Weighted HLS inequalities for radial functions and Strichartz estimates for wave and Schroedinger equations
aa r X i v : . [ m a t h . A P ] N ov WEIGHTED HLS INEQUALITIES FOR RADIALFUNCTIONS AND STRICHARTZ ESTIMATES FOR WAVEAND SCHR ¨ODINGER EQUATIONS
KUNIO HIDANO AND YUKI KUROKAWA
Abstract.
This paper is concerned with derivation of the global or lo-cal in time Strichartz estimates for radially symmetric solutions of thefree wave equation from some Morawetz-type estimates via weightedHardy-Littlewood-Sobolev (HLS) inequalities. In the same way we alsoderive the weighted end-point Strichartz estimates with gain of deriva-tives for radially symmetric solutions of the free Schr¨odinger equation.The proof of the weighted HLS inequality for radially symmetricfunctions involves an application of the weighted inequality due to Steinand Weiss and the Hardy-Littlewood maximal inequality in the weightedLebesgue space due to Muckenhoupt. Under radial symmetry we getsignificant gains over the usual HLS inequality and Strichartz estimate. Introduction
In this paper we discuss the roles of the weighted Hardy-Littlewood-Sobolev(HLS, for short) inequalities for radially symmetric functions in the derivationof the Strichartz estimates for the free wave equation and the free Schr¨odingerequation.In the first half of this paper we prove the weighted HLS inequality forradially symmetric functions on R n ( n ≥
2) (see (2.2) below). The proofproceeds by writing out the Riesz potentials in polar coordinates, integrat-ing out the angular coordinates, and reducing the argument to the one-dimensional setting. We then make use of the weighted inequality due toStein and Weiss [33] and the Hardy-Littlewood maximal inequality in theweighted Lebesgue space due to Muckenhoupt [26]. Naturally, the weightedHLS inequality thereby obtained has some similarity with the one-dimensionalpart of the weighted inequalities due to Stein and Weiss, except that the normon the right-hand side of (2.2) involves such a homogeneous weight functionas | x | − ( n − /p ) − (1 /q )) . At the cost of the presence of such a singular weightfunction on the right-hand side, the weighted radial HLS inequality (2.2)holds even for the Riesz potential whose kernel has a rather singular form | x | − n + µ with µ = α + β + (1 /p ) − (1 /q ). (Compare it with the kernel | x | − n +˜ µ , Mathematics Subject Classification . Primary 35L05, 35Q40. Secondary 35B65,35Q55. ˜ µ = α + β + ( n/p ) − ( n/q ), of the Riesz potential in the usual weighted HLSinequality (2.15) below.)In the second half of this paper we discuss how the weighted radial HLSinequality is used to prove the global (in space and time) or local (in time)Strichartz estimate for radially symmetric solutions. It is well-known thatthe range of admissible exponents in the global Strichartz estimate for thefree wave equation can be significantly improved in the radial setting. (SeeTheorem 6.6.2 of Sogge [31], Proposition 4 of Klainerman and Machedon[19], Theorem 1.3 of Sterbenz [34], and Theorem 4 of Fang and Wang [5].)Adapting an argument of Vilela [39], we explain how to derive the global radialStrichartz estimate in space dimension n ≥ L q estimate due to Sogge in space dimension n = 3 into the space-time mixed-norm estimate in space dimension n ≥ L -estimate (5.7) belowby combining it with the weighted radial HLS inequality. Such a method doesnot end with applications to the free wave equation. Combined with the global(in space and time) estimate of the local smoothing property (6.4) below, theweighted radial HLS inequality is useful in proving the weighted end-pointStrichartz estimate for radially symmetric solutions to the free Sch¨odingerequation (see (6.3) below). In the radial setting we observe a significant gainof regularity over the end-point estimate due to Keel and Tao [18].The authors have received a couple of very instructive suggestions from thereferee. One is concerned with the flexibility in our approach. The approachto proving the Strichartz estimates used here does not rely upon explicitrepresentations or parametrices for the solution. Therefore it can provideStrichartz-type estimates for the large family of equations with defocusingradial potentials. Another is concerned with the weighted versions of the in-homogeneous Strichartz estimates. As was first observed by Kato [16] andhas been explored by Oberlin [27], Harmse [8], Foschi [6] and Vilela [40], the(unweighted) inhomogeneous Strichartz estimates are known to hold for thelarger range of exponent pairs. We now enjoy the approach based upon thecelebrated lemma of Christ and Kiselev [4], and the referee has kindly sug-gested that radial weighted analogs thereby obtained may turn out to be veryuseful for certain nonlinear problems. Indeed, by virtue of the Christ-Kiselevlemma one of the present authors has obtained some radial weighted analogsin order to study global existence of small solutions to nonlinear wave equa-tions [12] and nonlinear Schr¨odinger equations [11] with radially symmetricdata of scale-critical regularity.We conclude this section by explaining the notation. By L p ( R n , ω ( x ) dx ) wemean the Lebesgue space of all µ -measurable functions ( dµ ( x ) = ω ( x ) dx ) on EIGHTED HLS INEQUALITY AND STRICHARTZ ESTIMATE 3 R n . We simply denote L p ( R n , dx ) by L p ( R n ). The mixed norm k u k L q ( R ; L p ( R n )) for functions u on R × R n is defined as k u k L q ( R ; L p ( R n )) = (cid:18)Z R (cid:18)Z R n | u ( t, x ) | p dx (cid:19) q/p dt (cid:19) /q with an obvious modification for q = ∞ or p = ∞ . By p ′ we denote theexponent conjugate to p , that is (1 /p ) + (1 /p ′ ) = 1. The operator | D x | s ( s ∈ R ) is defined by using the Fourier transform F and the inverse Fouriertransform F − , as usual. We denote by ˙ H s ( R n ) the homogeneous Sobolevspace | D x | − s L ( R n ). The free evolution operators for the wave equation andthe Schr¨odinger equation are defined as( W ϕ )( t, x ) = W ( t ) ϕ ( x ) = F − e it | ξ | F ϕ, (1.1) ( Sϕ )( t, x ) = S ( t ) ϕ ( x ) = F − e it | ξ | F ϕ, (1.2)respectively.This paper is organized as follows. In the next section we prove theweighted HLS inequality for radial functions. Section 3 is devoted to theproof of the global-in-time Strichartz estimate for radial solutions to the freewave equation. In Section 4 we draw our attention to the limiting case ofthe estimates obtained in Section 3. An adaptation of observations due toAgemi [1], Rammaha [29] and Takamura [37] shows the failure of such criticalestimates. In Section 5 we are concerned with the local-in-time Strichartzestimate for radial solutions to the free wave equation. In the final sectionwe revisit the problem of deriving the end-point Strichartz estimate for radialsolutions to the free Schr¨odinger equation from the global (in space) estimateof local smoothing property. Using the weighted radial HLS inequality, weshow the weighted end-point Strichartz estimate with gain of derivatives forradial free solutions. 2. Weighted HLS inequality
We let(2.1) ( T γ v )( x ) = Z R n v ( y ) | x − y | γ dy, < γ < n. The purpose of this section is to prove the weighted Hardy-Littlewood-Sobolev(HLS) inequalities for radially symmetric functions. We show the following:
Theorem 2.1.
