$[0,T]$. The last assertion is due to the fact that both $U(t)$ and $U(t)\chi_{[0,T]}$ satisfy the same basic assumptions. We shall also consider estimates for $U(t)$ for mixed $L^p$-data. $$\label{eq: hom Strich mix}
 \normQRP{U(t)f}{q}{r}{p} \lesssim \normBC{f}{b}{c}.$$

The inhomogeneous estimates have the form $$\label{eq: inhom Strich}
 \normQRP{W(t)F}{q}{r}{p} \lesssim \normQRP{F}{\tilde q'}{\tilde r'}{\tilde p'},$$ for some exponent triplets $\trip$ and $\tript$, the precise range of which shall be subject of theorem \[thm: infty case\]. The “$\lesssim$” sign means less or equal up to a constant that does not depend on the function $f$ or $F$ in the estimates (\[eq: hom Strich\]), (\[eq: inhom Strich\]), etc. We shall say that two non-negative quantities $a$ and $b$ are comparable, and write $a\sim b$, if $\exists C>0$, $C$=const: $a \leq Cb$ and $b \leq Ca$. We shall say that two functions $f$ and $g$, lying in the same normed space with norm ${\left\Vert\cdot\right\Vert}$ are approximately equal and write $f \approx g$, if ${\left\Vertf-g\right\Vert}$ is small enough.

Some definitions
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Following Keel and Tao [@KT], we shall call the quadruplet of exponents that appear in estimate (\[eq: hom Strich\]) [*admissible*]{} if (\[eq: hom Strich\]) holds for all $f \in L^a$. The range of validity of the inhomogeneous 