 q + σ r σ. (4) If equaliy holds in (4) we say ha (q, r) is sharp σ-admissible, oherwise nonsharp σ-admissible. In paricular, when σ > 1 he endpoin P = (, σ σ 1 ) is sharp σ-admissible.

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7 Theorem 1.. If U() obeys (1) and (), hen he esimaes R U()f q r x C f H, (5) U(s) F (s) ds H C F q, (6) r x 6 U()U(s) F (s) ds q r x C F eq er x hold for all sharp σ-admissible exponen pairs (q, r), ( q, r), where q is he Hölder conjugae of q (i.e. 1 q + 1 q = 1). Furhermore, if he decay hypohesis is srenghened o (3), hen (5) (7) hold for all (sharp and nonsharp) σ-admissible (q, r) and ( q, r). Resul on endpoin cases (i.e. σ > 1 and (q, r) or ( q, r) = P ) is new. (7)

8 Corollary 1.3 (wave case). Suppose ha n and (q, r) and (eq, er) are 7 n 1 -admissible pairs wih r, er <. If u is a (weak) soluion o he problem ( ( + )u(, x) = F (, x), (, x) [0, T ] R n, u(0, ) = f, u(0, ) = g for some daa f, g, F and ime 0 < T <, hen u q ([0,T ]; r ) + u C([0,T ]; Ḣ γ ) + u C([0,T ]; Ḣ γ 1 ) C` f Ḣγ + g Ḣγ 1 + F eq ([0,T ]; er ), (9) under he assumpion 1 q + n r = n γ = 1 eq + n. (10) er The consan C > 0 in (9) is independen of f, g, F, T. Conversely, if (9) holds for all f, g, F, T, hen (q, r) and (eq, er) mus be n 1 -admissible and (10) mus hold. Furhermore, when r = he esimae (9) holds wih he r wih he Besov norm Ḃ0 r,, and similarly for er =. norm replaced

9 Corollary 1.4 (Schrödinger case). Suppose ha n 1 and (q, r) and 8 ( q, r) are sharp n -admissible pairs. If u is a (weak) soluion o he problem { (i + )u(, x) = F (, x), (, x) [0, T ] R n, u(0, ) = f for some daa f, F and ime 0 < T <, hen u q ([0,T ]; r ) + u C([0,T ]; ) C( f + F eq ([0,T ]; er )), (11) where he consan C > 0 is independen of f, F, T. Conversely, if (11) holds for all f, F, T, hen (q, r) and ( q, r) mus be sharp n -admissible.

10 Conens of he paper 9 1. Inroducion. Ouline of paper 3. Proof of (5) and (6) for (q, r) P 4. Proof of (5) and (6) for (q, r) = P : Sep I 5. Proof of (5) and (6) for (q, r) = P : Sep II 6. Alernae proof for Sep II 7. Proof of (7) 8. Proof of Corollaries 1.3 and Applicaion o a semi-linear wave equaion 10. Applicaion o oher equaions

11 3. Proof of (5) and (6) for (q, r) P 10 Firs of all, we see ha he esimae (5) follows from (6) by dualiy. ( ) For any es funcion G : R X C, we have [U()f](x)G(, x) dx d R X = U()f, G() d = f, U() G() H d R R = f, U() G() d H f H U() G() d H R C f H G q r x which implies (5)., R

12 Since R U(s) F (s) d H = R R U(s) F (s), U() F () H ds d, (6) follows from he bilinear form esimae U(s) F (s), U() G() H ds d C F q R R In fac, (6) is equivalen o (13). (T T mehod) I hen suffices o prove (13) for r x 11 G q. (13) r x q, r, 1 q + σ r = σ, (q, r) (, σ σ 1 ) q, r, 1 q + σ r σ, (q, r) (, σ σ 1 ) under (1), () [Unruncaed], under (1), (3) [Truncaed].

