---
abstract: 'We study the range of validity of the Strichartz estimates for the kinetic transport equation. We are able to give the full range in spatial dimension $n=1$ while in higher dimensions we leave open some inhomogeneous estimates. Our results extend the previous works by Castella and Perthame [@CP], Keel and Tao [@KT] and Guo and Peng [@GP]. Based on a geometric interpretation of the kinetic transport propagator we construct counterexamples over Besicovitch sets to various endpoint Strichartz estimates containing $L^\infty$-norms. We also give an application to a nonlinear kinetic model of bacterial chemotaxis where we extend the result of Bournaveas et al. [@B].'
address: 'University of Edinburgh, School of Mathematics, JCMB, King’s Buildings, Edinburgh EH9 3JZ, UK'
author:
- Evgeni Ovcharov
bibliography:
- 'KTS-2.bib'
date: 25 March 2009
nocite: '[@HKS]'
title: Strichartz Estimates for the Kinetic Transport Equation
---

[^1]

Introduction
============

Historical Remarks
------------------

In this paper we prove Strichartz estimates for the kinetic transport (KT) equation $$\label{eq: KT}
 {\partial_t u(t,x,v) + v\cdot\nabla_x u(t,x,v)} = F\coor, \quad \coor \in {\mathbb R}\times{\mathbb R}^n\times {\mathbb R}^n,$$ to which we prescribe initial data $$\label{eq: KTf}
u(0,x,v) = f(x,v).$$

Strichartz estimates for the KT equation were first proved by Castella and 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 dimensions $n \geq 3$.

Scope of the paper
------------------

We try to give a full and systematic study of the Strichartz estimates for the KT equation with the known methods to us from the literature. We thus vastly improve the range of the known estimates compared to [@CP], [@KT], in several directions.

1\. We prove all (homogeneous and inhomogeneous) Strichartz estimates for admissible exponents apart from the homogeneous endpoint in $n>1$ and the inhomogeneous double endpoint in $n>1$, see theorem \[thm: Strich admiss\] and the remark thereafter.

2\. We also consider homogeneous estimates for mixed $L^p$-data. For comparison, these estimates correspond to homogeneous estimates for $L^p$-data for the equation. We find their full range in $n=1$ and leave open only some endpoint estimates in higher dimensions.

3\. We apply the techniques of Foschi [@F] to study the inhomogeneous estimates for non-admissible exponents. We find the full range in $n=1$ and leave open some estimates in higher dimensions. An analogous set of estimates is unresolved in the context of the equation too. Note that our result is not contained in the abstract estimates of Taggart [@Tg]. The reason for that is twofold. One is that the real method of interpolation used 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
---------------------------------------------

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 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 $[0,T]$. 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
----------------

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 estimates (\[eq: inhom Strich\]) is in some sense larger than that of the homogeneous estimates (\[eq: hom Strich\]). As in Foschi [@F], we shall call the exponents for which (\[eq: inhom Strich\]) holds [*acceptable*]{}. The precise definitions are given next.

We denote by $\operatorname{HM}(p,r)$ the harmonic mean of $p$ and $r$, i.e. $a= {\operatorname{HM}(p,r)}$ if and only if $$\rp a = \rp 2 \left( \rp r + \rp p \right).$$

\[dfn: r\*\] Set $$\begin{aligned}
\label{cond: r*}
\begin{cases}
 p^*(a) = \frac {na} {n+1}, \quad r^*(a) =  \frac {na} {n-1},   \quad \text{if} \quad \frac {n+1} n \leq a \leq \infty,\\
 p^*(a) = 1,   \qquad \;     r^*(a) =  \frac {a} {2-a},         \quad \text{if} \qquad  1\leq a\leq \frac {n+1} n.
\end{cases}\end{aligned}$$ When $a=2$ we simply write $p^*=p^*(2)$, $r^*=r^*(2)$.

\[dfn: admiss\] We say that the exponent quadruplet $\qlet$ is *KT-admissible* if $$\begin{aligned}
\label{eq: KT admiss}
 \rp q + \frac n r = \frac n a, \quad a = \operatorname{HM}(p,r), \qquad\qquad\\
  1\leq a\leq \infty, \quad a \leq r \leq r^*(a), \quad  p^*(a) \leq p \leq a,\end{aligned}$$ apart from the case $n=1$, $\qlet=(a, \infty, a/2, a)$, $2\leq a < \infty$.

The boundaries $r^*(a)$ and $p^*(a)$ are chosen to ensure that the Lebesgue exponents $a,p,q,r$ remain in the range $[1,\infty]$. Note that when $a \geq (n+1)/n$ this is always so and in this case definition \[dfn: admiss\] gives the range of all possible homogeneous estimates in the $L^p$-norms (up to an endpoint), see theorem \[thm: Strich admiss\]. In view of the power invariance , we can generalize these estimates to any $0<a<\infty$. However, the proof of the inhomogeneous estimates relies on the duality relations of the $L^p$-spaces, for $1\leq p \leq \infty$. This restriction is reflected in the second condition in . We next give an equivalent definition to \[dfn: admiss\], which is more convenient in the context of the inhomogeneous estimates.

\[dfn: admiss 2\] We say that the exponent triplet $\trip$ is *KT-admissible* if $$\begin{aligned}
\label{eq: KT admiss 2}
 \rp q &= \frac n 2 \left(\rp p - \rp r \right), \quad a \stackrel{def}{=} \operatorname{HM}(p,r),\\
 1\leq p,q, r &\leq \infty, \quad a \leq r \leq r^*(a), \quad  p^*(a) \leq p \leq a,\end{aligned}$$ apart from the case $n=1$, $\trip=(a, \infty, a/2)$.

