Bilinear identities involving the $k$-plane transform and Fourier extension operators
David Beltran Luis Vega
math.CA


Effective Entropy Production and Thermodynamic Uncertainty Relation
of Active Brownian Particles
    Zhonghuai Hou
    December 30, 2023
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We prove certain L^2(^n) bilinear estimates for Fourier extension operators associated to spheres and hyperboloids under the action of the k-plane transform. As the estimates are L^2-based, they follow from bilinear identities: in particular, these are the analogues of a known identity for paraboloids, and may be seen as higher-dimensional versions of the classical L^2(^2)-bilinear identity for Fourier extension operators associated to curves in ^2.











§ INTRODUCTION








For n ≥ 2, let U be an open subset in ^n-1 and ϕ:^n-1→ be a smooth function parametrising a hypersurface S={(ξ, ϕ(ξ)) : ξ∈ U}. Associated to S, define the Fourier extension operator

    Ef(z):=∫_U e^i (x ·ξ + t ϕ(ξ) ) f(ξ)  ξ,

where z=(x,t) ∈^n-1× and f ∈ L^1(U). The terminology extension comes from the fact that E is the adjoint operator to the restriction of the Fourier transform to S, that is E^*h(ξ)=h(ξ, ϕ(ξ)). Stein observed in the late 1960s that under certain curvature hypothesis on S it is possible to obtain L^p(U)-L^q(^n) estimates for E besides the trivial L^1(U)-L^∞(^n) estimate implied by Minkowski's inequality. In particular, the Fourier restriction conjecture asserts that if S is compact and has everywhere non-vanishing Gaussian curvature

    Ef _L^q(^n)≤ C f _L^p(U)

should hold for all q > 2n/n-1 and 1/q≤n-1/n+11/p'. This conjecture is fully solved for n=2 <cit.>, but is still open for n ≥ 3 and constitutes one of the main open problems in Euclidean Harmonic Analysis. The first fundamental result in this direction was the Stein–Tomas <cit.> restriction estimate

    Ef _L^2(n+1)/n-1(^n)≤ C  f _L^2(U);

note that this estimate is best possible in terms of the exponent q for f ∈ L^2(U). Over the last few years, there has been a great interest in establishing the sharp value of C and the existence and characterisation of extremisers in (<ref>) depending on the underlying surface S: see, for instance, <cit.> or the most recent survey <cit.>.

Substantial improvements on (<ref>) have been achieved over the last few decades. An important ingredient for this has been the bilinear and multilinear approach. Multilinear restriction estimates generally adopt the form

    ∏_j=1^k E_j f_j _L^q/k(^n)≤ C ∏_j=1^k  f_j _L^p(U_j),

where the E_j are associated to hypersurfaces S_j satisfying certain transversality hypotheses. A key feature of these inequalities is that, under such additional hypotheses, it is possible to obtain estimates for p=2 and 2n/n-1<q<2(n+1)/n-1. The interested reader is referred, for instance, to <cit.> for the theory of bilinear restriction estimates and to <cit.> for the multilinear approach; see also the survey papers <cit.>.

An elementary instance of a bilinear estimate is in fact the identity

    E_1 f_1 E_2 f_2 _L^2(^2)^2 = (2 π)^2 ∫_U_1 × U_2|f_1(ξ_1)|^2 |f_2(ξ_2)|^2/|ϕ_1'(ξ_1)-ϕ_2'(ξ_2)| ξ_1  ξ_2,

which follows from an application of Plancherel's theorem and a change of variables; note that under the transversality hypothesis |ϕ_1'(ξ_1) - ϕ_2'(ξ_2)|>c>0 for ξ_1 ∈ U_1, ξ_2 ∈ U_2, one may interpret the identity (<ref>) in the framework of (<ref>). Of course, the presence of L^2 on the left-hand side in (<ref>) is key for the use of Plancherel's theorem. This bilinear approach has its roots in the work of Fefferman <cit.> and may also be extended to higher dimensions. Identifying E_j f_j = g_j μ_j,[See <ref> for the non-standard normalisation chosen for the Fourier transform ·.] where g_j:^n → is the lift of f_j to S_j, i.e., g_j(ξ, ϕ_j(ξ))=f_j(ξ) and μ_j is the parametrised measure in S_j defined via 

    ∫_^n g(η)  μ(η) = ∫_U_j g(ξ, ϕ_j(ξ)) ξ,

one may obtain the L^2(^n) bilinear estimate

    E_1 f_1 E_2 f_2 _L^2(^n)^2   ≤ |g_1|^2 μ_1 ∗ |g_2|^2 μ_2 _L^1(^n)μ_1 ∗μ_2 _L^∞ (^n)≤ C f_1 _L^2(U_1)^2  f_2 _L^2(U_2)^2

after an application of Plancherel's theorem and the Cauchy–Schwarz inequality, provided one assumes the transversality condition μ_1 ∗μ_2 _L^∞ (^n)≤ C < ∞. It should be remarked that the Lebesgue exponent 2 on the left-hand side of (<ref>) corresponds to q=4≥2(n+1)/n-1 if n ≥ 3; note that  in a bilinear formulation the Lebesgue exponent is interpreted as q/2. This is very much in contrast to the setting described in (<ref>), in which the main goal is to obtain estimates when q<2(n+1)/n-1; bilinear and multilinear estimates of that type are deep and difficult and will not be explored in this paper.

It is interesting to compare (<ref>) and (<ref>). The first observation is that (<ref>) is an identity, whilst (<ref>) is an inequality. The second is the presence of the weight factor |ϕ_1'(ξ_1) - ϕ_2'(ξ_2)|^-1 in (<ref>); the transversality weight |μ_1 ∗μ_2| in (<ref>) does not necessarily have a closed form in terms of the variables of integration of f_1 and f_2.

The main purpose of this paper is to further exploit the elementary 2-dimensional analysis in (<ref>) in a higher dimensional setting. More precisely, we wish to obtain a bilinear identity in higher dimensions which incorporates an explicit weight factor amounting to some transversality condition; we note that an alternative higher dimensional version of (<ref>) has recently been obtained by Bennett and Iliopoulou <cit.> in a n-linear level. In our goal of obtaining bilinear identities, we shall replace the L^2(^n) in (<ref>) by a mixed-norm L^1(^n-2) × L^2(^2). Given x=(x̅, x”) ∈^n-2×^2, taking the L^1-norm in the x̅ variables will essentially reduce matters to a 2-dimensional analysis in the x”=(x_n-1,x_n) variables, where the resulting extension operators E_1 and E_2 will correspond to sections of the original surfaces by 2-dimensional planes parallel to ξ_1=… = ξ_n-2=0. The existence of such bilinear identities has already been established by Planchon and the second author <cit.> if the underlying hypersurfaces are paraboloids. The motivation in their work came from the relevant role played by these types of inequalities in the global behaviour of large solutions of nonlinear Schrödinger equations; see the next subsection for further details. 
Here we further explore whether bilinear identities hold for two other fundamental surfaces: the sphere and the hyperboloid.

Before describing our results in detail we shall first review the known results in the case of paraboloids, as they will provide the framework and context to understand our results.








 §.§ Estimates for paraboloids and connections to Schrödinger equations


In recent years, starting with the work of Ozawa and Tsutsumi <cit.> for the paraboloid S_1=S_2={(ξ, |ξ|^2) : ξ∈^n-1}, there has been an increasing interest in understanding the weight |μ_1 ∗μ_2| in (<ref>) so that a L^2-bilinear estimate

    E_1 f_1 E_2 f_2 _L^2(^n)^2 ≤ C ∫_U_1 × U_2 K_S_1,S_2(ξ_1,ξ_2) |f_1(ξ_1)|^2 |f_2(ξ_2)|^2  ξ_1  ξ_2

holds for some kernel K_S_1,S_2 and such that the constant C is best possible; in many cases, extremisers for the above kinds of inequalities have also been characterised. This has been mostly studied for paraboloids <cit.>, cones <cit.>, spheres <cit.> and hyperboloids <cit.>, with the corresponding natural interpretations in PDE.

It should be noted that the bilinear estimates (<ref>) and (<ref>) also hold when E_2 f_2 is replaced by its complex conjugate E_2 f_2. This is, of course, of interest when S_1=S_2 and f_1=f_2, as then the bilinear estimates can be reinterpreted as L^2 estimates of |Ef|^2. In particular, in the case of paraboloids, the identity (<ref>) may be reinterpreted as

    ∫_× |D_x |u|^2|^2   x   t = 1/2(2π)^2∫_× |ξ-η| |u_0(ξ)|^2 |u_0(η)|^2  ξ η

or simply

    ∫_× |D_x^1/2 |u|^2|^2   x   t = 1/2 u_0 _L^2()^2  u_0 _L^2()^2

in order to avoid the singularity of the resulting weight |ϕ'(ξ)-ϕ'(η)|=2|ξ-η|; here we interpret the extension operator u(x,t)=  Eu_0 (x,t) as the solution of the free Schrödinger equation i ∂_t u - Δ u =0 in ^d associated to the initial data u(x,0)=u_0(x), with the normalisation of the Fourier transform considered in <ref>. Note that, for this specific case, it is crucial that the multiplier associated to D_x coincides precisely with |ϕ'(ξ)-ϕ'(η)|. Moreover, Ozawa and Tsutsumi <cit.> made use of the Radon transform to obtain the higher dimensional version

    (-Δ)^(2-d)/4 |u|^2 _L^2_x,t(^d ×)^2 ≤OT(d)  u_0 _L^2(^d)^2  u_0 _L^2(^d)^2

where the constant OT(d)=2^-dπ^(2-d)/2/Γ(d/2) is sharp after verifying that for u_0(x)=e^-|x|^2 the inequality becomes an identity; see also <cit.>. 

The interest of Ozawa and Tsutsumi comes from the noninear Schrödinger equation

    i∂_tu+∂^2 u=iλ(∂|u|^2)u + f(u),

where λ∈ and f is a nonlinear interaction, which can be taken to be zero for simplicity of this exposition. In <cit.> the authors proved a well-posedness result in the Sobolev space H^1/2(). This was a non-trivial task due to the presence of the derivative term ∂|u|^2 on the right-hand side of the equation (<ref>). The advantage of (<ref>) (or (<ref>) when d=1) as opposed to an inequality of the type (<ref>) is the gain in derivatives of the solution with respect to the intial data, which allowed the authors to treat the term ∂ |u|^2 as a perturbation. 

