estimates (\[eq: inhom Strich\]) is in some sense larger than that of the homogeneous estimates (\[eq: hom Strich\]). As in Foschi [@F], we shall call the exponents for which (\[eq: inhom Strich\]) holds [*acceptable*]{}. The precise definitions are given next.

We denote by $\operatorname{HM}(p,r)$ the harmonic mean of $p$ and $r$, i.e. $a= {\operatorname{HM}(p,r)}$ if and only if $$\rp a = \rp 2 \left( \rp r + \rp p \right).$$

\[dfn: r\*\] Set $$\begin{aligned}
\label{cond: r*}
\begin{cases}
 p^*(a) = \frac {na} {n+1}, \quad r^*(a) =  \frac {na} {n-1},   \quad \text{if} \quad \frac {n+1} n \leq a \leq \infty,\\
 p^*(a) = 1,   \qquad \;     r^*(a) =  \frac {a} {2-a},         \quad \text{if} \qquad  1\leq a\leq \frac {n+1} n.
\end{cases}\end{aligned}$$ When $a=2$ we simply write $p^*=p^*(2)$, $r^*=r^*(2)$.

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

The boundaries $r^*(a)$ and $p^*(a)$ are chosen to ensure that the Lebesgue exponents $a,p,q,r$ remain in the 