A consequence of the above definition is the fact that if $\trip$ is KT-admissible then $a \leq q \leq \infty$ and $p\leq r$. Triplets of the form $\trip=(a,r^*(a), p^*(a))$, for $(n+1)/n\leq a<\infty$, shall be called endpoint. When $a=1$ the only admissible triplet is $(\infty, 1, 1)$, and similarly when $a=\infty$ the only admissible triplet is $(\infty, \infty, \infty)$.

We say that the exponent 5-tuple $\qtet$ is *mixed KT-\
admissible* if $$\begin{aligned}
\label{eq: KT admiss mix}
  \rp q + \frac n r = \frac n b,  \quad \operatorname{HM}(p,r)=\operatorname{HM}(b,c) \stackrel{def}{=} a \in [1, \infty],\\
 a \leq r < r^*(a), \quad p^*(a) < p \leq a, \quad p < b<a<c<r,\end{aligned}$$ or if $a=b=c$ and $\qlet$ is KT-admissible.

\[dfn: accept\] We say that the exponent triplet $\trip$ is *KT-acceptable* if $$\begin{aligned}
\label{cond: KT accept}
\rp q < n \left( \rp p - \rp r \right), \quad 1\leq q\leq\infty,\\
1 \leq p < r \leq \infty,\end{aligned}$$ or if $q=\infty$, $1 \leq p=r \leq \infty$.

Note that every KT-admissible triplet is KT-acceptable too. We introduce the following definition

\[dfn: joint accept\] We say that the two KT-acceptable exponent triplets $\trip$ and $\tript$ are [*jointly KT-acceptable*]{} if $$\begin{aligned}
\rp q + \rp {\tilde q} = n\left(1 - \rp r - \rp {\tilde r}\right), \quad \rp q + \rp {\tilde q} \leq 1, \label{cond: joint KT accept}\\
\operatorname{HM}(p,r)=\operatorname{HM}(\tilde p', \tilde r')=a,  \notag\end{aligned}$$ and if further the exponents satisfy the following restrictions

-   $r <\infty$, unless the point $\triptrip \in \Sigma_1 \cup B$, $$\begin{aligned}
      \Sigma_1 &={\left\{\left(\mu,0,\kappa,\nu,1 - \kappa,1\right)\;:\;  0 < \mu, \nu < 1, \; 0 < \kappa < 1/n, \;\mu + \nu =n\kappa\right\}},\\
           B  &=(0,0,0,0,1,1).\end{aligned}$$

-   $\tilde r <\infty$, unless the point $\triptrip \in \Sigma_2 \cup C$, $$\begin{aligned}
      \Sigma_2 &={\left\{\left(\mu,1-\kappa,1,\nu,0,\kappa \right)\;:\;   0 < \mu, \nu < 1, \; 0 < \kappa < 1/n, \; \mu + \nu =n\kappa\right\}},\\
            C &=(0,1,1,0,0,0).\end{aligned}$$

-   if $\tilde q=\infty$, then $a \leq r < r^*(a)$,

-   if $q=\infty$, then $a \leq \tilde r < r^*(a)$.

Main Results {#sect: main res}
============

Throughout this section we suppose that $1 \leq q,r,p,a \leq \infty$.

\[thm: Strich admiss\] Let $u$ be the solution to the IVP for , . Then the estimate $$\label{est: Strich admiss}
  \normQRP{u(t, x, v)}{q}{r}{p} \lesssim \normP{f}{a} + \normQRP{F}{\tilde q'}{\tilde r'}{\tilde p'},$$ holds for all $f \in L^a({\mathbb R}^{2n})$, $F \in L_t^{\tilde q'}L_x^{\tilde r'}L_v^{\tilde p'}$ if and only if $\trip$ and $\tript$ are two KT-admissible exponent triplets and $a=\operatorname{HM}(p,r)=\operatorname{HM}(\tilde p', \tilde r')$, apart from the case when $n>1$ and $\trip$ being an endpoint triplet, which remains open.

\[rem: admiss endpoint\] Note that in theorem \[thm: Strich admiss\] the KT-admissible exponent triplet $\tript$ is allowed to be endpoint in $n>1$ as long as the other KT-admissible exponent triplet $\trip$ remains non-endpoint. In such case $\trip$ and $\tript$ are jointly acceptable and satisfy the conditions of the next theorem.

\[thm: infty case\] The estimate $$\label{eq: Strich inhom}
\normQRP{W(t)F}{q}{r}{p} \lesssim \normQRP{F}{\tilde q'}{\tilde r'}{\tilde p'}$$ holds for all $F \in \spaceLqrpd$ in $n= 1$ if and only if $\trip$ and $\tript$ are two jointly KT-acceptable exponent triplets. If $n > 1$ we suppose that $\trip$ and $\tript$ are two jointly KT-acceptable exponent triplets and consider the cases

-   $q,\tilde q < \infty$, $1/q + 1/\tilde q < 1$, and $2\leq r, \tilde r < \infty$: then estimate (\[eq: Strich inhom\]) holds if additionally $$\label{cond: glob inhom}
            \frac {n -1}{p'} < \frac {n}{\tilde{r}}, \quad \frac {n-1}{\tilde{p}'} < \frac {n}{r}.$$

-   $r$, or $\tilde r=\infty$: estimate (\[eq: Strich inhom\]) holds only under the conditions of definition \[dfn: joint accept\], that is if $\triptrip \in \Sigma_1 \cup B$,\
    $\triptrip \in \Sigma_2 \cup C$, respectively.