The result by Ozawa and Tsutsumi was not further explored until <cit.>, where Planchon and the second author  established certain higher dimensional analogues of the ^1+1 identity (<ref>). Their identities also involved the Radon transform in the spatial variables[Note that the Radon transform in the spatial variables in u(x,t) amounts to a (n-2)-plane transform in the context of the extension operators Ef(z).], which in fact features in the statement. Recall that given a linear k-dimensional subspace π∈𝒢_k,n and y ∈π^⊥,  the k-plane transform of a function f belonging to a suitable a priori class is defined as

    T_k,n f (π,y):=∫_π f(x+y) λ_π (x),

where 𝒢_k,n denotes the Grassmanian manifold of all k-dimensional subspaces in ^n and λ_π is the induced Lebesgue measure on π. The cases k=1 and k=n-1 correspond to the X-ray transform X and the Radon transform[The Radon transform ℛf is identified with a function in ^n-1_+ × setting ℛf (ω, s) ≡ℛf (⟨ω⟩^⊥, s ω).] ℛ respectively. With this notation, the following was shown in <cit.>. 

Let n ≥ 2 and ω∈^d-1. Then,

    ∫_∫_ |∂_s ℛ (|u (·, t)|^2) (ω, s)|^2   s   t + J_ω(u)  =π/(2π)^2d+1∫_^d∫_^d |(ξ-η) ·ω| |u_0(ξ)|^2 |u_0(η)|^2  ξ η,

where

    J_ω(u) :=∫_∫_∫_(⟨ω⟩^⊥)^2|u(x + sω, t) ∂_s u (y + sω, t)-  u(y + sω, t) ∂_s u (x + sω, t)|^2  λ_(⟨ω⟩^⊥)^2(x,y)  s    t.



Note that fixing ω=e_d (or any other coordinate vector) in (<ref>) above, the first term on the left-hand side amounts to ∂_s  |u|^2 _L^1(^d-1)_L^2_x_d,t(^2)^2, which in the absence of the derivative ∂_s becomes u ^4_L^4_x_d,t(^2; L^2(^d-1)); note the contrast with the L^4-nature of (<ref>) and (<ref>).

The approach used in <cit.> to establish Theorem <ref> uses integration-by-parts arguments  
and extends to versions of (<ref>) for nonlinear Schrödinger equations with nonlinearity of the type ± |u|^p-1u, where p ≥ 1. The motivation in <cit.> is similar to that of Ozawa and Tsutsumi, and comes, more precisely, from the breakthrough result by Colliander, Keel, Staffilani, Takaoka, and  Tao in <cit.>, who established global well-posedness of the critical defocusing 3d nonlinear Schrödinger equation (NLS)

    i  ∂_t u + Δ u = u |u|^4

in the energy space. This builds up on a previous result of Bourgain  <cit.>, who showed the well-posedness under the assumption of radial symmetry.


Bourgain used an ad hoc modification of a well-known weighted estimate, typically referred to as Morawetz inequality, proved in <cit.>; see also <cit.>. The weights are of the type |x|^-1 and therefore not translation invariant, leading to the well-posedness only under the radial assumption. To overcome that obstacle, the authors in <cit.> established a bilinear Morawetz estimate that avoids the loss of the translation symmetry. Whilst their strategy was successful in dimension 3, the method has some obstructions when considering nonlinear Schrödinger equations in dimensions 1 and 2. The nonlinear versions of the identities (<ref>) were then used in <cit.> to prove certain lower dimensional well-posedness results, also obtained  independently by Colliander, Grillakis and Tzirakis <cit.> by different methods. It is remarked that (<ref>) has further applications, such as well-posedness in 3d for exterior domains, scattering of solutions (see also <cit.>) or recovering Bourgain's <cit.> bilinear refinement of the Strichartz estimate; the reader is referred to <cit.> for further details.


Unfortunately, the bilinear identities (<ref>) (or more precisely, the integration-by-parts proof method) are extremely rigid and they rely on the fact that the Schrödinger equation is a system with a quadratic dispersion relation. However, the connections of these estimates with Strichartz inequalities suggest that similar identities should also be true for general dispersion relations. This is what we start to explore in this paper for the particular case of the Helmholtz and Klein-Gordon equations. Our approach completely relies on Fourier Analysis techniques, after noting that (<ref>) can be obtained from applications of Plancherel's theorem, in the spirit of (<ref>) and (<ref>). Of course, such a proof method only applies to linear problems, and it is therefore more natural to understand our results in the context of the interaction of the k-plane transform and |Ef|^2, where the Fourier extension operator E is associated to spheres and hyperboloids; see also the recent paper <cit.> or the upcoming preprint <cit.> for further examples of this interaction. Despite the lack of nonlinear results, we expect that the identification of bilinear identities for the linear Helmholtz and Klein–Gordon equations presented in this article will provide some light to develop methods based on direct integration by parts, which would be more amenable to nonlinear counterparts. This will be explored somewhere else in the future.
















 §.§ Estimates for the sphere

In the case of the sphere ^n-1_r ≡ r ^n-1 of radius r in ^n, consider the more classical form of the extension operator

    g ↦g σ^n_r,

where σ^n_r denotes the induced normalised Lebesgue measure on ^n-1_r and g ∈ L^1(^n-1_r). The following L^2-identities for T_n-2,n(g_1σ^n_rg_2σ^n_r ) are obtained.

Let n ≥ 3. Let π∈𝒢_n-2,n and let π^⊥ denote the orthogonal subspace to π. For each z ∈^n, write z=z^π + z^⊥, where z^π is the orthogonal projection of z into π. Then

    ∫_π^⊥|  (-Δ_y     )^1/4 T_n-2,n  (g_1σ^n_rg_2σ^n_r) (π,y) |^2  λ_π^⊥(y)  
        = 𝐂_^n-1∫_ (^n-1_r)^2  K_π, ^n-1_r (ξ,ζ)g_1(ξ) g̅_2(ξ^π + ξ̃^⊥) g_2(ζ) g̅_1(ζ^π + ζ̃^⊥)   σ^n_r (ξ)  σ^n_r (ζ)

where

    K_π, ^n-1_r(ξ, ζ):= 2/|ξ^⊥ + ζ^⊥|,       𝐂_^n-1:=(2π)^2(n-1),

r_ξ=√(r^2- |ξ^π|^2) and ξ̃^⊥, ζ̃^⊥∈π^⊥ are the reflected points of ξ^⊥ and ζ^⊥ in π^⊥ with respect to the line passing through the origin and ξ^⊥ + ζ^⊥, that is ξ^⊥ + ζ^⊥ = ξ̃^⊥ + ζ̃^⊥ (see Figure <ref>).



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Of course, the L^2-nature of the inequality on its left-hand side allows one to take advantage of Plancherel's theorem.  As briefly described before <ref>, the key presence of the (n-2)-plane transform reduces the problem to a 2-dimensional analysis, and one is left to understand the convolution of two weighted measures associated to concentric circles of different radii in the subspace π^⊥≃^2. The main advantage with respect to (<ref>) is that in this setting it is possible to express h_1 σ_r_1^2 ∗ h_2 σ_r_2^2 (ξ^⊥ + ζ^⊥) as the weight σ_r_1^2 ∗σ_r_2^2(ξ^⊥ + ζ^⊥) times an evaluation of the functions h_1 and h_2 at points depending on ξ^⊥ and ζ^⊥. 

Several interesting corollaries can be deduced from Theorem <ref>; their short proofs will be given in <ref>. Given complex numbers a, b, c, d ∈, the well known identity

    a b̅c̅ d = 1/2( |ac|^2 + |bd|^2 - | a c̅ - b d̅|^2 ) + i   (a b̅c̅ d)

may be used in Theorem <ref> to replace the 4-linear wave interaction

    g_1(ξ) g̅_2(ξ^π + ξ̃^⊥) g_2(ζ) g̅_1(ζ^π + ζ̃^⊥)

in (<ref>) by an alternative expression involving |g_1(ξ)|^2 |g_2(ζ)|^2 and which is closer in spirit to (<ref>). 


Let n ≥ 3 and π∈𝒢_n-2,n. Then

    ∫_π^⊥| (-Δ_y )^1/4    T_n-2,n  (g_1σ^n_rg_2σ^n_r) (π,y) |^2   y   
        = 𝐂_^n-1∫_ (^n-1_r)^2  K_π, ^n-1_r (ξ,ζ) |g_1 ( ξ )|^2 |g_2 (ζ)|^2   σ^n_r (ξ)  σ^n _r(ζ) - I_π, ^n-1_r(g_1,g_2),

where

    I_π, ^n-1_r(g_1,g_2):=𝐂_^n-1/2∫_ (^n-1_r)^2  K_π, ^n-1_r (ξ,ζ) | g_1 ( ξ ) g_2(ζ) -  g_2 (ξ^π +  ξ̃^⊥ ) g_1(ζ^π + ζ̃^⊥) |^2    σ^n_r (ξ)  σ^n_r (ζ).


Of course, the term I_π, ^n-1_r(g_1,g_2) ≥ 0 and is identically zero if g_1 and g_2 are constant functions, so it may be dropped from (<ref>) at the expense of losing the identity, leading to a sharp inequality which fits in the context of (<ref>). Thus, the term I_π, ^n-1_r(g_1,g_2) may be interpreted as the distance of such a resulting inequality to become an identity.[The inequality resulting from dropping I_π, ^n-1_r(g_1,g_2) in (<ref>) may be obtained more directly by an application of the Cauchy–Schwarz inequality: see <ref>]

As the k-plane transform satisfies the Fourier transform relation

    ℱ_y T_k,n f (π, ξ) = f(ξ)      for ξ∈π^⊥,

one may easily obtain by means of Plancherel's theorem the relation

    f _L^2(^n)^2 = (2π)^-k/|𝒢_n-k-1, n-1| (-Δ_y)^k/4 T_k,n f _L^2(𝒢_k,n, L^2(π^⊥))^2;

see <ref> for further details. Thus, on averaging Theorem <ref> over all π∈𝒢_n-2,n one has the following.

Let n ≥ 3. Then

    (-Δ)^3-n/4 ( g_1σ^n_rg_2σ^n_r) ^2_L^2(^n)≤ (2π)^2-n𝐂_^n-1∫_(^n-1_r)^2 K_^n-1_r(ξ,ζ)  |g_1 ( ξ )|^2 |g_2 (ζ)|^2   σ^n_r (ξ)  σ^n_r (ζ)

where

    K_^n-1_r(ξ,ζ):=1/|𝒢_1,n-1|∫__n-2, n K_π, ^n-1_r(ξ,ζ)  μ_𝒢(π).


