the point $\triptrip \in \Sigma_1 \cup B$, $$\begin{aligned}
      \Sigma_1 &={\left\{\left(\mu,0,\kappa,\nu,1 - \kappa,1\right)\;:\;  0 < \mu, \nu < 1, \; 0 < \kappa < 1/n, \;\mu + \nu =n\kappa\right\}},\\
           B  &=(0,0,0,0,1,1).\end{aligned}$$

-   $\tilde r <\infty$, unless the point $\triptrip \in \Sigma_2 \cup C$, $$\begin{aligned}
      \Sigma_2 &={\left\{\left(\mu,1-\kappa,1,\nu,0,\kappa \right)\;:\;   0 < \mu, \nu < 1, \; 0 < \kappa < 1/n, \; \mu + \nu =n\kappa\right\}},\\
            C &=(0,1,1,0,0,0).\end{aligned}$$

-   if $\tilde q=\infty$, then $a \leq r < r^*(a)$,

-   if $q=\infty$, then $a \leq \tilde r < r^*(a)$.

Main Results {#sect: main res}
============

Throughout this section we suppose that $1 \leq q,r,p,a \leq \infty$.

\[thm: Strich admiss\] Let $u$ be the solution to the IVP for , . Then the estimate $$\label{est: Strich admiss}
  \normQRP{u(t, x, v)}{q}{r}{p} \lesssim \normP{f}{a} + \normQRP{F}{\tilde q'}{\tilde r'}{\tilde p'},$$ holds for all $f \in L^a({\mathbb R}^{2n})$, $F \in L_t^{\tilde q'}L_x^{\tilde r'}L_v^{\tilde p'}$ if and only if $\trip$ and $\tript$ are two KT-admissible exponent triplets and $a=\operatorname{HM}(p,r)=\operatorname{HM}(\tilde p', \tilde r')$, apart from the case when $n>1$ and $\trip$ being an endpoint triplet, which remains open.

\[rem: admiss endpoint\] Note that in theorem \[thm: Strich admiss\] the KT-admissible exponent triplet $\tript$ is allowed to be endpoint in $n>1$ as 