Equation 15.18.27E603AE0642F8

\begin{align} \lVert u_ j-u\rVert _\infty & \leq Cj^{-a}, \\ \sup _{\overline B}\operatorname {tr}_\beta (\vartheta +\mathrm{dd}^{\mathrm{c}}u_ j)& \leq Cj^ b, \tag{15.18.28}\\ \sup _ j\sup _ K\operatorname {tr}_\beta (\vartheta +\mathrm{dd}^{\mathrm{c}}u_ j)& <\infty \quad \text{for every }K\Subset \overline B\setminus \mathcal S. \tag{15.18.29} \end{align}