On Morawetz estimates with time-dependent weights for the Klein-Gordon equation

We obtain some new Morawetz estimates for the Klein-Gordon flow of the form \begin{equation*} \big\||\nabla|^{\sigma} e^{it \sqrt{1-\Delta}}f \big\|_{L^2_{x,t}(|(x,t)|^{-\alpha})} \lesssim \|f\|_{H^s} \end{equation*} where $\sigma,s\geq0$ and $\alpha>0$. The conventional approaches to Morawetz estimates with $|x|^{-\alpha}$ are no longer available in the case of time-dependent weights $|(x,t)|^{-\alpha}$. Here we instead apply the Littlewood-Paley theory with Muckenhoupt $A_2$ weights to frequency localized estimates thereof that are obtained by making use of the bilinear interpolation between their bilinear form estimates which need to carefully analyze some relevant oscillatory integrals according to the different scaling of $\sqrt{1-\Delta}$ for low and high frequencies.


Introduction
In this paper we are concerned with weighted L 2 estimates for the Klein-Gordon flow e it √ 1−∆ which gives a solution formula (see (1.5)) for the Klein-Gordon equation ∂ 2 t u − ∆u + u = 0. Let us first consider where σ ∈ R and α > 0. This type of estimates was firstly initiated by Morawetz [15] and is sometimes called Morawetz estimates. Especially, the case σ > 0 in the setting (1.1) is indicative of a smoothing effect. Let n ≥ 2. In [16], Ozawa and Rogers showed that (1.1) holds for −n/2 + 1 < σ < 1/2 and α = 2 − 2σ. Here, the region of (α, σ) when σ > 0 is given by the open segment (A, B) (see Figure 1 below). Recently, it was shown by D'Ancona [3] (see also [14]) that the flow can have much smoothness locally as At this point, it should be noted that this local smoothing estimate can be written in terms of a weighted L 2 setting as in (1.1). Indeed, if ρ is any function such that and then (1.2) implies a weaker estimate, A typical example of such ρ is given by ρ = (1 + (log |x|) 2 ) − 1 2 ( 1 2 +ε) , ε > 0, as mentioned in [3]. But this case does not cover the point A because the weight ρ 2 |x| −1 in (1.3) decays faster than |x| −1 . The smoothing effect in the setting (1.1) can occur due to the decay of the weight |x| −α in the spatial direction. As shown in Figure 1, the smoothing factor σ is indeed related with the decay factor α. At this point, we naturally further ask how much regularity we can expect when considering decay not in the spatial direction but in the space-time direction like |(x, t)| −α . In other words, we expect a smoothing effect above the line through the points A, B in this new setting (see Remark 1.2 below). Our result is the following theorem: Theorem 1.1. Let n ≥ 1 and σ ≥ 0. Then we have ≤ α < min{n + 1, 2(n+1) n } and 1 2 − n 4 < σ < 1 2 − nα 4(n+1) for n ≤ 3. Here, (α, σ) lies in some region inside the triangle with vertices C, B, D, as expected.
In general, there are two approaches to smoothing estimates with time-independent weights. One is spectral methods based on resolvent estimates, starting from Kato's work [6], and the other is to use Fourier restriction estimates from harmonic analysis (see e.g. [1,18] and references therein). A completely different approach was also recently developed by Ruzhansky and Sugimoto [17] using canonical transforms.
However, these are no longer available in the case of time-dependent weights. Here we instead combine the theory of dispersive estimates (T T * argument and bilinear approach) and the Littlewood-Paley theory with Muckenhoupt A p weights. This allows us to take advantage of localization on the frequency space. This more fruitful and direct approach has been developed by Koh and the third author [8,9,10,11] for the Schrödinger and wave flows. But here we cannot take advantage of scaling consideration any more. In general, the Klein-Gordon flow e it √ 1−∆ is more complicated by the different scaling of √ 1 − ∆ for low and high frequencies. Indeed, it behaves like the wave flow at high frequency and the Schrödinger flow at low frequency. Hence we need to modify delicately the approach to make it applicable to the Klein-Gordon flow according to low and high frequencies.
The solution u of the Klein-Gordon equation Thus the above theorem for the flow e it √ 1−∆ implies immediately the following corollary for (1.5).
Outline of paper. In Section 2 we prove the estimate (1.4) in Theorem 1.1 applying the Littlewood-Paley theory with Muckenhoupt A 2 weights to its frequency localized estimates in Proposition 2.1. These localized estimates are obtained in Section 3 by utilizing the T T * argument and then making use of the bilinear interpolation between their bilinear form estimates (see, for example, Proposition 3.4). We carry out this process by dividing cases into two parts according to the different scaling of √ 1 − ∆ for low ((2.4)) and high ((2.5)) frequencies. The proof of (2.5) is much more delicate. We generate further localizations in time and space and need to analyze some relevant oscillatory integrals more carefully (see Lemma 3.6) based on the stationary phase lemma, Lemma 3.3.
Throughout this paper, the letter C stands for a positive constant which may be different at each occurrence. We also denote A B to mean A ≤ CB with unspecified constants C > 0.