-   $\tilde q = \infty$: then we have the estimate $$\label{eq: Strich inhom a}
    \normQRP{W(t)F}{q, \tilde p'}{r}{p} \lesssim \normQRP{F}{1}{\tilde r'}{\tilde p'}.$$ Whenever $q \geq \tilde p'$ (\[eq: Strich inhom a\]) implies (\[eq: Strich inhom\]).

-   $q = \infty$: then we have the estimate $$\label{eq: Strich inhom b}
    \normQRP{W(t)F}{\infty}{r}{p} \lesssim \normQRP{F}{\tilde q', p}{\tilde r'}{\tilde p'}.$$ Whenever $\tilde q' \leq p$ (\[eq: Strich inhom b\]) implies (\[eq: Strich inhom\]).

Conversely, if estimate (\[eq: Strich inhom\]) holds for all $F \in \spaceLqrpd$ in some dimension $n>1$, then $\trip$ and $\tript$ must be two jointly KT-acceptable exponent triplets.

Apparently, theorem \[thm: infty case\] leaves open the existence of some inhomogeneous estimates in $n>1$. The analogous estimates for the equation are posed as unresolved in Foschi [@F]\[section Open Questions\]. The latter work addressed the observation by Keel and Tao [@KT] (and others) that the full range of validity of the inhomogeneous Strichartz estimates for a given dispersive equation is larger than the set of those that correspond to admissible exponent pairs (triplets) only. We hope that our results could be beneficial to the general question by providing a perspective to a specific context.

\[thm: Strich admiss mix\] Suppose that $\qtet$ is a mixed KT-admissible exponent 5-tuple. Then the estimate $$\label{eq: Strich hom mix}
\normQRP{U(t)f}{q,c}{r}{p} \lesssim \normBC{f}{b}{c}$$ holds for all $f \in L^b_xL^c_v$. Conversely, if estimate (\[eq: Strich hom mix\]) [^3] holds for all $f \in L^b_xL^c_v$, then $\qtet$ must be a mixed KT-admissible exponent 5-tuple.[^4]

The appearance of Lorentz norms in some of the estimates above is not a great obstacle to applications. For example, if we restrict to finite time intervals $[0, T]$, we have the continuous embeddings $$\begin{aligned}
    L^{q,r}([0, T]) \hookrightarrow L^{p}([0, T]),   \quad q > p, \; 1\leq q,p,r\leq \infty,\\
    L^{p}([0, T])   \hookrightarrow L^{q,r}([0, T]), \quad p > q, \; 1\leq q,p,r\leq \infty,\end{aligned}$$ see [@BS p. 217]. Let us recall also the continuous embedding $L^{q,c}({\mathbb R}^n) \hookrightarrow L^q({\mathbb R}^n)$ whenever $q\geq c$. For example, let $(\infty, r,p)$ and $\tript$ be such that estimate holds and let $1\leq \widetilde Q < \tilde q$. Then we have the local inhomogeneous estimate $$\begin{aligned}
      \normQRPloc{W(t)F}{\infty}{q}{r}{[0,T]} \lesssim_T \normQRPloc{F}{\widetilde Q'}{\tilde r'}{\tilde p'}{[0,T]}.\end{aligned}$$

Preliminaries
=============

The kinetic transport propagator $U(t)$ enjoys the estimate $$\label{eq: disp est 2}
  {\left\VertU(t)f\right\Vert}_{L^{\infty}_xL^{1}_v} \leq \frac 1 {{\left\vertt\right\vert}^{n}}{\left\Vertf\right\Vert}_{L^{1}_xL^{\infty}_v},$$ for all $f \in {L^{1}_xL^{\infty}_v}$.

$$\begin{split}
  \int_{{\mathbb R}^n} {\left\vertU(t)f\right\vert} dv = \int_{{\mathbb R}^n} {\left\vertf(x-tv,v)\right\vert}dv \leq \int_{{\mathbb R}^n} \sup_{y \in {\mathbb R}^n} {\left\vertf(x-tv,y)\right\vert} dv \\
  \leq \rp {{\left\vertt\right\vert}^n} \int_{{\mathbb R}^n} \sup_{y \in {\mathbb R}^n} {\left\vertf(z,y)\right\vert} dz = \rp {{\left\vertt\right\vert}^n} \normBC{f}{1}{\infty}.
 \end{split}$$

The kinetic transport propagator $U(t)$ enjoys the estimate $$\label{eq: tran est}
  \normQRP{U(t)f}{\infty}{a}{a} \leq \normA{f}{a}, \quad 0<a\leq \infty,$$ for all $f \in L^{a}_{x,v}$.

Trivial.

\[cor: dec est\] The kinetic transport propagator $U(t)$ enjoys the estimate $$\label{DispEst}
   {\left\VertU(t)f\right\Vert}_{L^r_xL^p_v} \leq \frac 1 {{\left\vertt\right\vert}^{n\left(\frac 1 p - \frac 1 r\right)}}{\left\Vertf\right\Vert}_{L^p_xL^r_v}, \quad 1 \leq p \leq r \leq \infty,$$ for all $f \in {L^{p}_xL^{r}_v}$.

Complex interpolation between the dispersive estimate (\[eq: disp est\]) and the two transport estimates (\[eq: tran est\]) with $a=1$ and $a=\infty$.

\[lem: adjoint\] The formal adjoint to $U(t)$ is the operator $U(t)^*=U(-t)$.