13 Case (i): Unruncaed decay () 1 By he energy esimae (1), we have U() F () H C F () uniformly in, which implies U()U(s) F (s) C F (s). Using Riesz-Thorin heorem o inerpolae his inequaliy and (), we have U()U(s) F (s) r C s σ(1 r ) F (s) r for any r. Therefore, HS of (13) U()U(s) F (s), G() ds d R R U()U(s) F (s) r G() r ds d R R C s σ(1 ) r F (s) r G() r ds d. (A) R R

14 Hardy-ilewood-Sobolev inequaliy (cf. Sein [], Secion V.1.) 13 e 1 < p 1, p <, 0 < λ < n be such ha 1 p p + λ n R n R n f(ξ)g(η) ξ η λ dξ dη C f p 1 g p. =. Then, Now, since (q, r) is sharp σ-admissible, we see ha 1 q + σ r = σ 1 q + 1 (1 q + σ ) r =. If q <, he nonendpoin assumpion (q, r) (, σ σ 1 ) implies ( 0 < σ 1 ) r < 1. We apply he Hardy-ilewood-Sobolev inequaliy o (A) and obain (13). The case q = follows direcly from (A).

15 Case (ii): Truncaed decay (3) 14 Since (3) (), proof is reduced o Case (i) if (q, r) is sharp σ-admissible. We consider nonsharp σ-admissible exponens, namely, q, r, 1 q + σ r < σ. The same argumen as Case (i) shows ( ) σ(1 HS of (13) C 1 + s r ) F (s) r G() r ds d R R for any r insead of (A). Applying Young s inequaliy, we have HS of (13) C (1 + ) σ(1 ) r F q/ (R) q G r x q r x whenever σ(1 r ) q > 1 1 q + σ r < σ. This concludes he proof of (5) and (6) when (q, r) P.

16 4. Proof of (5) and (6) for endpoin cases: Sep I 15 Now, we consider he remaining endpoin case (q, r) = P = (, σ ), σ > 1. (0) σ 1 Noe ha < r <. Since P is sharp σ-admissible and (3) implies (), we only consider he case of unruncaed decay (). The same argumen as in 3 is no valid. In fac, he Hardy-ilewood- Sobolev inequaliy is no applicable because σ(1 r ) = 1. To show (13), we firs decompose HS dyadically as HS of (13) U(s) F (s), U() G() H ds d. j Z j s < j+1

17 By symmery i suffices o show where T j (F, G) = j Z T j (F, G) C F q r x G q r x, 16 () j+1 <s j U(s) F (s), U() G() H ds d. (1) The goal of Sep I is he following wo-parameer family of esimaes: emma 4.1. Assume (0). The esimae T j (F, G) C j{σ 1 σ( 1 a + 1 b )} F a x G b x (3) holds (uniformly) for all j Z and all ( 1 a, 1 b ) in a neighborhood of ( 1 r, 1 r ). Since σ 1 σ( 1 r + 1 r ) = 0, we have () wih j replaced by sup j.

18 17 Firs of all, we noe ha for he esimae of T j (F, G) we may assume ha F, G are suppored on a ime inerval of lengh O( j ). ( ) We decompose F as F (s) = l Z χ [lj,(l+1) j )(s)f (s), and assume ha (3) holds (uniformly) for F, G suppored in ime on an O( j ) inerval. Then, is resriced o [(l+1) j, (l+3) j ) whenever s [l j, (l+1) j ), since he inegral in s, is resriced o { j+1 < s j }. Therefore, T j (F, G) X l Z Tj (χ [l j,(l+1) j )F, χ [(l+1) j,(l+3) j )G) C j{σ 1 σ( 1 a + 1 b )} X l Z χ [l j,(l+1) j )F χ a [(l+1) j,(l+3) j )G x b x C j{...} X l Z C j{σ 1 σ( 1 a + 1 b )} F a x χ [l j,(l+1) j )F «1/ X χ a [(l+1) j,(l+3) j )G «1/ x b x G. b x l Z

19 We shall prove (3) for he exponens 18 (i) a = b =, (ii) a < r, b = (iii) b < r, a = The lemma will hen follow by inerpolaion and he fac ha < r <.