In the particular case n=3 and after setting g_1=g_2, the right-hand side in Corollary <ref> amounts to a bilinear quantity appearing in the work of Foschi <cit.> on the sharp constant in the Stein–Tomas inequality (<ref>) for ^2. Thus, appealing to his work, one can deduce the following.



    g σ^3_L^4(^3)≤ 2π g _L^2(^2).


Besides the value for the sharp constant, Foschi <cit.> also showed that the only real valued extremisers are constant functions; the existence of extremisers was previously verified in <cit.>.




  §.§.§ Solution to the Helmholtz equation


Consider the Helmholtz equation Δ u + k^2 u=0 in ^n. If sup_R>01/R∫_B_R |u|^2 < ∞, then there exists g ∈ L^2(^n-1_k) such that u=g σ_^n-1_k. Theorem <ref> and the subsequent corollaries may be then interpreted in that context.















 §.§ Estimates for the  hyperboloid


A similar analysis to the one described for ^n-1 may be carried for one of the components of the elliptic hyperboloid in ^d+1, defined by

    ℍ^d_m := { (ξ, ξ_d+1) ∈^d × : ξ_d+1 = ϕ_m(ξ):= √(m^2+|ξ|^2)}

and equipped with the Lorentz invariant measure σ_ℍ_m^d (see <ref>), defined by

    ∫_ℍ_m^d g(ξ,ξ_d+1)  σ_ℍ_m^d(ξ,ξ_d+1) = ∫_^d g(ξ, ϕ_m(ξ)) ξ/ϕ_m(ξ).

A function f ∈ L^1(^d) is identified with its lift g to ℍ_m^d, given by g(ξ, ϕ_m(ξ)) = f(ξ), and note

    g σ_ℍ_m^d (x,t) = ∫_^d e^i x ·ξ e^i t √(m^2+|ξ|^2)f(ξ) ξ/√(m^2+|ξ|^2)

where (x,t) ∈^n = ^d ×. A natural reason to split into a space-time domain is in view of the connection of g σ_ℍ^d_m with the Klein–Gordon propagator e^i t √(m^2 - Δ)f; this will be further discussed below. Thus, considering the Radon transform in the space variables - as in (<ref>) and as opposed to Theorem <ref>, where no time role is given and therefore (n-2)-plane transform is taken - one obtains the following (see <ref> for the definition of Lorentz transformation).






Let d ≥ 2. Let ω∈^d-1_+ and let π:=⟨ω⟩^⊥∈𝒢_d-1,d be the orthogonal subspace to ⟨ω⟩. For each x ∈^d, write x=x^π + x^ωω, where x^ω= x ·ω. Then 

    ∫_∫_|  ∂_s^1/2ℛ  (  g_1  σ_ℍ_m^d    (·, t) g_2  σ_ℍ_m^d  (·, t) ) (ω, s)  |^2   s   t  
        = 𝐂_ℍ^d∫_(^d)^2 K_ω, ℍ_m^d (ξ, ζ) f_1(ξ) f̅_2 (ξ^π + ξ̃^ωω)  f_2(ζ) f̅_1 ( ζ^π + ζ̃^ωω)  ξ/ϕ_m(ξ) ζ/ϕ_m(ζ)

where

    K_ω, ℍ_m^d(ξ, ζ) := |ξ^ω  - ξ̃^ω |^1/2 |ζ^ω  - ζ̃^ω |^1/2/|ξ^ωϕ_m(ζ) - ζ^ωϕ_m(ξ)|    and    𝐂_ℍ^d = (2π)^2d.

Above, the points (ξ̃^ω, ϕ_m_ξ^π (ξ̃^ω) ) ∈^1_m_ξ^π and (ζ̃^ω, ϕ_m_ζ^π (ζ̃^ω)) ∈^1_m_ζ^π are the image under L^-1 of the reflected points of L( (ξ^ω, ϕ_m_ξ^π (ξ^ω))) and L( (ζ^ω, ϕ_m_ζ^π (ζ^ω) )) in ^2 with respect to the vertical axis respectively, where L is the unique Lorentz transformation mapping (ξ^ω + ζ^ω , ϕ_m_ξ^π (ξ^ω) + ϕ_m_ζ^π (ζ^ω) ) to the vertical axis and m_ξ^π:=√(m^2+|ξ^π|^2) (see Figure <ref>).




As in the case of the sphere, the use of the Radon transform in ^d and Plancherel's theorem reduces the above estimate to explicitly understand h_1 σ_ℍ^1_m_1∗ h_2 σ_ℍ^1_m_2 (ξ^ω + ζ^ω, ϕ_m_1(ξ^ω) + ϕ_m_2(ζ^ω)). In fact, note that the value of K_ω, ^d_m amounts to the expression | ( [             1             1; ϕ'_m_ξ^π(ξ^ω) ϕ'_m_ζ^π(ζ^ω) ]) |
corrected with the natural weight 1/ϕ_m_ξ^π(ξ^ω) ϕ_m_ζ^π(ζ^ω) coming from the definition of σ_^1_m. This should be compared with the elementary two-dimensional identity (<ref>). The presence of the numerator |ξ^ω - ξ̃^ω|^1/2 |ζ^ω - ζ̃^ω|^1/2 is due to the action of ∂_s^1/2 on ℛ  (  g_1  σ_ℍ_m^d  (·, t) g_2  σ_ℍ_m^d  (·, t) ). Moreover, one can explicitly write ξ̃^ω and ζ̃^ω in terms of ξ, ζ and ω, leading to the more compact expression

    K_ω, ^d_m(ξ, ζ)=  2(ϕ_m(ξ) + ϕ_m(ζ) )/(ϕ_m(ξ) + ϕ_m(ζ) )^2-( (ξ+ζ) ·ω )^2.


As in the case of the sphere, several corollaries can be deduced from Theorem <ref>. As for Corollary <ref>, one may use (<ref>) to rewrite Theorem <ref> in the spirit of Theorem <ref>.



Let d ≥ 2 and ω∈^d-1_+. Then 

    ∫_∫_|  ∂_s^1/2ℛ ( g_1  σ_ℍ_m^d    (·, t) g_2  σ_ℍ_m^d  (·, t) ) (ω,s)  |^2   s   t 
        = 𝐂_ℍ^d∫_(^d)^2 K_ω, ℍ^d_m (ξ, ζ) |f_1(ξ)|^2 |f_2(ζ)|^2  ξ/ϕ_m(ξ)ζ/ϕ_m(ζ) -  I_ω, ^d_m(f_1,f_2)

where 

    I_ω, ^d_m(f_1,f_2):= 𝐂_ℍ^d/2∫_(^d)^2 K_ω, ℍ^d_m (ξ, ζ) | f_1(ξ) f_2(ζ) - f_2(ξ^π + ξ̃^ωω ) f_1(ζ^π + ζ̃^ωω )|^2  ξ/ϕ_m(ξ) ζ/ϕ_m(ζ).



As ℛ=T_d-1,d, the use of the Plancherel's relation (<ref>) after averaging over ω∈^d-1_+ yields the following analogue of Corollary <ref> for hyperboloids.

Let d ≥ 2. Then

    (-Δ_x)^2-d/4 ( g_1  σ_ℍ_m^dg_2  σ_ℍ_m^d )  _L^2_x,t(^d ×)^2 ≤ (2π)^1-d𝐂_ℍ^d∫_(^d)^2 K_ℍ^d_m (ξ, ζ) |f_1(ξ)|^2 |f_2(ζ)|^2  ξ/ϕ_m(ξ) ζ/ϕ_m(ζ)

where

    K_ℍ^d_m (ξ, ζ):= 1/2∫_^d-1 K_ω, ℍ^d_m (ξ, ζ)  σ^d( ω).






  §.§.§ The Klein–Gordon propagator


The solution to the Klein–Gordon equation -∂^2_tu + Δ u = m^2 u in ^d×, with initial data u(x,0)=f_0(x), ∂_t u(x,0)=f_1(x) is given by

    u(x,t)=e^i t √(m^2-Δ) f_-(x) + e^-i t √(-Δ) f_+(x)

where f_+=1/2(f_0 + i (√(m^2-Δ))^-1 f_1) and f_-=1/2(f_0 - i (√(m^2-Δ))^-1 f_1) and 

    e^± i t √(m^2 - Δ) f(x) := 1/(2π)^d∫_^d e^i x ·ξ e^± i t √(m^2+ |ξ|^2)f (ξ)  ξ.

Note that e^± i t √(m^2 - Δ) f(x)= (2π)^-d (gσ_ℍ^d_m)    (x,t); where g is the lift of f√(m^2 + |·|^2) to ℍ^d_m. Thus, Theorem <ref> and Corollaries <ref> and <ref> may be re-interpreted in terms of e^i t √(m^2-Δ); in particular, setting 𝐊𝐆(𝐝)= (2π)^-4d𝐂_ℍ^d, the estimate in Corollary <ref> reads as

    (-Δ_x)^2-d/4 ( e^i t √(m^2-Δ) f_1e^i t √(m^2-Δ) f_2)  _L^2_x,t(^d+1)^2 ≤𝐊𝐆(𝐝)∫_(^d)^2  K_ℍ^d_m (ξ, ζ) |f_1(ξ)|^2 |f_2(ζ)|^2 ϕ_m(ξ) ϕ_m(ζ)  ξζ.






 §.§ Structure of the paper
 Section 2 contains some notation and standard observations which will be useful throughout the paper. In Section 3 we revisit the convolution of weighted measures of circles and hyperbolas. Section 4 contains the proofs of Theorems <ref> and <ref> whilst Section 5 is concerned with the derivation of the several corollaries. Finally, we provide a Fourier analytic proof of Theorem <ref> in Section 6, together with a further discussion on Fourier bilinear identities associated to paraboloids.



 §.§ Acknowledgements
 The authors would like to thank Pedro Caro for several discussions at early stages of this project and Jon Bennett, Giuseppe Negro and Mateus Sousa for stimulating conversations. They also would like to thank the anonymous referee for a careful reading and many valuable suggestions.













§ NOTATION AND PRELIMINARIES








 §.§ Fourier transform

We work with the normalisation of the Fourier transform

    ℱ(f)(ξ)=f(ξ)=∫_^n e^i z ·ξ f(z)   z     and    ℱ^-1(f)(z)=1/(2π)^n∫_^n e^-i z ·ξ f(ξ)  ξ.

