Lemma 2.8.4CAAD90CD1576

Proof. Suppose first that $F$ is continuously differentiable near $Q$. For $x,y\in Q$, the points $z^{(k)}=(y_1,\ldots ,y_ k,x_{k+1},\ldots ,x_ d)$, $0\leq k\leq d$, lead from $x$ to $y$ by changing one coordinate at a time. If $I_ k$ is the $k$th side interval of $Q$, the fundamental theorem of calculus gives

Integrate over $(x,y)\in Q\times Q$. The $k$th term does not depend on the $d+1$ coordinates $x_1,\ldots ,x_ k,y_ k,\ldots ,y_ d$, which contribute a factor $l^{d+1}$, and integration over the remaining coordinates and $t$ gives $\int _ Q|\partial _ kF|\, \mathrm dx$. Since $|F(x)-F_ Q|\leq |Q|^{-1}\int _ Q|F(x)-F(y)|\, \mathrm dy$ and $|Q|=l^ d$, the inequality follows. In general, the mollifications $F_\varepsilon $ are smooth on $V_\varepsilon =\{ x\in V:\operatorname {dist}(x,\mathbb R^ d\setminus V)>\varepsilon \} $, which is a neighborhood of $Q$ for small $\varepsilon $. By lemma 2.5.1(2) and proposition 2.5.3(1), $F_\varepsilon \to F$ and $\partial _ kF_\varepsilon =(\partial _ kF)_\varepsilon \to \partial _ kF$ in $L^1(Q)$. Since $|(F_\varepsilon )_ Q-F_ Q|\leq |Q|^{-1}\| F_\varepsilon -F\| _{L^1(Q)}$, the left sides of the inequality for $F_\varepsilon $ and for $F$ differ by at most $2\| F_\varepsilon -F\| _{L^1(Q)}$, and the inequality for $F_\varepsilon $ passes to the limit. $\square$