We denote by $\langle \cdot, \cdot \rangle$ the scalar product on $L^2({\mathbb R}^{2n})$. Thus, $$\begin{aligned}
\langle U(t)f,g\rangle &= \Int f(x-tv,v) \overline{g(x,v)}dxdv\\
          &=\Int f(x,v) \overline{g(y+tv,v)}dydv=\langle f,U(-t)g\rangle,
$$ where we have made the substitution $y=x-tv$.

\[lem: scaling\] The evolution operators $U(t)$ and $W(t)$ enjoy the following scaling properties $$\begin{aligned}
U(t)f_\lambda = f\left({x}/{\lambda} - {t v}/{\lambda} , v\right)= U(t/\lambda, x/\lambda, v)f,\\
\text{where  } f_\lambda(x,v) = f\left(x/{\lambda}, v\right),\\
U(t)f_\lambda = f\left(x / {\lambda} - {t v}/{\lambda} , {v}/{\lambda}\right)= U(t, x/\lambda, v/\lambda)f,\\
\text{where  } f_\lambda(x,v) = f\left(x/{\lambda}, v/\lambda \right),\\
W(t)F_\lambda = \lambda \int_{0}^{t / \lambda}F\left(s, x / {\lambda} - \left(t / \lambda - s \right) v, v\right)ds =  \lambda W\left(t/ \lambda,x/ \lambda,v\right)F,\\
\text{where  } F_\lambda(t,x,v) = F(t/\lambda, x/\lambda, v),\\
W(t)F_\lambda = \int_{0}^{t}F\left(s, x / {\lambda} - (t-s) v / {\lambda} , v / \lambda \right)ds= W(t,x/\lambda,v/\lambda)F,\\
\text{where  } F_\lambda(t,x,v) = F\left(t, x/ \lambda, v/ \lambda\right).\end{aligned}$$

Direct inspection.

\[lem: Christ-Kiselev\] Suppose that the integral operator $$\label{eq: Boch int}
  T[F](t)=\int_{-\infty}^{\infty} K(t,s)F(s)ds$$ is bounded from $L^p({\mathbb R};\mathcal B_1)$ to $L^q({\mathbb R};\mathcal B_2)$ for some Banach spaces ${\mathcal B}_1$, ${\mathcal B}_2$ and $1\leq p< q\leq \infty$. The operator-valued kernel $K(t,s)$ maps ${\mathcal B}_1$ to ${\mathcal B}_2$ for all $t,s \in {\mathbb R}$. Assume also that $K$ is regular enough to ensure that (\[eq: Boch int\]) makes sense as a ${\mathcal B}_2$-valued Bochner integral for almost all $t \in {\mathbb R}$. Then the operator $$\tilde{T}[F](t)=\int_{-\infty}^{t} K(t,s)F(s)ds$$ is also bounded on the same spaces.

\[prop: Horm\] Whenever a (non-trivial) linear and bounded operator maps a vector-valued $L^p$-space to another vector-valued $L^q$-space, $1\leq p,q \leq \infty$, and additionally this operator is translation invariant, then we must have that $p \leq q$.

The TT\*-principle.
-------------------

Consider the operator $T: \spaceLtwo \rightarrow \spaceLqrp$, given by $$\begin{aligned}
    T[f](t,x,v)=f(x-tv,v), \quad \forall f \in \spaceLtwo.\end{aligned}$$ Its formal adjoint $T^*:\spaceLqrpd \rightarrow \spaceLtwo$ has the form of an $L^2$-valued integral $$\begin{aligned}
    T^*[F](x,v)=\Int F(s, x + sv,v)ds, \quad \forall F \in \spaceLqrpd.\end{aligned}$$ Then the composition of the two $TT^*: \spaceLqrpd \rightarrow \spaceLqrp$ has the form $$\begin{aligned}
    TT^*[F](t,x,v)=\Int F(s, x - (t-s)v,v)ds, \quad \forall F \in \spaceLqrpd.\end{aligned}$$ By the TT\*-principle, $T$ and $TT^*$ are equally bounded with ${\left\VertT\right\Vert}^2={\left\VertTT^*\right\Vert}$. Thus, the two estimates are equivalent $$\begin{aligned}
 \normQRP{Tf}{q}{r}{r'}    &\leq C \normA{f}{2}, \quad\qquad \forall f \in \spaceLa{2}, \label{est: T}\\
 \normQRP{TT^*F}{q}{r}{r'} &\leq C^2 \normQRP{F}{q'}{r'}{r}, \quad \forall F \in \spaceLqrpd, \label{est: TT*}\end{aligned}$$ where ${\left\VertT\right\Vert} \leq C$. Moreover, if $\trip$ and $\tript$ are two exponent triplets for which holds, then by the factorization property of $TT^*$ we obtain the consequence $$\begin{aligned}
 \label{est: TT* 2}
 \normQRP{TT^*F}{q}{r}{p} \leq C^2 \normQRP{F}{\tilde q'}{\tilde r'}{ \tilde p'}, \quad \forall F \in \spaceLqrptd.\end{aligned}$$ By duality, is equivalent to $$\begin{aligned}
    \Int \langle TT^*[F](t), G(t) \rangle dt \leq C^2 \normQRP{F}{q'}{r'}{r}\normQRP{G}{q'}{r'}{r}, \quad \forall F, \; \forall G \in \spaceLqrpd,\end{aligned}$$ which in turn simplifies to $$\begin{aligned}
    \Int\Int \langle U(s)^*F, U(t)^*G \rangle dsdt \leq C^2 \normQRP{F}{q'}{r'}{r}\normQRP{G}{q'}{r'}{r},\end{aligned}$$ Consider the bilinear form $$B(F,G) = \iint_{s<t}  \langle U(s)^*F, U(t)^*G \rangle dsdt.$$ By symmetry, the last inequality simplifies to $$\begin{aligned}
 \label{eq: B 1}
   {\left\vertB(F,G)\right\vert} \leq C^2 \normQRP{F}{q'}{r'}{r}\normQRP{G}{q'}{r'}{r}, \quad \forall F, \; \forall G \in \spaceLqrpd.\end{aligned}$$

