the Hardy-Littlewood-Sobolev (HLS) theorem of fractional integration, see [@BS pp. 228-229], [@S], we obtain $$\begin{aligned}
    \normQRP{TT^*F}{q}{r}{r'} \lesssim \normQRP{F}{q'}{r'}{r},\end{aligned}$$ whenever $0<\beta(r)<1$, $1 + 1/q = 1/q' + \beta(r)$. The latter conditions are equivalent to $2 < r < r^*$, $1/q+n/r=n/2$. The left endpoint $r=2$ follows trivially from the transport estimate (\[eq: tran est\]). The right endpoint $r = r^*$ remains an open problem.

The inhomogeneous estimate is a consequence of estimate , the factorization , and the Christ-Kiselev lemma \[lem: Christ-Kiselev\]. Note that the latter is applicable only in the non-endpoint case $1/q+1/{\tilde q} <1$.

To complete the proof of the sufficient part of theorem \[thm: Strich admiss\] we only need to consider the case when $\trip$ is a non-endpoint admissible triplet while $\tript$ is an endpoint admissible triplet. It is easy to check that the two triplets are jointly acceptable with $1/q+1/{\tilde q} <1$. Indeed, the last assertion follows from the fact that $q>a$ while $\tilde q = a'$. Suppose that both $q, \tilde q < \infty$. Then condition is satisfied by the mere construction of the proof of theorem \[thm: infty case\]. We can give a direct proof of that claim as well. For example, the inequality $(n-1)/p' < 