20 Case (i) a = b = 19 From he esimae (A) (wih r = ) and he resricion o { j+1 < s j }, we have T j (F, G) C σj F 1 1 x G 1 1 x. Recall ha F, G are resriced in ime o an inerval of lengh O( j ). We apply Hölder s inequaliy in ime o obain T j (F, G) C (σ 1)j F 1 x G 1 x, which is he desired esimae.

21 Case (ii) a < r, b = 0 Noe ha Case (iii) is parallel o (ii). We bring he inegraion in s inside he inner produc in (1) o obain T j (F, G) R sup R R j U(s) F (s) ds, U() G() H d j+1 j U(s) F (s) ds H U() G() H d j+1 U(s) [ χ ( j+1, j ](s)f (s) ] ds H R R U() G() H d. Since a < r, we can ake q(a) such ha (q(a), a) is sharp σ-admissible and (q(a), a) P. By he nonendpoin Sricharz esimae (6) proved in 3 and Hölder s inequaliy in, we obain

22 R U(s) [ χ ( j+1, j ](s)f (s) ] ds H C χ ( j+1, j ]F q(a) a x uniformly in. C j( 1 q(a) 1 ) F, a x 1 By he energy esimae (1) and Hölder s inequaliy in, we have U() G() H d C G 1 C j/ G x x. R Combining hese esimaes, we have T j (F, G) C j/q(a) F a x This is nohing bu (3), since G x. 1 q(a) = 1 1 q(a) = 1 σ( 1 1 ) ( ( 1 = σ 1 σ a a + 1 )).

23 5. Proof of (5) and (6) for endpoin cases: Sep II If we apply emma 4.1 direcly for a = b = r, hen we obain T j (F, G) C F r x G r x (5) for each j Z (uniformly), which clearly won sum o give (). Observaion: To see how o sum up in j, we begin wih he model case. Assume ha F and g have he special form F (, x) = f() k/r χ E() (x), G(, x) = g() e k/r χ ee() (x), where k, k Z and E(), Ẽ() are ses of measure k and ek respecively for each. Noe ha F r x f, G r x g.

24 By emma 4.1, i holds ha T j (F, G) C j{σ 1 σ( 1 a + 1 b )} F a x G b x σ 1 jσ{ C σ ( 1 a + 1 b )} k a k r f k e b k e r g 3 (uniformly) for all j Z and ( 1 a, 1 b ) in a neighborhood of ( 1 r, 1 r ). Noe ha σ 1 σ = r, 1 a 1 r = 1 r 1 a, 1 b 1 r = 1 r 1 b. Then, he above esimae is simplified o T j (F, G) C (k jσ)( 1 r 1 a ) (e k jσ)( 1 r 1 b ) F r x G. (6) r x Take ε > 0 sufficienly small so ha he esimae is valid for 1 a, 1 b { 1 r ± ε}. Now, for each j Z we choose 1 a, 1 b { 1 r ± ε} appropriaely o obain T j (F, G) C ε k jσ ε e k jσ F G r x, r x which does imply ().

25 This observaion suggess ha (5) is only sharp when F and G are 4 boh concenraed in a se of size jσ. However, such funcions can only be criical for one scale of j. Tha s why we expec o obain () for general F, G from emma 4.1. Also noe ha his argumen requires a wo-parameer family of esimaes as emma 4.1, while he Sricharz esimaes for nonendpoin case was obained from a one-parameer family of esimaes (namely, a = b). To apply he above argumen in he general case, we use he following lemma o decompose F, G so ha each piece has a form similar o he above.

26 emma 5.1. e 0 < p < and f p. Then here exis 5 {c k } k Z [0, ), {χ k } k Z such ha (i) f(x) = c k χ k (x), k Z (ii) χ k k/p and meas { x χ k (x) 0 } k, (iii) c k l p 1+1/p f p. By applying emma 5.1 wih p = r o F () and G(), we have F (, x) = k Z c k ()χ k (, x), G(, x) = e k Z c ek () χ ek (, x), (9) where for each R and k Z he funcion χ k (, ) saisfies χ k (, ) k/r, meas { x χ k (, x) 0 } k, and similarly for χ ek. Moreover, c k () and c ek () saisfy he inequaliies c k () l r k C F, r x c ek () l r ek C G. (30) r x

27 We are now ready o prove (). By he decomposiion (9) we have T j (F, G) j Z k, e k Z j Z T j (c k χ k, c ek χ ek ). 6 Bu by Observaion a he sar of 5, emma 4.1 gives T j (c k χ k, c ek χ ek ) C ε( k jσ + e k jσ ) c k c ek for some ε > 0. Summing in j, we have T j (F, G) C j Z k, e k Z ( 1 + k k ) ε k e k c k c ek.

