then $$\begin{aligned}
    J_{k}& \equiv \int_{Q}^{\infty }s^{\omega _{k}}e^{-s}ds \\
    & \leq \int_{0}^{\infty }s^{\omega _{k}-1}e^{-s}ds=\Gamma (\omega _{k}) \\
    & =\Gamma (\gamma (n-3k)).\end{aligned}$$If $\omega _{k}<0$, i.e. $3k>n$, then $$J_{k}\leq Q^{\omega _{k}-1}\int_{Q}^{\infty }e^{-s}ds\leq Q^{\omega _{k}-1}.$$ Splitting the sum in accordingly, we see that the right hand side of is dominated by $$\begin{gathered}
      \label{eq5.8.1}
    \sum_{0\leq 3k\leq n}\binom{n}{k}(3t)^{-\frac{n-k}{2}}Q^{-\omega _{k}}\Gamma
    \left( \frac{n-k}{2}\right) \Gamma \left( \omega _{k}\right) +\sum_{n<3k\leq
    3n}\binom{n}{k}(3t)^{-\frac{n-k}{2}}\Gamma \left( \frac{n-k}{2}\right) \\
    \equiv S_{1}+S_{2}.\end{gathered}$$

    Analyze now $S_{1}$ and $S_{2}$. For $S_{1}$ we have $$\begin{aligned}
    S_{1}& \leq \Gamma \left( \frac{n}{2}\right) \Gamma (\gamma n)\sum_{0\leq
    3k\leq n}\binom{n}{k}(3t)^{-\frac{n-k}{2}}Q^{-\omega _{k}}  \notag \\
    & \leq \left( Q^{2\gamma }+\frac{1}{\sqrt{3t}Q^{\gamma }}\right) ^{n}\Gamma
    \left( \frac{n}{2}\right) \Gamma (\gamma n).  \label{eq5.9.1}\end{aligned}$$For $S_{2}$ we obtain $$S_{2}\leq \left( 1+\frac{1}{\sqrt{3t}}\right) ^{n}\Gamma \left( \frac{n}{3}\right)$$and hence the contribution from $S_{2}$ to produces a real analytic function. On the other hand, as it easily follows from , the contribution from $S_{1}$ produces a function from $G^{\gamma -1/2}=G^{\frac{1}{2\varepsilon }-1}$. Thus we have proven that if $0<\varepsilon <1/2$ then $$\mathbb{M}_{+}(x,t)=\mathbb{M}_{+}^{\left( 1\right) }(x,t)+\mathbb{M}_{+}^{\left( 2\right) }(x,t)$$where $\mathbb{M}_{+}^{\left( 1\right) }(x,t)$ is a real analytic $\mathfrak{S}_{1}$-valued function and $\mathbb{M}_{+}^{\left( 2\right) }(x,t)$ is a $\mathfrak{S}_{1}$-valued function from the Gevrey class $G^{\frac{1}{2\varepsilon }-1}$. Thus, we can pass to the limit as before. The limiting function is from the Gevrey class $G^{\frac{1}{2\varepsilon }-1}$ if 