Analogously, we have that the inhomogeneous estimate $$\begin{aligned}
    \normQRP{W(t)F}{q}{r}{p} \lesssim \normQRP{F}{\tilde q'}{\tilde r'}{\tilde p'}, \quad \forall F \in \spaceLqrptd,\end{aligned}$$ is equivalent to the estimate $$\begin{aligned}
 \label{eq: B 2}
   {\left\vertB(F,G)\right\vert} \lesssim \normQRP{F}{\tilde q'}{\tilde r'}{\tilde p'} & \normQRP{G}{q'}{r'}{r}, \\
   &\forall F \in \spaceLqrptd ,\; \forall G \in \spaceLqrpd. \notag\end{aligned}$$

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
==================================

Estimates for $L^p$-data
------------------------

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 the Hardy-Littlewood-Sobolev (HLS) theorem of fractional integration, see [@BS pp. 228-229], [@S], we obtain $$\begin{aligned}
    \normQRP{TT^*F}{q}{r}{r'} \lesssim \normQRP{F}{q'}{r'}{r},\end{aligned}$$ whenever $0<\beta(r)<1$, $1 + 1/q = 1/q' + \beta(r)$. The latter conditions are equivalent to $2 < r < r^*$, $1/q+n/r=n/2$. The left endpoint $r=2$ follows trivially from the transport estimate (\[eq: tran est\]). The right endpoint $r = r^*$ remains an open problem.

The inhomogeneous estimate is a consequence of estimate , the factorization , and the Christ-Kiselev lemma \[lem: Christ-Kiselev\]. Note that the latter is applicable only in the non-endpoint case $1/q+1/{\tilde q} <1$.

To complete the proof of the sufficient part of theorem \[thm: Strich admiss\] we only need to consider the case when $\trip$ is a non-endpoint admissible triplet while $\tript$ is an endpoint admissible triplet. It is easy to check that the two triplets are jointly acceptable with $1/q+1/{\tilde q} <1$. Indeed, the last assertion follows from the fact that $q>a$ while $\tilde q = a'$. Suppose that both $q, \tilde q < \infty$. Then condition is satisfied by the mere construction of the proof of theorem \[thm: infty case\]. We can give a direct proof of that claim as well. For example, the inequality $(n-1)/p' < n / {\tilde r}$ is equivalent to $a'/ p' < r^*(a') / \tilde r = 1$, i.e. to $p < a$. But if the last inequality is violated, i.e. $p=a$, then $p=r$ and $q=\infty$, contradiction. Similarly, $(n-1)/\tilde p' < n / {r}$ is congruent with $a/\tilde p' < r^*(a) /r$. For the left hand side we have that $a/\tilde p' <1$, while for the right hand side $r^*(a)/r >1$. Thus, we are left only with the case when either $q$ or $\tilde q$ is equal to infinity. Suppose, for example, that $q=\infty$. Then $r=p=a$, $\tilde q'=a$ and hence estimate implies the desired claim. Finally, the case when $\tilde q=\infty$ and $\tript$ is endpoint is only possible if $\tilde q= \tilde r=\tilde p = \infty$. Then $\tilde p'=1$ and estimate implies the desired claim.

Estimates for mixed $L^p$-data {#sect: admiss mix Lp}
------------------------------

We prove here theorem \[thm: Strich admiss mix\], that is estimate $$\begin{aligned}
\normQRP{U(t)f}{q,c}{r}{p} \lesssim \normBC{f}{b}{c} \tag{\ref{eq: Strich hom mix}}.\end{aligned}$$

To that end we recall two results from real interpolation that shall be used in the proof. By $L^p=L^p(A)$ and $L^{p,q}=L^{p,q}(A)$ we denote the $L^p$-space and the Lorentz space, respectively of functions that take their values in a fixed Banach space $A$. If $A_0$ and $A_1$ are two Banach spaces that are compatible, we denote by $\bar A$ the interpolation couple $(A_0, A_1)$.

\[prop: real interp 1\] For every $1 \leq p_0, p_1 < \infty$, $0 < \theta < 1$, $1/p = (1 - \theta)/p_0 + \theta/p_1$ and $p \leq q$ we have $$L^p (\bar A_{\theta ,q}) \hookrightarrow (L^{p_0}(A_0), L^{p_1}(A_1))_{\theta,q}.$$

\[prop: real interp 2\] Suppose that $0 < p_0,\, p_1,\, q_0,\, q_1 \leq \infty$, $0< \theta < 1$, and $p_0 \neq p_1$. Then $$\left( L^{p_0, q_0}, L^{p_1, q_1} \right)_{\theta, q} = L^{p, q}, $$ where $1/p = (1 - \theta)/p_0 + \theta/p_1$.

Now we are ready to commence in earnest.

