Pome370C4F1C4AF0Lemma 8.11.1.370C4F1C4AF0 Fix $K\Subset U$ and $M<\infty $. For smooth local $m$-subharmonic $f_1,\ldots ,f_ q$ with $|f_ i|\leq M$, $0\leq q\leq m$, their mixed trace has mass at most a constant depending only on $K,U,\omega ,m,M$. If also $f$ is smooth in this class and $|f|\leq M$, then, for $q\leq m-1$,
Proof. For one potential and a nonnegative compactly supported cutoff $\chi $, closedness of $\mathrm{dd}^{\mathrm{c}}f$ gives
For $q\geq 1$, the absolute-form estimate 8.2.7 applies at degree $q-1<m$. Nested cutoffs and induction, starting with volume, therefore bound the left side by $C M^ q$; replacing $M$ by $1+M$ includes $M=0$. For mixed factors, positivity of every term in the expansion of $(\sum _ i\mathrm{dd}^{\mathrm{c}}f_ i)^ q\omega ^{n-q}$ bounds the mixed trace by a fixed multiple of the single trace for $\sum _ i f_ i$. This proves the first assertion, including its degree-zero convention.
Replace $f$ by $f+M$, so that $0\leq f\leq 2M$ without changing its gradient or Hessian. The identity
also shows that $f^2$ belongs to the local cone. Wedge with the remaining $q$ Hessians and $\omega ^{n-q-1}$. The first term on the right is nonnegative, and the mixed mass estimate applied to $f^2$ and the other potentials bounds the second term. The degree used is $q+1\leq m$. A cutoff equal to one on $K$ gives the stated local estimate. $\square$
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