Some sharp null-form type estimates for the Klein--Gordon equation
aa r X i v : . [ m a t h . A P ] F e b SOME SHARP NULL-FORM TYPE ESTIMATES FOR THE KLEIN–GORDONEQUATION
JAYSON CUNANAN AND SHOBU SHIRAKI
Abstract.
We establish a sharp bilinear estimate for the Klein–Gordon propagator in the spirit of recentwork of Beltran–Vega. Our approach is inspired by work in the setting of the wave equation due to Bez,Jeavons and Ozawa. As a consequence of our main bilinear estimate, we deduce several sharp estimatesof null-form type and recover some sharp Strichartz estimates found by Quilodr´an and Jeavons. Introduction
Let d ≥ φ s ( r ) = √ s + r for r ≥ s >
0. We write D for the operator √− ∆ x , that is d Df ( ξ ) = | ξ | b f ( ξ )where b · denotes the (spatial) Fourier transform defined by b f ( ξ ) = Z R d e − ix · ξ f ( x ) d x for appropriate functions f on R d . Additionally, we define D ± by ] D ± f ( τ, ξ ) = || τ | ± | ξ || e f ( τ, ξ ) , where e · is the space-time Fourier transform of appropriate functions f on R × R d . The d’Alembertianoperator ∂ t − ∆ x will be denoted by (cid:3) , so that (cid:3) = D − D + . Our main object of interest is the Klein–Gordon propagator given by e itφ s ( D ) f ( x ) = 1(2 π ) d Z R d e i ( x · ξ + tφ s ( | ξ | )) b f ( ξ ) d ξ for sufficiently nice initial data f .As part of the study of sharp bilinear estimates for the Fourier extension operator and inspired by workof Ozawa–Tsutsumi [34], Beltran–Vega [3] very recently presented the following sharp estimate associatedto the Klein–Gordon propagator k D − d ( e itφ s ( D ) f e itφ s ( D ) g ) k L ( R d × R ) (1.1) ≤ (2 π ) − d Z ( R d ) | b f ( η ) | | g ( η ) | φ s ( | η | ) φ s ( | η | ) K BV ( η , η ) d η d η , where K BV ( η , η ) = Z S d − φ s ( | η | ) + φ s ( | η | )( φ s ( | η | ) + φ s ( | η | )) − (( η + η ) · θ ) d σ ( θ ) . Date : February 8, 2021.2010
Mathematics Subject Classification.
Key words and phrases.
Bilinear estimates, Klein–Gordon equation, optimal constants, extremisers, Strichartz estimate.
The estimate (1.1) has some interesting connections to well-known results. For example, as we shall see inmore detail later, (1.1) leads null-form type estimates by appropriately estimating the kernel. In particular,when d = 2 the Strichartz estimate(1.2) k e itφ ( D ) f k L ( R ) ≤ − k f k H ( R ) with the optimal constant quickly follows from (1.1). Here, where the inhomogeneous Sobolev norm oforder α is defined by k f k H α ( R d ) := k φ ( D ) α f k L ( R d ) . The estimate (1.2) with the optimal constant was first obtained by Quilodr´an [36]. Bilinear estimates whichbear resemblance to (1.1) for the Klein–Gordon equation, as well as the Schr¨odinger and wave equation,have often arisen in the pursuit of optimal constants for Strichartz estimates and closely related null-formtype estimates. As well as the aforementioned work of Beltran–Vega [3], estimates of the form (1.1) forthe Klein–Gordon propagator can be found in work of Jeavons [21] (see also [22]). For the Schr¨odingerequation, in addition to the Ozawa–Tsutsumi estimates in [34], estimates resembling (1.1) may be foundin work of Carneiro [13] and Planchon–Vega [35], with a unification of each of these results by Bennett etal. in [5]. For the wave equation, Bez–Rogers [9] and Bez–Jeavons–Ozawa [11] have established estimatesresembling (1.1). We also remark that the related literature on sharp Strichartz estimates is large. Inaddition to the papers already cited, this body of work includes, for example, [10, 14, 15, 17, 20, 28]; theinterested reader is referred to the survey article by Foschi–Oliveira e Silva [19] for further information.In the present paper, we establish the following new bilinear estimates for the Klein–Gordon propagator.Let K ba ( η , η ) := (cid:0) φ s ( | η | ) φ s ( | η | ) − η · η − s (cid:1) b ( φ s ( | η | ) φ s ( | η | ) − η · η + s ) a . Theorem 1.1.
For d ≥ and β > − d , we have the estimate k| (cid:3) − (2 s ) | β ( e itφ s ( D ) f e itφ s ( D ) g ) k L ( R d +1 ) (1.3) ≤ KG ( β, d ) Z R d | b f ( η ) | | b g ( η ) | φ s ( | η | ) φ s ( | η | ) K d − +2 β ( η , η ) d η d η , with the optimal constant KG ( β, d ) := 2 − d +12 +2 β π − d +12 Γ( d − + 2 β )Γ( d − β ) . In the case when s →
0, certain sharp bilinear estimates for solutions to the wave equation with theoperator | (cid:3) | β has been deeply studied by Bez–Jeavons–Ozawa [11]. One may note that, when d = 2, aslightly larger range of β is valid in Theorem 1.1 than one for the corresponding result (1.13) for the wavecase in [11]. In order to prove Theorem 1.1, we employ their argument and adapt it into the context ofKlein–Gordon equation. As a consequence of Theorem 1.1, we will generate null-form type estimates ofthe form(1.4) k| (cid:3) − (2 s ) | β | e itφ s ( D ) f | k L ( R d +1 ) ≤ C k φ s ( D ) α f k L ( R d ) for certain pairs ( α, β ) with the optimal constant. OME SHARP NULL-FORM TYPE ESTIMATES FOR THE KLEIN–GORDON EQUATION 3
Wave regime.
