range $[1,\infty]$. Note that when $a \geq (n+1)/n$ this is always so and in this case definition \[dfn: admiss\] gives the range of all possible homogeneous estimates in the $L^p$-norms (up to an endpoint), see theorem \[thm: Strich admiss\]. In view of the power invariance , we can generalize these estimates to any $0<a<\infty$. However, the proof of the inhomogeneous estimates relies on the duality relations of the $L^p$-spaces, for $1\leq p \leq \infty$. This restriction is reflected in the second condition in . We next give an equivalent definition to \[dfn: admiss\], which is more convenient in the context of the inhomogeneous estimates.

\[dfn: admiss 2\] We say that the exponent triplet $\trip$ is *KT-admissible* if $$\begin{aligned}
\label{eq: KT admiss 2}
 \rp q &= \frac n 2 \left(\rp p - \rp r \right), \quad a \stackrel{def}{=} \operatorname{HM}(p,r),\\
 1\leq p,q, r &\leq \infty, \quad a \leq r \leq r^*(a), \quad  p^*(a) \leq p \leq a,\end{aligned}$$ apart from the case $n=1$, $\trip=(a, \infty, a/2)$.

A consequence of the above definition is the fact that if $\trip$ is KT-admissible then $a \leq q \leq \infty$ and $p\leq r$. Triplets of the form $\trip=(a,r^*(a), p^*(a))$, for $(n+1)/n\leq a<\infty$, shall be called endpoint. When 