Hardy–Krause variation anchored at the upper corner

Fix $d\ge 1$ and a real function $f:[0,1{]}^d\to \mathbb{R}$. For each nonempty coordinate subset $u\subseteq \{1,\dots ,d\}$, restrict $f$ to the face adjacent to the all-ones corner by setting $f_u(x_u):=f(z),z_j=\{\begin{array}{ll}{x}_j,& j\in u,\\ 1,& j\mathrm{\notin }u\mathrm{.}\end{array}$ For a finite coordinate grid on $[0,1{]}^{\mathrm{\mid }u\mathrm{\mid }}$, write its cells as $R_k={\prod }_{j\in u}[t_{j,k_j},t_{j,k_j+1}]$. For a cell with lower and upper vertices $a,b$, define its alternating increment $\mathrm{\Delta }(f_u;R_k):={\sum }_{ϵ\in \{0,1{\}}^{\mathrm{\mid }u\mathrm{\mid }}}(-1{)}^{\mathrm{\mid }u\mathrm{\mid }-\mathrm{\mid }ϵ\mathrm{\mid }}{f}_u(z_{ϵ}),$ where $(z_{ϵ}{)}_j=a_j$ if ${ϵ}_j=0$ and $(z_{ϵ}{)}_j=b_j$ if ${ϵ}_j=1$. The Vitali variation of $f_u$ is $V_V(f_u):={\sup }_{\text{finite coordinate grids on }[0,1{]}^{\mathrm{\mid }u\mathrm{\mid }}}{\sum }_k\mathrm{\mid }\mathrm{\Delta }(f_u;R_k)\mathrm{\mid }\mathrm{.}$ The Hardy–Krause variation anchored at the upper corner is $V_{HK,1}(f):={\sum }_{\mathrm{\varnothing }\mathrm{\ne }u\subseteq \{1,\dots ,d\}}{V}_V(f_u),$ with value $+infty$ if any summand is infinite. For $d=1$ this is the ordinary total variation on $[0,1]$. The anchor is part of the convention: every coordinate outside $u$ is fixed at $1$. The smooth mixed-derivative formula for this convention is proved separately for $C^d$ functions.

The same Vitali-variation definition applies on a nondegenerate rectangle $R={\prod }_{j=1}^q[a_j,b_j]$: each coordinate grid starts at $a_j$ and ends at $b_j$. For a subrectangle $R\subseteq [0,1{]}^d$, write $V_{HK,1}(f;R)$ for the sum of the Vitali variations of its face restrictions with complementary coordinates fixed at the upper endpoints of $R$. If a varied side has length zero, that face's Vitali variation is defined to be zero. For $R=[0,1{]}^d$ this notation is exactly $V_{HK,1}(f)$.