rule$$\label{correspondence alpha}
(q, r, p, a) \rightarrow (\alpha q, \alpha r, \alpha p, \alpha a), \quad 0<\alpha<\infty.$$

But on the other hand, the KT equation enjoys a rather complex vector-valued dispersive inequality $$\label{eq: disp est}
  {\left\VertU(t)f\right\Vert}_{L^{\infty}_xL^{1}_v} \lesssim \frac 1 {{\left\vertt\right\vert}^{n}}{\left\Vertf\right\Vert}_{L^{1}_xL^{\infty}_v}.$$ This fact makes the endpoint Strichartz estimate intractable with the current methods, see section \[sect: counterex loc inhom\] for an example in support of that claim.

Let us fix the notation for the remainder of the paper. By $W(t)$ we shall denote the operator $$W(t)F=\intI U(t-\tau)F(\tau)d\tau $$ that gives the solution to the inhomogeneous equation (\[eq: KT\]) that decays as $t \rightarrow -\infty$. In particular, when supp $F \subseteq [0, \infty)$ this coincides with Duhamel’s formula, i.e. $W(t)$ is the operator that solves , with zero initial conditions. The estimates for $U(t)$ have the form $$\label{eq: hom Strich}
 \normQRP{U(t)f}{q}{r}{p} \lesssim \normA{f}{a},$$ where the subscripts indicate integration over the whole space and the shortcut $\spaceLqrp$ stands for $L^q({\mathbb R}; L^r({\mathbb R}^n; L^p({\mathbb R}^n)))$. Note that the global in time estimate (\[eq: hom Strich\]) trivially implies the local estimate $$\label{eq: hom Strich loc}
 \normQRPloc{U(t)f}{q}{r}{p}{[0,T]} \lesssim \normA{f}{a},$$ if one considers the localized operator $U(t)\chi_{[0,T]}$, where $\chi_{[0,T]}$ is the characteristic function of the interval 