Suppose n ≥ . Let p , q , α and β satisfy < p < q < ∞ , α < /p ′ , β < /q and α + β ≥ . Set µ = α + β + (1 /p ) − (1 /q ) . There existsa constant C depending only on n , p , q , α and β , and the inequality (2.2) k| x | − β T n − µ v k L q ( R n ) ≤ C k| x | α − ( n − /p − /q ) v k L p ( R n ) holds for radially symmetric v ∈ L p ( R n , | x | p ( α − ( n − /p − /q )) dx ) . KUNIO HIDANO AND YUKI KUROKAWA
Remark . Obviously, the number µ in Theorem 2.1 is strictly positive. More-over, we should note that µ is strictly smaller than one. Indeed, by theassumption α < /p ′ , β < /q we see µ < /p ′ + 1 /q + 1 /p − /q = 1. Proof of Theorem 2.1.
We start with the well-known formula:(2.3) ( T n − s v )( x ) = ω n − r Z ∞ λ n − w ( λ ) dλ Z r + λ | r − λ | ρ − n + s +1 h ( ρ, λ ; r ) ( n − / dρ (0 < s < n ) for radially symmetric function v ( x ) = w ( r ). Here and in whatfollows we use the notation r = | x | = p x + · · · + x n ,(2.4) h ( ρ, λ ; r ) = 1 − (cid:18) r + λ − ρ λr (cid:19) ,ω = 2, and ω n ( n = 2 , , . . . ) is the area of S n − = { x ∈ R n | | x | = 1 } .For the proof of (2.3) consult, e.g., John [15] on page 8. Since the function h ( ρ, λ ; r ) ( n − / causes another singularity in the case of n = 2, let us firststudy the case n ≥
3. By virtue of the following proposition our argument willbe reduced to the special case of the weighted estimate of Stein and Weiss.(See Lemma 2.3 below.)
Proposition 2.2.
Suppose n ≥ , < q < ∞ and < s < . The inequality (2.5) r ( n − /q ( T n − s v )( x ) ≤ C Z ∞ λ ( n − /q w ( λ ) | r − λ | − s dλ ( x ∈ R n ) holds for radially symmetric, non-negative function v ( x ) = w ( r ) .Proof of Proposition 2.2. Set I i = I i ( r ) ( i = 1 ,
2) as1 r Z r/ λ n − w ( λ ) dλ Z r + λr − λ ρ − n + s +1 h ( ρ, λ ; r ) ( n − / dρ (2.6) + 1 r Z ∞ r/ λ n − w ( λ ) dλ Z r + λ | r − λ | ρ − n + s +1 h ( ρ, λ ; r ) ( n − / dρ =: I ( r ) + I ( r ) . For the estimate of I we note that − ≤ ( r + λ − ρ ) / (2 λr ) ≤ | r − λ | ≤ ρ ≤ r + λ , which implies h ( ρ, λ ; r ) ( n − / ≤ n ≥
3. We therefore obtain I ( r ) ≤ r Z r/ λ n − w ( λ ) dλ Z r + λr − λ ρ − n + s +1 dρ (2.7) ≤ Cr n − s Z r/ λ n − w ( λ ) dλ. The last inequality is due to the fact that for 0 < s < ≤ λ ≤ r/ Z r + λr − λ ρ − n + s +1 dρ ≤ λ ( r − λ ) − n + s +1 ≤ Cλr − n + s +1 . EIGHTED HLS INEQUALITY AND STRICHARTZ ESTIMATE 5
For the estimate of I let us first observe h ( ρ, λ ; r ) = ( ρ /λ ) h ( λ, ρ ; r ). Indeed,we see that h ( ρ, λ ; r )(2.9) = 4 λ r − ( r + λ − ρ ) λ r = { ρ − ( r − λ ) }{ ( r + λ ) − ρ } λ r = ( ρ + r − λ )( ρ − r + λ )( r + λ + ρ )( r + λ − ρ )4 λ r = { ( ρ + r ) − λ }{ λ − ( ρ − r ) } λ r = ρ λ (cid:26) − (cid:18) r + ρ − λ ρr (cid:19) (cid:27) = ρ λ h ( λ, ρ ; r ) , as desired. Since − ≤ ( r + ρ − λ ) / (2 ρr ) ≤ | r − λ | ≤ ρ ≤ r + λ , wehave h ( ρ, λ ; r ) ≤ ρ /λ and therefore I ( r ) ≤ r Z ∞ r/ λ n − w ( λ ) dλ Z r + λ | r − λ | ρ − n + s +1 (cid:18) ρ λ (cid:19) ( n − / dρ (2.10) = 1 r Z ∞ r/ λw ( λ ) dλ Z r + λ | r − λ | ρ − s dρ. Keeping the assumption 0 < s < Z r + λ | r − λ | ρ − s dρ = 1(1 − s ) | r − λ | − s (cid:26) − (cid:18) | r − λ | r + λ (cid:19) − s (cid:27) (2.11) ≤ − s ) | r − λ | − s (cid:18) − | r − λ | r + λ (cid:19) = 2 min { λ, r } (1 − s ) | r − λ | − s ( r + λ ) . Combining (2.10) with (2.11), we get(2.12) I ( r ) ≤ Cr Z ∞ r/ r | r − λ | − s ( r + λ ) λw ( λ ) dλ ≤ C Z ∞ r/ w ( λ ) | r − λ | − s dλ. Therefore, we have obtained by (2.3), (2.6), (2.7) and (2.12)(2.13) ( T n − s v )( x ) ≤ Cr n − s Z r/ λ n − w ( λ ) dλ + C Z ∞ r/ w ( λ ) | r − λ | − s dλ. KUNIO HIDANO AND YUKI KUROKAWA
We are in a position to complete the proof of (2.5). It follows from (2.13) that r ( n − /q ( T n − s v )( x )(2.14) ≤ Cr (( n − /q ) − n + s Z r/ λ n − w ( λ ) dλ + Cr ( n − /q Z ∞ r/ w ( λ ) | r − λ | − s dλ ≤ C Z r/ r − s (cid:18) λr (cid:19) ( n − − (1 /q )) λ ( n − /q w ( λ ) dλ + C Z ∞ r/ λ ( n − /q w ( λ ) | r − λ | − s dλ ≤ C Z ∞ λ ( n − /q w ( λ ) | r − λ | − s dλ as desired. The proof of Proposition 2.2 has been finished. (cid:3) Once we have obtained the point-wise (in x ) estimate (2.5), the three orhigher dimensional part of Theorem 2.1 is an immediate consequence of thefollowing lemma due to Stein and Weiss [33]. Lemma 2.3.
Assume n ≥ , < γ < n , < p < ∞ , α < n/p ′ , β < n/q , α + β ≥ , and /q = (1 /p ) + (( γ + α + β ) /n ) − . If p ≤ q < ∞ , then theinequality (2.15) k| x | − β T γ v k L q ( R n ) ≤ C k| x | α v k L p ( R n ) holds for any v ∈ L p ( R n , | x | pα dx ) . It is easily seen that the three or higher dimensional part of Theorem 2.1is a consequence of (2.5) with s = µ and (2.15) with n = 1. The proof ofTheorem 2.1 has been finished for n ≥ n = 2 we set J i = J i ( r ) ( i = 1 , ,
3) forradially symmetric function v ( x ) = w ( r ):1 r Z r/ w ( λ ) dλ Z r + λr − λ ρ − s h ( ρ, λ ; r ) − / dρ (2.16) + 1 r Z rr/ w ( λ ) dλ Z r + λ | r − λ | ρ − s h ( ρ, λ ; r ) − / dρ + 1 r Z ∞ r w ( λ ) dλ Z λ + rλ − r ρ − s h ( ρ, λ ; r ) − / dρ =: J ( r ) + J ( r ) + J ( r ) . It is possible to show the counterpart of Proposition 2.2 for J and J . It is J that we must handle quite differently from before. Let us begin with theproof of the following: Proposition 2.4.
