A Sharp Sobolev--Strichartz estimate for the wave equation
aa r X i v : . [ m a t h . A P ] J u l A SHARP SOBOLEV–STRICHARTZ ESTIMATE FOR THE WAVE EQUATION
NEAL BEZ AND CHRIS JEAVONS
Abstract.
We calculate the the sharp constant and characterise the extremal initial data in ˙ H × ˙ H − for the L Sobolev–Strichartz estimate for the wave equation in four space dimensions. Introduction
For d ≥ s ∈ (cid:2) , d (cid:1) the well-known Sobolev–Strichartz estimate for the one-sided wave propagatorstates that, for some finite constant C > k e it √− ∆ f k L p ( R d +1 ) ≤ C k f k ˙ H s ( R d ) for each f in the homogeneous Sobolev space ˙ H s ( R d ), with norm given by k f k ˙ H s ( R d ) = k ( − ∆) s f k L ( R d ) ,and where p = 2( d + 1) d − s . The sharp constant in the estimate (1) given by W ( d, s ) := sup f ∈ ˙ H s \{ } k e it √− ∆ f k L p ( R d +1 ) k f k ˙ H s ( R d ) has attracted attention in recent years; however, to date, the value of W ( d, s ) and a full characteri-sation of extremisers (those f which attain the supremum) has been established only in some ratherisolated cases. It is known that, for all admissible ( d, s ), an extremiser exists (see [5], [7], [14]). Iden-tifying the exact shape of such extremisers appears to be a rather difficult problem, with prior resultsin this direction only available in the cases ( d, s ) equal to (2 , ) and (3 , ), due to Foschi [8], and thecase ( d, s ) equal to (5 ,
1) in [3]. In each of these cases, the initial datum f ⋆ whose Fourier transformis given by b f ⋆ ( ξ ) = e −| ξ | | ξ | is extremal; in fact, these works also gave a full characterisation of the extremal data by showing thatany extremiser f coincides with f ⋆ up to the action of a certain group of transformations (which areslightly different when s = and s = 1). Based on these results, it is tempting to boldly conjecturethat such f are extremisers for all admissible ( d, s ). Whilst this is premature, the purpose of thisshort paper is to add further weight and show that this is indeed the case for ( d, s ) = (4 , ). Theorem 1.1.
The one-sided wave propagator satisfies the estimate (2) k e it √− ∆ f k L ( R ) ≤ W (4 , ) k f k ˙ H ( R ) Date : October 27, 2018.2010
Mathematics Subject Classification.
Primary 35B45; Secondary 35L05. with constant W (4 , ) = (cid:18) π (cid:19) . The constant is sharp and is attained if and only if b f ( ξ ) = e a | ξ | + ib · ξ + c | ξ | , where a, c ∈ C such that Re( a ) < , and b ∈ R d . Our proof of Theorem 1.1 relies on a sharp estimate for the one-sided wave propagator from [3]; thisis followed by a further argument using spherical harmonics inspired by recent work of Foschi [9] onthe sharp Stein–Tomas adjoint Fourier restriction theorem for the sphere S in R . We also show thatsuch an approach may be used to recover in a new manner the characterisation of extremisers in [3]for the case ( d, s ) = (5 , Corollary 1.2.
The solution of the wave equation ∂ tt u = ∆ u on R × R with initial data ( u (0) , ∂ t u (0)) satisfies k u k L ( R ) ≤ (cid:18) π (cid:19) (cid:18) k u (0) k H ( R ) + k ∂ t u (0) k H − ( R ) (cid:19) and the constant is sharp. Furthermore, the initial data given by ( u (0) , ∂ t u (0)) = (0 , (1 + | x | ) − ) , is extremal and generates the set of all extremal initial data under the action of the group generatedby the transformations: • space-time translations u ( t, x ) → u ( t + t , x + x ) with ( t , x ) ∈ R d +1 ; • parabolic dilations u ( t, x ) → u ( µ t, µx ) with µ > ; • change of scale u ( t, x ) → µu ( t, x ) with µ > ; • phase shift u ( t, x ) → e iθ u ( t, x ) with θ ∈ R . Our results here also fit into a broader collection of recent papers on sharp Sobolev–Strichartz estimatesfor dispersive propagators, where, broadly speaking, the question of existence of extremisers is well-understood yet the identification of their shape has only been established in rather special cases (see,for example, [7], [8], [11], [13], [15]).In Section 2 we prove Theorem 1.1 and Corollary 1.2, and in Section 3 we adapt our method to obtainan alternative proof of the analogous result from [3] for the case ( d, s ) = (5 , Acknowledgement.
The authors express their thanks to Jon Bennett for helpful conversations.2.