We interpolate between the estimates $$\begin{aligned}
  \normQRP{U(t)f}{\tilde q}{r}{p} &\lesssim \normA{f}a, \qquad \qquad \quad\label{eq: KT 3}\\
  \normQRP{U(t)f}{\xi, \infty}rp &\lesssim \normBC{f}pr, \quad 1 / \xi = n\left(1/ p - 1/ r\right), \label{DispEst2}\end{aligned}$$ where $(\tilde q,r,p,a)$ is an admissible exponent quadruplet. In view of the real method with interpolation functor $K_{\theta, c}$ and propositions \[prop: real interp 1\], \[prop: real interp 2\], we obtain (\[eq: Strich hom mix\]), where $$\begin{aligned}
    0 < \theta < 1, \qquad \rp c = \frac {1-\theta}{a} + \frac {\theta}{r},\\
    \rp q = \frac {1-\theta}{\tilde q} + \frac {\theta}{\xi}, \qquad \rp b = \frac {1-\theta}{ a} + \frac {\theta}{p}.\end{aligned}$$ We can eliminate $\theta$ to obtain that estimate (\[eq: Strich hom mix\]) holds under the conditions $$\begin{aligned}
    \rp q + \frac n r = \frac n b,  \quad \operatorname{HM}(p,r)=\operatorname{HM}(b,c) = a,\\
 a < r < r^*(a), \quad p^*(a) < p < a, \quad p < b<a<c<r,\end{aligned}$$ or if $a=b=c$ and $\qlet$ is an admissible exponent quadruplet (corresponds to $\theta=0$).

We next prove the two cases of theorem \[thm: infty case\] when either $\tilde q=\infty$ or $q=\infty$, that is estimates $$\begin{aligned}
\normQRP{W(t)F}{q, \tilde p'}{r}{p} \lesssim \normQRP{F}{1}{\tilde r'}{\tilde p'}, \tag{\ref{eq: Strich inhom a}}\\
\normQRP{W(t)F}{\infty}{r}{p} \lesssim \normQRP{F}{\tilde q', p}{\tilde r'}{\tilde p'}\tag{\ref{eq: Strich inhom b}},\end{aligned}$$ where $\trip$ and $\tript$ are two jointly acceptable exponent triplets.

The homogeneous estimate (\[eq: Strich hom mix\]) implies the inhomogeneous estimate (\[eq: Strich inhom a\]), see [@O theorem 2.9] for proof. We need only to verify that the range of validity of (\[eq: Strich inhom a\]) is precisely described by the requirement that $\trip$ and $\tript$ be two jointly acceptable exponent triplets. In the table below we give the conditions of a mixed KT-admissible 5-tuple in the second column and we reformulate them as in the definition of two jointly acceptable exponent triplets in the first column.

$$\begin{aligned}
    \rp q + \rp \infty = n\left(1-\rp r-\rp {\tilde r}\right),& &    \rp q + \frac n r = \frac n b,\qquad\\
    \rp q < n\left( \rp p - \rp r \right),\qquad&                        &\rp b < \rp p,\qquad\quad\\
    q=\infty, \quad 1\leq p=r\leq\infty,&                          &a=b=c=p=r,\quad\\
    \operatorname{HM}(p,r)=\operatorname{HM}(\tilde p', \tilde r')&                              &\operatorname{HM}(p,r)=\operatorname{HM}(b,c),\\
    \tilde q=\infty,\quad \trip \text{ is KT-admissible}&               &a=b=c,\qquad\\
    a < r < r^*(a)\qquad&    & \begin{cases}  \quad a < r < r^*(a),&\\
                                        \quad p^*(a) < p < a,&\\
                                        \quad p < b<a<c<r.&
                         \end{cases}\end{aligned}$$

Estimate follows from by duality, see [@O Duality lemma] for proof. Note also that the tricky cases which cannot be treated by duality, e.g. the dual estimate of , where $\tilde p'=\infty$, $\tilde r' < \infty$, are precluded by our definition of joint acceptability in the case when $\tilde q=\infty$.

Estimates for acceptable exponents
==================================

Local Inhomogeneous Estimates
-----------------------------

Following Foschi [@F], we want to find the range of local estimates for $W(t)$ that are invariant to the scaling $$\begin{aligned}
 \label{eq: local scal}
 \normLL{W(t)[\chi_{\lambda J}F]} \lesssim  \lambda^{\bbeta}\normRL{F}, \quad \forall \lambda >0,
 $$ where $I$ and $J$ are two unit intervals separated by a unit distance and $\chi_{\lambda J}$ is the characteristic of the scaled-out interval $\lambda J$. Note that $$\bbeta=\frac 1 q + \frac 1 {\tilde{q}} - \frac n 2 \left(\rp p - \frac 1 r + \rp {\tilde p} - \rp {\tilde{r}}\right),$$ due to the scaling condition $\operatorname{HM}(p,r)=\operatorname{HM}(\tilde p', \tilde r')$. We set $$\beta(q,r,\tilde q, \tilde r) = \bbeta.$$

\[lem: local estim 1\] Estimate holds for any two non-endpoint KT-admissible triplets $\trip$ and $\tript$ with $a=\tilde a'$.

The proof follows trivially from theorem \[thm: Strich admiss\] due to the fact that $\beta(q,r,\tilde q, \tilde r)=0$ under the hypothesis of the lemma.

\[lem: local estim 2\] Estimate holds with $\trip=(\infty, r, p)$ and $\tript=(\infty, p', r')$, where $1 \leq p \leq r \leq \infty$.

Due to the decay estimate (\[DispEst\]) we have that $$\begin{aligned}
  \sup_{t \in \lambda I} {\left\VertW(t)[\chi_{\lambda J}F]\right\Vert}_{ L^{r}_xL^{p}_v} &\lesssim \sup_{t \in \lambda I}\int_{\lambda J} \frac {{\left\VertF(\tau)\right\Vert}_{ L^{p}_xL^{r}_v}}             {{\left\vertt-\tau\right\vert}^\betta}d\tau \\
                                                                          &\lesssim  \lambda^{\beta(\infty, r, \infty, p')}{\left\VertF\right\Vert}_{L^1(\lambda J; L^{p}_xL^{r}_v)}.\end{aligned}$$

\[lem: local estim 3\] Whenever $\trip$ and $\tript$ are exponent triplets for which estimate holds, we have that also holds with $(Q,r,p)$ and $(\tilde Q, \tilde r, \tilde p)$, where $1\leq Q \leq q$, $1 \leq \tilde{Q} \leq \tilde{q}$.