Suppose < q < ∞ and < s < . The inequality (2.17) r /q J i ( r ) ≤ C Z ∞ λ /q w ( λ ) | r − λ | − s dλ ( i = 1 , holds for non-negative w . EIGHTED HLS INEQUALITY AND STRICHARTZ ESTIMATE 7
Proof of Proposition 2.4.
We use the property of the beta function B ( · , · ):(2.18) Z ba ρ p ρ − a p b − ρ dρ = B (cid:18) , (cid:19) = π. Observing 1 r Z r + λ | r − λ | ρ − s h ( ρ, λ ; r ) − / dρ (2.19) = λ Z r + λ | r − λ | ρ − s ρ p ρ − ( r − λ ) p ( r + λ ) − ρ dρ ≤ λ | r − λ | − s B (cid:18) , (cid:19) , we are led to(2.20) r /q J ( r ) ≤ C Z r/ (cid:18) r /q λ − (1 /q ) r − λ (cid:19) λ /q w ( λ )( r − λ ) − s dλ and(2.21) r /q J ( r ) ≤ C Z ∞ r (cid:18) r /q λ − (1 /q ) λ − r (cid:19) λ /q w ( λ )( λ − r ) − s dλ. Since r /q λ − (1 /q ) / ( r − λ ) ≤ C for 0 ≤ λ ≤ r/ r /q λ − (1 /q ) / ( λ − r ) ≤ C for 2 r ≤ λ , the inequality (2.17) is a consequence of (2.20)–(2.21). We havefinished the proof of Proposition 2.4. (cid:3) It remains to show the bound for J . Proposition 2.5.
Let p , q , α , β and µ be the same as in Theorem . . Theinequality (cid:13)(cid:13)(cid:13)(cid:13) r (1 /q ) − Z rr/ w ( λ ) dλ Z r + λ | r − λ | ρ − µ h ( ρ, λ ; r ) − / dρ (cid:13)(cid:13)(cid:13)(cid:13) L q ((0 , ∞ ) ,r − qβ dr ) (2.22) ≤ C k r /q w k L p ((0 , ∞ ) ,r pα dr ) holds.Proof of Proposition 2.5. Without loss of generality we may assume that w is non-negative. Identifying the dual space of L q ((0 , ∞ ) , r − qβ dr ) with KUNIO HIDANO AND YUKI KUROKAWA L q ′ ((0 , ∞ ) , r q ′ β dr ) and reversing the order integration twice, we have (cid:13)(cid:13)(cid:13)(cid:13) r (1 /q ) − Z rr/ w ( λ ) dλ Z r + λ | r − λ | ρ − µ h ( ρ, λ ; r ) − / dρ (cid:13)(cid:13)(cid:13)(cid:13) L q ((0 , ∞ ) ,r − qβ dr ) (2.23)= sup Z ∞ r (1 /q ) − g ( r ) dr Z rr/ w ( λ ) dλ Z r + λ | r − λ | ρ − µ h ( ρ, λ ; r ) − / dρ = sup Z ∞ w ( λ ) dλ Z λλ/ r (1 /q ) − g ( r ) dr Z r + λ | r − λ | ρ − µ h ( ρ, λ ; r ) − / dρ = sup (cid:18)Z ∞ w ( λ ) dλ Z λ/ ρ − µ dρ Z λ + ρλ − ρ r (1 /q ) − h ( ρ, λ ; r ) − / g ( r ) dr + Z ∞ w ( λ ) dλ Z λλ/ ρ − µ dρ Z λ + ρλ/ r (1 /q ) − h ( ρ, λ ; r ) − / g ( r ) dr + Z ∞ w ( λ ) dλ Z λ/ λ ρ − µ dρ Z λλ/ r (1 /q ) − h ( ρ, λ ; r ) − / g ( r ) dr + Z ∞ w ( λ ) dλ Z λ λ/ ρ − µ dρ Z λρ − λ r (1 /q ) − h ( ρ, λ ; r ) − / g ( r ) dr (cid:19) =: sup( L + L + L + L ) . Here the supremum is taken over all non-negative g ∈ L q ′ ((0 , ∞ ) , r q ′ β dr ) with k g k L q ′ ((0 , ∞ ) ,r q ′ β dr ) = 1.In what follows we shall often use the identity(2.24) r − h ( ρ, λ ; r ) − / = 2 λ p ( ρ − r + λ )( ρ + r − λ )( r + λ − ρ )( r + λ + ρ )as well as the inequality 2 λ ≤ r + λ + ρ ≤ λ for λ/ ≤ r ≤ λ , | r − λ | ≤ ρ ≤ r + λ . To begin with, we first estimate L . Observing r + λ − ρ ≤ ( λ + ρ ) + λ − ρ = 2 λ , r + λ − ρ ≥ ( λ − ρ ) + λ − ρ ≥ λ for λ − ρ ≤ r ≤ λ + ρ EIGHTED HLS INEQUALITY AND STRICHARTZ ESTIMATE 9 and 0 ≤ ρ ≤ λ/
2, we obtain for 0 ≤ ρ ≤ λ/ Z λ + ρλ − ρ r (1 /q ) − h ( ρ, λ ; r ) − / g ( r ) dr (2.25) ≤ Cλ /q Z λ + ρλ − ρ p ( ρ − r + λ )( ρ + r − λ ) g ( r ) dr ≤ Cλ /q (cid:18)Z λλ − ρ p ρ ( r − λ + ρ ) g ( r ) dr + Z λ + ρλ p ρ ( λ + ρ − r ) g ( r ) dr (cid:19) ≤ Cλ /q (cid:18) ρ Z ( λ − ρ )+ ρ ( λ − ρ ) − ρ (cid:12)(cid:12)(cid:12)(cid:12) ρη − ( λ − ρ ) (cid:12)(cid:12)(cid:12)(cid:12) / g ∗ ( η ) dη + 12 ρ Z ( λ + ρ )+ ρ ( λ + ρ ) − ρ (cid:12)(cid:12)(cid:12)(cid:12) ρλ + ρ − η (cid:12)(cid:12)(cid:12)(cid:12) / g ∗ ( η ) dη (cid:19) ≤ Cλ /q (cid:18) sup σ> σ Z ( λ − ρ )+ σ ( λ − ρ ) − σ (cid:12)(cid:12)(cid:12)(cid:12) ση − ( λ − ρ ) (cid:12)(cid:12)(cid:12)(cid:12) / g ∗ ( η ) dη + sup σ> σ Z ( λ + ρ )+ σ ( λ + ρ ) − σ (cid:12)(cid:12)(cid:12)(cid:12) σλ + ρ − η (cid:12)(cid:12)(cid:12)(cid:12) / g ∗ ( η ) dη (cid:19) = Cλ /q ( M / g ∗ )( λ − ρ ) + Cλ /q ( M / g ∗ )( λ + ρ ) . Here we have set g ∗ ( η ) = g ( η ) for η ≥ g ∗ ( η ) = g ( − η ) for η <
0, and for t ∈ R (2.26) ( M / f )( t ) = sup σ> σ Z t + σt − σ (cid:12)(cid:12)(cid:12)(cid:12) ση − t (cid:12)(cid:12)(cid:12)(cid:12) / | f ( η ) | dη. It follows from the observation of Lindblad and Sogge that the maximal func-tion M / f , which is a singular variant of the Hardy-Littlewood maximalfunction(2.27) ( M f )( t ) = sup σ> σ Z t + σt − σ | f ( η ) | dη, has the point-wise estimate(2.28) ( M / f )( t ) ≤ C ( M f )( t ) (see page 1062 of [23]). Combining (2.25) with (2.28), we see that L has thebound such as L ≤ C Z ∞ λ /q w ( λ ) dλ Z λ/ ρ − µ ( M g ∗ )( λ − ρ ) dρ (2.29) + C Z ∞ λ /q w ( λ ) dλ Z λ/ ρ − µ ( M g ∗ )( λ + ρ ) dρ ≤ C k λ /q w k L p ((0 , ∞ ) ,λ pα dλ ) k T − µ ( M g ∗ ) k L p ′ ((0 , ∞ ) ,λ − p ′ α dλ ) ≤ C k λ /q w k L p ((0 , ∞ ) ,λ pα dλ ) kM g ∗ k L q ′ ( R , | t | q ′ β dt ) . At the last inequality we have used the one-dimensional part of Lemma 2.3.To finish the estimate of L we need: Lemma 2.6.
