in lemma}
  \begin{split}
   \left(\kappa,0,\mu,\nu,1 - \mu,1\right), \qquad 0 \leq \kappa, \mu, \nu \leq 1, \qquad\\
   \left(\kappa,1-\mu,1,\nu,0,\mu \right),  \qquad 0 \leq \kappa, \mu, \nu \leq 1.\qquad
  \end{split}$$

We apply the Riesz-Thorin convexity theorem to interpolate between the already proven local estimates. In essence, we find the convex hull of the locally acceptable sets associated with lemmas \[lem: local estim 1\] and \[lem: local estim 2\] and then expand that set by the rule given in lemma \[lem: local estim 3\].

The set of acceptability $S_1$ of the local estimates in lemma \[lem: local estim 1\] is given by the system $$\begin{aligned}
  0 \leq \rp q, \rp {\tilde q}, \rp p, \rp {\tilde p}, \rp r, \rp {\tilde r} \leq 1,\label{sys: 1 line 1}\\
  \frac 1 q = \frac n 2 \left(\rp p - \rp r \right), \quad \frac 1 {\tilde q} = \frac n 2 \left(\rp {\tilde p} - \rp {\tilde r} \right), \label{sys: 1 line 2}\\
  \rp r + \rp p + \rp {\tilde r} + \rp {\tilde p} = 2, \label{sys: 1 line 3}\\
  \frac {n-1}{p} < \frac {n+1}{r}, \qquad \frac {n-1}{\tilde p}  < \frac {n+1}{\tilde r},\label{sys: 1 line 4}\end{aligned}$$ or if $(1/q, 1/r, 1/p, 1/\tilde q, 1/ \tilde 