$$\mathbb{H}=\int_{0}^{h}\mathbb{H}_{\beta ,1}\mathbb{H}_{\beta ,2}d\beta,
\label{eq5.2.3}$$where $\mathbb{H}_{\beta ,1}$ and $\mathbb{H}_{\beta ,2}$ and integral (but not Hankel) operators on $L^{2}\left( \mathbb{R}\right) $ with the kernels $$\begin{aligned}
H_{\beta ,1}(x,s)& =\chi (x)e^{-\beta x}\widehat{F_{\beta }^{1/2}}(x+s), \\
H_{\beta ,2}(s,y)& =\chi (y)e^{-\beta y}\widehat{F_{\beta }^{1/2}}(y-s)\end{aligned}$$respectively.

It follows from that $$\label{eq5.3.1too}
\left\Vert \mathbb{H} \right\Vert_{\mathfrak{S}_1} \le \int_0^h \left\Vert 
\mathbb{H}_{\beta,1} \right\Vert_{\mathfrak{S}_2} \cdot\left\Vert \mathbb{H}_{\beta,2} \right\Vert_{\mathfrak{S}_2}d\beta.$$

Evaluate now the Hilbert-Schmidt norms of $\mathbb{H}_{\beta ,1}$ and $\mathbb{H}_{\beta ,2}$. By the Plancherel equation we have $$\begin{aligned}
\left\Vert \mathbb{H}_{\beta ,1}\right\Vert _{\mathfrak{S_{2}}}^{2}&
=\int_{0}^{\infty }\int_{-\infty }^{\infty }\left\vert H_{\beta
,1}(x,s)\right\vert ^{2}ds\;dx \\
& =\int_{0}^{\infty }dxe^{-2\beta x}\int_{-\infty }^{\infty }ds\left\vert 
\widehat{F_{\beta }^{1/2}}(x+s)\right\vert ^{2} \\
& =\frac{\left\Vert F_{\beta }^{1/2}\right\Vert _{L^{2}\left( \mathbb{R}\right) }^{2}}{2\beta }=\frac{1}{2\beta }\left\Vert F_{\beta }\right\Vert
_{L^{1}\left( \mathbb{R}\right) }.\end{aligned}$$That is $$\left\Vert H_{\beta ,1}\right\Vert _{\mathfrak{S}_{2}}\leq \frac{1}{\sqrt{2\beta }}\left\Vert F_{\beta }\right\Vert _{L^{1}\left( \mathbb{R}\right)
}^{1/2}.$$Similarly, $$\left\Vert H_{\beta ,2}\right\Vert _{\mathfrak{S}_{2}}\leq \frac{1}{\sqrt{2\beta }}\left\Vert F_{\beta }\right\Vert _{L^{1}\left( \mathbb{R}\right)
}^{1/2}$$and yields $$\left\Vert \mathbb{H}\right\Vert _{\mathfrak{S}_{1}}\leq \frac{1}{2}\int_{0}^{h}\left\Vert F_{\beta }\right\Vert _{L^{1}}\frac{d\beta }{\beta }.$$The lemma is proven.

Let us find suitable bounds on $\mathbb{R}\times \lbrack 0,\varkappa /2]$ for the functions involved in $F$: $$\begin{aligned}
\left\vert \zeta _{z,t}(\lambda )\right\vert & =\left\vert e^{8i\lambda
^{3}t+2i\lambda z}\right\vert =e^{8\beta ^{3}t-2\beta {\operatorname{Re}}z}\cdot
e^{-24\beta t\alpha ^{2}-2\alpha {\operatorname{Im}}z} \\
& \leq e^{\varkappa (\varkappa ^{2}t+\left\vert z\right\vert )}\cdot \exp
\left\{ -\left( \sqrt{24\beta t}\alpha +\frac{{\operatorname{Im}}z}{\sqrt{24\beta t}}\right) ^{2}+\frac{{\operatorname{Im}}^{2}z}{24\beta t}\right\} , \\
\left\vert \lambda ^{n-1}B^{-1}(\lambda )S(\lambda )\right\vert & 