there does not interact well with mixed $L^p$-spaces and obtains estimates in non-$L^p$ spaces. The other reason is that has a specific invariance, see below, that is not taken into account in the abstract setting.

4\. We devise a number of counterexamples to show sharpness to our results. We present a new approach to treat the various endpoint estimates that contain $L^\infty$-norms by considering them over Besicovitch sets, thus including and extending the counterexamples of Guo and Peng [@GP].

In the final section of this paper we consider the Othmer-Dunbar-Alt kinetic model of chemotaxis - and prove the existence of global weak solutions for small data in dimensions $n \geq 3$ by making use of the Strichartz estimates of theorem \[thm: Strich admiss\].

Basic Facts and Properties of the KT equation
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The KT equation is a very interesting model for studying Strichartz estimates. First, the kinetic transport propagator $U(t)$ has a simple explicit form $$U(t)f = f(x-tv, v).$$ Second, the homogeneous equation [^2] is invariant to the transformation $$\label{power transform}
f \rightarrow f^\alpha, \quad U(t)f \rightarrow (U(t)f)^\alpha,$$ which allows us to derive from a known Strichartz estimate a new estimate with different exponents. The exponents transform according to the 