Proof of Theorem 1.1
In this section we prove Theorem 1.1 by making use of the Littlewood-Paley theory on weighted L 2 spaces.
Let us first recall that a weight 1 w : R n → [0, ∞] is said to be in the Muckenhoupt (See e.g. [4] for details.) We then observe that the weight |(x, t)| −α for −(n + 1) < α < n + 1 is in If we denote Q = Q × I with a cube Q ⊂ R n and an interval I ⊂ R, then (2.1) can be rewritten as Since |(x, t)| ±α are L 1 loc (R n ) functions for −(n + 1) < α < (n + 1), by Lebesgue's differentiation theorem, we can shrink |I| to zero so that |(x, t)| −α ∈ A 2 (R n ) uniformly in almost every t. Now we are in a good light that the Littlewood-Paley theory (see Theorem 1 in [12]) on weighted L 2 space with A 2 weight is applied to get Here, we used the fact that It is a locally integrable function that is allowed to be zero or infinite only on a set of Lebesgue measure zero.
To bound the right-hand side of (2.3), we use the following frequency localized estimates which will be obtained in the next section. and By combining (2.3), (2.6) and (2.7), we conclude that Here we recall the Besov space B s r,m for s ∈ R and 1 ≤ r, m ≤ ∞ equipped with the norm In this section we obtain the frequency localized estimates in Proposition 2.1. To do so, we first utilize the standard T T * argument and then make use of the bilinear interpolation between their bilinear form estimates. We carry out this process by dividing cases into two parts according to the different scaling of √ 1 − ∆ for low ((2.4)) and high ((2.5)) frequencies.
3.1. Proof of (2.4). First we consider an operator and its adjoint operator Again by duality, it suffices to show the following bilinear form estimate To show this, we first write with a smooth cut-off function ϕ : R n → [0, 1] supported in {ξ ∈ R n : |ξ| ≤ 2}. Next we decompose the kernel K in the following way For this, we assume for the moment that for α < (n + 1)/p and use the following bilinear interpolation lemma (see [2], Section 3.13, Exercise 5(b)). if 0 < θ i < θ = θ 0 + θ 1 < 1 and 1/q + 1/r ≥ 1 for 1 ≤ q, r ≤ ∞.
Now it remains to show the estimates (3.2), (3.3) and (3.4). For j ≥ 0, let {Q λ } λ∈2 j Z n+1 be a collection of cubes Q λ ⊂ R n+1 centered at λ with side length 2 j . Then by disjointness of cubes, we see that where Q λ denotes the cube with side length 2 j+2 and the same center as Q λ . By the Young and Cauchy-Schwarz inequalities, it follows that Now we need to bound the terms For the first term, we use the stationary phase lemma, Lemma 3.3, which is essentially due to Littman [13] (see also [19], Chap. VIII). Indeed, by applying the lemma with ψ(ξ) = 1 + |ξ| 2 , it follows that since σ ≥ 0 and rank Hψ = n for each ξ ∈ {ξ ∈ R n : |ξ| ≤ 2}. Thus we get Lemma 3.3. Let Hψ be the Hessian matrix given by ( ∂ 2 ψ ∂ξi∂ξj ). Suppose that ϕ is a compactly supported smooth function on R n and ψ is a smooth function which satisfies rank Hψ ≥ k on the support of ϕ. Then, for (x, t) ∈ R n+1 , For the second term, we have for α < (n + 1)/p, while Similarly for λ1∈2 j Z n+1 Gχ Q λ 1

3.2.
Proof of (2.5). The proof of the high frequency part (2.5) is much more delicate. We generate further localizations in time and space to make (2.5) sharp as far as possible, and need to analyze some relevant oscillatory integrals more carefully (see Lemma 3.6) based on the stationary phase lemma, Lemma 3.3.
By the T T * argument as before, we may show By dividing the integral ∞ −∞ into two parts t −∞ and ∞ t , and then using duality, we are reduced to showing the following bilinear form estimate For this, we first decompose dyadically the left-hand side of (3.12) in time; for fixed j ≥ 1, define intervals I j = [2 j−1 , 2 j ) and I 0 = [0, 1). Then we may write For these dyadic pieces B j , we will obtain the following estimates in the next subsection.
3.2.1. Proof of Proposition 3.4. Now we prove the estimates (3.14) and (3.15) in Proposition 3.4. Compared to the former that is relatively easy to prove, the proof of the latter needs further localizations in space and to carefully analyze some relevant oscillatory integrals under the localization (see Lemma 3.6) based on the stationary phase lemma, Lemma 3.3.
Proof of (3.14). For fixed j ≥ 0, we first decompose R into intervals of length 2 j to have By Hölder's inequality and Plancherel's theorem in x-variable, we then see that Using Hölder's inequality again and then applying the Cauchy-Schwarz inequality in ℓ, we obtain Combining (3.16), (3.17) and (3.18) yields the first estimate (3.14).