Filtered spectral generators have summable local shells

For each finite nonempty $Y\subseteq {\mathrm{\Lambda }}_n$, set $B_{Y,n}(s)={\int }_{\mathbb{R}}{W}_{\gamma }(t){\tau }_t^{{\mathrm{\Lambda }}_n,{\mathrm{\Phi }}_s}({\mathrm{\Phi }}_s^{\mathrm{\prime }}(Y))dt\mathrm{.}$ For fixed $n,Y,t$, the finite matrix integrand is continuous in $s$, and it is bounded by ${\sup }_s\mathrm{\parallel }{\mathrm{\Phi }}_s^{\mathrm{\prime }}(Y)\mathrm{\parallel }\mathrm{\mid }{W}_{\gamma }(t)\mathrm{\mid }$, an integrable majorant. Dominated convergence makes $B_{Y,n}(s)$ norm-continuous. Linearity of the finite sum and [F2] give $D_{{\mathrm{\Lambda }}_n}(s)={\sum }_{\mathrm{\varnothing }\mathrm{\ne }Y\subseteq {\mathrm{\Lambda }}_n}{B}_{Y,n}(s)$. Define $Y_k=\{x:d_{\mathrm{\infty }}(x,Y)\le k\}$, $E_{Y_k}^{(n)}:=E_{Y_k\cap {\mathrm{\Lambda }}_n}$, and shells ${\mathrm{\Delta }}_{Y,n}^0=E_{Y_0}^{(n)}(B_{Y,n}),{\mathrm{\Delta }}_{Y,n}^k=E_{Y_k}^{(n)}(B_{Y,n})-E_{Y_{k-1}}^{(n)}(B_{Y,n})(k\ge 1)\mathrm{.}$ For all sufficiently large $k$, $Y_k\cap {\mathrm{\Lambda }}_n={\mathrm{\Lambda }}_n$, so the sum of these shells is $B_{Y,n}$. Each shell is self-adjoint and supported in $Y_k\cap {\mathrm{\Lambda }}_n$. Grouping the finitely many shells by this support defines ${\mathrm{\Psi }}_{{\mathrm{\Lambda }}_n}(Z,s)$ and proves the displayed decomposition. Each expectation is a fixed linear map, so the shell and grouped coefficients are norm-continuous in $s$.

Let $a>0$ be the weight parameter and $v=2C_{F_a}{\sup }_s\mathrm{\parallel }{\mathrm{\Phi }}_s{\mathrm{\parallel }}_{F_a}$, with $V=1+v$. For $z\in {\mathrm{\Lambda }}_n\setminus Y_k$, the supports $Y$ and $\{z\}$ are disjoint; [F3] and the local-expectation estimate [F4] give $\parallel {\tau }_t^{{\mathrm{\Lambda }}_n,{\mathrm{\Phi }}_s}({\mathrm{\Phi }}_s^{\mathrm{\prime }}(Y))-E_{Y_k}^{(n)}{\tau }_t^{{\mathrm{\Lambda }}_n,{\mathrm{\Phi }}_s}({\mathrm{\Phi }}_s^{\mathrm{\prime }}(Y))\parallel \le C\mathrm{\mid }Y\mathrm{\mid }\mathrm{\parallel }{\mathrm{\Phi }}_s^{\mathrm{\prime }}(Y)\mathrm{\parallel }{e}^{v\mathrm{\mid }t\mathrm{\mid }}{e}^{-ak\mathrm{/}2},$ where the lattice shell count and the exponential factor in $F_a$ give one constant $C$ independent of $Y,n,s,t$. The same difference is at most $2\mathrm{\parallel }{\mathrm{\Phi }}_s^{\mathrm{\prime }}(Y)\mathrm{\parallel }$ by contractivity. Split the filtered integral at $T_k=ak\mathrm{/}(4V)$. On $\mathrm{\mid }t\mathrm{\mid }\le T_k$ the first bound is at most $C\mathrm{\mid }Y\mathrm{\mid }\mathrm{\parallel }{\mathrm{\Phi }}_s^{\mathrm{\prime }}(Y)\mathrm{\parallel }{e}^{-ak\mathrm{/}4}$; on $\mathrm{\mid }t\mathrm{\mid }>T_k$, the Schwartz-tail construction in [F2] gives ${\int }_{\mathrm{\mid }t\mathrm{\mid }>T_k}\mathrm{\mid }{W}_{\gamma }(t)\mathrm{\mid }dt\le C_N(1+k{)}^{-N}$ for each $N$. Since ${\mathrm{\Delta }}_{Y,n}^k$ is a difference of the (k)- and $(k-1)$-expectations, its norm is bounded by the sum of these two expectation errors. Thus, after increasing $C_N$, for every $N$ there is a constant $C_N$ independent of $Y,n,s,k$ such that $\mathrm{\parallel }{\mathrm{\Delta }}_{Y,n}^k(s)\mathrm{\parallel }\le C_N\mathrm{\mid }Y\mathrm{\mid }\mathrm{\parallel }{\mathrm{\Phi }}_s^{\mathrm{\prime }}(Y)\mathrm{\parallel }(1+k{)}^{-N}\mathrm{.}$ For $k=0$, the same estimate with an enlarged $C_N$ follows from $\mathrm{\parallel }{E}_Y(B_{Y,n})\mathrm{\parallel }\le \mathrm{\parallel }{W}_{\gamma }{\mathrm{\parallel }}_1\mathrm{\parallel }{\mathrm{\Phi }}_s^{\mathrm{\prime }}(Y)\mathrm{\parallel }$.