Proof of Theorem 1.1 and Corollary 1.2
A key ingredient in the proof of Theorem 1.1 is the following sharp inequality proved in [3]. Here weuse the notation y ′ = y | y | , for y ∈ R d \ { } . SHARP SOBOLEV–STRICHARTZ ESTIMATE FOR THE WAVE EQUATION 3
Theorem 2.1.
Let d ≥ . Then k e it √− ∆ f k L ( R d +1 ) ≤ C ( d ) Z R d | b f ( y ) | | b f ( y ) | | y | d − | y | d − (1 − y ′ · y ′ ) d − d y d y (3) holds with sharp constant C ( d ) = 2 − d − (2 π ) − d +1 | S d − | which is attained if and only if b f ( ξ ) = e a | ξ | + b · ξ + c | ξ | , where a, c ∈ C , b ∈ C d with | Re( b ) | < − Re( a ) . The one-sided wave propagator is given by e it √− ∆ f ( x ) = 1(2 π ) d Z R d e ix · ξ + it | ξ | b f ( ξ ) d ξ, ( x, t ) ∈ R d × R , for appropriate functions f , and the Fourier transform we use is b f ( ξ ) = Z R d f ( x ) e − ix · ξ d x, ξ ∈ R d . Our observation is that if we introduce polar coordinates for y and y in (3), then we are led toreal-valued functionals of the form H λ ( g ) = Z S d − × S d − g ( η ) g ( η ) | η − η | − λ d η d η for g ∈ L ( S d − ) and λ ≤
0. This is reminiscent of recent work of Foschi [9] where a sharp upperbound for H − was established for d = 3. For Theorem 1.1 we need an analogous result for d = 4;this is contained in the subsequent proposition, which we state more generally to highlight why ourapproach only works as it stands for d = 4 , x, y ) = Z t x − (1 − t ) y − d t, x, y > , and µ g to denote the average value of g on the sphere. Also, we use for the function which isidentically equal to one on the sphere. Proposition 2.2.
Let − < λ < , and let g be any L function on S d − . Then, H λ ( g ) ≤ H λ ( µ g ) = 2 d − − λ B( d − − λ , d − ) | S d − || S d − | (cid:12)(cid:12)(cid:12)(cid:12)Z S d − g (cid:12)(cid:12)(cid:12)(cid:12) . Further, equality holds if and only if g is constant. Following Foschi [9], our proof of Proposition 2.2 is based on a spectral argument using a sphericalharmonic decomposition of g and the Funk–Hecke formula to obtain explicit expressions for the eigen-values. We remark that similar types of arguments have proved profitable in understanding sharpforms of other important estimates; see, for example, [2], [4] and [10]. The connection to the latterpaper deserves a further remark; indeed, in [10], Frank and Lieb provide a reproof of the sharp Hardy–Littlewood–Sobolev inequality on the sphere, originally due to Lieb [12], which gives the sharp upperbound on H λ for 0 < λ < d − L p norm of g , where p = d − d − − λ . NEAL BEZ AND CHRIS JEAVONS
The information we need concerning the eigenvalues is contained in the following lemma. Here we use P k,d to denote the Legendre polynomial of degree k in d dimensions, which may be defined using thegenerating function 1(1 + r − rt ) d − = ∞ X k =0 (cid:18) k + d − d − (cid:19) r k P k,d ( t ) , | r | < , | t | ≤ . Lemma 2.3.
Let − < λ < , and define I k ( d, λ ) = | S d − | Z − (1 − t ) − λ P k,d ( t )(1 − t ) d − d t. Then I ( d, λ ) = | S d − | d − − λ B( d − − λ , d − ) > and I k ( d, λ ) < for all k ≥ .Remark. The inequality in Proposition 2.2 is false if λ < −
2. This is because ( − k I k ( d, λ ) > k ≥ ( d, λ ) > λ . This is the reason why our approachdoes not allow us to prove a generalisation of Theorem 1.1 to dimension 6 and above (for general d , weshould take λ = 3 − d ). A similar obstacle arises in [6] when generalising Foschi’s argument to obtainthe result in [9] in higher dimensions. At the endpoint λ = − g . This turns out not to matterfor our application and we can recover the sharp inequality and characterisation of extremisers for (1)in the case ( d, s ) = (5 ,
1) first proved in [3]; we expand upon this point in Section 3.Assume Lemma 2.3 to be true for the moment, then to prove Proposition 2.2, we first observe thatit suffices by density and continuity of the functional H λ on L ( S d − ) to consider g ∈ L ( S d − ). Wemay then write g = P k ≥ Y k as a sum of orthogonal spherical harmonics; upon which it follows that(4) H λ ( g ) = 2 − λ X k ≥ Z S d − g ( η ) Z S d − Y k ( η )(1 − η · η ) − λ d η d η . To deal with the inner integral in (4) we use the Funk–Hecke formula for the spherical harmonics,which states that Z S d − Y k ( η ) F ( ω · η ) d η = Λ k Y k ( ω )for ω ∈ S d − and k ∈ N , whereΛ k := | S d − | Z − F ( t ) P k,d ( t )(1 − t ) d − d t, provided that F ∈ L ([ − , , (1 − t ) d − ) (see [1], pp. 35–36). It then follows that the inner integralin (4) evaluates to a (positive) constant multiple of I k ( d, λ ) Y k ( η ). Precisely, using the orthogonalityof the spherical harmonics of different degrees and Lemma 2.3, H λ ( g ) = 2 − λ X k ≥ I k ( d, λ ) Z S d − | Y k ( η ) | d η ≤ − λ I ( d, λ ) Z S d − | Y | d η = H λ ( µ g ) . Equality is clearly satisfied for g = Y or equivalently g which are constant. There are no further casesof equality since I k ( d, λ ) is strictly negative for k ≥
1, by Lemma 2.3.Using the expression for I ( d, λ ) in Lemma 2.3 and the definition of µ g , it is then easy to derive theclaimed expression for H λ ( µ g ), which completes the proof of Proposition 2.2. SHARP SOBOLEV–STRICHARTZ ESTIMATE FOR THE WAVE EQUATION 5
Proof of Lemma 2.3.