For d ≥
4, the kernel K BV can be estimated as K BV d ( η , η ) ≤ | S d − | φ s ( | η | ) + φ s ( | η | ) Z − − (cid:12)(cid:12)(cid:12)(cid:12) η + η φ s ( | η | ) + φ s ( | η | ) (cid:12)(cid:12)(cid:12)(cid:12) λ ! − (1 − λ ) d − d λ ≤ Cφ s ( | η | ) + φ s ( | η | )for some absolute constant C since | η + η | ≤ φ s ( | η | ) + φ s ( | η | ). Then, it follows from the arithmetic-geometric mean that φ s ( | η | ) φ s ( | η | ) K BV d ( η , η ) ≤ Cφ s ( | η | ) φ s ( | η | ) , and hence the null-form type estimate(1.5) k D − d ( e itφ s ( D ) f e itφ s ( D ) g ) k L ( R d × R ) ≤ C k φ s ( D ) f k L ( R d ) k φ s ( D ) g k L ( R d ) holds. When s →
0, the estimate (1.5) yields(1.6) k D β D β − − D β + + ( e itD f e itD g ) k L ( R d +1 ) ≤ C k f k ˙ H α k g k ˙ H α , in the case of ( β , β − , β + , α − , α + ) = ( − d , , , , ) for the propagator e itD associated with the waveequation. The estimate (1.6), as well as the corresponding (++) case (while (1.6) is (+ − ) case),(1.7) k D β D β − − D β + + ( e itD f e itD g ) k L ( R d +1 ) ≤ C k f k ˙ H α k g k ˙ H α has found important applications in study of nonlinear wave equations. This type of estimate has beenstudied back in work of Beals [2] and Klainerman–Machedon [23, 24, 25]. A complete characterizationof the admissible exponents ( β , β − , β + , α − , α + ) for (1.6) and (1.7) were eventually obtained by Foschi–Klainerman [18]. Such a characterization when the L t,x norm on the left-hand side of (1.6) is replaced by L qt L rx has also drawn great attention. Using bilinear Fourier restriction techniques, Bourgain [12] madea breakthrough contribution, then Wolff [41] and Tao [39] (in the endpoint case; see also Lee [29] andTataru [40]) completed the diagonal case q = r . For the non-diagonal case we refer readers to [31] due toLee–Vargas for a complete characterization when d ≥ d = 2, 3. Soon later Lee–Rogers–Vargas [30] completed d = 3, but a gap between necessary and sufficient conditions still remainswhen d = 2.As a means of comparing our bilinear estimate (1.3) with (1.1), we note that using the trivial bound(1.8) φ s ( | η | ) φ s ( | η | ) − η · η − s φ s ( | η | ) φ s ( | η | ) − η · η + s ≤ , we estimate our kernel as(1.9) K d − +2 β ( η , η ) ≤ K d − +2 β ( η , η ) . For β ≥ − d , it follows that k| (cid:3) − (2 s ) | β ( e itφ s ( D ) f e itφ s ( D ) g ) k L ( R d +1 ) ≤ C k φ s ( D ) d − + β f k L ( R d ) k φ s ( D ) d − + β g k L ( R d ) (1.10)for some absolute constant C , which, as in the discussion for the Beltran–Vega bilinear estimate, placesTheorem 1.1 in the framework of null-form type estimates. If we formally set β = − d in (1.10) to getdata with regularity whose order is as in (1.5), the order of “smoothing” from | (cid:3) − (2 s ) | β becomes2 β = − d , which is compatible with (1.5). Unfortunately, − d is outside the range β ≥ − d and, in fact, aswe shall see in Proposition 3.4, β ≥ − d is a necessary condition for (1.10). Nevertheless, as an application We observe that d ≥ d = 3, the estimate (1.5) actually does not hold. The counterexample hasbeen given by Foschi [16] for the wave equation, and the same argument appropriately adapted works for the Klein–Gordonpropagator. JAYSON CUNANAN AND SHOBU SHIRAKI of Theorem 1.1 one can widen the range to, at least, β > − d if one considers radially symmetric data. Weshall state this result as part of the forthcoming Corollary 1.2. In addition, for a large range of β we shallin fact obtain the optimal constant for such null-form type estimates; to state our result, we introduce theconstant F ( β, d ) := 2 d − β π − d Γ( d )Γ( d − + 2 β )( d − β )Γ( d − + 2 β ) . Corollary 1.2.
Let d ≥ , β ≥ − d . Then, there exists a constant C > such that (1.10) holds whenever f and g are radially symmetric. Moreover, for β ∈ [ − d , − d ] ∪ [ − d , ∞ ) , the optimal constant in (1.10) for radially symmetric f and g is F ( β, d ) , but there does not exist a non-trivial pair of functions ( f, g ) that attains equality. In the case of the wave propagator when s →
0, the estimate (1.10) becomes(1.11) k| (cid:3) | β ( e itD f e itD g ) k L ( R d +1 ) ≤ F ( β, d ) k f k ˙ H d −
14 + β ( R d ) k g k ˙ H d −
14 + β ( R d ) and, in certain situation, it is known that the constant F ( β, d ) is optimal. In the case β = 0 and d = 3,pioneering work of Foschi [17] established the optimality of the constant F (0 , . The constant F ( β, d ) isalso known to be optimal when ( β, d ) = (0 ,
4) and ( β, d ) = (0 , k e itD f e itD g k L ( R d +1 ) ≤ W (0 , d ) Z ( R d +1 ) | b f ( η ) | | b g ( η ) | | η || η | K BR0 ( η , η ) d η d η . Here, W ( β, d ) turns out to be KG ( β, d ) and K BR β formally coincides with the special case of our kernel K d − +2 β when s = 0. The optimality of F (0 , in (1.11) was proved by Bez–Jeavons [10] by making useof (1.12), polar coordinates and techniques from the theory of spherical harmonics.Also we note that Bez–Jeavons–Ozawa [11] established the bilinear estimates(1.13) k| (cid:3) | β ( e itD f e itD g ) k L ( R d +1 ) ≤ W ( β, d ) Z ( R d +1 ) | b f ( η ) | | b g ( η ) | | η || η | K BR β ( η , η ) d η d η , from which it quickly follows (via polar coordinates) that F ( β, d ) is the optimal constant in (1.11) when-ever d ≥ β > β d := max { − d , − d } if we restrict to radially symmetric data f and g . We remarkthat it is a result of the homogeneity of the kernel K d − + β when s = 0 that one can immediately deducethe optimality of F ( β, d ) in (1.11) for radial data from (1.13). In contrast, our concern is the case s > − d , − d ) in the range of β in Corollary 1.2. We prove Corollary 1.2 by firstmaking use of our bilinear estimate (1.3); somewhat surprisingly given that (1.3) is a sharp inequality, weshall prove (Proposition 3.3) that it is impossible to obtain the optimality of F ( β, d ) in (1.10) for radialdata and β ∈ ( − d , − d ) once one makes use of (1.3) as a first step. We also expect that (1.10) holds with C = F ( β, d ) for β ∈ ( β d , − d ) but we do not pursue this here.There are some special cases of β ; the endpoints − d and − d of the gap, at which we can remove theradial symmetry hypothesis on the initial data and still keep the optimal constants. Corollary 1.3.
Let d ≥ . Then, the estimate (1.4) holds with the optimal constant C = F ( β, d ) for ( α, β ) = ( , − d ) and ( α, β ) = (1 , − d ) , but there are no non-trivial extremisers. Furthermore, when ( α, β ) = (1 , − d ) , we have the refined Strichartz estimate k| (cid:3) − (2 s ) | − d | e itφ s ( D ) f | k L ( R d +1 ) ≤ F ( − d , d ) (cid:16) k φ s ( D ) f k L ( R d ) − s k φ s ( D ) f k L ( R d ) (cid:17) , (1.14) OME SHARP NULL-FORM TYPE ESTIMATES FOR THE KLEIN–GORDON EQUATION 5 where the constant is optimal and there are no non-trivial extremisers.
Corollary 1.3 generalizes the following recent results. In the context of the Klein–Gordon equation,Quilodr´an [36] appropriately developed Foschi’s argument in [17] and proved the sharp Strichartz estimate(1.15) k e itφ ( D ) f k L q ( R d +1 ) ≤ H ( d, q ) k f k H ( R d ) for ( d, q ) = (2 , , , q , namely,2( d + 2) d ≤ q ≤ d + 1) d − . The constant H ( d, q ) denotes the optimal constant so that (1.15) in the case ( d, q ) = (3 ,
4) is recoveredby Corollary 1.3 in the case ( α, β ) = ( , − d ) and F (0 , = H (4 ,
3) holds. In [36], Quilod´an also provedthat there is no extremiser which attains (1.15) for ( d, q ) = (2 , , , d = 1, 2, by answering questions raised in[36]; in particular, they found the best constant in (1.15) for ( d, q ) = (1 ,
6) and absence of the extremisers(the case ( d, q ) = (1 ,
6) is the endpoint of the admissible range of 6 ≤ q ≤ ∞ when d = 1). Meanwhile,they also established there exist extremisers in the non-endpoint cases in low dimensions d = 1, 2. Asubsequent study by the same authors in collaboration with Stovall [15] proved the analogous results inthe non-endpoint cases for higher dimensions d ≥ k e itφ s ( D ) f k L ( R ) ≤ F (0 , (cid:16) k φ s ( D ) f k L ( R ) − s k φ s ( D ) f k L ( R ) (cid:17) , (1.16)which recovers the inequality (1.11) when ( β, d ) = (0 ,
5) in the limit s →
0. Moreover, by simply omittingthe negative second term, it follows that k e itφ ( D ) f k L ( R ) ≤ F (0 , k f k H ( R ) , where the constant F (0 , = (24 π ) − is still sharp. These results are recovered too by Corollary 1.3 inthe case ( α, β ) = (1 , − d ).1.2. Non-wave regime.
