= BV jump-product limit with one bounded factor
{c}
{title2=$\lim_{t\downarrow0}t^{-1}\int\delta_ta\,\delta_tb=\int_{J_b}[a][b]\varphi|\nu_b\cdot e_j|\,d\mathcal H^{n-1}$}
Let $a\in BV(\Omega)$, $b\in BV(\Omega)\cap L^\infty(\Omega)$ and $\delta_tc=c\circ\Phi_t-c$, where $\Phi_t$ is the <local flow> of $\varphi e_j$ with nonnegative $\varphi\in C_c^\infty(\Omega)$. The formula uses common oriented <BV traces on a hypersurface>; the trace difference of $a$ is zero almost everywhere on $J_b\setminus J_a$. It also holds for both negative-time increments with denominator positive $t$.
The <BV slicing theorem> reduces the calculation to one dimension. With a right-continuous representative of $a$ and $\mu=Da$, <Fubini's theorem> yields
$$
\frac1t\int\delta_ta\,\delta_tb=\int\left(\frac1t\int_{\Phi_{-t}(s)}^s\delta_tb(x)\,dx\right)d\mu(s).
$$
The inner factor tends to $\varphi(s)[b](s)$ and is bounded in absolute value by $2\|b\|_\infty\|\varphi\|_\infty$. <Dominated convergence> against $|\mu|$ leaves only its <measure atoms> at the countable slice jumps of $b$. That same bound times $|Da_y|$ permits integration over the transverse coordinates. No essential bound on $a$ is used, even if its slice values grow without bound. The formula is useful for <opposite-flow fidelity identity for quadratic data> with unbounded data and a bounded comparison function.
Back to article page
