Remark 4.19.1 . [052X]

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Remark 4.19.1.

The function $e^{\delta\cdot U_{T}(\bm{x})}$ is the dominating term at large scales on $\mathcal{M}_{T}$ which behaves like an exponential function. The term $U_{T}(\bm{x})$ is defined by (4.275) just for unifying the weighted analysis for different “large scales” on $\mathcal{M}_{T}$ , which will be seen in the proof of Proposition 6.10 in Section 6. For intuition, there are two cases in which $U_{T}$ has simple expressions:

(4.276)

\displaystyle\begin{cases}U_{T}(\bm{x})=-L_{0}(z),&n=2,\\ U_{T}(\bm{x})\approx-\frac{n}{2}\cdot L_{0}(z),&n>2,\ |z(\bm{x})|\ll T.\end{cases}