By a simple change of variables, it is easily checked that I ( d, λ ) satisfies theclaimed equality in terms of the beta function. To prove the strict negativity of I k ( d, λ ) for k ≥
1, wefirst use the Rodrigues formula for P k,d (see [1], pp. 37), which states(1 − t ) d − P k,d ( t ) = ( − k R k,d d k d t k (1 − t ) k + d − , t ∈ [ − , , with R k,d = Γ( d − )2 k Γ( k + d − ) > , to obtain that I k ( d, λ ) = ( − k R k,d Z − (1 − t ) − λ d k d t k (1 − t ) k + d − d t. Integrating by parts, the boundary terms disappear and we obtain(5) I k ( d, λ ) = ( − k R k,d (cid:18) − λ (cid:19) Z − (1 − t ) − λ − d k − d t k − (1 − t ) k + d − d t. Since − λ >
0, the sign of the constant in front of the integral in (5) does not change at the firstintegration by parts. However, since − λ − <
0, at every integration by parts step after the first, wewill incur a sign change. Hence, integrating by parts a total of k times, we see that I k ( d, λ ) evaluatesto − C k ( d, λ ) Z − (1 − t ) − λ − k (1 − t ) k + d − d t for some strictly positive constant C k ( d, λ ). Hence I k ( d, λ ) < (cid:3) Proof of Theorem 1.1.
If we set d = 4 and write the integral on the right-hand side of (3) using polarcoordinates, we get Z (0 , ∞ ) Z ( S ) | b f ( r η ) | | b f ( r η ) | r r (1 − η · η ) d η d r = 1 √ Z S × S g ( η ) g ( η ) | η − η | d η d η , where(6) g ( η ) := Z ∞ | b f ( rη ) | r d r, for η ∈ S . By Plancherel’s theorem, Z S g = (2 π ) k f k H ( R ) . If we then apply (3) and take λ = − k e it √− ∆ f k L ( R ) ≤ C (4) √ H − ( g ) ≤ C (4) √ H − ( ) | µ g | = 415 π k f k H ( R ) , as claimed. The first inequality in (7) is an equality when f extremises inequality (3), and the secondis an equality when the function g defined by (6) is constant on S . In particular, equality holds inboth cases for f given by b f ( ξ ) = e a | ξ | + ib · ξ + c | ξ | , where a, c ∈ C such that Re( a ) <
0, and b ∈ R . Note that, for such f , we have that | b f | is radial (andhence g is constant). NEAL BEZ AND CHRIS JEAVONS
On the other hand, if f is an extremiser for (2), then we must have equality at both of the inequalitiesin (7). From the first inequality, using Theorem 2.1, we see that necessarily b f ( ξ ) = e a | ξ | + b · ξ + c | ξ | , where a, c ∈ C , b ∈ C and Re( a ) < −| Re( b ) | . However, in this case, g ( η ) = e c ) Z ∞ e r (Re( a )+Re( b ) · η ) r d r = e c ) R ∞ e − r r d r [ − a ) + Re( b ) · η )] and for this to be constant in η , we must have Re( b ) = 0. This completes the proof of Theorem1.1. (cid:3) Proof of Corollary 1.2.