Suppose that < p < ∞ and − < a < p − . The oparator M enjoys the boundedness (2.30) kM f k L p ( R , | t | a dt ) ≤ C k f k L p ( R , | t | a dt ) . The original proof of (2.30) is due to Muckenhoupt [26]. See also Chapter5 of Stein [32] for further references. Before we use Lemma 2.6 to bound kM g ∗ k L q ′ ( R , | t | q ′ β dt ) , let us see that the condition − < q ′ β < q ′ − q ′ β < q ′ − β < /q which is supposed inTheorem 2.1. Moreover, to see that the condition − < q ′ β is also satisfied,we note that the assumption (1 /q ) − (1 /p ) − β + µ = α < /p ′ implies µ < (1 /q ′ ) + β . Since µ is positive, we finally find that − /q ′ < β , as desired. Wemay therefore use Lemma 2.6 to proceed as L ≤ C k λ /q w k L p ((0 , ∞ ) ,λ pα dλ ) k g ∗ k L q ′ ( R , | t | q ′ β dt ) (2.31) ≤ C k λ /q w k L p ((0 , ∞ ) ,λ pα dλ ) k g k L q ′ ((0 , ∞ ) ,r q ′ β dr ) . The estimate of L has been completed.We next consider the estimate of L . Observing r + λ − ρ ≤ ( ρ + λ )+ λ − ρ =2 λ , r + λ − ρ ≥ ( λ/
2) + λ − λ = λ/ λ/ ≤ r ≤ λ + ρ and λ/ ≤ ρ ≤ λ , EIGHTED HLS INEQUALITY AND STRICHARTZ ESTIMATE 11 we obtain for λ/ ≤ ρ ≤ λ Z λ + ρλ/ r (1 /q ) − h ( ρ, λ ; r ) − / g ( r ) dr (2.32) ≤ Cλ /q Z λ + ρλ/ p ( ρ − r + λ )( ρ + r − λ ) g ( r ) dr ≤ Cλ (1 /q ) − (cid:18)Z λλ/ s λr − ( λ − ρ ) g ( r ) dr + Z λ + ρλ s λ ( λ + ρ ) − r g ( r ) dr (cid:19) ≤ Cλ /q (cid:18) ρ Z ( λ − ρ )+ ρ ( λ − ρ ) − ρ (cid:12)(cid:12)(cid:12)(cid:12) ρη − ( λ − ρ ) (cid:12)(cid:12)(cid:12)(cid:12) / g ∗ ( η ) dη + 12 ρ Z ( λ + ρ )+ ρ ( λ + ρ ) − ρ (cid:12)(cid:12)(cid:12)(cid:12) ρ ( λ + ρ ) − η (cid:12)(cid:12)(cid:12)(cid:12) / g ∗ ( η ) dη (cid:19) ≤ Cλ /q ( M / g ∗ )( λ − ρ ) + Cλ /q ( M / g ∗ )( λ + ρ )as in (2.25). Note that, at the second inequality, we have used λ ≤ ρ − λ + λ ≤ ρ − r + λ ≤ λ − λ λ = 32 λ for λ/ ≤ r ≤ λ and λ/ ≤ ρ ≤ λ , and λ ≤ ρ + λ − λ ≤ ρ + r − λ ≤ λ + ( λ + ρ ) − λ ≤ λ for λ ≤ r ≤ λ + ρ and λ/ ≤ ρ ≤ λ . By virtue of the estimate (2.32) we canobtain(2.33) L ≤ C k λ /q w k L p ((0 , ∞ ) ,λ pα dλ ) k g k L q ′ ((0 , ∞ ) ,r q ′ β dr ) as in (2.29), (2.31). The estimate of L has been completed.Next let us consider the estimate of L . Note that ρ + r − λ ≤ λ/ ρ + r − λ ≥ λ + r − λ ≥ λ/ λ/ ≤ r ≤ λ and λ ≤ ρ ≤ λ/
2. Using (2.24), we hence have for λ ≤ ρ ≤ λ/ Z λλ/ r (1 /q ) − h ( ρ, λ ; r ) − / g ( r ) dr (2.34) ≤ Cλ /q Z λλ/ p ( ρ − r + λ )( r + λ − ρ ) g ( r ) dr ≤ Cλ (1 /q ) − (cid:18)Z λλ/ s λr − ( ρ − λ ) g ( r ) dr + Z λλ s λ ( ρ + λ ) − r g ( r ) dr (cid:19) ≤ Cλ /q (cid:18) λ Z ( ρ − λ )+ λ ( ρ − λ ) − λ (cid:12)(cid:12)(cid:12)(cid:12) λη − ( ρ − λ ) (cid:12)(cid:12)(cid:12)(cid:12) / g ∗ ( η ) dη + 13 λ Z ( ρ + λ )+(3 λ/ ρ + λ ) − (3 λ/ (cid:12)(cid:12)(cid:12)(cid:12) λ/ ρ + λ ) − η (cid:12)(cid:12)(cid:12)(cid:12) / g ∗ ( η ) dη (cid:19) ≤ Cλ /q ( M / g ∗ )( ρ − λ ) + Cλ /q ( M / g ∗ )( ρ + λ ) . This leads us to the estimate(2.35) L ≤ C k λ /q w k L p ((0 , ∞ ) ,λ pα dλ ) k g k L q ′ ((0 , ∞ ) ,r q ′ β dr ) as before. The estimate of L has been completed.It remains to bound L . Note that, for ρ − λ ≤ r ≤ λ and 3 λ/ ≤ ρ ≤ λ ,we have ρ − r + λ ≤ ρ − ( ρ − λ ) + λ = 2 λ , ρ − r + λ ≥ (3 λ/ − λ + λ = λ/ ρ + r − λ ≤ λ + 2 λ − λ = 4 λ , ρ + r − λ ≥ ρ + ( ρ − λ ) − λ ≥ λ . Wetherefore obtain for 3 λ/ ≤ ρ ≤ λ Z λρ − λ r (1 /q ) − h ( ρ, λ ; r ) − / g ( r ) dr (2.36) ≤ Cλ (1 /q ) − Z λρ − λ s λr − ( ρ − λ ) g ( r ) dr ≤ Cλ /q λ Z ( ρ − λ )+2 λ ( ρ − λ ) − λ (cid:12)(cid:12)(cid:12)(cid:12) λη − ( ρ − λ ) (cid:12)(cid:12)(cid:12)(cid:12) / g ∗ ( η ) dη ≤ Cλ /q ( M / g ∗ )( ρ − r ) , which yields(2.37) L ≤ C k λ /q w k L p ((0 , ∞ ) ,λ pα dλ ) k g k L q ′ ((0 , ∞ ) ,r q ′ β dr ) as before. Combining (2.31), (2.33), (2.35), (2.37) with (2.23), we have shown(2.22). The proof of Proposition 2.5 has been finished. (cid:3) We are in a position to complete the proof of Theorem 2.1 for n = 2. Thisis a direct consequence of (2.3), (2.17), (2.15) with n = 1, and (2.22). Theproof of Theorem 2.1 has been completed for all n ≥ (cid:3) EIGHTED HLS INEQUALITY AND STRICHARTZ ESTIMATE 13
Remark . The inequality (2.2) with α = β = 0 is just the one Vilela has usedin [39]. Vilela has shown the inequality by employing some ideas in Stein andWeiss [33]. (See [39] on page 369.) Now that we have completed the proof ofTheorem 2.1, it is obvious that we can show (2.2) for α = β = 0 by employingthe classical Hardy-Littlewood inequality and the Hardy-Littlewood maximalinequality in the standard L p ( R n ) space. Hence it is also possible to show(2.2) without results in [33], as far as the case α = β = 0 is concerned. It isin the case α = 0 or β = 0 that our proof of (2.2) essentially relies upon theresult of Stein and Weiss [33].3. Strichartz estimates for radial solutions
Adapting an argument of Vilela [39], we explain how the weighted Hardy-Littlewood-Sobolev inequality (2.2) is used to prove the Strichartz estimatefor the free wave equation with radially symmetric data. Let us start ourconsideration with global-in-time estimates. Recalling the definition of theoperator W (see (1.1)), we shall show Theorem 3.1.