One may examine the Beltran–Vega bilinear estimate (1.1) from a somewhatdifferent perspective to that taken in our earlier discussion which led to (1.5). For d ≥ K BV can also be reinterpreted by K BV ( η , η ) = | S d − | Z π − π φ s ( | η | ) + φ s ( | η | )2( φ s ( | η | ) φ s ( | η | ) − η · η + s ) + | η + η | cos θ (cos θ ) d − d θ, then by applying the trivial bound cos θ ≤ − π , π ], and another key relationship(1.17) φ s ( | η ) φ s ( | η | ) − η · η ≥ s , we have K BV ( η , η ) ≤ − π | S d − | s − . Thus, the inequality (1.1) directly implies k D − d | e itφ s ( D ) f | k L ( R d +1 ) ≤ − s − k φ s ( D ) f k L ( R d ) . (1.18)By comparison with (1.5), the regularity level on the initial data has increased to H but this has allowedfor a wider range of d which, in particular, includes d = 2 in which case (1.18) coincides with the sharp H → L x,t Strichartz estimate (1.2) obtained by Quilodr´an. Note that, in the non-wave regime, we arenot allowed to let s → s − appearing in the constant. JAYSON CUNANAN AND SHOBU SHIRAKI
On the other hand, Theorem 1.1 also yields (1.2) as a special case of the following family of sharpnull-form type estimates valid in all dimensions d ≥
2. Indeed, since we have another kernel estimate(1.19) K d − +2 β ( η , η ) ≤ − K d − +2 β ( η , η ) s − via (1.17), we immediately deduce the following from Theorem 1.1. Corollary 1.4.
Let d ≥ . Then the estimate (1.4) holds with the optimal constant C = − d +1 π − d +22 s Γ( d ) ! for ( α, β ) = ( , − d ) , but there are no non-trivial extremisers. Furthermore, when ( α, β ) = (1 , − d ) , wehave the refined Strichartz estimate k| (cid:3) − (2 s ) | − d | e itφ s ( D ) f | k L ( R d +1 ) ≤ − d +1 π − d +22 s Γ( d +22 ) ! (cid:16) k φ s ( D ) f k L ( R d ) − s k φ s ( D ) f k L ( R d ) (cid:17) , (1.20) where the constant is optimal and there are no non-trivial extremisers. One may note that (1.20) provides a sharp form of the following refined Strichartz inequality in theanalogous manner of (1.16) when d = 4: k e itφ ( D ) f k L ( R ) ≤ (16 π ) − ( k f k H ( R ) − k f k H ( R ) ) , however we have unable to conclude whether the constant (16 π ) − continues to be optimal if we drop thesecond term on the right-hand side.For solutions u of certain PDE, in addition to the null-form estimates (1.6), estimates which controlquantities like | u | , through its interplay with other types of operators have appeared numerous times in theliterature. In particular, we note that the approach taken by Beltran–Vega [3], which in turn built on workof Planchon–Vega [35] rested on interplay with geometric operators such as the Radon transform or, moregenerally, the k -plane transform. For related work in this context of interaction with geometrically-definedoperators, we also refer the reader to work of Bennett et. al [6] and Bennett–Nakamura [8].Our approach to proving Theorem 1.1 more closely follows the argument in [11] and does not appearto fit into such a geometric perspective.Throughout the paper, we denote A & B if A ≥ CB , A . B if A ≤ CB and A ∼ B if C − B ≤ A ≤ CB for some constant C >
0. In the subsequent Section 2, we first prove Theorem 1.1 by adapting theargument of [11]. In Section 3 we prove (1.10) for radial data and show that for β ∈ [ − d , − d ] ∪ [ − d , ∞ )we have (1.10) with C = F ( β, d ) . We also make an observation that suggests it may be difficult to obtainthe optimal constant in (1.10) for β ∈ ( − d , − d ) even for radially symmetric data (see Proposition 3.3).Finally, we show β ≥ − d is necessary for (1.10) to hold for general data. The sharpness of the constantsin Corollary 1.2, Corollary 1.3 and Corollary 1.4 are considered in Section 4 and proved by a differentmethod from [21] or [11] in order to deal with the more delicate situation of the non-wave regime. Lastly,we end the paper with Section 5 by discussing analogous results for the (++) case. As [11] has alreadyobserved, the (++) case is far easier than the (+ − ) case, and this will become clear from our argument inthis section. OME SHARP NULL-FORM TYPE ESTIMATES FOR THE KLEIN–GORDON EQUATION 7 Proof of Theorem 1.1
Although some steps require additional care due to the extra parameter s , broadly speaking Theorem1.1 can be proved by adapting the argument for wave propagators presented in [11], whose techniquesare originated in [4] (see also [5]). The key tool here is the following Lorentz transform given by L ; for( t, x ) ∈ R × R d L (cid:18) tx (cid:19) = (cid:18) γ ( t − ζ · x ) x + ( γ − | ζ | ζ · x − γt ) ζ (cid:19) , where ζ := − ξτ and γ := τ ( τ −| ξ | ) . Let us first introduce two lemmas whose proof come later in thissection. Lemma 2.1.
For η , η ∈ R d and β > − d , define J β ( η , η ) := Z R d | φ s ( | η | ) φ s ( | η | ) − η · η − s | β φ s ( | η | ) φ s ( | η | ) δ (cid:18) τ − φ s ( | η | ) − φ s ( | η | ) ξ − η − η (cid:19) d η d η , (2.1) where τ = φ s ( | η | ) + φ s ( | η | ) and ξ = η + η . Then, we have J β ( η , η ) = (2 π ) d − Γ( d − + 2 β )Γ( d − β ) K d − +2 β ( η , η ) . Lemma 2.2.
Let η , η ∈ R d . Set ξ = η + η , τ = φ s ( | η | ) + φ s ( | η | ) and η ∈ R d satisfying φ s ( | η | ) = ( τ − | ξ | ) . Then, there exists ω ∗ ∈ S d − depending only on η , η and | η | such that (cid:18) φ s ( | η | ) − η (cid:19) · L (cid:18) φ s ( | η | ) η (cid:19) − s = | η | (cid:18) η | η | · ω ∗ (cid:19) . Proof of Theorem 1.1.