With this normalisation,

    f ∗ g = f·g,     fg(ξ)=(2π)^-nf∗g (ξ),     f(z)=(2π)^nf̃(z),     f̅(z)=(2π)^nf̅(z),

where f̃(z):=f(-z), Plancherel's theorem adopts the form

    f_L^2(^n) = (2 π)^n/2 f_L^2(^n).

The n-dimensional Dirac delta, denoted by δ_n is understood as

    δ_n(a) = 1/(2π)^n∫_^n e^i a · z  z .









 §.§ k-plane transform

The Grassmannian manifold _k,n of all k-dimensional subspaces of ^n is equipped with an invariant measure μ_ under the action of the orthogonal group. This measure is unique up to a constant, and is chosen to be normalised as

    |𝒢_k,n|=∫_𝒢_k,n μ_𝒢(π) = |^n-1| ⋯ |^n-k| /|^k-1| ⋯ |^0|.

Given π∈𝒢_k,n and ξ∈π^⊥, the relation (<ref>) between the k-plane transform T_k,n and the Fourier transform easily follows from the definition

    _y T_k,n f (π, ξ)= ∫_π^⊥ e^i y ·ξ T_k,n f (π, y)  λ_π^⊥(y) = ∫_π^⊥ e^i y ·ξ∫_π f (x+ y)  λ_π(x)  λ_π^⊥( y )= f(ξ)

after changing variables z=x+y and noting that ξ· x =0 for ξ∈π^⊥. This and the known identity (see, for instance, <cit.>)

    ∫_^n-1 f(ω)  σ^n(ω)  =  1/|_n-k-1,n-1|∫__k,n∫_^n-1∩π^⊥ f(ω)  σ^n_π^⊥(ω)  μ_𝒢(π),

yield via Plancherel's theorem and a change to polar coordinates the Plancherel-type identity (<ref>) for the k-plane transform.












 §.§ Lorentz transformations


The Lorentz group ℒ is defined as the group of invertible linear transformations in ^d+1 preserving the bilinear form

    (z, u ) ↦ z_d+1u_d+1 - z_d u_d - ⋯ - z_1u_1.

It is well-known that the measure σ_^d_m is invariant under the action of the subgroup of ℒ that preserves the hyperboloid ^d_m, denoted by ℒ^+. More precisely,

    ∫_^d_m f ∘ L  σ_^d_m = ∫_^d_m f   σ_^d_m

for all L ∈ℒ^+. It is also a well-known fact that given P=(ξ, τ) ∈^d+1 with τ > |ξ|, there exists a Lorentz transformation L ∈ℒ^+ such that L(ξ, τ)=(0, √(τ^2-|ξ|^2)); see, for instance, <cit.>. For d=1, this transformation is given by

    L ≡ L_γ_P:=[   coshγ_P - sinhγ_P;  -sinhγ_P   coshγ_P ],     where  γ_P:=ln√(τ+ξ/τ-ξ);

recall that P may be expressed in hyperbolic coordinates as P=(ξ,τ)=(r_P sinhγ_P, r_P coshγ_P), where r_P:= √(τ^2-ξ^2). The inverse Lorentz transformation that maps (0, r_P) back to (ξ, τ) is given by L_-γ_P.











§ CONVOLUTION OF WEIGHTED MEASURES


As is discussed in the introduction, a key ingredient in the proofs of Theorems <ref> and <ref> is to understand convolutions of two weighted measures associated to concentric circles of different radii in ^2 and to hyperbolas in ^2 with the same perpendicular asymptotes and foci lying on the same line but with different major axis. The computation of such convolutions is standard; see, for instance, <cit.> for the circular case or <cit.> for the hyperbolic case. The main feature here is that the convolution is carried with respect to weighted measures, and, since the analysis is restricted to ^2, one can give a precise evaluation of such weights at certain points.








 §.§ Circles











Given r ∈_+, let σ^2_r denote the normalised Lebesgue measure of ^1_r≡ r ^1, that is

    ∫_^1_r g(ω)  σ_r^2 (ω) = ∫_^1 g(r ω)  σ^2 (ω),

and recall that σ^2(ω) = δ_1 (1 - |ω|)  ω = 2δ_1 (1-|ω|^2)  ω, where ω denotes the Lebesgue measure on ^2 and δ_n denotes the n-dimensional Dirac delta.

Given 0 < r_1 ≤ r_2,  the domain of integration in σ_r_1^2 ∗σ^2_r_2(x) is _r_2^1 ∩ ({x} + _r_1^1). This set is non-empty if and only if |x|∈ [r_2-r_1, r_2+r_1] and consists of one point in the tangent case |x|=r_2-r_1 or |x|=r_2+r_1 and of two points otherwise. In the non-empty case, let v_x ∈^1 denote the π/2 degrees rotation of x/|x| in the anti-clockwise direction, and let P_2^+(x) and P_2^-(x) denote the points in _r_2^1 ∩ ({x} + ^1_r_1) such that P_2^+(x) · v_x ≥ 0 and P_2^-(x) · v_x ≤ 0 respectively; note that P_2^+(x)=P_2^-(x) in the tangent case. Define P_1^-(x):=x-P_2^+(x) ∈^1_r_1 and P_1^+(x):=x-P_2^-(x) ∈^1_r_1; note that P_1^+(x), P_1^-(x) ∈𝕊^1_r_1∩ ({x} + 𝕊_r_2^1). Observe that P_j^+(x) and P_j^-(x) are reflected points one another with respect to the line passing through the origin containing x:  see Figure <ref>.



Let r_1, r_2 ∈ such that 0 < r_1 ≤ r_2. 
Then

    g_1 σ_r_1^2∗ g_2 σ^2_r_2 (x) = 2 g_1(P_1^+(x)) g_2(P_2^-(x)) + 2 g_1(P_1^-(x)) g_2(P_2^+(x)) /√( -(|x|^2 - (r_2+r_1)^2) (|x|^2-(r_2-r_1)^2))

if |x| ∈ [r_2-r_1, r_2+r_1].



A standard computation shows (see for instance <cit.> or <cit.> for similar arguments)

    g_1 σ_r_1^2 ∗ g_2  σ_r_2^2 (x)     =  ∫_^1∫_^1 g_1(r_1 ω_1) g_2(r_2ω_2) δ_2(x-r_1ω_1 - r_2ω_2)  σ^2(ω_1) σ^2(ω_2) 
        = 2/r_2^2∫_^1∫_^2 g_1(r_1 ω_1) g_2(r_2 ω_2) δ_2 (x/r_2-r_1/r_2ω_1 - ω_2 ) δ_1(1-|ω_2|^2)  σ^2(ω_1)  ω_2 
        = 1/r_1 |x|∫_^1 g_1(r_1 ω_1) g_2(x-r_1ω_1)  δ_1(r_2^2/2r_1|x|- |x|/2r_1 - r_1/2|x| +  x/|x|·ω_1)  σ^2(ω_1) 
        =:I^+(x) + I^-(x),

where I^+(x) corresponds to the integration over ^1_+(x):={ω∈^1 : x ·ω≥0} and I^-(x) to the integration over ^1_-(x):= ^1 \^1_+(x) = {ω∈^1 : x ·ω≤ 0}.

Denoting by α_x the clockwise angle between e_1 and x and P_x(u)=(cos(α_x+arccos(u)), sin(α_x + arccos(u)) ), the expression for I^+(x) becomes, after a change of variable,

    I^+(x)    = 1/r_1 |x|∫_-1^1 δ_1 (r_2^2/2r_1|x|- |x|/2r_1 - r_1/2|x| + u) (1-u^2)^-1/2 g_1(r_1 P_x(u) ) g_2(x- r_1P_x(u) )   u 
        = 1/r_1 |x|(1-(|x|^2+r_1^2 - r_2^2/2r_1|x|)^2 )^-1/2g_1(P_1^+(x)) g_2(P_2^-(x))χ_{r_2-r_1 ≤ |x| ≤ r_2+r_1} (x)
        = 2g_1(P_1^+(x)) g_2(P_2^-(x))/√(-(|x|^2 - (r_2+r_1)^2) (|x|^2-(r_2-r_1)^2))χ_{r_2-r_1 ≤ |x| ≤ r_2+r_1} (x)

noting that r_1P_x(u)=P_1^+(x) after integrating in u. Indeed, if y ∈𝕊_r_1^1 ∩ ({x} + 𝕊^1_r_2 ) one has |y-x|^2=r_2^2 and |y|^2=r_1^2. Therefore y/|y|·x/|x|= |x|^2+r_1^2-r_2^2/2 r_1 |x| = u, and r_1 P_x(u)=P_1^+(x) follows from noting that P_1^+(x) ∈𝕊_r_1^1 ∩ ({x} + 𝕊^1_r_2 ) and P_1^+(x) · v_x ≥ 0. Arguing similarly for I^-(x) concludes the proof.











 §.§ Hyperbolas










Consider the Lorentz invariant measure σ_ℍ^1_m defined in (<ref>). Given 0 < m_1 ≤ m_2, the domain of integration in σ_ℍ^1_m_1∗  σ_ℍ^1_m_2 (x) is ℍ^1_m_2∩ ({x} - ℍ^1_m_1). Reasoning as in the previous case, this set is non-empty if and only if √(x_2^2-x_1^2)≥ m_1+m_2 and consists of one single point in the tangent case √(x_2^2-x_1^2) = m_1+m_2 and of two points otherwise; here x=(x_1,x_2) ∈^2. In the non-empty case, let Q_2^+(x) and Q_2^-(x) denote the points in ℍ^1_m_2∩ ({x} - ℍ^1_m_1) such that (Q_2^+(x) - x)· e_1 ≥ 0 and (Q_2^-(x) - x)· e_1 ≤ 0 respectively; of course, Q_2^+(x)=Q_2^-(x) in the tangent case. Define Q_1^+(x)=x-Q_2^-(x) ∈ℍ^1_m_1 and Q_1^-(x)=x-Q_2^+(x) ∈ℍ^1_m_1 (see Figure <ref>).




