Graph-patch extension family: dispersive and L2 slice bounds

The dispersive bound. Since $K(x^{\mathrm{\prime }},t)=\check{\mu }(x^{\mathrm{\prime }},t)$ and $⟨t⟩\le 1+\mathrm{\mid }(x^{\mathrm{\prime }},t)\mathrm{\mid }$, [F1] gives $\mathrm{\mid }K(x^{\mathrm{\prime }},t)\mathrm{\mid }\le C_a(1+\mathrm{\mid }(x^{\mathrm{\prime }},t)\mathrm{\mid }{)}^{-(n-1)\mathrm{/}2}\le C_a⟨t{⟩}^{-(n-1)\mathrm{/}2}$. Substituting this bound into the convolution of [F2], $\mathrm{\mid }U(t)g(x^{\mathrm{\prime }})\mathrm{\mid }\le \int \mathrm{\mid }K(x^{\mathrm{\prime }}-y^{\mathrm{\prime }},t)\mathrm{\mid }\mathrm{\mid }g(y^{\mathrm{\prime }})\mathrm{\mid }d{y}^{\mathrm{\prime }}\le C_a⟨t{⟩}^{-(n-1)\mathrm{/}2}\mathrm{\parallel }g{\mathrm{\parallel }}_1\mathrm{.}$ This proves (i) together with the absolute convergence of the defining integral.

The multiplier form. For $g\in \mathcal{S}({\mathbb{R}}^{n-1})$ insert the definition of $K$ into the convolution and apply Fubini (the absolute double integral is $\mathrm{\parallel }{F}_t{\mathrm{\parallel }}_1\mathrm{\parallel }g{\mathrm{\parallel }}_1<\mathrm{\infty }$): $U(t)g(x^{\mathrm{\prime }})=\int \text{\hspace{0.17em}⁣}\text{\hspace{0.17em}⁣}\int e^{2\pi i((x^{\mathrm{\prime }}-y^{\mathrm{\prime }})\cdot \eta +th(\eta ))}a(\eta )\sqrt{1+\mathrm{\mid }\mathrm{\nabla }h(\eta ){\mathrm{\mid }}^2}g(y^{\mathrm{\prime }})d\eta d{y}^{\mathrm{\prime }}=\int e^{2\pi i{x}^{\mathrm{\prime }}\cdot \eta }{F}_t(\eta )\widehat{g}(\eta )d\eta ,$ where $F_t(\eta )=e^{2\pi ith(\eta )}a(\eta )\sqrt{1+\mathrm{\mid }\mathrm{\nabla }h(\eta ){\mathrm{\mid }}^2}$ is supported on $\mathrm{supp}a$ and $\mathrm{\mid }\mathrm{\nabla }h\mathrm{\mid }$ is bounded there. Thus $U(t)g$ is the everywhere-defined inverse transform of the Schwartz function $F_t\widehat{g}$, so $\widehat{U(t)g}=F_t\widehat{g}$; equivalently, the kernel transform identity $\widehat{K(\cdot ,t)}=F_t$ holds pointwise in the Schwartz sense: $F_t$, extended by zero outside $U$, is smooth with compact support, so $K(\cdot ,t)={\check{F}}_t$ is Schwartz for each fixed $t$. By the Plancherel isometry [F3], $\mathrm{\parallel }U(t)g{\mathrm{\parallel }}_2=\mathrm{\parallel }{F}_t\widehat{g}{\mathrm{\parallel }}_2\le \mathrm{\parallel }{F}_t{\mathrm{\parallel }}_{\mathrm{\infty }}\mathrm{\parallel }\widehat{g}{\mathrm{\parallel }}_2=\mathrm{\parallel }{F}_t{\mathrm{\parallel }}_{\mathrm{\infty }}\mathrm{\parallel }g{\mathrm{\parallel }}_2\le B_a\mathrm{\parallel }g{\mathrm{\parallel }}_2$ with $B_a:=\mathrm{\parallel }{F}_0{\mathrm{\parallel }}_{\mathrm{\infty }}<\mathrm{\infty }$, uniformly in $t$ because $\mathrm{\mid }{e}^{2\pi ith}\mathrm{\mid }=1$; enlarge the constant $C_a$ from step 1.1 to be at least $B_a$. This is (ii).

Interpolation. For a finite simple $g$ of finite-measure support, $g\in L^1\cap L^2$. Its convolution equals the Plancherel multiplier: choose smooth compactly supported approximants converging in both $L^1$ and $L^2$ (truncate the support to balls and mollify; the density proof applies in both norms). The convolution converges uniformly by the bounded kernel, while the multipliers converge in $L^2$ by Plancherel, so their limits agree almost everywhere. Thus $U(t)$ defines a compatible complex-linear operator on the finite simple core and satisfies the $L^1\to L^{\mathrm{\infty }}$ bound $A(t)=C_a⟨t{⟩}^{-(n-1)\mathrm{/}2}$ by step 1.1 and the $L^2\to L^2$ bound $B=C_a$ by step 2.1. Applying the Riesz–Thorin corollary [F4] with $1<p<2$ gives $\mathrm{\parallel }U(t)g{\mathrm{\parallel }}_{p^{\mathrm{\prime }}}\le A(t{)}^{2\mathrm{/}p-1}{C}_a^{2-2\mathrm{/}p}\mathrm{\parallel }g{\mathrm{\parallel }}_p=C_a^{2\mathrm{/}p-1+2-2\mathrm{/}p}⟨t{⟩}^{-\frac{n-1}{2}(2\mathrm{/}p-1)}\mathrm{\parallel }g{\mathrm{\parallel }}_p=C_a⟨t{⟩}^{-(n-1)(1\mathrm{/}p-1\mathrm{/}2)}\mathrm{\parallel }g{\mathrm{\parallel }}_p,$ because $2\mathrm{/}p-1=2(1\mathrm{/}p-1\mathrm{/}2)$; the endpoint cases $p=1$ and $p=2$ are the bounds of steps 1.1 and 2.1. The unique compatible bounded extensions to the full $L^p$ spaces exist by [F4]. This proves (iii).

Conclusion. Step 1.1 gives the kernel decay and the $L^1\to L^{\mathrm{\infty }}$ slice bound, step 2.1 identifies the slice operator as the Fourier multiplier by the bounded symbol $F_t$ and gives the uniform $L^2$ bound, and step 3.1 interpolates to the full range $1\le p\le 2$ with the exponent $(n-1)(1\mathrm{/}p-1\mathrm{/}2)$. All constants depend only on $a,h,n$ and not on $t$.