n / {\tilde r}$ is equivalent to $a'/ p' < r^*(a') / \tilde r = 1$, i.e. to $p < a$. But if the last inequality is violated, i.e. $p=a$, then $p=r$ and $q=\infty$, contradiction. Similarly, $(n-1)/\tilde p' < n / {r}$ is congruent with $a/\tilde p' < r^*(a) /r$. For the left hand side we have that $a/\tilde p' <1$, while for the right hand side $r^*(a)/r >1$. Thus, we are left only with the case when either $q$ or $\tilde q$ is equal to infinity. Suppose, for example, that $q=\infty$. Then $r=p=a$, $\tilde q'=a$ and hence estimate implies the desired claim. Finally, the case when $\tilde q=\infty$ and $\tript$ is endpoint is only possible if $\tilde q= \tilde r=\tilde p = \infty$. Then $\tilde p'=1$ and estimate implies the desired claim.

Estimates for mixed $L^p$-data {#sect: admiss mix Lp}
------------------------------

We prove here theorem \[thm: Strich admiss mix\], that is estimate $$\begin{aligned}
\normQRP{U(t)f}{q,c}{r}{p} \lesssim \normBC{f}{b}{c} \tag{\ref{eq: Strich hom mix}}.\end{aligned}$$

To that end we recall two results from real interpolation that shall be used in the proof. By $L^p=L^p(A)$ and $L^{p,q}=L^{p,q}(A)$ we denote the $L^p$-space and the Lorentz space, respectively of functions that take their values in a fixed 