Let u ( t, x ) = e itφ s ( √− ∆) f ( x ) and v ( t, x ) = e itφ s ( √− ∆) g ( x ). By the expressions e u ( τ, ξ ) = 2 πδ ( τ − φ s ( | ξ | )) b f ( ξ ) and e v ( τ, ξ ) = 2 πδ ( τ + φ s ( | ξ | )) b g ( − ξ ), Plancherel’s theorem, and appropriatelyrelabeling the variables, one can deduce(2 π ) d +1) k| (cid:3) − (2 s ) | β ( uv ) k L ( R d +1 ) = Z R d +1 | τ − | ξ | − (2 s ) | β | e u ∗ e v ( ξ, τ ) | d τ d ξ = 2 β Z R d | φ s ( | η | ) φ s ( | η | ) − η · η − s | β F ( η , η ) F ( η , η )( φ s ( | η | ) φ s ( | η | ) φ s ( | η | ) φ s ( | η | )) × δ (cid:18) φ s ( | η | ) − φ s ( | η | ) − φ s ( | η | ) + φ s ( | η | ) η + η − η − η (cid:19) d η d η d η d η . Here, F ( η , η ) := b f ( η ) b g ( η ) φ s ( | η | ) φ s ( | η | ) . If we define Ψ = Ψ s ( η , η , η , η ) = (cid:16) φ s ( | η | ) φ s ( | η | ) φ s ( | η | ) φ s ( | η | ) (cid:17) , then by the arithmetic-geometric mean we have F ( η , η )Ψ F ( η , η )Ψ − ≤ (cid:0) | F ( η , η ) | Ψ + | F ( η , η ) | Ψ − (cid:1) so that(2.2) F ( η , η ) F ( η , η )( φ s ( | η | ) φ s ( | η | ) φ s ( | η | ) φ s ( | η | )) ≤ (cid:18) | F ( η , η ) | φ s ( | η | ) φ s ( | η | ) + | F ( η , η ) | φ s ( | η | ) φ s ( | η | ) (cid:19) . JAYSON CUNANAN AND SHOBU SHIRAKI
Here, the equality holds if and only if φ s ( | η | ) φ s ( | η | ) b f ( η ) b g ( η ) = φ s ( | η | ) φ s ( | η | ) b f ( η ) b g ( η )almost everywhere on the support of the delta measure, which is satisfied by, for instance, f = g = f a with a > b f a ( ξ ) = e − aφ s ( | ξ | ) φ s ( | ξ | ) . Therefore, (cid:0) d +3+2 β π d +3 (cid:1) − k| (cid:3) − (2 s ) | β ( uv ) k L ( R d +1 ) ≤ Z R d F ( η , η ) J β ( η , η ) d η d η + 12 Z R d F ( η , η ) J β ( η , η ) d η d η , which implies (1.3) by applying Lemma 2.1. One may note that the constant in (1.3) is sharp since weonly apply the inequality (2.2) in the proof. (cid:3) We now prove the aforementioned lemmas.
Proof of Lemma 2.1.
Let τ = φ s ( | η | ) + φ s ( | η | ) and ξ = η + η . It is well known that the measure δ ( σ − φ s ( | η | )) φ s ( | η | ) for ( σ, η ) ∈ R × R d is invariant under the Lorentz transform L , | det L | = 1, and L (cid:18) ( τ − | ξ | ) (cid:19) = (cid:18) τξ (cid:19) . The change of variables (cid:0) σ j η j (cid:1) L (cid:0) σ j η j (cid:1) for j = 3 , J β ( η , η ) = Z R d +1) (cid:12)(cid:12)(cid:12)(cid:12)(cid:18) φ s ( | η | ) − η (cid:19) · (cid:18) σ η (cid:19) − s (cid:12)(cid:12)(cid:12)(cid:12) β × δ ( σ − φ s ( | η | )) φ s ( | η | ) δ ( σ − φ s ( | η | )) φ s ( | η | ) δ (cid:18) τ − σ − σ ξ − η − η (cid:19) d σ d σ d η d η = Z R d (cid:12)(cid:12)(cid:12)(cid:12)(cid:18) φ s ( | η | ) − η (cid:19) · L (cid:18) φ s ( | η | ) η (cid:19) − s (cid:12)(cid:12)(cid:12)(cid:12) β φ s ( | η | ) δ (2 φ s ( | η | ) − ( τ − | ξ | ) ) d η. By Lemma 2.2 and switching to polar coordinates, J β ( η , η ) = Z ∞ (cid:18)Z S d − (1 + θ · ω ∗ ) β d σ ( θ ) (cid:19) r β φ s ( r ) δ (2 φ s ( r ) − ( τ − | ξ | ) ) r d − d r. Now, one can find a rotation R such that R T ω ∗ = e = (1 , , . . . , T so that Z S d − (1 + θ · ω ∗ ) β d θ = 2 d − β | S d − | B (cid:0) d − + 2 β, d − (cid:1) , where B denotes the beta function given by B ( z, w ) = Z λ z − (1 − λ ) w − d λ for z , w ∈ C whose real parts are strictly positive. For the remaining radial integration, one can performthe change of variables 2 φ s ( r ) ν in order to get Z ∞ r β φ s ( r ) δ (2 φ s ( r ) − ( τ − | ξ | ) ) r d − d r = 2 − d +12 − β K d − +2 β ( η , η ) OME SHARP NULL-FORM TYPE ESTIMATES FOR THE KLEIN–GORDON EQUATION 9 and hence J β ( η , η ) = 2 d − | S d − | B ( d − + 2 β, d − ) K d − +2 β ( η , η ) . Finally, simplifying the constant by the formulae(2.4) | S d − | = 2 π d Γ( d )and B ( d − + 2 β, d − ) = Γ( d − + 2 β )Γ( d − )Γ( d − β ) , we are done. (cid:3) Proof of Lemma 2.2.
Observe first that L (cid:18) φ s ( | η | ) η (cid:19) = 12 (cid:18) τ + ξ · ηφ s ( | η | ) η + ξ (1 + ξ · η ( τ +2 φ s ( | η | )) φ s ( | η | ) ) (cid:19) . Then, a direct calculation gives (cid:18) φ s ( | η | ) − η (cid:19) · L (cid:18) φ s ( | η | ) η (cid:19) = ( φ s ( | η | )) (cid:18) η | η | · | η | z (cid:19) , where z = ( φ s ( | η | ) + φ s ( | η | )) η − ( φ s ( | η | ) + φ s ( | η | )) η φ s ( | η | ) ( φ s ( | η | ) + φ s ( | η | ) + 2 φ s ( | η | )) . Since we have the relation 2 φ s ( | η | ) = φ s ( | η | ) φ s ( | η | ) − η · η + s , the numerator of z can be simplified by([ φ s ( | η | ) + φ s ( | η | )] η − [ φ s ( | η | ) + φ s ( | η | )] η ) = [ φ s ( | η | ) + φ s ( | η | )] | η | + [ φ s ( | η | ) + φ s ( | η | )] | η | − φ s ( | η | ) + φ s ( | η | )][ φ s ( | η | ) + φ s ( | η | )] η · η = [ φ s ( | η | ) + φ s ( | η | )] φ s ( | η | ) + [ φ s ( | η | ) + φ s ( | η | )] φ s ( | η | ) − φ s ( | η | ) + φ s ( | η | )][ φ s ( | η | ) + φ s ( | η | )]( φ s ( | η | ) φ s ( | η | ) − φ s ( | η | ) ) − s ([ φ s ( | η | ) + φ s ( | η | )] + [ φ s ( | η | ) + φ s ( | η | )] + 2[ φ s ( | η | ) + φ s ( | η | )][ φ s ( | η | ) + φ s ( | η | )])= (cid:0) φ s ( | η | ) − s (cid:1) [ φ s ( | η | ) + φ s ( | η | ) + 2 φ s ( | η | )] , and so it follows that | z | = | η | φ s ( | η | ) . Therefore, (cid:18) φ s ( | η | ) − η (cid:19) · L (cid:18) φ s ( | η | ) η (cid:19) − s = | η | (cid:18) η | η | · ω ∗ (cid:19) , where we have set ω ∗ = z | z | . (cid:3) On estimate (1.10)3.1.
Estimate (1.10) with explicit constant.
In this subsection, we prove (1.10) for radially symmetricdata f and g for β > − d and an explicit constant C < ∞ ; for β = [ − d , − d ] ∪ [ − d , ∞ ), this explicitconstant coincides with F ( β, d ) . In order to complete the proof of Corollary 1.2, we need to showthe sharpness of F ( β, d ) for β ∈ [ − d , − d ] ∪ [ − d , ∞ ), and the non-existence of extremisers; for thesearguments, we refer the reader to Section 4. Lemma 3.1.