Let m_1,m_2 ∈ such that 0< m_1 ≤ m_2. For each x=(x_1,x_2) ∈^2 such that x_2^2 ≥ x_1^2 one has

    g_1 σ_ℍ^1_m_1∗ g_2 σ_ℍ^1_m_2 (x) = 2 g_1(Q_1^+(x)) g_2(Q_2^-(x)) + 2g_1(Q_1^-(x)) g_2(Q_2^+(x)) /√((x_2^2-x_1^2)^2-2(x_2^2-x_1^2)(m_1^2+m_2^2) + (m_1^2-m_2^2)^2)χ_{√(x_2^2-x_1^2)≥ m_1+m_2 }(x).






By invariance of the measure σ_ℍ^1_m under Lorentz transformations, it suffices to prove the above identity for x=(0,z). Indeed, note that if L_x ∈ℒ^+ is the Lorentz transformation satisfying L_x (x)=(0, z)=(0,√(x_2^2-x_1^2)), then

    g_1 σ_ℍ^1_m_1∗ g_2 σ_ℍ^1_m_2 (x)      = ∫_ℍ_m_1^1∫_ℍ_m_2^1 g_1(ω) g_2(ν)  δ_2( x - ω - ν )  σ_ℍ^1_m_1(ω)  σ_ℍ^1_m_2 (ν)
        = ∫_ℍ_m_1^1∫_ℍ_m_2^1 g_1(L_x^-1(ω)) g_2(L_x^-1(ν)) δ_2( (0,z) - ω - ν )   σ_ℍ^1_m_1(ω)  σ_ℍ^1_m_2 (ν) 
        = h_1 σ_ℍ^1_m_1∗ h_2 σ_ℍ^1_m_2 (0,z)

where h_j = g_j ∘ L_x^-1; the reduction to the vertical axis then follows from noting that h_j (Q_ℓ^± (0,z))= g_j (Q_ℓ^±(x)) for j, ℓ=1,2. Next,

    h_1 σ_ℍ^1_m_1∗ h_2 σ_ℍ^1_m_2 (0,z)      = ∫_ℍ_m_1^1∫_ℍ_m_2^1 h_1(ω) h_2(ν) δ_2( (0,z) - (ω_1,ω_2) - (ν_1,ν_2) )  σ_ℍ^1_m_1(ω)  σ_ℍ^1_m_2 (ν)
        =  ∫_ h_1(ω_1,ϕ_m_1(ω_1)) h_2(-ω_1, ϕ_m_2( ω_1))  δ_1(z-ϕ_m_1(ω_1) - ϕ_m_2(ω_1))/ϕ_m_1(ω_1) ϕ_m_2(ω_1) ω_1.

Splitting =_- ∪_+ and doing the change of variables

    v= ϕ_m_1(ω_1) + ϕ_m_2(ω_1),     with  ω_1/ϕ_m_1(ω_1) ϕ_m_2(ω_1) =  v/ω_1 v,

on each half-line one has

    h_1 σ_ℍ^1_m_1∗ h_2 σ_ℍ^1_m_2 (0,z) = ∫_m_1 +m_2^∞( h_1(    ω_1,ϕ_m_1  (ω_1)) h_2(-ω_1, ϕ_m_2(ω_1)) 
       + h_1(-ω_1,ϕ_m_1(ω_1)) h_2(ω_1, ϕ_m_2(ω_1)) )   δ_1(z-v)  v/ω_1 v

where ω_1 above is the function of v

    ω_1=ω_1(v):= √(v^4 -2v^2(m_1^2+m_2^2) + (m_1^2-m_2^2)^2)/2v.

Noting that Q_1^±(0,v)=( ±ω_1(v), ϕ_m_1 (ω_1(v)) ) and  Q_2^±(0,v)=( ∓ω_1(v), ϕ_m_2 (ω_1(v))), one has

    h_1 σ_ℍ^1_m_1∗ h_2 σ_ℍ^1_m_2 (0,z)     = 2 h_1(Q^+_1(0,z)) h_2(Q^-_2(0,z)) + 2 h_1(Q^-_1(0,z)) h_2(Q^+_2(0,z)) /√(z^4-2z^2(m_1^2+m_2^2) + (m_1^2-m_2^2)^2) 1_{z ≥ m_1+m_2},

completing the proof.
















§ THE PROOFS OF THEOREMS <REF> AND <REF>










 §.§ Proof of Theorem <ref>


For simplicity we work on the unit sphere r=1; the result for ^n-1_r follows analogously. Given π∈_n-2, n, let π^⊥ denote its orthogonal subspace. For each ξ∈^n, write ξ=ξ^π + ξ^⊥, where ξ^π∈π and ξ^⊥∈π^⊥, and let r_ξ^π:=√(1-|ξ^π|^2). Given x ∈π and y ∈π^⊥,

    g_j σ^n (x+y)    = ∫_^n-1 e^i (x+y) ·ξ g_j(ξ)  σ^n (ξ) 
        = ∫_|ξ^π| ≤ 1 e^i x ·ξ^π∫_r_ξ^π^1 e^i y ·ξ^⊥ g_j(ξ^π+ξ^⊥)  σ^⊥_r_ξ^π (ξ^⊥)  λ_π (ξ^π) 
       =  ∫_|ξ^π| ≤ 1 e^i x ·ξ^πℱ^⊥ ( g_j,ξ^πσ_r_ξ^π^⊥  ) (y)  λ_π (ξ^π)

where g_j,ξ^π(ω):= g_j (ξ^π + ω), ℱ^⊥ denotes the Fourier transform in π^⊥ and σ_r_ξ^π^⊥ denotes the induced normalised Lebesgue measure of r_ξ^π^1 in π^⊥, which can be, of course, identified with σ_r_ξ^π^2. Then, by Plancherel's theorem in π (with the  normalisation for the Fourier transform taken  in <ref>),

    T_n-2,n (g_1 σ^ng_2 σ^n) (π, y)    = ∫_πg_1 σ^n(x+y) g_2 σ^n(x+y)  λ_π (x) 
        = (2π)^n-2∫_|ξ^π| ≤ 1ℱ^⊥ ( g_1,ξ^πσ_r_ξ^π^⊥  ) (y) ℱ^⊥ ( g_2,ξ^πσ_r_ξ^π^⊥  ) (y)   λ_π (ξ^π).

A further application of Plancherel's theorem in π^⊥ yields

    ∫_π^⊥   |(-Δ_y)^1/4    T_n-2,n (g_1 σ^ng_2 σ^n) (π, y)|^2  λ_π^⊥ (y)  
       = (2π)^2(n-1)∫_|ξ^π| ≤ 1∫_|ζ^π| ≤ 1 A_𝕊^n-1, π (ξ^π, ζ^π )  λ_π (ξ^π)  λ_π (ζ^π).

where 

    A_𝕊^n-1, π (ξ^π, ζ^π ):= ∫_π^⊥ |v| ( g̃_1,ξ^πσ_r_ξ^π^⊥∗^⊥g̅_2,ξ^πσ_r_ξ^π^⊥)   (v) ( g̃_1,ζ^πσ_r_ζ^π^⊥∗^⊥g̅_2,ζ^πσ_r_ζ^π^⊥)   (v)    λ_π^⊥ (v)

and g̃(·) = g(- ·). For fixed ξ^π, ζ^π with |ξ^π| ≤ 1 and |ζ^π| ≤ 1, A_𝕊^n-1, π(ξ^π, ζ^π) equals to

    ∫_(r_ξ^π^1)^2 × (r_ζ^π^1)^2  |ξ^⊥ -η^⊥|^1/2 |ζ^⊥-μ^⊥|^1/2 g_1, ξ^π (ξ^⊥) g̅_2, ξ^π (η^⊥)  g_2, ζ^π(ζ^⊥) g̅_1, ζ^π(μ^⊥)  Σ^⊥_ξ^π, ζ^π(ξ^⊥, η^⊥,ζ^⊥,μ^⊥)

where

    Σ^⊥_ξ^π, ζ^π(ξ^⊥, η^⊥,ζ^⊥,μ^⊥) := δ (ξ^⊥ + ζ^⊥ -η^⊥-μ^⊥)  σ_r_ξ^π^⊥(ξ^⊥)  σ_r_ξ^π^⊥(η^⊥)  σ_r_ζ^π^⊥(ζ^⊥)  σ_r_ζ^π^⊥(μ^⊥).

Observe that one may rewrite the above integral as

    A_𝕊^n-1, π(ξ^π, ζ^π)= ∫_r_ξ^π^1 × r_ζ^π^1  g_1,ξ^π (ξ^⊥) g_2, ζ^π(ζ^⊥) ( h_2, ξσ^⊥_r_ξ^π∗^⊥ h_1, ζσ^⊥_r_ζ^π) (ξ^⊥ + ζ^⊥)  σ_r_ξ^π^⊥(ξ^⊥)    σ_r_ζ^π^⊥(ζ^⊥)

where h_2, ξ(η^⊥):= g̅_2, ξ^π(η^⊥) |ξ^⊥ - η^⊥|^1/2 and similarly for h_1, ζ. As π^⊥≅^2,  assuming without loss of generality that r_ξ^π≤ r_ζ^π, one can appeal to Lemma <ref> to evaluate

    ( h_2, ξσ^⊥_r_ξ^π∗^⊥ h_1,ζσ^⊥_r_ζ^π) (ξ^⊥ + ζ^⊥)  =  2 h_2,ξ(ξ^⊥) h_1,ζ(ζ^⊥) + 2 h_2,ξ(ξ̃^⊥)h_1,ζ(ζ̃^⊥) /√( -(|ξ^⊥ + ζ^⊥|^2 - (r_ζ^π+r_ξ^π)^2) (|ξ^⊥ + ζ^⊥|^2-(r_ζ^π-r_ξ^π)^2))

after noting that if x=ξ^⊥ + ζ^⊥ then (P_1^+(x), P_1^-(x), P_2^+(x), P_2^-(x))=(ξ^⊥, ξ̃^⊥, ζ^⊥, ζ̃^⊥), where ξ̃^⊥, ζ̃^⊥∈π^⊥ are the reflected points of ξ^⊥ and ζ^⊥ with respect to ξ^⊥ + ζ^⊥. Note that the implicit support condition r_ζ^π - r_ξ^π≤ |ξ^⊥+ζ^⊥| ≤ r_ζ^π + r_ξ^π in (<ref>) always holds under the assumption r_ξ^π≤ r_ζ^π. Observe that h_2,ξ(ξ^⊥)=h_1,ζ(ζ^⊥)=0, so manipulating the denominator one has

