18 1. Simple Examples of Propagation one in (t, x) and satisfy 1 |∇ξφ|2 ∂φ ∂ξj C(t + |x|)−1 for (t, x, ξ) Γμ × supp a . The identity eiφ = eiφ implies a(ξ) eiφ = a(ξ) N eiφ . Denote by the transpose of and integrate by parts to find a(ξ) eiφ = ( )Na(ξ) eiφ . The operator ( )N = |α|≤N cα(t, x, ξ) ∂α ξ with coefficients smooth on a neighborhood of Γμ and homogeneous of degree −N in t, x. Therefore, |cα(t, x)| C(α)(1 + t + |x|)−N for (t, x, ξ) Γμ × supp a . It follows that a(ξ) eiφ C (1 + t + |x|)−N . Since the t, x derivatives of this integral are again integrals of the same form, this suffices to prove the proposition. Example 1.3.1. Introduce for 0 μ 1, ˜ μ := Rd \ an open set slightly larger than V. For t virtually all the energy of a solution is contained in the cone (t, x) : x/t ˜ . This is particularly interesting when are supported in a small neighborhood of a fixed ξ. For large times virtually all the energy is localized in a small conic neighborhood of the pair of lines x = −t ∇ξτ±(ξ) that travel with the group velocities associated to ξ. The integration by parts method introduced in this proof is very impor- tant. The next estimate for nonstationary oscillatory integrals is a straight- forward application. The fact that the estimate is uniform in the phases is useful. Lemma of Nonstationary Phase 1.3.2. Suppose that Ω is a bounded open subset of Rd, m N, and C1 1. Then there is a constant C2 0 so that for all f Cm(Ω) 0 and φ Cm(Ω R) satisfying ∀|α| m , ∂αφ L∞ C1 , and ∀x Ω , C−1 1 |∇xφ| C1 ,
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