PomeLemma 8.7.2.21F540C3F7D1 If $u\leq v$ are continuous admissible functions and the remaining $m-1$ factors are fixed continuous admissible functions, then on $\{ u=v\} $ the mixed measure with first factor $\omega +\mathrm{dd}^{\mathrm{c}}u$ is at most the one with first factor $\omega +\mathrm{dd}^{\mathrm{c}}v$. In particular $\mathbf1_{\{ u=v\} }H_ m(u)\leq \mathbf1_{\{ u=v\} }H_ m(v)$.
Lemma 8.7.221F540C3F7D1
Proof. Write $M(w)$ for the mixed measure with first potential $w$. Positivity and locality give, for $t>0$,
\[ M(\max (v,u+t))\geq \mathbf1_{\{ v>u+t\} }M(v)+\mathbf1_{\{ u\geq v\} }M(u). \]
The first indicators increase to $\mathbf1_{\{ v>u\} }$ as $t\downarrow 0$, and the potentials on the left converge uniformly to $v$. Integrate against nonnegative continuous tests and pass to the limit. Subtract the first restricted measure to obtain the claimed contact inequality. Replacing one factor at a time proves the pure-power claim and the analogous contact inequalities for all mixed powers. $\square$
Comments
Corrections, questions, and mathematical discussion are welcome. Sign in with GitHub to comment. Comments are public.