    ( h_ξσ^⊥_r_ξ^π∗^⊥ h_ζσ^⊥_r_ζ^π) (ξ^⊥ + ζ^⊥)      = (   |ξ^⊥- ξ̃^⊥| |ζ^⊥- ζ̃^⊥|/(r_ξ^π r_ζ^π)^2 -(ξ^⊥·ζ^⊥)^2 )^1/2g̅_2,ξ^π(ξ̃^⊥)g̅_1,ζ^π(ζ̃^⊥)

for all ξ^⊥∈ r_ξ^π^1 and ζ^⊥∈ r_ζ^π^1. Next note that |ξ^⊥∧ζ^⊥|^2 = (r_ξ^π r_ζ^π)^2 -(ξ^⊥·ζ^⊥)^2, but also |ξ^⊥∧ζ^⊥|^2 = 1/4 |ξ^⊥ + ζ^⊥|^2  |ξ^⊥- ξ̃^⊥| |ζ^⊥- ζ̃^⊥|, as the points satisfy the relation ξ^⊥+ζ^⊥= ξ̃^⊥ + ζ̃^⊥. Thus,

    (   |ξ^⊥- ξ̃^⊥| |ζ^⊥- ζ̃^⊥|/(r_ξ^π r_ζ^π)^2 -(ξ^⊥·ζ^⊥)^2 )^1/2 = 2/|ξ^⊥+ ζ^⊥|,

and combining the above estimates one obtains

    ∫_π^⊥  |(-Δ_y)^1/4  T_n-2,n (g_1 σ^ng_2 σ^n) (π, y)|^2  λ_π^⊥ (y) 
       = (2π)^2(n-1)∫_|ξ^π| ≤ 1∫_|ζ^π| ≤ 1∫_r_ξ^π^1 × r_ζ^π^1 K_π, ^n-1(ξ,ζ)g_1,ξ^π (ξ^⊥) g_2,ζ^π(ζ^⊥)g̅_2,ξ^π(ξ̃^⊥)g̅_1,ζ^π(ζ̃^⊥)   Σ_π (ξ, ζ) 
        = (2π)^2(n-1)∫_(^n-1)^2 K_π, ^n-1(ξ,ζ)g_1 (ξ) g_2(ζ)g̅_2(ξ^π+ξ̃^⊥)g̅_1(ζ^π+ζ̃^⊥)   σ^n(ξ)    σ^n(ζ) ,

completing the proof of Theorem <ref>; above Σ_π (ξ, ζ):=  σ_r_ξ^π^⊥(ξ^⊥)    σ_r_ζ^π^⊥(ζ^⊥)   λ_π (ξ^π)  λ_π (ζ^π). 










 §.§ Proof of Theorem <ref>
 Given ω∈^d-1_+ and π= ⟨ω⟩^⊥∈𝒢_d-1,d write, for each ξ∈^d, ξ= ξ^π + ξ^ωω, where ξ^ω = ξ·ω and let m_ξ^π:=√(m^2+|ξ^π|^2). Given s ∈ and x ∈π,

    g_j σ__m^d (x+ s ω, t)    = ∫_^d e^i (x+sω) ·ξ + i t √(m^2+|ξ|^2) f_j(ξ) ξ/√(m^2+|ξ|^2)
        = ∫_π e^i x ·ξ^π∫_  e^i s ξ^ω  + i t √(m^2+|ξ^π|^2 + |ξ^ω|^2) f_j ( ξ^π + ξ^ωω) ξ^ω/√(m^2+ |ξ^π|^2 + |ξ^ω|^2) λ_π(ξ^π) 
        = ∫_π e^i x ·ξ^πℱ^2 (g_j,ξ^πσ_^1_m_ξ^π) (s,t)  λ_π(ξ^π),

where f_j,ξ^π (ν) := f_j (ξ^π+νω) for all ν∈ and g_j, ξ^π denotes the lift of f_j,ξ^π to ^1_m_ξ^π, and ℱ^2 denotes the 2-dimensional Fourier transform. Reasoning as in the proof of Theorem <ref>,

    ∫_   ∫_  |∂_s^1/2ℛ( g_1 σ__m^d)(·, t) g_2 σ__m^d(·, t)) (ω, s) |^2   s   t  =(2π)^2d∫_π∫_π A_ℍ^d_m, π(ξ^π, ζ^π)   λ_π(ξ^π)  λ_π(ζ^π),

where 

    A_ℍ^d_m, π(ξ^π, ζ^π):= ∫_∫_ |v|(g̃_1,ξ^πσ_^1_m_ξ^π∗^2 g̅_2,ξ^πσ_^1_m_ξ^π)(v,τ)  (g̃_1,ζ^πσ_^1_m_ζ^π∗^2 g̅_2,ζ^πσ_^1_m_ζ^π) (v,τ)   v  τ.

Note that

    A_ℍ^d_m, π (ξ^π, ζ^π)= ∫_∫_ f_1,ξ^π(ξ^ω) f_2,ζ^π (ζ^ω) ( H_2,ξσ_^1_m_ξ^π∗ H_1,ζσ_^1_m_ζ^π) (P_ξ,ζ,ω )  ξ^ω/ϕ_m_ξ^π(ξ^ω) ζ^ω/ϕ_m_ζ^π(ζ^ω),

where H_2,ξ is the lift of h_2,ξ(η):= f̅_2,ξ^π(η)|ξ^ω-η|^1/2 to ^1_m_ξ^π (similarly for H_1,ζ) and P_ξ, ζ, ω denotes the point

    P_ξ,ζ,ω:=(ξ^ω+ζ^ω, ϕ_m_ξ^π(ξ^ω) + ϕ_m_ζ^π(ζ^ω)).

Denoting by r_P the hyperbolic radius of P_ξ,ζ,ω, that is, r_P^2=(ϕ_m_ξ^π(ξ^ω) + ϕ_m_ζ^π(ζ^ω) )^2 - (ξ^ω+ζ^ω)^2, Lemma <ref> yields

    ( H_2,ξσ_^1_m_ξ^π∗ H_1,ζσ_^1_m_ζ^π) (P_ξ,ζ,ω) = 2 h_2,ξ(ξ^ω) h_1,ζ(ζ^ω) + 2h_2,ξ(ξ̃^ω) h_1,ζ(ζ̃^ω) /√( r_P^4 - 2r_P^2 ( (m_ξ^π)^2 + (m_ζ^π)^2  ) + ( (m_ξ^π)^2 - (m_ζ^π)^2  )^2)

where (ξ̃^ω, ϕ_m_ξ^π (ξ̃^ω)) = Q_1^-( P_ξ,ζ,ω ) ∈^1_m_ξ^π and (ζ̃^ω, ϕ_m_ζ^π (ζ̃^ω)) = Q_2^+( P_ξ,ζ,ω) ∈^1_m_ζ^π. After an algebraic manipulation and noting that h_1,ζ(ζ^ω)=h_2,ξ(ξ^ω)=0, (<ref>) becomes

    ( H_2,ξσ_^1_m_ξ^π∗ H_1,ζσ_^1_m_ζ^π) (P_ξ,ζ,ω) = |ξ^ω-ξ̃^ω|^1/2f̅_2,ξ^π(ξ̃^ω) |ζ^ω -ζ̃^ω|^1/2f̅_1,ζ^π (ζ̃^ω)/|ξ^ωϕ_m_ζ^π(ζ^ω) - ζ^ωϕ_m_ξ^π(ξ^ω)|.

Putting all the estimates together as in the proof of Theorem <ref> concludes the proof. 



As the points in the pairs (Q_1^+(0,z), Q_1^-(0,z)) and (Q_2^+(0,z), Q_2^-(0,z)) are symmetric with respect to the vertical axis, it is a simple exercise to obtain an expression for ξ̃^ω and ζ̃^ω via Lorentz transformations. Indeed, let γ_P denote the hyperbolic angle of P_ξ,ζ,ω and let L_γ_P denote, as in (<ref>), the Lorentz transformation such that L_γ_P (P_ξ,ζ,ω)=(0,r_P). Then

    Q_1^+(0,r_P)   =L_γ_P (ξ^ω, ϕ_m_ξ^π(ξ^ω))=(m_ξ^πsinh (γ_ξ - γ_P) , m_ξ^πcosh(γ_ξ-γ_P)) 
    
     Q_2^-(0,r_P)   =L_γ_P (ζ^ω, ϕ_m_ζ^π(ζ^ω))=(m_ζ^πsinh (γ_ζ - γ_P), m_ζ^πcosh(γ_ζ-γ_P) ).

Clearly, 

    Q_1^-(0,r_P)   =(-m_ξ^πsinh (γ_ξ - γ_P), m_ξ^πcosh(γ_ξ-γ_P) ) 
    
     Q_2^+(0,r_P)   =(-m_ζ^πsinh (γ_ζ - γ_P), m_ζ^πcosh(γ_ζ-γ_P) )

and

    Q_1^-(P_ξ,ζ,ω)     = L_-γ_P (Q_1^-(0,r_P)) = (m_ξ^πsinh(2γ_P-γ_ξ), m_ξ^πcosh(2γ_P - γ_ξ)) 
    
    Q_2^+(P_ξ,ζ,ω)     = L_-γ_P (Q_2^+(0,r_P)) = (m_ζ^πsinh(2γ_P-γ_ζ), m_ζ^πcosh(2γ_P - γ_ζ)),

so ξ̃^ω = m_ξ^πsinh(2γ_P-γ_ξ) and ζ̃^ω=m_ζ^πsinh(2γ_P-γ_ζ). In particular, this allows one to rewrite the kernel as

    K_ω, ^d_m(ξ,ζ)= ( m_ξ^ω m_ζ^ω |sinh(γ_ξ-γ_P)| |cosh(γ_ξ+γ_P)| |sinh(γ_ζ-γ_P)| |cosh(γ_ζ+γ_P)|   )^1/2/m_ξ^ω m_ζ^ω |sinh(γ_ξ-γ_ζ)|.





Note that

    |ξ^ω - ξ̃^ω| = |(L_γ_P^-1 (L_γ_P(ξ^ω, ϕ_m_ξ^π(ξ^ω)) - L_γ_P((ξ̃^ω, ϕ_m_ξ^π(ξ̃^ω)) )  )_1| = | (L_γ_P^-1 (2a,0) )_1| = 2|a| cosh(γ_P),

where a:=m_ξ^πsinh(γ_ξ - γ_P). As |ξ^ω - ξ̃^ω|=|ζ^ω - ζ̃^ω| and the denominator in (<ref>) is easily seen to be equal to | a| r_P (see the proof of Lemma <ref>), the kernel K_ω, ^d_m may then be expressed as

    K_ω, ^d_m=2(ϕ_m_ξ^π (ξ^ω) + ϕ_m_ζ^π(ζ^ω)  )/ (  ϕ_m_ξ^π(ξ^ω) + ϕ_m_ζ^π (ζ^ω)  )^2-(ξ^ω + ζ^ω)^2

after noting that cosh(γ_P)= (ϕ_m_ξ^π(ξ^ω) + ϕ_m_ζ^π(ζ^ω) )/r_P.