Let a + b > − , b > − and κ ∈ [0 , . Define h a,b ( κ ) := Z − (1 − κλ ) a (1 − λ ) b d λ. Then, sup κ ∈ [0 , h a,b ( κ ) < ∞ . Moreover, for a ∈ ( −∞ , ∪ [1 , ∞ )sup κ ∈ [0 , h a,b ( κ ) = h a,b (1) = 2 a +2 b +1 B ( a + b + 1 , b + 1) . Proof of Lemma 3.1.
By the Lebesgue dominated convergence theorem,dd κ h a,b ( κ ) = − aκ Z − (1 − κλ ) a − λ (1 − λ ) d λ = aκ Z (cid:0) (1 + κλ ) a − − (1 − κλ ) a − (cid:1) λ (1 − λ ) b d λ Thus, ( dd κ h a,b ( κ ) ≥ a ∈ ( −∞ , ∪ [1 , ∞ ) , dd κ h a,b ( κ ) < a ∈ (0 , . For a ∈ ( −∞ , ∪ [1 , ∞ ), sup κ ∈ [0 , h a,b ( κ ) = h a,b (1) = Z − (1 − λ ) a (1 − λ ) b d λ and the change of variables 1 + λ λ gives Z − (1 − λ ) a (1 − λ ) b d λ = 2 a +2 b +1 B ( a + b + 1 , b + 1) < ∞ if a + b > b > −
1. Similarly, for a ∈ (0 , h a,b ( κ ) ≤ h a,b (0) = 2 b +1 B ( b + 1 , b + 1) < ∞ if b > − (cid:3) Let f , g be radially symmetric. By Theorem 1.1, we have k| (cid:3) − (2 s ) | β ( e itφ s ( √− ∆) f e itφ s ( √− ∆) g ) k L ( R d +1 ) ≤ KG ( β, d ) Z ∞ Z ∞ | b f ( r ) | | b g ( r ) | φ s ( r ) d − +2 β φ s ( r ) d − +2 β Θ d − +2 β ( r , r ) r d − r d − d r d r , (3.1)where Θ ba ( r , r ) := Z ( S d − ) (cid:16) − r r θ · θ φ s ( r ) φ s ( r ) − s φ s ( r ) φ s ( r ) (cid:17) b (cid:16) − r r θ · θ φ s ( r ) φ s ( r ) + s φ s ( r ) φ s ( r ) (cid:17) a d σ ( θ )d σ ( θ ) . We divide the range of β into β ∈ [ − d , − d ] and β ∈ [ − d , ∞ ) and treat these cases differently. First, letus consider β ∈ [ − d , ∞ ) as the easier case. By applying the fundamental kernel estimate (1.9), we haveΘ d − +2 β ( r , r ) ≤ Θ d − +2 β ( r , r ) = | S d − || S d − | h d − +2 β, d − ( κ )with κ = r r φ s ( r ) φ s ( r ) . Since d − β ≥
1, Lemma 3.1 implies thatsup κ ∈ [0 , h d − +2 β, d − ( κ ) = h d − +2 β, d − (1) , OME SHARP NULL-FORM TYPE ESTIMATES FOR THE KLEIN–GORDON EQUATION 11 and hence sup r ,r > Θ d − +2 β ( r , r ) ≤ d − +2 β | S d − || S d − | B (cid:0) d − β, d − (cid:1) , which yields (1.10) with C = F ( β, d ) .For β ∈ [ − d , − d ], in which case d − + 2 β ∈ [0 , ], the basic idea of our argument is the same as abovebut it requires a few more steps. LetΞ( ν, υ ) := Z − (1 − ν − √ − ν − υ λ ) d − +2 β (1 + ν − √ − ν − υ λ ) d µ ( λ )with ν and υ satisfying ν ∈ [0 , , υ ≤ − ν , and d µ ( λ ) = (1 − λ ) d − d λ . Then, from (3.1), it suffices to show(3.2) Ξ( ν, υ ) ≤ Ξ(0 , υ ) ≤ Ξ(0 , . In order to show the first inequality of (3.2), we establish monotonicity in ν on (cid:2) , q − υ (cid:3) , and calculatedirectly for ν ∈ (cid:2)q − υ , √ − υ (cid:3) . Indeed, it simply follows that ∂ ν Ξ( ν, υ ) ≤ − (cid:18) d −
22 + 2 β (cid:19) Z (1 − ν − √ − ν − υ λ ) d − +2 β (1 + ν − √ − ν − υ λ ) (cid:18) − ν √ − ν − υ λ (cid:19) d µ ( λ ) − Z (1 − ν − √ − ν − υ λ ) d − +2 β (1 + ν − √ − ν − υ λ ) (cid:18) ν √ − ν − υ λ (cid:19) d µ ( λ ) − (cid:18) d −
22 + 2 β (cid:19) Z (1 − ν + √ − ν − υ λ ) d − +2 β (1 + ν + √ − ν − υ λ ) (cid:18) ν √ − ν − υ λ (cid:19) d µ ( λ ) − Z (1 − ν + √ − ν − υ λ ) d − +2 β (1 + ν + √ − ν − υ λ ) (cid:18) − ν √ − ν − υ λ (cid:19) d µ ( λ ) , which is non-positive since 1 − ν √ − ν − υ λ ≥ ν ∈ (cid:2) , q − υ (cid:3) . On the other hand, for ν ∈ (cid:2)q − υ , √ − υ (cid:3) , which imposes 0 ≤ √ − ν − υ ≤ ν ,it follows thatΞ( ν, υ ) = Z (1 − ν − √ − ν − υ λ ) d − +2 β (1 + ν − √ − ν − υ λ ) d µ ( λ ) + Z (1 − ν + √ − ν − υ λ ) d − +2 β (1 + ν + √ − ν − υ λ ) d µ ( λ ) ≤ Z µ ( λ ) ≤ Z (1 − p − υ λ ) d − +2 β d µ ( λ ) + Z (1 + p − υ λ ) d − +2 β d µ ( λ )= Ξ(0 , υ ) . Here, the last inequality is given by the arithmetic-geometric mean:12 (cid:16) (1 − p − υ λ ) d − +2 β + (1 + p − υ λ ) d − +2 β (cid:17) ≥ (cid:0) − (1 − υ ) λ (cid:1) d − + β ≥ . Since the second inequality of (3.2) can be readily proved by Lemma 3.1, we have (1.10) with C = F ( β, d ) for β ∈ [ − d , − d ] as well. Threshold of our argument for β ∈ ( − d , − d ) . Although C = F ( β, d ) will be shown to beoptimal for β ∈ [ − d , − d ] ∪ [ − d , ∞ ) in the case of radial data, it remains unclear whether this continuesto be true for β ∈ ( − d , − d ); here we establish that there is no way to obtain the constant F ( β, d ) if onefirst makes use of Theorem 1.1. In order to show that, we shall invoke the following useful result for thebeta function due to Agarwal–Barnett–Dragmir [1]: Lemma 3.2 ([1]) . Let m , p and k ∈ R satisfy m , p > , and p > k > − m . If we have k ( p − m − k ) > then B ( p, m ) > B ( p − k, m + k ) holds. Proposition 3.3.
