{\theta}{\xi}, \qquad \rp b = \frac {1-\theta}{ a} + \frac {\theta}{p}.\end{aligned}$$ We can eliminate $\theta$ to obtain that estimate (\[eq: Strich hom mix\]) holds under the conditions $$\begin{aligned}
    \rp q + \frac n r = \frac n b,  \quad \operatorname{HM}(p,r)=\operatorname{HM}(b,c) = a,\\
 a < r < r^*(a), \quad p^*(a) < p < a, \quad p < b<a<c<r,\end{aligned}$$ or if $a=b=c$ and $\qlet$ is an admissible exponent quadruplet (corresponds to $\theta=0$).

We next prove the two cases of theorem \[thm: infty case\] when either $\tilde q=\infty$ or $q=\infty$, that is estimates $$\begin{aligned}
\normQRP{W(t)F}{q, \tilde p'}{r}{p} \lesssim \normQRP{F}{1}{\tilde r'}{\tilde p'}, \tag{\ref{eq: Strich inhom a}}\\
\normQRP{W(t)F}{\infty}{r}{p} \lesssim \normQRP{F}{\tilde q', p}{\tilde r'}{\tilde p'}\tag{\ref{eq: Strich inhom b}},\end{aligned}$$ where $\trip$ and $\tript$ are two jointly acceptable exponent triplets.

The homogeneous estimate (\[eq: Strich hom mix\]) implies the inhomogeneous estimate (\[eq: Strich inhom a\]), see [@O theorem 2.9] for proof. We need only to verify that the range of validity of (\[eq: Strich inhom a\]) is precisely described by the requirement that $\trip$ and $\tript$ be two jointly acceptable exponent triplets. In the table below we give the conditions of a mixed KT-admissible 5-tuple in the second column and we reformulate them as in the definition of 