Let $G^{\alpha}_k$ with $\alpha\in\Bbb C$ et $k\in\Bbb N^*$ be analytic family of operators such that: For $\tau\in\Bbb R$
$$\|G^{i\tau}_k f\|_{\infty}\leq C(1+|\tau|)^{\frac{1}{2}}\|f\|_1 $$
and
$$\|G^{1+i\tau}_k f\|_{2}\leq C(1+|\tau|)^{n}k^{-n}\|f\|_2$$
So by Stein's interpolation theorem , we have for $\theta\in [0,1]$ $$\|G^{\theta}_k f\|_{\frac{2n}{n-1}}\leq C\|f\|_{\frac{2n}{n+1} } $$ and hence for $\theta=\frac{2n}{n-1}$, we get $$\|G^{\frac{n-1}{n}}_k f\|_{\frac{2n}{n-1}}\leq C\|f\|_{\frac{2n}{n+1} }$$
Now put $T_k f=\frac{\Gamma(k+n)\Gamma(n)}{\Gamma(k+1)}G^{\frac{n-1}{n}}_k f$.
I have read in a paper the following : as $\frac{\Gamma(k+n)\Gamma(n)}{\Gamma(k+1)}<C k^{n-1}$ then $$\|T_k f\|_{\frac{2n}{n-1}}\leq C_1\|f\|_{\frac{2n}{n+1} }$$
My question why $C_1$ is independent of $k$.
Thanks in advance