A trivial application of Hölder’s inequality $$\begin{split}
  {\left\VertW(t)[\chi_{\lambda J}F]\right\Vert}_{L^{Q}(\lambda I; L^{r}_xL^{p}_v)} \lesssim
  \lambda^{\rp Q - \rp q} {\left\VertW(t)[\chi_{\lambda J}F]\right\Vert}_{L^{q}(\lambda I; L^{r}_xL^{p}_v)} \qquad\\
  \lesssim \lambda^{\beta(Q,r,\tilde q , \tilde r)}{\left\VertF\right\Vert}_{L^{\tilde{q}'}(\lambda J; L^{\tilde{r}'}_xL^{\tilde{p}'}_v)}
  \lesssim  \lambda^{\beta(Q,r,\tilde Q , \tilde r)}{\left\VertF\right\Vert}_{L^{\tilde{Q}'}(\lambda J; L^{\tilde{r}'}_xL^{\tilde{p}'}_v)}. \qedhere
 \end{split}$$

Let us define the range of validity of the local estimates as the set $\mathcal{E}$ in ${\mathbb R}^6$. Each point in $\mathcal{E}$ corresponds to a 6-tuple of exponents $(1/q, 1/r, 1/p, 1/\tilde q, 1/ \tilde r, 1/ \tilde p)$. Below we find the convex hull $\mathcal{E}^*$ ($\mathcal{E}^* \subseteq \mathcal{E}$) of the points in ${\mathbb R}^6$ that correspond to the estimates in the three lemmas above. We shall call any point or collection of points in $\mathcal{E}$ [*acceptable*]{}.

\[lem: local inhom estim\] Estimate holds whenever the exponent triplets $\trip$, $\tript$ satisfy the following conditions $$\begin{aligned}
0 \leq \rp q, \rp {\tilde{q}}, \rp p, \rp {\tilde{p}}, \rp r, \rp {\tilde{r}} \leq 1,\quad
\rp r \leq \rp p, &\quad \rp {\tilde r} \leq \rp {\tilde p}, \quad \operatorname{HM}(p,r)=\operatorname{HM}(\tilde p', \tilde r'),\\
\qconds, &\quad \qcondts, \qquad \label{cond: loc inhom 2}\\
\frac {n -1}{p'} < \frac {n}{\tilde{r}}, &\quad \frac {n-1}{\tilde{p}'} < \frac {n}{r},\qquad\qquad\qquad\end{aligned}$$ or if the point $(1/q, 1/r, 1/p, 1/\tilde q, 1/ \tilde r, 1/ \tilde p)$ lies inside one of the cubes in ${\mathbb R}^6$ below $$\label{omitted points in lemma}
  \begin{split}
   \left(\kappa,0,\mu,\nu,1 - \mu,1\right), \qquad 0 \leq \kappa, \mu, \nu \leq 1, \qquad\\
   \left(\kappa,1-\mu,1,\nu,0,\mu \right),  \qquad 0 \leq \kappa, \mu, \nu \leq 1.\qquad
  \end{split}$$

We apply the Riesz-Thorin convexity theorem to interpolate between the already proven local estimates. In essence, we find the convex hull of the locally acceptable sets associated with lemmas \[lem: local estim 1\] and \[lem: local estim 2\] and then expand that set by the rule given in lemma \[lem: local estim 3\].

The set of acceptability $S_1$ of the local estimates in lemma \[lem: local estim 1\] is given by the system $$\begin{aligned}
  0 \leq \rp q, \rp {\tilde q}, \rp p, \rp {\tilde p}, \rp r, \rp {\tilde r} \leq 1,\label{sys: 1 line 1}\\
  \frac 1 q = \frac n 2 \left(\rp p - \rp r \right), \quad \frac 1 {\tilde q} = \frac n 2 \left(\rp {\tilde p} - \rp {\tilde r} \right), \label{sys: 1 line 2}\\
  \rp r + \rp p + \rp {\tilde r} + \rp {\tilde p} = 2, \label{sys: 1 line 3}\\
  \frac {n-1}{p} < \frac {n+1}{r}, \qquad \frac {n-1}{\tilde p}  < \frac {n+1}{\tilde r},\label{sys: 1 line 4}\end{aligned}$$ or if $(1/q, 1/r, 1/p, 1/\tilde q, 1/ \tilde r, 1/ \tilde p) \in \{B,C | B=(0,0,0,0,1,1),  C=(0,1,1,0,0,0)\}$.

Note that $S_1$ is a convex polyhedron in ${\mathbb R}^6$ and the two points $B$ and $C$ lie on its boundary. The set of acceptability $S_2$ of the local estimates in lemma \[lem: local estim 2\] is the convex hull (in fact a triangle) of the three points $$\label{triangle}
 A=(0,0,1,0,0,1), \qquad B=(0,0,0,0,1,1), \qquad C=(0,1,1,0,0,0).$$ Vertices $B$ and $C$ are already included in $S_1$, thus it would suffice to take only the vertex $A$. Hence, we obtain the following set $$\begin{aligned}
 \rp Q = \frac \theta q, \qquad \rp R = \frac \theta r, \qquad \rp P = 1-\theta + \frac \theta p,\qquad\qquad\qquad\;\;\;\\
 \rp {\tilde Q} = \frac \theta {\tilde q}, \qquad \rp {\tilde R} = \frac \theta {\tilde r}, \qquad \rp {\tilde P} = 1-\theta + \frac \theta {\tilde p}, \qquad 0 \leq \theta \leq 1,\end{aligned}$$ where $\Triptrip$ are the coordinates of the new set $S_3$ written in terms of $\triptrip$ and $\theta$. Of course, to $S_3$ we must also add the line segments $[A,B]$ and $[A,C]$. We shall treat this case separately at the end.