28 7 Noe ha he quaniy w k := (1 + k ) ε k is summable, and RHS of he above esimae has he form k c ) k ( w( ) c ( ) k. We apply Young s inequaliy: k f k(w g) k w l 1 f l g l o obain j Z T j (F, G) C c k l c ek k l ek. Inerchanging he and l norms and using l r j Z T j (F, G) C c k () l r k l, we obain c ek () l r ek. () hen follows from (30), concluding he proof of (5), (6) for endpoin.

29 8 We now proceed o he proof of emma 5.1. e f p, 0 < p <. Define he disribuion funcion λ(α) of f for α 0 by λ(α) = meas { x f(x) > α }. Noe ha λ(α) is non-increasing and righ-coninuous. For each k Z, we se α k = inf { α > 0 λ(α) < k }. From definiion we see ha 0 α k <, α k is non-increasing in k, lim k α k = f λ(α k ) k, and λ(α k 0) k if α k > 0. [0, ], (B)

30 Finally, we define 9 c k = k/p α k, c 1 k χ k (x) = χ (α k+1,α k ]( f(x) )f(x) if α k > 0, 0 if α k = 0 (= α k+1 ). Propery (i) is sraighforward. For (ii), he bound is easily verified. Since {χ k 0} { f(x) > α k+1 }, we have meas{χ k 0} λ(α k+1 ) k+1, where we have used (B).

31 I remains o verify (iii). If we know a priori ha c p k = k α p k <, k Z k Z 30 hen we have k Z k α p k = k Z( k+1 k )α p k = k Z k+1 (α p k αp k+1 ). e us ake a non-increasing sequence {α k } [0, ) such ha α k > α k > α k+1, α k α k/ if α k > α k+1, α k = α k if α k = α k+1. Noe ha α k α k α k+1 and α k α k for all k Z. Furhermore, from (B) we have k λ(α k ) k+1 whenever α k > α k+1, which implies

32 k (α p k αp k+1 ) λ(α k)(α p k αp k+1 ) k Z k Z k Z Since RHS is absoluely summable, we have k α p k λ(α k)(α p k αp k+1 ) k Z k Z = ( λ(α k ) λ(α k 1) ) α p k k Z k+1 (α p k αp k+1 ). 31 p k Z ( λ(α k ) λ(α k 1) ) (α k) p ( α k α k) = 1+p (α k) p dx k Z {α k < f(x) α k 1 } 1+p f p, p which shows (iii).

33 3 In he general case, we firs define a non-negaive funcion f K for K N by I is easy o see ha f K (x) = α K if f(x) > α K, f(x) if α K f(x) > α K, 0 if f(x) α K. f K (x) f(x), K N.

34 33 If we define he disribuion funcion λ K (α) and he sequence {αk K} k Z for each f K as before, hen λ K (α) = so ha 0 if α α K, λ(α) if α K > α α K, αk K = λ(α K ) if α < α K, 0 if k > K, α k if K k > K, if k K, α K k (αk K ) p = k α p k + α K k < k Z K<k K k K for any K N. Therefore, we can apply he previous argumen and obain k α p k k (αk K ) p 1+p f K p 1+p f p. p p K<k K k Z eing K verifies (iii), which concludes he proof of emma 5.1.

35 6. Alernae proof for Sep II 34 In 5, we have deduced he srong (l 1 j ) summabiliy in r x r x from he weak (l j ) summabiliy in a x b x wih (a, b) around (r, r). Such a siuaion is similar o he Marcinkiewicz inerpolaion heorem, which assers ha he srong boundedness T : p p of an operaor T is deduced from he weak boundedness if 1 p 1 < p < p. T : p 1 p 1,, T : p p, Marcinkiewicz inerpolaion heorem is one of he simples consequences in real inerpolaion heory. In fac, we can rephrase he previous derivaion of () from emma 4.1 by using exising resuls in real inerpolaion heory.