Suppose n ≥ and / < ( n − / − (1 /p )) < ( n − / .There exists a constant C depending on n and p , and the estimate k W ϕ k L ( R ; L p ( R n )) ≤ C k| D x | s ϕ k L ( R n ) , (3.1) 12 + np = n − s holds for radially symmetric ϕ ∈ ˙ H s ( R n ) . It should be mentioned that Sterbenz has proved (3.1) in a completelydifferent way (see Proposition 1.2 of [34]). As has been done in [34], we canactually obtain the following result by the interpolation between (3.1) and theenergy identity. For any integer n ≥ D n := (cid:26) ( x, y ) ∈ R (cid:12)(cid:12)(cid:12) < x ≤ , < y ≤ , (3.2) n − (cid:18) − y (cid:19) < x < ( n − (cid:18) − y (cid:19) (cid:27) and(3.3) A n := D n ∪ (cid:26) ( x, y ) ∈ R (cid:12)(cid:12)(cid:12) x = 0 and y = 12 (cid:27) . Corollary 3.2.
Suppose n ≥ and (1 /q, /p ) ∈ A n . There exists a constant C depending on n , p , q , and the estimate k W ϕ k L q ( R ; L p ( R n )) ≤ C k| D x | s ϕ k L ( R n ) , (3.4) 1 q + np = n − s holds for radially symmetric ϕ ∈ ˙ H s ( R n ) .Remark . Without the assumption of radial symmetry the Strichartz esti-mate (3.4) holds, provided that n ≥ , ≤ q ≤ , ≤ p ≤ , (cid:18) q , p (cid:19) = (0 , , q ≤ ( n − (cid:18) − p (cid:19) , (3.5) (cid:18) q , p (cid:19) = (cid:18) , (cid:19) if n = 2 , (cid:18) q , p (cid:19) = (cid:18) , (cid:19) if n ≥ . See [35], [28], [22], [7], [20], and [18] for the proof. We note that the condition2 /q ≤ ( n − / − /p ) of (3.5) is necessary. Otherwise, it is well-knownthat, using the method of Knapp, one can indeed choose a sequence { ϕ j } ⊂S ( R n ) of non-radial data for which the existence of such a uniform constant C = C ( n, p, q ) as in (3.4) is forbidden. Keeping in mind that some non-radialsolutions yield this counterexample, we mention that radial symmetry vastlyimproves on the range of the admissible pairs (1 /q, /p ). Indeed, it has turnedout by the works of Klainerman and Machedon [19], Sterbenz [34], and Fangand Wang [5] (see also Sogge [31] on page 125) that one actually has theStrichartz estimate (3.4) under the assumption of radial symmetry in the caseof(3.6) n ≥ , ≤ q ≤ , ≤ p ≤ , (cid:18) q , p (cid:19) = (0 , , q < ( n − (cid:18) − p (cid:19) , in addition to the obvious case (1 /q, /p ) = (0 , / /q < ( n − / − (1 /p )) of(3.6) is necessary for the global-in-time estimate (3.4) to hold for radiallysymmetric data. The prime purpose of this section is to explain how we canprove Proposition 1.2 of Sterbenz [34] using the weighted Hardy-Littlewood-Sobolev inequality (2.2). Proof of Theorem 3.1.
We use the following result which is a generalizationof the classical estimate of Morawetz [25].
Lemma 3.3.
Suppose n ≥ and / < α < n/ . There exists a constant C depending on n and α , and the estimate (3.7) k| x | − α W ϕ k L ( R × R n ) ≤ C k| D x | α − (1 / ϕ k L ( R n ) holds for ϕ ∈ ˙ H α − (1 / ( R n ) . The proof of (3.7) uses the trace inequality in the Fourier space(3.8) sup λ> λ ( n/ − s Z S n − | ˆ w ( λω ) | dσ ≤ C k| D ξ | s ˆ w k L ( R n ) = C ′ k| x | s w k L ( R n ) which holds for 1 / < s < n/
2. See Ben-Artzi [2], Ben-Artzi and Klainerman[3], Hoshiro [13] for the proof of (3.7) via the trace inequality such as (3.8)
EIGHTED HLS INEQUALITY AND STRICHARTZ ESTIMATE 15 and the duality argument. For the proof of (3.8) see, e.g., (2.45) of Li andZhou [21] and Appendix of Hidano [9].We are in a position to complete the proof of Theorem 3.1. We follow theargument of Vilela [39]. Fix any p satisfying 1 / < ( n − / − (1 /p )) < ( n − /
2. It follows from Theorem 2.1 with α = β = 0 that the Sobolev-typeinequality(3.9) k v k L p ( R n ) ≤ C k| x | − ( n − / − (1 /p )) | D x | (1 / − (1 /p ) v k L ( R n ) holds for radially symmetric v . The estimate (3.1) is an immediate conse-quence of (3.7) and (3.9). Indeed, we see that k W ϕ k L ( R ; L p ( R n )) (3.10) ≤ C k| x | − ( n − / − (1 /p )) | D x | (1 / − (1 /p ) W ϕ k L ( R × R n ) = C k| x | − ( n − / − (1 /p )) W ( | D x | (1 / − (1 /p ) ϕ ) k L ( R × R n ) ≤ C k| D x | (( n − / − ( n/p ) ϕ k L ( R n ) as desired. The proof of Theorem 3.1 has been finished. (cid:3) Failure of the critical estimate
The problem to be discussed in this section is whether the Strichartz esti-mate (3.4) holds under the assumption of radial symmetry of data even for thelimiting pair (1 /q, /p ) ∈ (0 , / × [0 , /
2) with 1 /q = ( n − / − (1 /p )).If it were true, we would enjoy(4.1) k W ϕ k L q ( R ; L p ( R n )) ≤ C k| D x | (1 / − (1 /p ) ϕ k L ( R n ) for radially symmetric data ϕ , and the estimate (4.1) would imply the estimate k u k L q ( R ; L p ( R n )) (4.2) ≤ C (cid:0) k| D x | (1 / − (1 /p ) f k L ( R n ) + k| D x | − (1 / − (1 /p ) g k L ( R n ) (cid:1) for the solution u to the wave equation (cid:3) u = 0 in R × R n with radiallysymmetric data ( f, g ). We shall show that the estimate (4.2) is false in thelimiting case (1 /q, /p ) ∈ (0 , / × [0 , /
2) with 1 /q = ( n − / − (1 /p )),though S ( R n ) ⊂ ˙ H − (1 / − (1 /p )2 ( R n ) ( n ≥ Lemma 4.1.
