{\left( t \log \frac{At}{\beta} \right)}^{1/\beta}, \quad A \text{ an absolute constant, }\label{eq:xtest}$$ if $t$ is large enough.

If we put into or elementary simplifications give the inequality .

Inequalities and guarantee that we can use the representation and eventually “asymptotics” if $$C_0 M(\alpha) \frac{\log e n}{n^{2\alpha}} \leq \frac{1}{2}, \quad C_0 \in \text{\eqref{eq:wcond}} \label{eq:somebd}$$ and $M(\alpha)$ by is chosen as $$M(\alpha) = \left[   6 + \frac{4/3}{1- 2 \alpha} + \frac{1}{3\alpha} \right]. \label{eq:malphhelp}$$ Then , with $\beta = 2\alpha < 1$, $t = 2 C_0 M(\alpha)$, implies that $N_*$ can be chosen as $$N_* = N_*(C_0; \alpha) = \left[ 2 C_0 M(\alpha) \log \left(\frac{A}{2\alpha} 2 C_0 M(\alpha)  \right) \right]^{1/(2\alpha)} \label{eq:nstarchoice}$$ Now if $\alpha$ is fixed we are interested in the dependence of $N_*$ on $C_0 \in $.

Recall that if $W$ is a multiplier-operator $$Wf = w(x) f(x), \text{ with } w \in L^p({\mathbb{R}}^1), \quad 1 \leq p < \infty, \label{eq:wmultdef}$$ then as we observed and used in [@MiSi] $$w_{jk} = {\left\langle W h_j, h_k \right\rangle} = \int_{-\infty}^{\infty} w(x) h_j(x) h_k(x) \, dx \label{eq:wmatr}$$ so by Hölder inequality $$\vert w_{jk} \vert \leq {\left\Vert w \right\Vert}_p \cdot {\left\Vert h_j \right\Vert}_{2q} {\left\Vert h_k \right\Vert}_{2q}, \quad \frac{1}{p} + \frac{1}{q} = 1. \label{eq:wmatrbd}$$ with $$q > 