36 Real inerpolaion mehod 35 e (A 0, A 1 ) be a pair of compaible Banach spaces. For parameers 0 < θ < 1 and 1 q, we define he real inerpolaion spaces (A 0, A 1 ) θ,q as he spaces of all he elemens a in A 0 + A 1 = { } a 0 + a 1 a 0 A 0, a 1 A 1 such ha he norm a (A0,A 1 ) θ,q = ( 0 [ θ ( ) ] ) q 1/q d inf a0 A0 + a 1 A1 a=a 0 +a 1 is finie, wih he usual modificaion when q =.

37 We will need he following wo inerpolaion space ideniies. 36 [Triebel [3], Secions and ] ( ) p 0 x, p 1 x θ, = p, x for 1 p 0 p 1, min{p 0, p 1 } <, 0 < θ < 1, 1 p = 1 θ p 0 + θ p 1. [Bergh, öfsröm [1], Secion 5.6] ( l s0, l s 1 ) θ,1 = l1 s for s 0 s 1 R and s = (1 θ)s 0 + θs 1. Here, p,q denoes orenz spaces which saisfy p,p = p, p,q 1 p,q for 1 p, 1 q 1 q, and l q s sands for weighed sequence spaces defined via he norm {a j } j Z l q s = { sj a j } j Z l q.

38 Wih he noion of weighed sequence spaces l q s, he esimae (3) can be regarded as boundedness T : a x b x l β(a,b), (31) 37 where T = {T j } j Z is he vecor-valued bilinear operaor and ( 1 β(a, b) = σ 1 σ a + 1 ), b and he claimed esimae () is rewrien as T : r x r x l 1 0. (C)

39 We will use he following bilinear inerpolaion heorem: 38 emma 6.1. ([1], Exercise 5(b) of Secion 3.13) e (A 0, A 1 ), (B 0, B 1 ), (C 0, C 1 ) be compaible pairs of Banach spaces. Suppose ha he bilinear operaor T is bounded as T : A 0 B 0 C 0, A 1 B 0 C 1, A 0 B 1 C 1. Then, we have boundedness T : ( ) A 0, A 1 θ A,p A q ( ) B 0, B 1 θ B,p B q ( C 0, C 1 )θ,q whenever 0 < θ A, θ B < θ = θ A + θ B < 1, 1 p A, p B, q, 1 p A + 1 p B 1.

40 We se, wih ε > 0 sufficienly small, A 0 = B 0 = a 0 x, A 1 = B 1 = a 1 x, 1 a 0 = 1 r ε, 1 a 1 = 1 r + ε. 39 Then, by emma 4.1 (or (31)) he assumpion in emma 6.1 is saisfied wih C 0 = l σε, C 1 = l σε. We apply emma 6.1 wih θ A = θ B = 1 3, p A = p B =, q = 1 o obain he boundedness T : r, x r, x l 1 0, which implies he claim (C) because of he embedding r r,.

41 There is also a one-parameer real inerpolaion heorem as follows: 40 emma. ([1], Exercise 4 of Secion 3.13) If he bilinear operaor T is bounded as hen we have whenever T : A 0 B 0 C 0, A 1 B 1 C 1, T : ( A 0, A 1 ) θ,q ( B 0, B 1 ) 0 < θ < 1, 1 q. However, we only deduce from his lemma T : r x r x l r 0, and his is no sufficien because r = σ σ+1 > 1. θ,q ( C 0, C 1 )θ,q

42 41 References [1] J. Bergh and J. öfsröm, Inerpolaion Spaces: An Inroducion, Springer-Verlag, New York, [] E. Sein, Singular Inegrals and Differeniabiliy Properies of Funcions, Princeon Universiy Press, [3] H. Triebel, Inerpolaion Theory, Funcion Spaces, Differenial Operaors, Norh-Holland, New York, 1978.

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