Write the solution of the wave equation u as e it √− ∆ f + + e − it √− ∆ f − , wherethe functions f + and f − are defined using the initial data by u (0) = f + + f − , ∂ t u (0) = i √− ∆( f + − f − ) . Using orthogonality and the Cauchy–Schwarz inequality on L ( R ), we get k u k L ( R ) = k e it √− ∆ f + k L ( R ) + k e − it √− ∆ f − k L ( R ) + 4 k e it √− ∆ f + e − it √− ∆ f − k L ( R ) ≤ k e it √− ∆ f + k L ( R ) + k e − it √− ∆ f − k L ( R ) + 4 k e it √− ∆ f + k L ( R ) k e − it √− ∆ f − k L ( R ) . The basic inequality 2( X + Y + 4 XY ) ≤ X + Y ) and Theorem 1.1, which clearly also holds for e − it √− ∆ , now yield k u k L ( R ) ≤ W (4 , ) (cid:18) k u (0) k H ( R ) + k ∂ t u (0) k H − ( R ) (cid:19) which gives the claimed inequality in Corollary 1.2.The above argument was used by Foschi in [8] when ( d, s ) = (3 , ) and in [3] when ( d, s ) = (5 , (cid:3) Five spatial dimensions
We conclude by presenting an alternative derivation of the sharp constant and characterisation ofextremisers for the estimate (1) in the case ( d, s ) = (5 , d = 5 and λ = −
2. However, it is straightforward to see thatI k (5 , −
2) = | S | Z − (1 − t ) P k, ( t )(1 − t ) d t satisfies I (5 , − >
0, I (5 , − < k (5 , −
2) vanishes for all k ≥
2. Thus H − ( g ) = I (5 , − k Y k L ( S ) + I (5 , − k Y k L ( S ) ≤ H − ( ) | µ g | , where g = P k ≥ Y k is the expansion of g into spherical harmonics. Here, equality holds if g is constant,but unlike the estimates in Proposition 2.2, there are further cases of equality. SHARP SOBOLEV–STRICHARTZ ESTIMATE FOR THE WAVE EQUATION 7
Taking f ∈ ˙ H ( R ) and applying this with g given by(8) g ( η ) := Z ∞ | b f ( rη ) | r d r for η ∈ S , we have k e it √− ∆ f k L ( R ) ≤ C (5)2 H − ( g ) ≤ C (5)2 H − ( ) | µ g | = 124 π k f k H ( R ) . As before, equality holds in both inequalities for f given by b f ( ξ ) = e a | ξ | + ib · ξ + c | ξ | , where a, c ∈ C such that Re( a ) <
0, and b ∈ R .Conversely, if f is an extremiser, then Theorem 2.1 implies that b f ( ξ ) = e a | ξ | + b · ξ + c | ξ | , where a, c ∈ C , b ∈ C and Re( a ) < −| Re( b ) | . Substituting our function f into (8), we see that itsuffices to consider g ( η ) = e c ) Z ∞ e r (Re( a )+Re( b ) · η ) r d r = e c ) − Re( a ) − Re( b ) · η ) Z ∞ e − r r d r. Since I (5 , − <
0, we must have that k Y k L = 0. On the other hand, using the projection Π ontothe space of spherical harmonics of degree one given byΠ g ( η ) | S | Z S P , ( η · ω ) g ( ω ) d ω for each η ∈ S , it follows that(9) Y ( η ) = C Z S P , ( η · ω )( − Re( a ) − Re( b ) · η ) d ω for some absolute constant C >
0. If we suppose, for a contradiction, that Re( b ) = 0, then anapplication of the Funk–Hecke formula implies that(10) Y ( η ) = CP , ( η · Re( b ) ′ ) Z − t (1 − t )(1 + At ) d t for each η ∈ S , where A := | Re( b ) | Re( a ) ∈ ( − , C > Y vanishes almost everywhere on S , it follows that the integral on the right-handside of (10) vanishes. This forces A = 0, which gives the desired contradiction.The above argument provides an alternative proof of the following, and at the level of the proof,unifies it with Theorem 1.1. Theorem 3.1 ([3], Corollary 2.2) . The one-sided wave propagator satisfies the estimate k e it √− ∆ f k L ( R ) ≤ W (5 , k f k ˙ H ( R ) with constant W (5 ,
1) = (cid:18) π (cid:19) . The constant is sharp and is attained if and only if b f ( ξ ) = e a | ξ | + ib · ξ + c | ξ | , NEAL BEZ AND CHRIS JEAVONS where a, c ∈ C such that Re( a ) < , and b ∈ R . References [1] K. Atkinson and W. Han,
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Neal Bez, Department of Mathematics, Graduate School of Science and Engineering, Saitama University,Saitama 338-8570, Japan
E-mail address : [email protected] Chris Jeavons, School of Mathematics, University of Birmingham, Edgbaston, Birmingham, B15 2TT, UK
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