ON DISPERSION DECAY FOR 3D KLEIN-GORDON EQUATION

. We improve previous results on dispersion decay for 3D Klein-Gordon equation with generic potential. We develop a novel approach, which allows us to establish the decay in more strong norms and to weaken assumptions on the potential.

(1) In vector form where Our goal is the improvement of previous results on dispersion decay of solutions. We suggest a novel approach which allows us to establish the decay in more strong norms and to weaken assumptions on the potential. We assume that V (x) is a continuous real function, and for some β > 3. We restrict ourselves to the "regular case" when the point 0 is neither eigenvalue nor resonance for the Schrödinger operator H = −∆ − V (x). Equivalently, the truncated resolvent of the operator H is bounded at the point 0.
For the 3D Schrödinger equation the dispersion decay in weighted norms has been established first by Jensen and Kato [8]: for σ > 5/2 in the regular case. Later, Goldberg and Schlag [6] proved the dispersion decay in L 1 → L ∞ norm: which implies (5) with σ > 3/2. The approach [6,8] to the Schrödinger equation relies on the spectral Fourier-Laplace representation where R(ω) = (H − ω) −1 is the resolvent of the Schrödinger operator H. Low energy and high energy parts are considered separately. The required dispersion decay follows from the regularity of resolvent for small ω and its decay in weighted norms for large ω (see [1,8]). The methods [8] cannot be directly applied to the Klein-Gordon equation, since the results of [1,8] imply the boundedness of the corresponding resolvent R(ω) = (K − ω) −1 only (see the discussion in Introduction of [9]). Similarly, the methods [6] are also inapplicable since the L 1 → L ∞ bound of type (6) fails for the Klein-Gordon equation (see below).
Our approach relies on the Born expansion for the dynamical group U(t), where U 0 (t) is the dynamical group of the free equation and V is the matrix (40). The decay of type (4) for the first term U 0 (t) in weighted energy norms was established in [7] using an analog of the strong Huygens principle. Then the decay for the next terms follows by estimates for convolutions and by the condition (3) on the potential.

DISPERSION DECAY FOR KLEIN-GORDON EQUATION 5767
The main difficulty is to prove the decay of type (4) for the remainder U k (t), , and R 0 (ω) stands for the free resolvent corresponding to V = 0. We derive this decay proving the time decay of U k (t) and of ∇U k (t) in For the proof we develop a streamlined version of the approach [6].
In conclusion, we note that for the free Schrödinger operator the decay of type (6) in L 1 → L ∞ norm follows from the boundedness of its integral kernel for |t| ≥ 1 and from the uniform decay of the kernel. Namely, On the other hand, for U 0 (t) the decay in L 1 ⊕ L 1 → L ∞ ⊕ L ∞ norm does not hold since its kernel is unbounded, This difference reflects the distinct character of the wave propagation for the relativistic and nonrelativistic equations. Namely, the singularities of solutions to the Schrödinger equation are concentrated at t = 0 and disappear at infinity for t = 0 due to infinite speed of propagation. On the other hand, in the case of Klein-Gordon equation, the singularities move with bounded velocities, thus they are present forever in the space. Our paper is organised as follows. In Section 2 we recall the known spectral properties of the Schrödinger resolvent. In Sections 3 and 4 we derive some properties of the finite Born series. In Section 5 we prove our main result. In Section 6 we apply our approach to the Schrödinger equation.
Let us note that the dispersion decay in weighted norms plays an important role in proving asymptotic stability of solitons in associated nonlinear equations [4,5,7,11,12].

DISPERSION DECAY FOR KLEIN-GORDON EQUATION 5773
The estimates forΛ N can been obtained similarly.

5.
Dispersion decay for the Klein-Gordon equation. The required dispersion decay for the free Klein-Gordon equation has been obtained in [7, Lemma 18.2] (see also [9,10]). Namely, Now consider the perturbed Klein-Gordon equation. We use the representation Substituting the Born series we obtain Proposition 1. Let the conditions of Theorem 1.2 hold. Then where the integral converges in E −σ with σ > 3/2.
It remains to prove the decay of type (4) for W(t).
Theorem 5.2. Let the conditions of Theorem 1.2 hold. Then We prove this theorem in the next two subsections.

The decay in
Proposition 2. Let the conditions of Theorem 1.2 hold. Then Proof. The resolvent R(ω) = (K−ω) −1 can be expressed in terms of the Schrödinger resolvent R as follows . Then where Λ ± 6 (λ) is defined in (17), and Due to Lemma 4.1, the integrand in (44) is a differentiable operator function of λ ≥ 0 with values in the space of bounded operators mapping L 1 into L ∞ . Moreover, due to Lemmas 3.3 and 4.1, we can integrate by parts in (44): In the case when the derivative falls on M ± (λ), we can integrate by parts one more time and get the factor t −2 then. Hence, it suffices to consider the case when the derivative falls on Λ ± 6 . More precisely, it suffices to prove that where All other combination of signs "+" and "-" in (45) can be considered in the same way. Denote Then the integral kernel of I(t) reads, We will use the following version of Van der Corput lemma where φ(λ) is real-valued function and f ∈ C 1 ([a, b]). If φ (λ) = 0 for λ ∈ [a, b] then Note, that the lemma remains valid for a = −∞ and b = ∞. The second derivative of the phase functions φ j (λ), j = 1, 2, 3, defined in (46), satisfies Moreover, |M ± (λ)| + |∂ λ M ± (λ)| ≤ C(1 + λ) 2 . Then Corollary 1 and Lemma 5.3 imply

5.2.
The decay of the derivatives.
Proposition 3. Let the conditions of Theorem 1.2 hold. Then Here W ij denotes the ij entry of the matrix operator W.
Proof. Similarly to (44) By Lemmas 3.3 and Lemma 4.2, we can integrate by parts and obtain In the cases when the derivative falls on M 1j ± (λ), we can integrate by parts one more time and get the factor t −2 then. Hence, it suffices to consider the case when the derivative falls on ∇Λ ± 6 . As before, we consider only the "++" case, and prove the decay ∼ |t| −1/2 in L ∞ (R 6 ) for the integral kernels of the operators The corresponding integral kernel reads + e −iφ3(λ)t ∇Λ + 6 (λ, x, y)) dλ. Applying Corollary 2 and Lemma 5.3, we obtain Theorem 5.2 is completely proved.
6. Application to the Schrödinger equation. We apply our technique, giving a short proof of the result of Goldberg and Schlag [6].
Proof. Using the definition (17), we represent L 2 (t) as Due to Lemma 4.1, the integrand is a differentiable operator function of λ ≥ 0 with values in the space of bounded operators mapping L 1 into L ∞ . Moreover, due to Lemmas 3.3 and 4.1 with N = 2, we can integrate by parts, .