Let n ≥ and r = | x | . Suppose g ( x ) is a smooth, non-negativefunction with supp g ⊂ { x ∈ R n | | x | ≤ R } for some R > . Suppose alsothat g is a radially symmetric function written as g ( x ) = ψ ( r ) for an evenfunction ψ ∈ C ∞ ( R ) . Let u be the solution to (cid:3) u = 0 in R × R n with data (0 , g ) at t = 0 . There exists a positive constant δ depending only on n suchthat the estimate (4.3) u ( t, x ) ≥ r ( n − / Z min { R,r + t } r − t λ ( n − / ψ ( λ ) dλ holds for any ( t, x ) with R/ (1 + δ ) ≤ r − t ≤ R , t > . Let us postpone the proof of Lemma 4.1 for the moment and see how itcan be used to prove
Theorem 4.2.
Let n ≥ and fix the constant δ > given by Lemma . .Suppose that g ( x ) ≥ is a smooth, radially symmetric function with supp g ⊂{ x ∈ R n | | x | ≤ } which is written as g ( x ) = ψ ( r ) for an even function ψ ∈ C ∞ ( R ) satisfying the condition that the function Ψ defined as (4.4) Ψ( ρ ) := Z ρ λ ( n − / ψ ( λ ) dλ does not vanish identically for ρ ∈ (1 / (1 + δ ) , .Let (1 /q, /p ) ∈ (0 , / × [0 , / satisfy /q = ( n − / − /p ) . Then,for the solution to (cid:3) u = 0 with data (0 , g ) at t = 0 , (4.5) lim T → + ∞ k u k L q ((0 ,T ); L p ( R n )) = + ∞ . Proof of Theorem 4.2.
We separate two cases: (1 /q, /p ) ∈ (0 , / × (0 , / /q = ( n − / − /p ) for n ≥ /q, /p ) = (1 / ,
0) for n =2. We start with the former. Employing (4.3) and writing u ( t, x ) = v ( t, r ),we have for t ∈ ( δ/ (2(1 + δ )) , T ) by the change of variables ρ = r − t Z t +1 t +(1 / (1+ δ )) v p ( t, r ) r n − dr (4.6) ≥ p Z t +1 t +(1 / (1+ δ )) (cid:18) r ( n − / Z r − t λ ( n − / ψ ( λ ) dλ (cid:19) p r n − dr = 14 p Z / (1+ δ ) t + ρ ) ( n − ) p − ( n − Ψ p ( ρ ) dρ ≥ p t + 1) ( n − ) p − ( n − Z / (1+ δ ) Ψ p ( ρ ) dρ. Setting a strictly positive constant A as A := (cid:18)Z / (1+ δ ) Ψ p ( ρ ) dρ (cid:19) /p , we then find k u k qL q ((0 ,T ); L p ( R n )) ≥ Z Tδ/ (2(1+ δ )) (cid:18)Z t +1 t +(1 / (1+ δ )) v p ( t, r ) r n − dr (cid:19) q/p dt (4.7) ≥ A q q Z Tδ/ (2(1+ δ )) (cid:18) t + 1) n − − n − p (cid:19) q dt = A q q Z Tδ/ (2(1+ δ )) t + 1 dt = A q q log T + 1 δ δ ) + 1 EIGHTED HLS INEQUALITY AND STRICHARTZ ESTIMATE 17 for all T ≥ δ/ (2(1 + δ )).It remains to deal with (1 /q, /p ) = (1 / ,
0) for n = 2. We naturally modifythe argument in (4.6)–(4.7) as follows. Fix a constant c satisfying 1 / (1+ δ )
0. We see, noting k v ( t, · ) k L ∞ ( t +(1 / (1+ δ )) Let n ≥ and (1 /q, /p ) ∈ (0 , / × [0 , / satisfy /q =( n − / − /p ) . Then, for the solution u h to (cid:3) u = 0 with radially symmetricdata (0 , h ) at t = 0sup (cid:26) k u h k L q ((0 , L p ( R n )) k| D x | − (1 / − (1 /p ) h k L ( R n ) (cid:12)(cid:12) h ∈ S ( R n ) \ { } and h is radially symmetric (cid:27) = + ∞ . This shows that the estimate (4.2) is false even if the global-in-time normis replaced by the local-in-time norm on the left-hand side. The proof ofCorollary 4.3 is straightforward, and therefore we leave it to the reader. Proof of Lemma 4.1. We must establish Lemma 4.1. The proof is essentiallybased on Rammaha’s way for Lemma 2 of [29] together with the treatmentof fundamental solutions in even space dimensions in Agemi [1], which issummarized in Takamura [38]. Following [1], [29] and [37]–[38], we showLemma 4.1.By the representations (6a) and (6b) of radial solutions in [29], u is ex-pressed as(4.9) u ( t, x ) = 12 r m Z r + t | r − t | λ m ψ ( λ ) P m − (cid:18) λ + r − t rλ (cid:19) dλ, if n = 2 m + 1, and u ( t, x )(4.10) = 2 πr m − Z t ρdρ p t − ρ Z r + ρ | r − ρ | λ m ψ ( λ ) p G ( λ, r, ρ ) T m − (cid:18) λ + r − ρ rλ (cid:19) dλ = 2 πr m − Z r + tr − t λ m ψ ( λ ) dλ Z t | r − λ | ρ p G ( ρ, r, λ ) p t − ρ × T m − (cid:18) λ + r − ρ rλ (cid:19) dρ, if n = 2 m and r > t , where G ( λ, r, ρ ) = ( λ − ( r − ρ ) )(( r + ρ ) − λ ) = G ( ρ, r, λ ) , and P k , T k are the Legendre and Tschebyscheff polynomials, respectively,defined by P k ( z ) = 12 k k ! d k dz k ( z − k ,T k ( z ) = ( − k (2 k − − z ) / d k dz k (1 − z ) k − / . See also [37] for details.As is well-known, P k and T k have the properties: | P k ( z ) | , | T k ( z ) | ≤ | z | ≤ 1) and P k (1) = T k (1) = 1 for all k = 1 , , . . . (see Magnus, Oberhettinger andSoni [24], p. 227, p. 237, pp. 256–267). By these properties together with thecontinuity of the two functions, one can choose a small constant δ dependingon n so that(4.11) P m − ( z ) , T m − ( z ) ≥ 12 for 11 + δ ≤ z ≤ , m ∈ N , in the same manner as Takamura did in Lemma 2.5 of [37].In what follows we assume that R/ (1 + δ ) ≤ r − t ≤ R with t > 0. Notethat the upper limit of the λ -integrals in (4.9) and (4.10) can be replaced withmin { R, r + t } by virtue of the support property of data. We then have1 ≥ λ + r − ρ rλ ≥ λ + r − t rλ (4.12) ≥ ( r − t ) + r − t rR = r − tR ≥ 11 + δ EIGHTED HLS INEQUALITY AND STRICHARTZ ESTIMATE 19 for r − t ≤ λ ≤ min { R, r + t } and 0 ≤ ρ ≤ t . It therefore follows from(4.9)–(4.11) that(4.13) u ( t, x ) ≥ r m Z min { R,r + t } r − t λ m ψ ( λ ) dλ if n = 2 m + 1 , πr m − Z min { R,r + t } r − t λ m ψ ( λ ) dλ Z t | r − λ | ρdρ p G ( ρ, r, λ ) p t − ρ if n = 2 m, provided ψ ≥ ρ -integral in (4.13) is estimatedas follows: ( ρ -integral) ≥ √ rλ Z t | r − λ | ρdρ p ρ − ( r − λ ) p t − ρ = B (2 − , − )4 √ rλ = π √ rλ . Here by B ( · , · ) we have meant the beta function as in Section 2. This completesthe proof for the even dimensional case. (cid:3) Remark . During the preparation of this article, the authors found that,arguing in a way similar to Takamura [37], Jiao and Zhou had already obtainedan estimate which is a bit less precise than (4.3) (see Lemma 2 of [14]).5. Local-in-time Strichartz estimates For any integer n ≥ n := (cid:26) ( x, y ) ∈ R (cid:12)(cid:12)(cid:12) < x ≤ , < y ≤ , x > ( n − (cid:18) − y (cid:19) (cid:27) and(5.2) Λ n := Ω n ∪ (cid:26) ( x, y ) ∈ R (cid:12)(cid:12)(cid:12) x = 0 and y = 12 (cid:27) The main result of this section is the following. Theorem 5.1. Suppose n ≥ and (1 /q, /p ) ∈ Λ n . Let T be an arbitrarypositive number. There exists a constant C depending only on n , p and q , andthe estimate k W ϕ k L q ((0 ,T ) ,L p ( R n )) ≤ CT θ k| D x | (1 / − (1 /p ) ϕ k L ( R n ) , (5.3) 1 q + np = θ + n − (cid:18) − p (cid:19) holds for radially symmetric data ϕ ∈ ˙ H (1 / − (1 /p )2 ( R n ) . Theorem 5.1 is an extension of the intriguing result of Sogge (Proposition6.3 on the page 125 of [31]) who proved the estimate (5.3) for n = 3 and (cid:18) q , p (cid:19) ∈ (cid:26) ( x, y ) ∈ R (cid:12)(cid:12)(cid:12) < x ≤ y ≤ , x > (cid:18) − y (cid:19)(cid:27) (5.4) ∪ (cid:26) ( x, y ) ∈ R (cid:12)(cid:12)(cid:12) x = 0 and y = 12 (cid:27) . Actually, Sogge himself proved the estimate (5.3) for n = 3, 1 / < /q =1 /p ≤ / 2. By the interpolation between his estimate and the energy estimatewe easily get (5.3) for n = 3 and (1 /q, /p ) satisfying (5.4).We should explain the significance of the local-in-time estimate (5.3). Ifthe radially symmetric estimate (4.1) were true even for the limiting pair(1 /q, /p ) ∈ (0 , / × [0 , / 2) with 1 /q = ( n − / − (1 /p )), our estimate(5.3) would be a trivial consequence of (4.1) and the H¨older inequality in time.The fact is that the estimate (4.1), even if localized in time, is false for anylimiting pair (1 /q, /p ) ∈ (0 , / × [0 , / 2) with 1 /q = ( n − / − (1 /p ))as we have seen in Section 4, and one can get nothing but a coarse estimate k W ϕ k L q ((0 ,T ) ,L p ( R n )) ≤ CT θ k| D x | (1 / − (1 /p )+ ε ϕ k L ( R n ) , (5.5) 1 q + np = θ + n − (cid:18) − p + ε (cid:19) , ε > /q, /p ) ∈ Ω n with (1 /q, /p ) ∈ (0 , / × (0 , / 2) by using boththe Strichartz estimate (3.4) for (1 /q, /p ) permitted in (3.6) and the H¨olderinequality in time. As we have just mentioned, Sogge proved the sharperestimate (5.3) in the case of n = 3, 1 / < /q = 1 /p ≤ / 2, and the key tohis proof was a clever use of the identity c dσ ( | ξ | ) = Z S e − iω · ξ dσ = 4 π sin | ξ || ξ | (5.6) (cid:0) ω ∈ S = { x ∈ R | | x | = 1 } , dσ = dσ ( ω ) (cid:1) . Though the formula of c dσ ( | ξ | ) in terms of the Bessel function is well-known for n = 2 or n ≥ 4, the authors do not know whether such a formula is useful inproving our estimate (5.3). In the rest of this section we see how the weightedinequality (2.2) is used to prove the local-in-time estimate (5.3). Proof of Theorem 5.1. We use the following result. Lemma 5.2. Suppose n ≥ and ≤ α < / . Let T be an arbitrary positivenumber. There exists a constant C depending on n and α , and the estimate (5.7) k| x | − α W ϕ k L ((0 ,T ) × R n ) ≤ CT (1 / − α k ϕ k L ( R n ) holds for all ϕ ∈ L ( R n ) . EIGHTED HLS INEQUALITY AND STRICHARTZ ESTIMATE 21 By scaling the proof of (5.7) can be reduced to the case T = 1. For T = 1 the estimate (5.7) has been shown in [10] as a direct consequence ofintegrability (in time) of the local energy [30](5.8) k W ϕ k L ( R ×{ x ∈ R n || x | < } ) ≤ C k ϕ k L ( R n ) , scaling, and the energy estimate.We are in a position to complete the proof of Theorem 5.1. Fix any p (0 < /p ≤ / 2) satisfying 1 / > ( n − / − (1 /p )). It follows from theSobolev-type estimate (3.9) and (5.7) that k W ϕ k L ((0 ,T ); L p ( R n )) (5.9) ≤ C k| x | − ( n − / − (1 /p )) | D x | (1 / − (1 /p ) W ϕ k L ((0 ,T ) × R n ) ≤ CT (1 / − ( n − / − (1 /p )) k| D x | (1 / − (1 /p ) ϕ k L ( R n ) . Our estimate (5.3) is a consequence of the interpolation between (5.9) and theenergy estimate. We have finished the proof of Theorem 5.1. (cid:3) End-point estimates for Schr¨odinger equations The final section is devoted to the study of the Strichartz estimate for theSchr¨odinger equation(6.1) i∂ t u − ∆ u = 0 in R × R n subject to the initial data u (0 , x ) = ϕ ( x ). The estimate(6.2) k Sϕ k L ( R ; L n/ ( n − ( R n )) ≤ C k ϕ k L ( R n ) ( n ≥ , which was proved by Keel and Tao [18], is called an end-point estimate. (See(1.2) for the definition of the operator S .) As Vilela has explained in Section3 of [39], it is possible to prove (6.2) for radially symmetric data via theweighted inequality (2.2) with α = β = 0. We revisit the problem of showing(6.2) for radially symmetric data. Using our weighted inequality (2.2) with − β = α , we prove Theorem 6.1. Suppose n ≥ and − (1 / 2) + (1 /n ) < α < (1 / − (1 /n ) .There exists a constant C depending on n , α , and the estimate (6.3) k| x | α | D x | α Sϕ k L ( R ; L n/ ( n − ( R n )) ≤ C k ϕ k L ( R n ) holds for radially symmetric data ϕ ∈ L ( R n ) .Proof of Theorem 6.1. We need the following lemma. Lemma 6.2. Suppose n ≥ and / < γ < n/ . There exists a constant C depending on n , γ , and the estimate (6.4) k| x | − γ | D x | − γ Sϕ k L ( R × R n ) ≤ C k ϕ k L ( R n ) holds. Large part of Lemma 6.2 was proved by Kato and Yajima [17], Ben-Artziand Klainerman [3], independently. Later their results were not only comple-mented but also generalized by Sugimoto [36] and Vilela [39]. For the proofof (6.4) see Section 4 of Sugimoto [36] or Section 1 of Vilela [39].In what follows we denote 2 n/ ( n − 2) by p . We note that12 < ( n − (cid:18) − p (cid:19) − α < n ⇐⇒ − n − n < α < − n and that the inequality 1 − n − n ≤ − 12 + 1 n is true for all n ≥ n ≥ − α < p ⇐⇒ − 12 + 1 n < α, we can employing (2.2) with − β = α first and (6.4) secondly to have forradially symmetric ϕ k| x | α | D x | α Sϕ k L ( R ; L p ( R n )) (6.5) ≤ C k| x | − ( n − / − (1 /p ))+ α | D x | α +(1 / − (1 /p ) ϕ k L ( R × R n ) ≤ C k ϕ k L ( R n ) . It is only at the last inequality above that the choice of p = 2 n/ ( n − 2) isessential. The proof of Theorem 6.1 has been completed. (cid:3) Acknowledgements. The authors are grateful to Professors RentaroAgemi and Hiroyuki Takamura for their comments. They also thank the ref-eree for reading the manuscript carefully and making a number of invaluablesuggestions. The first author was partly supported by the Grant-in-Aid forYoung Scientists (B) (No. 15740092 and 18740069), The Ministry of Educa-tion, Culture, Sports, Science and Technology, Japan. The second author waspartly supported by the Grant-in-Aid for Young Scientists (B) (No. 18740096),The Ministry of Education, Culture, Sports, Science and Technology, Japan. References [1] R. Agemi, Blow-up of solutions to nonlinear wave equations in two space dimensions ,Man. Math. (1991), 153–162.