§ COROLLARIES








 §.§ Proof of Corollary <ref>


By (<ref>) it is clear that the expression

    g_1(ξ) g̅_2(ξ^π + ξ̃^⊥) g_2(ζ) g̅_1(ζ^π + ζ̃^⊥)

on its right-hand side is real and positive. The identity (<ref>) then yields that (<ref>) equals to

    1/2( |g_1(ξ) g_2(ζ) |^2 + |g_2(ξ^π + ξ̃^⊥)  g_1(ζ^π + ζ̃^⊥)|^2 - |g_1(ξ) g_2(ζ)  -  g_2(ξ^π + ξ̃^⊥)  g_1(ζ^π + ζ̃^⊥)|^2 ).

The negative term above immediately gives raise to the expression I_π, ^n-1(g_1,g_2), whilst the positive terms amount to the same expression over the integral sign, finishing the proof. 


Observe that the resulting sharp inequality 

    ∫_π^⊥| (-Δ_y )^1/4  T_n-2,n  (g_1σ^ng_2σ^n) (π,y) |^2   y  ≤𝐂_^n-1∫_ (^n-1)^2  K_π, ^n-1 (ξ,ζ) |g_1 ( ξ )|^2 |g_2 (ζ)|^2  σ^n  (ξ)  σ^n  (ζ)

obtained from dropping the negative term in (<ref>) may be deduced more directly via a simple application of the Cauchy–Schwarz inequality. Note that (<ref>) is a positive quantity, so in particular equals to its modulus. By the triangle inequality, the left-hand side of (<ref>) is controlled by

    ∫_|ξ^π| ≤ 1∫_|ζ^π| ≤ 1∫_(r_ξ^π^1)^2 × (r_ζ^π^1)^2   |ξ^⊥ -η^⊥|^1/2 |ζ^⊥-μ^⊥|^1/2|     g_1,ξ^π (ξ^⊥)| | g_2,ξ^π (η^⊥)| | g_2,ζ^π(ζ^⊥)| | g_1,ζ^π(μ^⊥)| 
        Σ^⊥_ξ^π, ζ^π(ξ^⊥, η^⊥,ζ^⊥,μ^⊥)  λ_π (ξ^π)  λ_π (ζ^π).

Applying the Cauchy–Schwarz inequality with respect to the measure Σ_ξ^π, ζ^π^⊥ λ_π (ξ^π)  λ_π (ζ^π), the above is further controlled by

    ∫_|ξ^π| ≤ 1∫_|ζ^π| ≤ 1∫_r_ξ^π^1 × r_ζ^π^1 |g_1,ξ^π (ξ^⊥)|^2 |g_2,ζ^π(ζ^⊥)|^2 ( h_ξσ^⊥_r_ξ^π∗^⊥ h_ζσ^⊥_r_ζ^π) (ξ^⊥ + ζ^⊥)  σ_r_ξ^π^⊥(ξ^⊥)    Σ_π(ξ, ζ)

where h_ξ(η^⊥):=|ξ^⊥-η^⊥|^1/2 and similarly for h_ζ; above Σ_π(ξ, ζ):= σ_r_ζ^π^⊥(ζ^⊥)  λ_π (ξ^π)  λ_π (ζ^π). Evaluation of the innermost convolution as in (<ref>) yields then the desired inequality (<ref>). 










 §.§ Proof of Corollary <ref>


Given π∈𝒢_n-2,n, Plancherel's theorem and the relation (<ref>) yields

    ∫_π^⊥| (-Δ_y)^1/4 T_n-2,n h (π,y) |^2  λ_π^⊥ (y) = (2π)^-2∫_π^⊥ |ξ^⊥| | h ( ξ^⊥)|^2   λ_π^⊥ (ξ^⊥).

Averaging over all π∈_n-2,n, and using (<ref>) and polar coordinates

    ∫__n-2,n∫_π^⊥ |ξ^⊥|^3-n | h ( ξ^⊥)|^2      |ξ^⊥|^n-2λ_π^⊥(ξ^⊥)   μ_( π)  
        = ∫__n-2,n∫_0^∞∫_^n-1∩π^⊥ r^3-n |h(rω)|^2 r^n-2 r   r  σ^n,⊥ (ω)  μ_(π) 
        = |𝒢_1,n-1| ∫_0^∞∫_^n-1 r^3-n |h(rω)|^2 r^n-1  r  σ^n(ω) 
        =  |𝒢_1,n-1|  (2π)^n ∫_^n ||∇|^3-n/2 h (x) |^2   x,

which completes the proof on taking h=g_1 σ^ng_2σ^n. 











 §.§ Proof of Corollary <ref>


Recall ξ=ξ^π + ξ^⊥. For n=3, π=⟨ω⟩, where ω∈𝒢_1,3≃^2_+. Then ξ^π = (ξ·ω) ω and ξ^⊥ = ξ - (ξ·ω) ω, so

    |ξ^⊥ + ζ^⊥|^2 = |ξ + ζ|^2 + |(ξ + ζ) ·ω|^2 - 2 ((ξ + ζ) ·ω)^2 = |ξ + ζ|^2 (1- ( (ξ + ζ)/|ξ + ζ|·ω)^2).

Noting that |𝒢_1,2| = π,

    K_^n-1(ξ,ζ)=
    2/|𝒢_1,2|∫_^2_+σ^3_+(ω)/|ξ^⊥ + ζ^⊥| = 2 π/π |ξ + ζ|∫_-1^1  u/√(1-u^2) = 2π/|ξ + ζ| .

Thus

    g σ^3_L^4(^3)^4 ≤ (2π)^4 ∫_^2∫_^21/|ξ + ζ| |g(ξ)|^2 |g(ζ)|^2  σ^3(ξ)  σ^3(ζ)

and the desired sharp Stein–Tomas inequality for the sphere follows from the following fact due to Foschi <cit.>:

    ∫_^2∫_^21/|ξ + ζ| |g(ξ)|^2 |g(ζ)|^2  σ^3(ξ)  σ^3(ζ) ≤ g _L^2(^2)^4,

which holds for g antipodally symmetric. The reduction to the antipodally symmetric case may be done as in <cit.>, using the Cauchy–Schwarz inequality for real numbers 

    ac+bd ≤√(a^2+b^2)√(c^2+d^2).

Indeed, note that in the proof of (<ref>) via the Cauchy–Schwarz inequality given in Section <ref>, one may replace |g_ξ^π(ξ^⊥)| |g_ξ^π (η^⊥)| in the innermost integral in (<ref>) by

    |g_ξ^π(ξ^⊥)| |g_ξ^π (η^⊥)| + |g_ξ^π( - ξ^⊥)| |g_ξ^π ( - η^⊥)|/2,

and using (<ref>) this is bounded by |g_ξ^π^#(ξ^⊥)| |g_ξ^π^#(η^⊥)|, where for any function h, the function h^# denotes h^#(ξ):=√( (h(ξ) + h(-ξ))/2), which is antipodally symmetric. One can argue similarly to replace |g_ζ^π(ζ^⊥)| |g_ζ^π (μ^⊥)| by |g^#_ζ^π(ζ^⊥)| |g^#_ζ^π (μ^⊥)|. Thus, the right-hand side in (<ref>) is replaced by

    𝐂_^n-1∫_|ξ^π| ≤ 1∫_|ζ^π| ≤ 1∫_r_ξ^π^1 × r_ζ^π^12/|ξ^⊥ + ζ^⊥|  |g_ξ^π^# ( ξ^⊥ )|^2 |g_ζ^π^# (ζ^⊥)|^2   σ^1_r_ξ^π (ξ^⊥)  σ^1_r_ζ^π (ζ^⊥)  λ_π (ξ^π)  λ_π (ζ^π).

One desires, however, to have g^# rather than g^#_ξ^π and g_ζ^π^#. By a change of variables, the integrand 4 |g_ξ^π^# ( ξ^⊥ )|^2 |g_ζ^π^# (ζ^⊥)|^2 may be further replaced by

    ( |g_ξ^π^# ( ξ^⊥ )|^2  + |g_-ξ^π^# ( ξ^⊥ )|^2 )  ( |g_ζ^π^# (ζ^⊥)|^2 + |g_-ζ^π^# (ζ^⊥)|^2 ),

which equals

    |g^#(ξ)|^2 |g^#(ζ)|^2 + |g^#(ξ)|^2 |g^#(ζ^⊥ - ζ^π)|^2 + |g^#(ξ^⊥ - ξ^π)|^2 |g^#(ζ)|^2 + |g^#(ξ^⊥ - ξ^π)|^2 |g^#(ζ^⊥ - ζ^π)|^2.

A further change of variables in each of the terms allows one to see that (<ref>) equals

    𝐂_^n-1∫_ (^n-1)^2  K_π, ^n-1 (ξ,ζ) |g^# ( ξ )|^2 |g^# (ζ)|^2   σ^n (ξ)  σ^n (ζ),

as desired for the later application of Foschi's identity (<ref>) on antipodally symmetric functions. 









 §.§ Proof of Corollary <ref>


This follows the same argument as that of Corollary <ref>. 



 §.§ Proof of Corollary <ref>


The proof follows from the same argument as in  <ref>. Indeed, the elementary argument therein yields the relation

    (-Δ)^ℓ/2 f _L^2(^d)^2 = (2π)^-(d-1)∂_s^d-1/2 + ℓ f _L^2_ω, s(^d-1_+, )^2,

from which Corollary <ref> follows from taking ℓ=(2-d)/2 after averaging over ω∈^d-1_+; note that ω in the Radon transform  only runs over ^d-1_+ ≃𝒢_d-1,d. 













§ THE BILINEAR IDENTITY (<REF>) FOR PARABOLOIDS REVISITED


The purpose of this final section is to provide an alternative proof of Theorem <ref> via Fourier analysis. The proof follows the same scheme as those of Theorems <ref> and <ref> with a little twist, which is available when taking one full derivative in the s-variable in the case of paraboloids. 

