Perthame [@CP] in 1996. The following year appeared the seminal paper by Keel and Tao [@KT] where they introduce some novel techniques in the abstract setting. Although they improve the range of validity of the homogeneous Strichartz estimates compared to [@CP], the authors point out the intractability of the endpoint Strichartz estimate with the current methods. The next result appeared in 2007 when Guo and Peng [@GP] demonstrated the expected failure of the endpoint estimate in spatial dimension $n=1$. Interestingly enough, they also showed that the estimate holds if one replaces the $L^\infty$-norm in $x$ by the BMO norm.

In the meantime, several authors extended the results of [@KT] in the inhomogeneous setting, see Foschi [@F], Vilela [@V], Taggart [@Tg], Ovcharov [@O]. We shall apply their methods to the context of the KT equation to derive inhomogeneous Strichartz estimates for non-admissible exponents.

A recent survey article to the field of nonlinear kinetic equations with references to and the estimates in [@CP] can be found in Perthame [@P]. An application of the Strichartz estimates to prove existence of global solutions to a kinetic model of chemotaxis appeared in Bournaveas et al. [@B]. In section \[sect: applic\] we extend their result to 