[2] M. Ben-Artzi, Regularity and smoothing for some equations of evolution , in “Nonlin-ear Partial Differential Equations and Applications” (H. Brezis and J.L. Lions, eds.),Vol. 11, Pittman, London, 1994, pp. 1–12.[3] M. Ben-Artzi and S. Klainerman, Decay and regularity for the Schr¨odinger equation ,J. Anal. Math., (1992), 25–37.[4] M. Christ and A. Kiselev, Maximal functions associated to filtrations , J. Funct. Anal., (2001), 409–425. EIGHTED HLS INEQUALITY AND STRICHARTZ ESTIMATE 23 [5] D. Fang and C. Wang, Some remarks on Strichartz estimates for homogeneous waveequation , Nonlinear Anal., (2006), 697–706.[6] D. Foschi, Inhomogeneous Strichartz estimates , J. Hyperbolic Differ. Equ., (2005),1–24.[7] J. Ginibre and G. Velo, Generalized Strichartz inequalities for the wave equation , J.Funct. Anal., (1995), 50–68.[8] J. Harmse, On Lebesgue space estimates for the wave equation , Indiana Univ. Math. J., (1990), 229–248.[9] K. Hidano, Small data scattering and blow-up for a wave equation with a cubic con-volution , Funkcial. Ekvac., (2000), 559–588.[10] K. Hidano, Morawetz-Strichartz estimates for spherically symmetric solutions to waveequations and applications to semi-linear Cauchy problems , Differential Integral Equa-tions (2007), 735–754.[11] K. Hidano, Nonlinear Schr¨odinger equations with radially symmetric data of criticalregularity , Accepted for publication in Funkcialaj Ekvacioj.[12] K. Hidano, Small solutions to semi-linear wave equations with radial data of criticalregularity , Submitted for publication.[13] T. Hoshiro, On weighted L estimates of solutions to wave equations , J. Anal. Math., (1997), 127–140.[14] H. Jiao and Z. Zhou, An elementary proof of the blow-up for semilinear wave equationin high space dimensions , J. Differential Equations (2003), 355–365.[15] F. John, Plane waves and spherical means : applied to partial differential equations ,Interscience Publishers, NY, 1955.[16] T. Kato, An L q,r -theory for nonlinear Schr¨odinger equations , in “Spectral and scat-tering theory and applications” (K. Yajima, ed.), Adv. Stud. Pure Math., Vol. 23,Math. Soc. Japan, Tokyo, 1994, pp. 223–238.[17] T. Kato and K. Yajima, Some examples of smooth operators and the associatedsmoothing effect , Rev. Math. Phys., (1989), 481–496.[18] M. Keel and T. Tao, Endpoint Strichartz estimates , Amer. J. Math., (1998), 955–980.[19] S. Klainerman and M. Machedon, Spacetime estimates for null forms and the localexistence theorem , Comm. Pure Appl. Math., (1993), 1221–1268.[20] S. Klainerman and M. Machedon, On the algebraic properties of the H n/ , / spaces ,Int. Math. Res. Not., (1998), 765–774.[21] T.T. Li and Y. Zhou, A note on the life-span of classical solutions to nonlinear waveequations in four space dimensions , Indiana Univ. Math. J., (1995), 1207–1248.[22] H. Lindblad and C.D. Sogge, On existence and scattering with minimal regularity forsemilinear wave equations , J. Funct. Anal., (1995), 357–426.[23] H. Lindblad and C.D. Sogge, Long-time existence for small amplitude semilinear waveequations , Amer. J. Math., (1996), 1047–1135.[24] W. Magnus, F. Oberhettinger and R.P. Soni, Formulas and Theorems for the SpecialFunctions of Mathematical Physics, Third Edition , Springer-Verlag, Berlin, Heidel-berg, New York, 1966.[25] C.S. Morawetz, Time decay for the Klein-Gordon equation , Proc. Roy. Soc. A, (1968), 291–296.[26] B. Muckenhoupt, Weighted norm inequalities for the Hardy maximal function , Trans.Amer. Math. Soc., (1972), 207–226.[27] D.M. Oberlin, Convolution estimates for some distributions with singularities on thelight cone , Duke Math. J., (1989), 747–757.[28] H. Pecher, Nonlinear small data scattering for the wave and Klein-Gordon equation ,Math. Z. (1984), 261–270. [29] M.A. Rammaha, Finite-time blow-up for nonlinear wave equations in high dimensions ,Comm. Partial Differential Equations , (1987), 677–700.[30] H. Smith and C.D. Sogge, Global Strichartz estimates for nontrapping perturbationsof the Laplacian , Comm. Partial Differential Equations, (2000), 2171–2183.[31] C.D. Sogge, Lectures on nonlinear wave equations , Int. Press, Cambridge, MA, 1995.[32] E.M. Stein, Harmonic Analysis: Real-variable Methods, Orthogonality, and Oscilla-tory Integrals , Princeton University Press, Princeton, NJ, 1993.[33] E.M. Stein and G. Weiss, Fractional integrals on n -dimensional Euclidean space , J.Math. Mech., (1958), 503–514.[34] J. Sterbenz, Angular regularity and Strichartz estimates for the wave equation , Int.Math. Res. Not. (2005), 187–231.[35] R.S. Strichartz, Restrictions of Fourier Transforms to quadratic surfaces and decay ofsolutions of wave equations , Duke Math. J., (1977), 705–714.[36] M. Sugimoto, Global smoothing properties of generalized Schr¨odinger equations , J.Anal. Math., (1998), 191–204.[37] H. Takamura, Blow-up for semi-linear wave equations with slowly decaying data inhigh dimensions , Differential Integral Equations (1995), 647–661.[38] H. Takamura, Nonexistence of global solutions to semilinear wave equations , Thesis,Hokkaido University, 1995.[39] M.C. Vilela, Regularity of solutions to the free Schr¨odinger equation with radial initialdata , Illinois J. Math., (2001), 361-370.[40] M.C. Vilela, Inhomogeneous Strichartz estimates for the Schr¨odinger equation , Trans.Amer. Math. Soc., (2007), 2123–2136. Department of Mathematics, Faculty of Education, Mie University, 1577 Kurima-machiya-cho, Tsu, Mie 514-8507, Japan E-mail address : [email protected] General Education, Yonago National College of Technology, 4448 Hikona-cho, Yonago, Tottori 683-8502, Japan E-mail address ::