Given any compact set $K$ in $B\times\mathbb{C}^{n-m}$ and any $k\geqslant 0$ there exists a constant $C$ independent of $t>0$ such that

(4.16)

$$\|p^{*}T_{-\sigma}^{*}\tilde{\omega}_{t}\|_{C^{k}(K,\delta)}\leqslant C,$$

where $\delta$ is the Euclidean metric on $B\times\mathbb{C}^{n-m}$.