\notag\end{aligned}$$

In our proofs of the Strichartz estimates below we shall work with estimate and to treat the case of admissible exponents, and with the bilinear formulation (\[eq: B 2\]) in the case of acceptable exponents.

Estimates for admissible exponents
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Estimates for $L^p$-data
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We prove here the sufficient part of theorem \[thm: Strich admiss\]. In particular, we prove separately the homogeneous and the inhomogeneous estimate $$\begin{aligned}
  \normQRP{U(t)f}{q}{r}{p} \lesssim& \normP{f}{a}, \quad 1\leq a<\infty, \tag{\ref{est: Strich admiss}a} \label{est: Strich admiss a}\\
  \normQRP{W(t)F}{q}{r}{p} \lesssim&  \normQRP{F}{\tilde q'}{\tilde r'}{\tilde p'}, \tag{\ref{est: Strich admiss}b}\label{est: Strich admiss b}\end{aligned}$$ respectively, where $\trip$ and $\tript$ are two non-endpoint KT-admissible triplets subject to the scaling condition $a=\operatorname{HM}(p,r)=\operatorname{HM}(\tilde p', \tilde r')$.

It suffices to consider only estimate $$\begin{aligned}
  \normQRP{U(t)f}{q}{r}{r'} \lesssim& \normP{f}{2},\end{aligned}$$ since the general case of will follow by the power invariance . By the TT\*-principle, the last inequality is equivalent to .

In view of the decay estimate $$\begin{aligned}
    \normBC{U(t)f}{r}{r'} \lesssim \rp {{\left\vertt\right\vert}^{\beta(r)}} \normBC{f}{r'}{r}, \quad 2\leq r \leq \infty,
    $$ where $\beta(r) = n(1-2/r)$, cf. corollary \[cor: dec est\], we obtain the following estimate for $TT^*F$ $$\begin{aligned}
    \normBC{TT^*F}{r}{r'} \lesssim \Int \normBC{U(t-s)F(s)}{r}{r'} ds \lesssim \Int \frac {\normBC{F(s)}{r'}{r}} {{\left\vertt-s\right\vert}^{\beta(r)}}  ds.\end{aligned}$$ We take the $L^q$-norm in $t$ and in view of 