$a=1$ the only admissible triplet is $(\infty, 1, 1)$, and similarly when $a=\infty$ the only admissible triplet is $(\infty, \infty, \infty)$.

We say that the exponent 5-tuple $\qtet$ is *mixed KT-\
admissible* if $$\begin{aligned}
\label{eq: KT admiss mix}
  \rp q + \frac n r = \frac n b,  \quad \operatorname{HM}(p,r)=\operatorname{HM}(b,c) \stackrel{def}{=} a \in [1, \infty],\\
 a \leq r < r^*(a), \quad p^*(a) < p \leq a, \quad p < b<a<c<r,\end{aligned}$$ or if $a=b=c$ and $\qlet$ is KT-admissible.

\[dfn: accept\] We say that the exponent triplet $\trip$ is *KT-acceptable* if $$\begin{aligned}
\label{cond: KT accept}
\rp q < n \left( \rp p - \rp r \right), \quad 1\leq q\leq\infty,\\
1 \leq p < r \leq \infty,\end{aligned}$$ or if $q=\infty$, $1 \leq p=r \leq \infty$.

Note that every KT-admissible triplet is KT-acceptable too. We introduce the following definition

\[dfn: joint accept\] We say that the two KT-acceptable exponent triplets $\trip$ and $\tript$ are [*jointly KT-acceptable*]{} if $$\begin{aligned}
\rp q + \rp {\tilde q} = n\left(1 - \rp r - \rp {\tilde r}\right), \quad \rp q + \rp {\tilde q} \leq 1, \label{cond: joint KT accept}\\
\operatorname{HM}(p,r)=\operatorname{HM}(\tilde p', \tilde r')=a,  \notag\end{aligned}$$ and if further the exponents satisfy the following restrictions

-   $r <\infty$, unless 