Let d ≥ and β ∈ ( − d , − d ) . Then there exist radially symmetric f and g such that KG ( β, d ) Z R d | b f ( η ) | | b g ( η ) | φ s ( | η | ) φ s ( | η | ) K d − +2 β ( η , η ) d η d η > F ( β, d ) k φ s ( D ) d − + β f k L ( R d ) k φ s ( D ) d − + β g k L ( R d ) holds.Proof of Proposition 3.3. Let 0 < δ ≪ A = (cid:26) ξ ∈ R d : 12 < | ξ | < (cid:27) . Define f = f A and g = g A so that for ξ ∈ R d c f A ( ξ ) = χ A ( ξδ ) and c g A ( ξ ) = χ A ( δξ ) , where χ A is the characteristic function of A . By use of polar coordinates Z R d | b f ( η ) | | b g ( η ) | φ s ( | η | ) φ s ( | η | ) K d − +2 β ( η , η ) d η d η = Z O δ | b f ( r ) | | b g ( r ) | ( φ s ( r ) φ s ( r )) d − +2 β Θ d − +2 β ( r r ) d − d r d r .O r r ∼ δ ∼ δ O δ Figure 1.
The set O δ along the curve r = r − . OME SHARP NULL-FORM TYPE ESTIMATES FOR THE KLEIN–GORDON EQUATION 13
Here, the set O δ is defined by O δ = (cid:26) ( r , r ) : 12 δ < r < δ , δ < r < δ (cid:27) . Now, for ( r , r ) ∈ O δ , taking the limit δ → φ s ( r ) → ∞ and φ s ( r ) → s and invoking theLegendre duplication formula(3.3) Γ( z )Γ( z + ) = 2 − z π Γ(2 z ) , we obtain Θ d − +2 β ( r , r ) → | S d − | . Therefore, for sufficiently large δ > KG ( β, d ) Z R d | b f ( η ) | | b g ( η ) | φ s ( | η | ) φ s ( | η | ) K d − +2 β ( η , η ) d η d η = (2 π ) d KG ( β, d ) k φ s ( √− ∆) d − + β f k L ( R d ) k φ s ( √− ∆) d − + β g k L ( R d ) , and it is enough to show (2 π ) d KG ( β, d ) > F ( β, d ) . (3.4)By the formula (2.4) and the definitions of constants, this can be simplified as B ( d − + β, d − + β ) > B ( d − β, d )which, if fact, follows from Lemma 3.2 by letting p = d − + β , m = d − + β and k = − d − β . (cid:3) Contributions of radial symmetry.
Here, we observe for general (not necessary radially sym-metric) data f and g the inequality (1.10) holds only if β ≥ − d , in other words, the radial symmetrycondition on f and g widens the range of the regularity parameter β . The proof is based on the Knapptype argument in [18] where they proved β − ≥ − d is necessary for (1.6) to hold. Proposition 3.4.
Let β < − d . For any C ∗ > , there exists f, g ∈ H d − + β ( R d ) such that k| (cid:3) − (2 s ) | β ( e itφ s ( √− ∆) f e itφ s ( √− ∆) g ) k L ( R d +1 ) (3.5) > C ∗ k φ s ( √− ∆) d − + β f k L ( R d ) k φ s ( √− ∆) d − + β g k L ( R d ) . Proof of Proposition 3.4.
For η ∈ R d (similarly, for η ∈ R d ), we set indices (1) , . . . , ( d ) to indicatecomponents of vectors, namely, η = ( η , . . . , η d ) ). Also, denote η ′ = ( η , . . . , η d ) ) ∈ R d − and η ′′ = ( η , . . . , η d ) ) ∈ R d − . Now, for large L >
0, eventually sent to infinity, define sets F and G by F = { η ∈ R d : L ≤ η (1) ≤ L, ≤ η (2) ≤ , | η ′′ | ≤ } and G = { η ∈ R d : L ≤ η (1) ≤ L, − ≤ η (2) ≤ − , | η ′′ | ≤ } . For such f and g (cid:12)(cid:12)(cid:12) | (cid:3) − (2 s ) | β ( e itφ s ( D ) f ( x ) e itφ s ( D ) g ( x )) (cid:12)(cid:12)(cid:12) ∼ (cid:12)(cid:12)(cid:12)(cid:12)Z F Z G e i Φ s ( x,t : η ,η ) K − β ( η , η ) d η d η (cid:12)(cid:12)(cid:12)(cid:12) , where Φ s ( x, t : η , η ) = x · ( η − η ) + t ( φ s ( | η | ) − φ s ( | η | )) . Now, we follow the idea of Knapp’s example to derive a lower bound. From the setting (see also Figure 2)we have | η | ∼ | η | ∼ φ s ( | η | ) ∼ φ s ( | η | ) ∼ | η + η | ∼ L , θ ∼ L − for ( η , η ) ∈ F × G , | φ s ( | η | ) − η | ∼ η (1) η (2) η ′′ θ FG L O Figure 2.
The sets F and G , which are sent away from the origin along η (1) -axis. | η ′ | | η | − ∼ | φ s ( | η | ) − η | ∼ | η ′ | | η | − ∼ L − , | η − η | ∼ | η ′ + η ′ | .
1. Then, it followsthat ( φ s ( | η | ) φ s ( | η | )) − ( η · η − s ) ∼ s | η + η | + | η | | η | sin θ ∼ L (3.6)and hence K − β ( η , η ) ∼ (cid:18) ( φ s ( | η | ) φ s ( | η | )) − ( η · η − s ) φ s ( | η | ) φ s ( | η | ) + η · η + s (cid:19) β ∼ . Moreover, for the phase, then it follows that | Φ s ( x, t : η , η ) | = | t ( φ s ( | η | ) − η − φ s ( | η | ) + η ) + ( x + t )( η − η ) + x ′ · ( η ′ + η ′ ) |≤ | t | L − + | x + t | L + | x ′ | < π x, t ) in a slab R = [ − L − , L − ] × [ − , d − × [ − L, L ] whose volume is the order of 1. Hence, | (cid:3) − (2 s ) | β ( e itφ s ( D ) f ( x ) e itφ s ( D ) g ( x )) & | F || G | χ R ( x, t )and so k| (cid:3) − (2 s ) | β ( e itφ s ( D ) f ( x ) e itφ s ( D ) g ( x )) & | F | | G | | R | ∼ | F | | G | . On the other hand, we have k φ s ( √− ∆) d − + β f k L ( R d ) k φ s ( √− ∆) d − + β g k L ( R d ) . L d − β | F || G | . Therefore, it is implied that | F | | G | . L d − β | F || G | . The fact | F | ∼ | G | ∼ L and letting L → ∞ result in the desired necessary condition3 − d ≤ β. (cid:3) OME SHARP NULL-FORM TYPE ESTIMATES FOR THE KLEIN–GORDON EQUATION 15 Sharpness of constants
It is straightforward that the estimates (1.4) with claimed constants in Corollaries 1.3 and 1.4 when( α, β ) = ( , − d ) and ( α, β ) = ( , − d ) coincides with the results obtained by applying the kernel estimates(1.9) and (1.19) to (1.3), respectively. We will see in the forthcoming sections the sharpness of thoseconstants. To obtain the estimate (1.14) and (1.20), we require the additional fact that Z R d f ( x ) f ( y ) x · y d x d y ≥ . Indeed, in the wave regime, after we apply the kernel estimate (1.9) to (1.3), it follows that Z R d | b f ( η ) | | b f ( η ) | φ s ( | η | ) φ s ( | η | ) K ( η , η ) d η d η ≤ Z R d | b f ( η ) | | b f ( η ) | φ s ( | η | ) φ s ( | η | )( φ s ( | η | ) φ s ( | η | ) − s ) d η d η , which immediately yields (1.14). Similarly, one can deduce (1.20) in the non-wave regime. Finally, theestimate (1.4) with C = F ( − d , d ) when ( α, β ) = (1 , − d ) is obtained by further estimating the kernel of(1.14) as φ s ( | η | ) φ s ( | η | ) − s ≤ φ s ( | η | ) φ s ( | η | ) . Again, we will see the sharpness of constants below. Of course, by a similar argument to the above, onecan easily obtain the estimate (1.4) with(4.1) C = 2 − d +1 π − d +22 s Γ( d +22 )when ( α, β ) = (1 , − d ) from (1.20) in the non-wave regime, and it is natural to hope that the constant isstill optimal. We do not, however, know whether or not the constant (4.1) is optimal, which will becomeclear from the following argument on the sharpness of constants.In the rest of Section 4, we focus on completing our proof of Corollaries 1.2, 1.3 and 1.4 by proving thatthe stated constants are optimal and non-existence of non-trivial extremisers. We achieve optimality ofconstants by considering the functions f a given by (2.3); this is a natural guess given that such functionsare extremisers for (1.3), as shown in our proof of Theorem 1.1. Before proceeding, we introduce thefollowing useful notation.N a ( β ) := Z ∞ as e − ρ Z (2 a ) − √ ρ − (4 as ) ( ρ − (2 ar ) − (4 as ) ) d − +2 β ρ − (2 ar ) r d − d r d ρ and D a ( β, b ) := (cid:18)Z ∞ as e − ρ ρ b ( ρ − (2 as ) ) d − d ρ (cid:19) . Wave regime.
