Fixed-pivot projection has first-order bounded-Lipschitz kernel quadrature error

By [F5] and [F6], $F(\mu )\in {\mathcal{M}}_w(E)$, and [F4] makes $P_{\mathrm{\Delta }}F(\mu )$ well-defined; [F7] gives $F_{\mathrm{\Delta }}(P_{\mathrm{\Delta }}\mu )\in {\mathcal{M}}_w(E)$. Fix a test $\phi$ admissible in [F3] and set $\bar{\phi }(z)=\int \phi (u)P_{\mathrm{\Delta }}(z)(du)$. On each cell this is the linear interpolation of endpoint values; its slope has absolute value at most $1$, so $\bar{\phi }$ is also admissible. The displacement estimate [F4] gives $\int \mathrm{\mid }u-z\mathrm{\mid }{P}_{\mathrm{\Delta }}(z)(du)\le \mathrm{\Delta }\mathrm{/}2$.

For the coagulation term, sample $X\sim P_{\mathrm{\Delta }}(x)$ and $Y\sim P_{\mathrm{\Delta }}(y)$ independently. The grid-field test integrand averaged over $X,Y$ has gain $\mathbb{E}\bar{\phi }(X+Y)$ and losses $\bar{\phi }(x)+\bar{\phi }(y)$; the projected continuous-field integrand has gain $\bar{\phi }(x+y)$ and the same losses. The gain difference is at most $\mathbb{E}(\mathrm{\mid }X-x\mathrm{\mid }+\mathrm{\mid }Y-y\mathrm{\mid })\le \mathrm{\Delta }$. Also $\mathbb{E}\mathrm{\mid }K(X,Y)-K(x,y)\mathrm{\mid }\le L_K\mathrm{\Delta }$, and the event bracket has absolute value at most $3$. Thus the absolute difference of the two coagulation integrals is at most $\frac{1}{2}(3L_K+K_{\ast })\mathrm{\Delta }{m}^2$.

For fragmentation, write $G(x)=\int \bar{\phi }(z)B_x(dz)$. Then $\mathrm{\mid }G(x)\mathrm{\mid }\le D$, $\mathrm{\mid }G(x)-\phi (x)\mathrm{\mid }\le D+1$, and $\mathrm{\mid }G(X)-G(x)\mathrm{\mid }\le L_B\mathrm{\mid }X-x\mathrm{\mid }$ by [F2] and [F3]. Since $\mathbb{E}\phi (X)=\bar{\phi }(x)$, the integrand difference equals $\mathbb{E}\{\beta (X)[G(X)-\phi (X)]\}-\beta (x)[G(x)-\bar{\phi }(x)]=\mathbb{E}[\beta (X)(G(X)-G(x))]+\mathbb{E}[(\beta (X)-\beta (x))G(x)]+\mathbb{E}[(\beta (x)-\beta (X))\phi (X)]$. The three terms are bounded by ${\beta }_{\ast }{L}_B\mathbb{E}\mathrm{\mid }X-x\mathrm{\mid }$, $L_{\beta }D\mathbb{E}\mathrm{\mid }X-x\mathrm{\mid }$, and $L_{\beta }\mathbb{E}\mathrm{\mid }X-x\mathrm{\mid }$, respectively. Using $\mathbb{E}\mathrm{\mid }X-x\mathrm{\mid }\le \mathrm{\Delta }$ proves the displayed bound after integrating over $\mu$.

Adding the coagulation and fragmentation bounds holds for every admissible test in [F3]. Taking their supremum gives the asserted bounded-Lipschitz norm inequality.