To see this, let ℙ^d_a:= { (ξ, |ξ|^2+ a) : ξ∈^d } denote the paraboloid in (x,t) ∈^d× with tangent plane t=a at its vertex; if a=0 we simply denote it by ^d. Let σ_^d_a denote the parametrised measure on ^d_a, which satisfies g σ_^d_a(x,t)=Ef(x,t) where E is the extension operator associated to ϕ(ξ):=|ξ|^2+a and g is the lift of the function f:^d → to ^d_a.

Given ω∈^d-1_+ and π= ⟨ω⟩^⊥∈𝒢_d-1,d write, for each ξ∈^d, ξ= ξ^π + ξ^ωω, where ξ^ω= ξ·ω. Given s ∈ and x ∈π,

    g_j σ_^d (x+ s ω, t)    = ∫_^d e^i (x+sω) ·ξ + i t |ξ|^2 f_j(ξ)  ξ
        = ∫_π e^i x ·ξ^π∫_  e^i s ξ^ω  + i t |ξ^π|^2 + i t |ξ^ω|^2 f_j ( ξ^π + ξ^ωω )  ξ^ω λ_π (ξ^π)  
        = ∫_π e^i x ·ξ^πℱ^2 (g_j,ξ^πσ_^2_|ξ^π|^2) (s,t)  λ_π(ξ^π),

where f_j,ξ^π (ν) := f_j (ξ^π+νω), ℱ^2 denotes the 2-dimensional Fourier transform and g_j,ξ^π is the lift of f_j,ξ^π to ^2_|ξ^π|^2. Reasoning as in the proof of Theorem <ref>,

    ∫_   ∫_  |∂_s ℛ( g_1 σ_^d(·, t) g_2 σ_^d(·, t)) (ω, s) |^2   s   t 
        = (2π)^2d∫_π∫_π∫_^4 |ξ^ω-η^ω||ζ^ω-μ^ω| f_1,ξ^π (ξ^ω) f̅_2,ξ^π(η^ω) f̅_1,ζ^π(μ^ω) f_2,ζ^π (ζ^ω)  Σ_ξ^π, ζ^π(ξ^ω,η^ω,μ^ω, ζ^ω)

where

    Σ_ξ^π, ζ^π(ξ^ω,η^ω,μ^ω , ζ^ω ):=δ (ξ^ω-η^ω + ζ^ω - μ^ω)  δ ( (ξ^ω)^2-(η^ω)^2 + (ζ^ω)^2 - (μ^ω)^2)  ξ^ωη^ωμ^ω ζ^ωλ_π (ξ^π) λ_π (ζ^π).

Arguing similarly,

    J_ω (g_1 σ_^d, g_2 σ_^d) =  (2π)^2d∫_π∫_π∫_^4  (ζ^ωμ^ω- ζ^ωη^ω - ξ^ωμ^ω + ξ^ωη^ω) f_1,ξ^π (ξ^ω) f̅_2,ξ^π(η^ω) f̅_1,ζ^π(μ^ω) f_2,ζ^π (ζ^ω)

with respect to the measure Σ_ξ^π, ζ^π(ξ^ω,η^ω,μ^ω, ζ^ω), where J_ω(G_1,G_2) is the bilinearisation of J_ω(u); namely the integrand is replaced by

    G_1     (x + sω,t)  ∂_s G_2 (y+ sω,t) ( ∂_s G̅_̅1̅ (y+ sω,t) G̅_̅2̅(x+ sω,t) -  G̅_̅1̅ (y+ sω,t) ∂_s G̅_̅2̅(x+ sω,t)  ) 
         - G_2 (y+ sω,t) ∂_s G_1 (x+ sω,t) ( G̅_̅2̅ (x+ sω,t) ∂_s G̅_̅1̅ (y+ sω,t) - ∂_s G̅_̅2̅ (x+ sω,t)  G̅_̅1̅ (y+ sω,t) ) .

Noting that

    |ξ^ω-η^ω||ζ^ω-μ^ω| + (ζ^ωμ^ω- ζ^ωη^ω - ξ^ωμ^ω + ξ^ωη^ω) = |ξ^ω - μ^ω|^2

if (ξ^ω, η^ω, μ^ω, ζ^ω) ∈  ( Σ_ξ^π, ζ^π), one can combine the two terms above to obtain

    ∫_   ∫_  |∂_s ℛ(  g_1 σ_^d(·, t) g_2 σ_^d(·, t)) (s,ω) |^2   s   t  + J_ω(g_1 σ_^d,g_2 σ_^d)  
        =  (2π)^2d∫_π∫_π∫_^4 |ξ^ω-μ^ω|^2 f_1,ξ^π (ξ^ω) f̅_2,ξ^π(η^ω) f̅_1,ζ^π(μ^ω) f_2,ζ^⊥ (ζ^ω)  Σ_ξ^π, ζ^π(ξ^ω,η^ω,μ^ω, ζ^ω).

For fixed ξ^ω and μ^ω, the only solution for the equations in the δ function is η^ω=ξ^ω and ζ^ω=μ^ω. Thus, the right-hand side above equals

    (2π)^2d/2∫_ξ^π∫_ζ^π∫_^2  |ξ^ω-μ^ω| f_1,ξ^π (ξ^ω) f̅_2,ξ^π(ξ^ω) f̅_1,ζ^π(μ^ω) f_2,ζ^π (μ^ω)  ξ^ω μ^ω λ_π (ξ^π)  λ_π (ζ^π)

and if f_1=f_2,

    (2π)^2d/2∫_π∫_π∫_^2  |ξ^ω-μ^ω| |f_ξ^π (ξ^ω)|^2 |f_ζ^π (μ^ω)|^2  ξ^ω μ^ω λ_π (ξ^π)  λ_π (ζ^π)

which, of course, is

    (2π)^2d/2∫_^d∫_^d  |(ξ-η) ·ω| |f(ξ)|^2 |f(η)|^2  ξ η.

In the language of the Schrödinger equation, u=Eu_0, so the right-hand side is

    π/(2π)^d+1∫_^d∫_^d  |(ξ-η) ·ω| |u_0(ξ)|^2 |u_0(η)|^2  ξ η

and one obtains the desired identity (<ref>), finishing the proof of Theorem <ref>. 


Averaging over all ω∈_+^d-1 after dropping the term J_ω(u) from the obtained identity and noting that

    ∫_^d-1 |(ξ-η) ·ω|  σ^n(ω)=2 |ξ-η| ∫_0^1 u (1-u^2)^d-3/2  u = 2 |ξ-η| π^d-1/2/Γ((d+1)/2),

one has

    (-Δ_x)^3-d/4 ( |u|^2) ^2_L^2_x,t(^d ×)≤ (2π)^1-dπ/(2π)^d+1π^d-1/2/Γ((d+1)/2)∫_^d∫_^d  |ξ-η| u_0(ξ)|^2 |u_0(η)|^2  ξ η

and the constant simplifies as PV(d):=2^-3dπ^1-5d/2/Γ(d+1/2); this inequality was also obtained in <cit.> in a more direct way.


Finally, it is noted that  the honest analogue of Theorems <ref> and <ref> in the context of paraboloids is given by the following bilinear identity.

Let d ≥ 2 and ω∈^d-1_+. Then

    ∫_∫_    |∂_s^1/2ℛ( g_1 σ_^d(·, t) g_2 σ_^d(·, t)) (s,ω) |^2   s   t 
       = (2π)^2d/2∫_π∫_π∫_^2 f_1,ξ^π (ξ^ω) f̅_2,ξ^π(ζ^ω) f̅_1,ζ^π(ξ^ω) f_2,ζ^π (ζ^ω)   ξ^ω ζ^ω λ_π (ξ^π)  λ_π (ζ^π).


The proof of Theorem <ref> is a minor variant of the one for Theorem <ref> exposed above: the main difference is that here one solves the equations in the δ functions in terms of ξ^ω and ζ^ω; the solution in terms of ξ^ω and μ^ω is now degenerate in terms of the weight |ξ^ω-η^ω|^1/2 |ζ^ω-μ^ω|^1/2, which vanishes in this case. Note that, in (<ref>), the fact of taking one full derivative with respect to s and adding the term J_ω (g_1 σ_^d, g_2 σ_^d) had the effect of replacing the weight |ξ^ω-η^ω|  |ζ^ω-μ^ω| by |ξ^ω - μ^ω|^2 thanks to the algebraic identity (<ref>), allowing one to solve in those variables.

Corollaries in the spirit of those obtained for spheres and hyperboloids can also be deduced from Theorem <ref>. In particular, the identity for complex numbers (<ref>) allows one to rewrite (<ref>) as the following.
 Let d ≥ 2 and ω∈𝕊^d-1_+. Then

    ∫_∫_  |∂_s^1/2ℛ(    g_1 σ_^d(·, t) g_2 σ_^d(·, t)) (s,ω) |^2   s   t  
       =  (2π)^2d/2∫_π∫_π∫_^2 | f_1,ξ^π (ξ^ω)|^2 |f_2,ζ^π(ζ^ω)|^2    ξ^ω ζ^ω λ_π (ξ^π)  λ_π (ζ^π) - I_ω(f_1,f_2)

where

    I_ω(f_1,f_2):= (2π)^2d/4∫_π∫_π∫_^2 |f_1,ξ^π (ξ^ω) f_2,ζ^π(ζ^ω)  - f_1,ζ^π(ξ^ω) f_2,ξ^π (ζ^ω)|^2
       ξ^ω ζ^ω λ_π (ξ^π)  λ_π (ζ^π).


Note that, unlike J_ω(f), the term I_ω(f,f) does not have an obvious closed expression in terms of physical variables. Setting f_1=f_2 and averaging over all ω∈^d-1_+ after dropping I_ω(f,f) one obtains 

    (-Δ_x)^2-d/4 ( |u|^2) ^2_L^2_x,t(^d ×)≤ (2π)^1-d(2π)^2d/2|^d-1|/2u_0_L^2(^d)^4 = 2^-dπ^2-d/2/Γ(d/2) u_0 _L^2(^d)^4,

which is the Ozawa–Tsutsumi estimate (<ref>); note that for d=2 this amounts to the L^4(^2+1) Strichartz estimate. The interested reader should look at the work of Bennett, Bez, Jeavons and Pattakos <cit.> for a unified treatment of the Ozawa–Tsutsumi estimates (<ref>), the inequalities deduced from (<ref>), and a more general case with an arbitrary number of derivatives on the left-hand-side of such inequalities.













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