Quadratic free-packet decay

[Exact Fourier normalization and initial packet] Put $g(x)=\int e^{ik\cdot x}a(k)dk$. Differentiation in $x$ multiplies the compact smooth integrand by powers of $ik$, and integration by parts in $k$ turns multiplication by powers of $x$ into derivatives of that compact smooth integrand. Hence every $x^{\alpha }{\mathrm{\partial }}_x^{\beta }g$ is bounded, so $g$ is Schwartz and in $L^1\cap L^2$. More explicitly $(1+\mathrm{\mid }x{\mathrm{\mid }}^2{)}^{2}g(x)=\int e^{ik\cdot x}(1-{\mathrm{\Delta }}_k{)}^{2}a(k)dk$, so $\mathrm{\parallel }g{\mathrm{\parallel }}_1\le \mathrm{\parallel }(1-{\mathrm{\Delta }}_k{)}^{2}a{\mathrm{\parallel }}_1\int (1+\mathrm{\mid }x{\mathrm{\mid }}^2{)}^{-2}dx<\mathrm{\infty }$. Substituting $k=2\pi \xi$ into the defining integral and Fourier inversion gives ${\mathcal{F}}_{2}g(\xi )=(2\pi {)}^{3}a(2\pi \xi )$. Thus the inverse Fourier transform of $e^{-4{\pi }^{2}it\mathrm{\mid }\xi {\mathrm{\mid }}^2}{\mathcal{F}}_{2}g$ is exactly ${\psi }_t$, with no additional $2\pi$ in its phase.

[Uniform reciprocal-gradient coefficients] Suppose $a\mathrm{\ne }0$ and $K^{\mathrm{\prime }}\mathrm{\ne }\mathrm{\varnothing }$; the zero amplitude or empty velocity set gives the assertion immediately. Since the two compact sets $K^{\mathrm{\prime }}$ and $2\mathrm{supp}a$ are disjoint, $d:=\inf \{\mathrm{\mid }v-2k\mathrm{\mid }:v\in K^{\mathrm{\prime }},k\in \mathrm{supp}a\}>0$. Choose a fixed bounded open neighborhood $O$ of $\mathrm{supp}a$ small enough that $\mathrm{\mid }v-2k\mathrm{\mid }\ge d\mathrm{/}2$ for $v\in K^{\mathrm{\prime }}$ and $k\in O$. Let ${\varphi }_v(k)=k\cdot v-\mathrm{\mid }k{\mathrm{\mid }}^2$, $c_v(k)=(v-2k)\mathrm{/}\mathrm{\mid }v-2k{\mathrm{\mid }}^2$, and $B_{v}f=-{\mathrm{div}}_k(c_{v}f)$ on $O$. All $k$ derivatives of $c_v$ are uniformly bounded on $K^{\mathrm{\prime }}\times O$: its numerator is affine in $k$, its denominator is bounded away from zero, and $v,k$ range over bounded sets. Iterated product differentiation therefore gives, for every nonnegative integer $N$, $\mathrm{\parallel }{B}_v^{N}a{\mathrm{\parallel }}_1\le C_{K^{\mathrm{\prime }},N}<\mathrm{\infty }$, uniformly in $v\in K^{\mathrm{\prime }}$. More explicitly $B_v^{N}a$ is a finite sum of derivatives ${\mathrm{\partial }}^{\alpha }a$, $\mathrm{\mid }\alpha \mathrm{\mid }\le N$, multiplied by finite products of derivatives of $c_v$ of order at most $N$, with uniform coefficient bounds. Its support remains inside $\mathrm{supp}a$, so it has no boundary contribution on $O$.

[Global Gaussian estimate for both time signs] By [F1] and step 1.1, ${\psi }_t(x)=(4\pi it{)}^{-3\mathrm{/}2}\int e^{i\mathrm{\mid }x-y{\mathrm{\mid }}^2\mathrm{/}(4t)}g(y)dy$ almost everywhere. Both this representative and the compact-momentum defining integral are continuous in $x$ by dominated convergence, so equality holds everywhere. Taking absolute values gives ${\sup }_x\mathrm{\mid }{\psi }_t(x)\mathrm{\mid }\le (4\pi {)}^{-3\mathrm{/}2}\mathrm{\parallel }g{\mathrm{\parallel }}_1\mathrm{\mid }t{\mathrm{\mid }}^{-3\mathrm{/}2}$, for $t>0$ and $t<0$. In particular the first assertion holds with $C_K=(4\pi {)}^{-3\mathrm{/}2}\mathrm{\parallel }g{\mathrm{\parallel }}_1$. This argument proves exactly the promised estimate on the stated $K$; it needs no parameter-dependent stationary-phase constants.

[All-order nonstationary integration by parts] The operator $L_v=c_v\cdot {\mathrm{\nabla }}_k$ satisfies $L_v{e}^{it{\varphi }_v}=it{e}^{it{\varphi }_v}$ for every real nonzero $t$. Repeated integration by parts on the compact amplitude, using the formal adjoint $B_v$, gives ${\psi }_t(tv)=(it{)}^{-N}{\int }_O{e}^{it{\varphi }_v(k)}{B}_v^{N}a(k)dk$ for every $N\ge 0$. Step 1.2 yields ${\sup }_{v\in K^{\mathrm{\prime }}}\mathrm{\mid }{\psi }_t(tv)\mathrm{\mid }\le C_{K^{\mathrm{\prime }},N}\mathrm{\mid }t{\mathrm{\mid }}^{-N}$, independently of the sign of $t$. If the Statement's “every integer $N$” includes $N<0$, the elementary bound $\mathrm{\mid }{\psi }_t(tv)\mathrm{\mid }\le \mathrm{\parallel }a{\mathrm{\parallel }}_1\le \mathrm{\parallel }a{\mathrm{\parallel }}_1\mathrm{\mid }t{\mathrm{\mid }}^{-N}$ for $\mathrm{\mid }t\mathrm{\mid }\ge 1$ supplies those cases as well. Combining this with step 2.1 proves both unchanged public estimates.