We shall consider (1.4) with ( α, β ) = ( d − + β, β ) for β ∈ ( β d , ∞ ). Let f a satisfy(2.3). Then, we have k| (cid:3) − (2 s ) | β | e itφ s ( D ) f a | k L ( R d +1 ) = 2 − d +72 − β | S d − | KG ( β, d )(2 a ) − d +5 − β N a ( β )and(4.2) k φ s ( D ) d − + β f a k L ( R d ) = (2 π ) − d | S d − | (2 a ) − d +5 − β D a ( β, d − + 2 β ) , and so it is enough to show(4.3) lim a → k| (cid:3) − (2 s ) | β | e itφ s ( D ) f a | k L ( R d +1 ) k φ s ( D ) d − + β f a k L ( R d ) = lim a → (2 a ) d C ( β, d ) N a ( β )D a ( β, d − + 2 β ) = F ( β, d ) , where C ( β, d ) = 2 − d − π − d +12 Γ( d − + 2 β )Γ( d − β ) . Since we have, by appropriate change of variables,N a ( β ) = e − as (2 a ) − d Z ∞ e − ρ ρ d − β ( ρ + 8 as ) d − β Z (1 − ν ) d − β ν d − ( ρ + 4 as ) (1 − ν ) + (4 as ) ν d ν d ρ and D a ( β, d − + 2 β ) = e − as (cid:18)Z ∞ e − ρ ( ρ + 2 as ) d − +2 β ρ d − ( ρ + 4 as ) d − d ρ (cid:19) , one may deduce lim a → (2 a ) d N a ( β )D a ( β, d − + 2 β ) = Γ(3 d − β ) B ( d − β, d ))2Γ( d − + 2 β ) , which leads to (4.3).In order to show the constant F ( − d , d ) is sharp in (1.14), we apply a similar calculation. In particular,one may note that the right-hand side of (1.14) can be written as(4.4) (2 π ) − d | S d − | (2 a ) − d (D a ( β, − (2 as ) D a ( β, , instead of (4.2). One can also see the second term is negligible in the sense of the optimal constant sinceit vanishes while a tends to 0.4.2. Non-wave regime.
Let f a satisfy (2.3). Note that in the non-wave regime the right-hand side of(1.4) is expressed as(4.5) k φ s ( D ) d + β f a k L ( R d ) = (2 π ) − d | S d − | (2 a ) − d +4 − β D a ( β, d − + 2 β ) . Then, as we have done above, reform N a ( β ) and D a ( β, d − + 2 β ) as follows by some appropriate changeof variables:N a ( β ) = e − as (2 a ) d − β Z ∞ e − ρ ρ d − β ( ρ a + 4 s ) d − β Z (1 − ν ) d − β ν d − ( ρ a + 2 s ) (1 − ν ) + (2 s ) ν d ν d ρ and D a ( β, d − + 2 β ) = e − as (2 a ) d − β (cid:18)Z ∞ e − ρ ( ρ a + s ) d − +2 β ρ d − ( ρ a + 2 s ) d − d ρ (cid:19) . First, we shall consider (1.4) with ( α, β ) = (0 , − d ). By a similar argument to the wave regime above, onecan easily check that lim a →∞ (2 a ) d +1 N a ( − d )D a ( − d ,
0) = 2 d − s − holds, from which it follows thatlim a →∞ k| (cid:3) − (2 s ) | − d | e itφ s ( D ) f a | k L ( R d +1 ) k φ s ( D ) f a k L ( R d ) = lim a →∞ (2 a ) d +1 C ( − d , d ) N a ( − d )D a ( − d ,
0) = 2 − d +1 π − d +22 s Γ( d ) . OME SHARP NULL-FORM TYPE ESTIMATES FOR THE KLEIN–GORDON EQUATION 17
Similarly, for the case when ( α, β ) = (1 , − d ), it is enough to showlim a →∞ k| (cid:3) − (2 s ) | − d | e itφ s ( D ) f a | k L ( R d +1 ) k φ s ( D ) f a k L ( R d ) − (2 as ) k φ s ( D ) f a k L ( R d ) = lim a →∞ (2 a ) d +1 C ( − d , d ) N a ( − d )D a ( − d , − (2 as ) D a ( − d , − d +1 π − d +22 s Γ( d +22 ) . (4.6)In fact,((2 a ) d − e − as ) − (cid:0) D a ( − d , − (2 as ) D a ( − d , (cid:1) = (cid:18)Z ∞ e − ρ ( ρ + 2 as ) ρ d − (cid:16) ρ a + 2 s (cid:17) d − d ρ (cid:19) − (cid:18) (2 as ) Z ∞ e − ρ ρ d − (cid:16) ρ a + 2 s (cid:17) d − d ρ (cid:19) = (2 a ) (cid:18)Z ∞ e − ρ ρ d − (cid:16) ρ a + 2 s (cid:17) d d ρ (cid:19) (cid:18)Z ∞ e − ρ ρ d (cid:16) ρ a + 2 s (cid:17) d − d ρ (cid:19) implies lim a →∞ (2 a ) d +1 N a ( − d )D a ( − d , − (2 as ) D a ( − d ,
0) = 2 d − s − and so (4.6) follows.In the contrast to the wave regime, here a is sent to ∞ and the second term of the denominator of (4.6)does not vanish so that we cannot follow the argument for the wave regime and do not know whether theconstant (4.1) is still optimal for (1.4) when ( α, β ) = (1 , − d ).4.3. Non-existence of an extremiser.
Suppose there were non-trivial f and g that satisfy any of thestatements in Corollary 1.4 with equality. From our proof via Theorem 1.1, it would be required that Z R d | b f ( η ) | | b g ( η ) | φ s ( | η | ) φ s ( | η | ) K d − +2 β ( η , η ) d η d η = 2 − s − Z R d | b f ( η ) | | b g ( η ) | φ s ( | η | ) φ s ( | η | ) K d − +2 β ( η , η ) d η d η holds. Then, Z R d | b f ( η ) | | b g ( η ) | φ s ( | η | ) φ s ( | η | ) (cid:16) K d − +2 β ( η , η ) − − s − K d − +2 β ( η , η ) (cid:17) d η d η = 0would hold. Since f , g are assumed to be non-trivial b f , b g = 0 on some set F × G ⊆ R d with | F | , | G | > K d − +2 β ( η , η ) − − s − K d − +2 β ( η , η ) = 0on ( F × G ) \ N where N ⊆ R d is a null set. However, (4.7) would hold only on the diagonal line { ( η , η ) : η = η } (the equality condition of (1.19)), which is a null set and so is { ( η , η ) : η = η } ∩ ( F × G ). Thisis a contradiction.For Corollary 1.2, Corollary 1.3, similar arguments above can be carried. In particular, for equality inthe wave regime, the formula (4.7) might be replaced by K d − +2 β ( η , η ) − K d − +2 β ( η , η ) = 0on ( F × G ) \ N , which would only occur when s = 0 (the equality condition of (1.8)). Analogous results for (++) case
Here we note the analogous versions of our main results in the (++) case.
