1.3. Group velocity and the method of nonstationary phase 17 Consider the behavior for large times. The phases φ±(t, x, ξ) := ±ξ t + have gradients ∇ξφ±(t, x, ξ) := ∇ξ ± ξ t + = ±tξ ξ + x = t ±ξ ξ + x t . At space-time points (t, x) with t 1 and ±ξ ξ + x t = 0 , the phase oscillates rapidly and the contribution to the integral is expected to be small. The contribution to the integral from ξ ξ is felt predomi- nantly at points where x/t ∓ξ/ ξ . Setting τ±(ξ) := ±ξ , one has ∓ξ ξ = −∇ξτ±(ξ) . This agrees with the formula for the group velocity (re)introduced on purely geometric grounds in §2.4. For t the contributions of the plane waves a±(ξ)ei(τ±(ξ)t+xξ) with ξ ξ are expected to be felt at points with x/t −∇ξτ±(ξ). A precise version is proved using the method of nonstationary phase. Proposition 1.3.1. Suppose that a±(ξ) S(Rd), and define V := ± v : v = −∇ξτ±(ξ) for some ξ supp to be the closed set of group velocities that appear in the plane wave decom- position of u. For μ 0, let Rd denote the set of points at distance μ from V. Denote by Γμ the cone Γμ := (t, x) : t 0 and x/t . Then for all N 0 and α, (1 + t + |x|)N ∂α t,x u(t, x) L∞(Γμ) . Proof. The solution u is smooth with values in S so one need only consider {t 1}. We estimate the u+ summand. The u− summand is treated similarly. The subscript + is suppressed in u+, φ+, a+ and τ+. Introduce the first order differential operator (1.3.2) (t, x, ∂) := 1 i|∇ξφ|2 j ∂φ ∂ξj ∂ξj , so (t, x, ∂ξ) eiφ = eiφ . The operator is only defined where ∇ξφ = 0. The coefficients are smooth functions on a neighborhood of Γμ, and are homogeneous of degree minus
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