be rewritten as follows: if $k \geq
    35$ and $j-k \geq 24$, then $$\label{eq:6}
        {{\left\lVert \tilde{f}_{k,j} \right\rVert}_{\infty}}^2 \leq 2888 \pi^2
        (j+1)^{\frac12} k^{-\frac16}.$$ In consequence, for these values of $k$ and $j$, $$\label{eq:3}
        {{\left\lVert \tilde{f}_{k,j} \right\rVert}_{2}}^2 \leq C (j k^{-\frac16} +
        j^{\frac12} k^{\frac13} ).$$

On the other hand, a classical bound on Laguerre polynomials found in [@Szego:1959uq] yields that, for fixed values of $j-k$, ${{\left\lVert \tilde{f}_{k,j} \right\rVert}_{\infty}}^2 \leq C k^{\frac13}$, hence for all $k \geq 35$ and $j-k < 24$, $$\label{eq:4}
        {{\left\lVert \tilde{f}_{k,j} \right\rVert}_{2}}^2 \leq C (j^{\frac12}k^{\frac13} +
        k^{\frac56}).$$

When $k < 35$, we can use another result in [@Krasikov:2005db] which gives ${{\left\lVert \tilde{f}_{k,j} \right\rVert}_{\infty}}^2      \leq C k^{\frac16}j^{\frac12}$ independently of $j-k$, thus $$\label{eq:5}
        {{\left\lVert \tilde{f}_{k,j} \right\rVert}_{2}}^2 \leq C j.$$ Comparing (\[eq:3\]), (\[eq:4\]) and (\[eq:5\]) we see that when $N$ is large enough, in the sum over $0 \leq j, k        \leq N$, the terms $k \geq 35$, $j-k \geq 24$ dominate and (\[eq:3\]) yields the first inequality.

The second inequality is obtained by doing a similar computation, starting with ${{\left\lVert f_{j,k} \right\rVert}_{\infty}} \leq
    {{\left\lVert \tilde{f}_{j,k} \right\rVert}_{1}}$ and using (\[eq:6\]) to bound $${{\left\lVert \tilde{f}_{j,k} \right\rVert}_{1}}^2 \leq C(j^{\frac32} k^{-\frac16}
        + j^{\frac12} k^{\frac56})$$ when $k \geq 35$ and $j-k \geq 24$.

In the presence 