Theorem 5.1.
For d ≥ and β > − d , we have the sharp estimate k| (cid:3) − (2 s ) | β ( e itφ s ( D ) f e itφ s ( D ) g ) k L ( R d +1 ) (5.1) ≤ KG (++) ( β, d ) Z R d | b f ( η ) | | b g ( η ) | φ s ( | η | ) φ s ( | η | ) K d − +2 β ( η , η ) d η d η , where KG (++) ( β, d ) := 2 − d +12 +2 β π − d +12 Γ( d − )Γ( d − . Moreover, the constant KG (++) ( β, d ) is sharp. One may note that by invoking (3.3) KG (++) ( β, d ) = 2 − d +52 +2 β π − d +22 Γ( d )holds, which is the same constant introduced in [11].An extra symmetry in the (++) case allows (5.1) a wider range of β than β > − d for (1.3). Indeed, thecondition β > − d is no longer imposed because of the form of the sharp constant KG (++) ( β, d ), and thealternative lower bound of β emerges from the kernel K d − +2 β . To see this, let us consider an extremiser f = g = f in (2.3) and after applying polar coordinates to the right-hand side of (5.1) (without theconstant) we obtain Z Z e − ( φ s ( r )+ φ s ( r )) ( φ s ( r ) φ s ( r )) − ( r r ) d − (5.2) × Z − φ s ( r ) φ s ( r ) − s − r r λ ) d − +2 β ( φ s ( r ) φ s ( r ) + s − r r λ ) (1 − λ ) d − d λ d r d r . By (1.17), one may observe that (5.2) is essentially bounded by s − Z Z e − ( φ s ( r )+ φ s ( r )) ( φ s ( r ) φ s ( r )) d − +2 β Z − (cid:18) − r r φ s ( r ) φ s ( r ) − s λ (cid:19) d − +2 β (1 − λ ) d − d λ d r d r and finite whenever β > − d by applying Lemma 3.1.In order to state the various results which follow from Theorem 5.1, here we introduce the followingconstant for the wave regime F (++) ( β, d ) = 2 − β π − d +12 Γ( d − β )Γ( d − + 2 β ) . Corollary 5.2.
Let d ≥ , β ≥ − d . Then, there exists a constant C > such that k| (cid:3) − (2 s ) | β ( e itφ s ( D ) f e itφ s ( D ) g ) k L ( R d +1 ) ≤ C k φ s ( D ) d − + β f k L ( R d ) k φ s ( D ) d − + β g k L ( R d ) (5.3) holds whenever f and g are radially symmetric. Moreover, for β ∈ [ − d , − d ] ∪ [ − d , ∞ ) , the optimalconstant in (5.3) for radially symmetric f and g is F (++) ( β, d ) , but there does not exist a non-trivial pairof functions ( f, g ) that attains equality. We remark that, following the argument in Section 3.2, once we apply Theorem 5.1 as a first step, it isnot possible to obtain the constant F (++) ( β, d ) for β ∈ ( − d , − d ). OME SHARP NULL-FORM TYPE ESTIMATES FOR THE KLEIN–GORDON EQUATION 19
Corollary 5.3.
Let d ≥ .(i) The estimate (5.4) k| (cid:3) − (2 s ) | β ( e itφ s ( D ) f ) k L ( R d +1 ) ≤ C k φ s ( D ) α f k L ( R d ) holds with the optimal constant C = F (++) ( β, d ) for ( α, β ) = ( , − d ) and ( α, β ) = (1 , − d ) , butthere no non-trivial extremisers. Furthermore, when ( α, β ) = (1 , − d ) , we have the refined Strichartzestimate k| (cid:3) − (2 s ) | − d ( e itφ s ( D ) f ) k L ( R d +1 ) ≤ F (++) ( − d , d ) (cid:16) k φ s ( D ) f k L ( R d ) − s k φ s ( D ) f k L ( R d ) (cid:17) , where the constant is optimal and there are no non-trivial extremisers.(ii) The estimate (5.4) holds with the optimal constant C = − d π − d +12 Γ( d − )Γ( d ) s − ! for ( α, β ) = ( , − d ) , but there are no non-trivial extremisers. Furthermore, when ( α, β ) = (1 , − d ) ,we have the refined Strichartz estimate k| (cid:3) − (2 s ) | − d ( e itφ s ( D ) f ) k L ( R d +1 ) ≤ − d π − d +12 Γ( d − )Γ( d +22 ) s − ! (cid:16) k φ s ( D ) f k L ( R d ) − s k φ s ( D ) f k L ( R d ) (cid:17) , where the constant is optimal and there are no non-trivial extremisers. Theorem 5.1 follows from adapting the calculation by Jeavons [21] without any major difficulty. Indeed,he has computed by the Cauchy–Schwarz inequality that | g ( uv )( τ, ξ ) | ≤ J ( τ, ξ )(2 π ) d − Z ( R d ) | F ( η , η ) | δ (cid:18) τ − φ s ( | η | ) − φ s ( | η | ) ξ − η − η (cid:19) d η d η . Here, u ( t, x ) = e itφ s ( √− ∆) f ( x ), v ( t, x ) = e itφ s ( √− ∆) g ( x ), F ( η , η ) = b f ( η ) b g ( η ) φ s ( | η | ) φ s ( | η | ) , and J β is given by (2.1). In terms of the Lorentz invariant measure d σ s ( t, x ) = δ ( t − φ s ( | x | )) φ s ( | x | ) d x d t , J ( τ, ξ )is also written as J ( τ, ξ ) = σ s ∗ σ s ( τ, ξ ) . Invoking Lemma 1 in [21], we have J ( τ, ξ ) = | S d − | d − ( τ − | ξ | − (2 s ) ) d − ( τ − | ξ | ) so that k| (cid:3) − (2 s ) | β ( e itφ s ( D ) f e itφ s ( D ) g ) k L ( R d +1 ) = (2 π ) − ( d +1) Z R d +1 | τ − | ξ | − (2 s ) | β | g ( uv )( τ, ξ ) | d ξ d τ ≤ − d +32 +2 β π − d +1 Γ( d ) | S d − | Z ( R d ) | b f ( η ) | | b g ( η ) | φ s ( | η | ) φ s ( | η | ) K d − +2 β ( η , η ) d η d η , which is what we desired. (cid:3) Acknowledgment.
The first author was supported by JSPS Postdoctoral Research Fellowship (No.18F18020), and the second author was supported by JSPS KAKENHI Grant-in-Aid for JSPS Fellows(No. 20J11851). Authors express their sincere gratitude to Neal Bez, second author’s adviser, for intro-ducing the problem, sharing his immense knowledge and continuous support.
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Department of Mathematics, Graduate school of Science and Engineering, Saitama Univer-sity, Saitama, 338-8570, Japan
Email address : [email protected] (Shobu Shiraki) Department of Mathematics, Graduate school of Science and Engineering, Saitama University,Saitama, 338-8570, Japan
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