The finite-range path has a uniform range $R$, so $x,y\in Y_k$ implies $d_{\mathrm{\infty }}(x,y)\le R+2k$. Also [F1] at $x=y$ gives ${\sup }_s{\sum }_{Y\ni x}\mathrm{\mid }Y\mathrm{\mid }\mathrm{\parallel }{\mathrm{\Phi }}_s^{\mathrm{\prime }}(Y)\mathrm{\parallel }<\mathrm{\infty }$. Since $x\in Y_k$ forces $Y\cap B_k(x)\mathrm{\ne }\mathrm{\varnothing }$, the sum of $\mathrm{\mid }Y\mathrm{\mid }\mathrm{\parallel }{\mathrm{\Phi }}_s^{\mathrm{\prime }}(Y)\mathrm{\parallel }$ over such $Y$ is at most $C(k+1{)}^d$. For each fixed $s$, summing the shell estimate from step 2.1 over all $Y,k$ with $x,y\in Y_k$ gives ${\sum }_{\begin{array}{c}Z\subseteq {\mathrm{\Lambda }}_n\\ x,y\in Z\end{array}}\mathrm{\parallel }{\mathrm{\Psi }}_{{\mathrm{\Lambda }}_n}(Z,s)\mathrm{\parallel }\le C_N{\sum }_{k\ge (d_{\mathrm{\infty }}(x,y)-R)\mathrm{/}2}(1+k{)}^{d-N}\mathrm{.}$ For any prescribed $\rho >d$, choose $N>\rho +d+1$; the last tail is bounded by $C_{\rho }(1+d_{\mathrm{\infty }}(x,y){)}^{-\rho }$. The bound is independent of $s$ and $n$, so taking their suprema outside the support sum proves the stated uniform $F_{\rho }$-norm bound. No sum of separate time suprema is asserted; all finite-volume sums are finite and their nonnegative shell bounds are dominated by the displayed convergent scalar series.

Fix a finite support $Z$. Once ${\mathrm{\Lambda }}_n$ contains the $(2\mathrm{diam}Z+1)$-neighborhood of $Z$, any shell with support $Y_k\cap {\mathrm{\Lambda }}_n=Z$ has $Y\subseteq Z$ and $k\le \mathrm{diam}Z$; otherwise $Y_k\cap {\mathrm{\Lambda }}_n$ would contain a point in that neighborhood outside $Z$. Thus only finitely many pairs $(Y,k)$ contribute to ${\mathrm{\Psi }}_{{\mathrm{\Lambda }}_n}(Z,s)$, and their shell projections are the fixed $E_{Y_k}$ once $n$ is large. By [A1], [F6], and [F5], for each fixed $s$, ${\tau }_t^{{\mathrm{\Lambda }}_n,{\mathrm{\Phi }}_s}(A)$ converges locally uniformly on compact $t$-intervals. The same boundary estimate in the proof of [F5] is uniform in $s$, because the path has uniform finite range and interaction bounds. For $A={\mathrm{\Phi }}_s^{\mathrm{\prime }}(Y)$, uniformity over $s$ follows by covering the compact finite-dimensional set $\{{\mathrm{\Phi }}_s^{\mathrm{\prime }}(Y):s\in [0,1]\}$ by a finite norm net and using contractivity of the dynamics. The tails $\mathrm{\mid }t\mathrm{\mid }>T$ are bounded uniformly by $2{\sup }_s\mathrm{\parallel }{\mathrm{\Phi }}_s^{\mathrm{\prime }}(Y)\mathrm{\parallel }{\int }_{\mathrm{\mid }t\mathrm{\mid }>T}\mathrm{\mid }{W}_{\gamma }(t)\mathrm{\mid }dt$, which tends to zero by [F2]. Hence every contributing filtered term converges uniformly in $s$, defining ${\mathrm{\Psi }}_{\mathrm{\infty }}(Z,s)$. The $F_{\rho }$-norm bound passes to the limit by taking the limit on each finite set of support terms and then the supremum.