Given fixed real parameters $n\geq 2$, $T>10^{3}$, $\delta>0$, $\nu,\mu\in\mathbb{R}$ and $\alpha\in(0,1)$. For each $k\in\mathbb{N}$, the weight function $\rho_{\delta,\nu,\mu}^{(k+\alpha)}$ is defined as follows,

(4.273)

$\displaystyle\rho_{\delta,\nu,\mu}^{(k+\alpha)}(\bm{x})=e^{\delta\cdot U_{T}(\bm{x})}\cdot\mathfrak{s}(\bm{x})^{\nu+k+\alpha}\cdot T^{\mu},$

where $\mathfrak{s}(\bm{x})$ is the regularity scale at $\bm{x}$ given by Proposition 4.18 and

(4.274)		$\displaystyle U_{T}(\bm{x})$	$\displaystyle\equiv T\Big(1-(\frac{L_{T}(\bm{x})}{T})^{\frac{n}{2}}\Big),$
(4.275)		$\displaystyle L_{T}(\bm{x})$	$\displaystyle\equiv L_{T}(z(\bm{x}))=T+L_{0}(z(\bm{x})),$

where the functions $L_{T}$ and $L_{0}$ are defined in (4.12).