\widehat{g}\left( 2\lambda \right)  \label{Gg}$$where $\widehat{g}$ is the Fourier transform of $g$ which due to satisfies Condition 2 of Hypothesis \[hyp1.1\] with some $\widetilde{c}<c$. I.e. $$\int_{x}^{\infty }\left\vert g\right\vert \lesssim _{g}e^{-\widetilde{c}x^{\varepsilon }}.  \label{est_g}$$For any $\lambda \in \mathbb{C}^{+}$ define $$\widetilde{G}\left( \lambda \right) =\sum_{n\geq 1}\theta \left(
r^{\varepsilon +2}x_{n}^{1-\varepsilon }\frac{{\operatorname{Im}}\lambda }{\widetilde{c}}\right) G_{n}(\lambda ),  \label{eq6''.1too}$$where $\theta $ is a smooth on $\mathbb{R}_{+}$ function such that: $$\begin{aligned}
\theta (x)& =1,\;x\in \lbrack 0,1], \\
\theta (x)& =0,\;x\geq r,\end{aligned}$$$r>1$, $x_{n}=r^{n}$ and $$G_{n}(\lambda )=\int_{x_{n-1}}^{x_{n}}e^{-i\lambda x}g(x)dx.$$The formula clearly defines an extension of $\widehat{g}\left( \lambda \right) $ to complex $\lambda $. We next show that $\widetilde{G}$ is uniformly bounded on $\mathbb{C}^{+}$. Bound $G_{n}$ first. By $$\left\vert G_{n}(\lambda )\right\vert \lesssim e^{\left\vert {\operatorname{Im}}\lambda
\right\vert \cdot x_{n}}\int_{x_{n-1}}^{x_{n}}\left\vert g\right\vert
\lesssim _{g}\exp \left\{ \left\vert {\operatorname{Im}}\lambda \right\vert \cdot x_{n}-\widetilde{c}x_{n-1}^{\varepsilon }\right\}$$and one has $$\left\vert \widetilde{G}(\lambda )\right\vert \lesssim _{g}\sum_{n\geq
1}\sum_{n\geq 1}\theta \left( r^{\varepsilon +2}x_{n}^{1-\varepsilon }\frac{{\operatorname{Im}}\lambda }{\widetilde{c}}\right) \exp \left\{ \left\vert {\operatorname{Im}}\lambda
\right\vert \cdot x_{n}-\widetilde{c}x_{n-1}^{\varepsilon }\right\} .
\label{eq6'''.1}$$ In many terms are in fact zero and nontrivial ones are subject to $$r^{\varepsilon +2}x_{n}^{1-\varepsilon }\frac{\left\vert {\operatorname{Im}}\lambda
\right\vert }{\widetilde{c}}\leq r.$$I.e. only nonzero terms in are the ones obeying $$x_{n}^{1-\varepsilon }\leq \frac{\widetilde{c}}{r^{\varepsilon +1}}\cdot 
\frac{1}{\left\vert {\operatorname{Im}}\lambda \right\vert }.  \label{eq6'''.2}$$

Under the condition , for the argument of the exponential in , we have ($1/r<\delta <1$) $$\begin{aligned}