Finally, we apply the rule given in lemma \[lem: local estim 3\] and thus we replace the equations for $Q$ and $\tilde Q$ above with the following inequalities $$\begin{aligned}
 1 \geq \rp Q \geq \frac \theta q,\qquad 1 \geq \rp {\tilde Q} \geq \frac \theta {\tilde q},\end{aligned}$$ plus the restrictions $$\begin{aligned}
 \label{sys: 1 line 5}
 \rp r \leq \rp p, \quad \rp {\tilde r} \leq \rp {\tilde p},\end{aligned}$$ which were implicitly assumed in .

[**1.**]{} We first eliminate $q$ and $\tilde q$ from the system for $S_1$ to obtain $$\begin{aligned}
 \rp Q \geq \frac n 2 \left( \frac \theta p - \frac \theta r \right), \quad &\Leftrightarrow \quad \rp Q \geq \frac n 2 \left( \theta - 1 + \rp P - \rp R \right),\\
 &\Leftrightarrow \quad \theta \leq \rp {P'} + \rp R + \frac 2 {nQ}.\end{aligned}$$ Similarly, $$\qquad \quad\theta \leq \rp {\tilde P'} + \rp {\tilde R} + \frac 2 {n\tilde Q}, \qquad \rp Q, \rp {\tilde Q} \leq 1.\quad$$

[**2.**]{} As expected, condition (\[sys: 1 line 3\]) is invariant $$\scondAB.$$

[**3.**]{} Reworking condition (\[sys: 1 line 4\]), we obtain $$\theta < \frac {n+1}{n-1} \rp R + \rp {P'}, \qquad \theta < \frac {n+1}{n-1} \rp {\tilde R} + \rp {\tilde P'}.$$

[**4.**]{} Condition (\[sys: 1 line 5\]) is replaced by $$\rp {P'} + \rp R \leq \theta, \quad \rp {\tilde P'} + \rp {\tilde R} \leq \theta.$$

[**5.**]{} Finally, conditions (\[sys: 1 line 1\]) are transformed into $$\rp {P'}, \rp {\tilde{P}'}, \rp R, \rp {\tilde R} \leq \theta, \quad 0 \leq \rp Q, \rp {\tilde Q}, \rp P, \rp {\tilde{P}}, \rp R, \rp {\tilde{R}}\leq 1.$$

[**6.**]{} We group all conditions obtained in the previous 5 steps according to their type $$\begin{aligned}
  0,\, \rp {P'},\, \rp {\tilde{P}'},\, \rp R,\, \rp {\tilde{R}}, \, \rp {P'} + \rp R, \, \rp {\tilde P'} + \rp {\tilde R}  \leq \theta. \qquad\qquad\qquad\label{sys: 2 line 1}\\
   \theta \leq \rp R + \rp {P'} + \frac 2 {nQ},\,  \rp {\tilde{R}} + \rp {\tilde{P}'} + \frac 2 {n\tilde{Q}},
  \frac {n+1}{n-1} \rp R + \rp {P'},\, \frac {n+1}{n-1} \rp {\tilde{R}} + \rp {\tilde{P}'},\, 1. \label{sys: 2 line 2}\\
  0 \leq \rp Q,\, \rp {\tilde{Q}},\, \rp P,\, \rp {\tilde{P}},\, \rp R,\, \rp {\tilde{R}} \leq 1,\, \qquad \rp P + \rp R + \rp {\tilde{R}} + \rp {\tilde{P}} = 2.\qquad \label{sys: 2 line 3}\end{aligned}$$

[**7.**]{} We discard the redundant conditions like $$0,\, \rp {P'},\, \rp {\tilde{P}'}, \rp {R},\, \rp {\tilde R} \leq \theta,$$ which are all weaker than the other two in .

There exists $\theta$ solving all inequalities in , , if and only if every quantity in is bounded from above by any quantity in . Thus we form all possible combinations between the quantities in the two types of (reduced) inequalities to obtain the following set of conditions $$\begin{aligned}
 \rp R + \rp {P'} \leq \rp {\tilde{R}} + \rp {\tilde{P}'} + \frac 2 {n\tilde{Q}}, &\quad \rp {\tilde{R}} + \rp {\tilde{P}'} \leq \rp R + \rp {P'} + \frac 2 {nQ},\\
 \rp R + \rp {P'} \leq \frac {n+1}{n-1} \rp {\tilde{R}} + \rp {\tilde{P}'},  &\quad \Leftrightarrow \quad \frac {n -1}{P'} < \frac {n}{\tilde{R}},\\
 \rp {\tilde{R}} + \rp {\tilde{P}'} \leq  \frac {n+1}{n-1} \rp R + \rp {P'},  &\quad \Leftrightarrow \quad  \frac {n-1}{\tilde{P}'} < \frac {n}{R},\\
 \rp R \leq \rp P, &\quad \rp {\tilde R} \leq \rp {\tilde P},\end{aligned}$$ describing the region $S_3$.

[**8.**]{} We apply the rule given in lemma \[lem: local estim 3\] to the two line segments $[A,B]$ and $[A, C]$ to obtain the following two cubes in ${\mathbb R}^6$ $$\label{omitted points}
  \begin{split}
 \left(\mu,0,\kappa,\nu,1 - \kappa,1\right), \qquad 0 \leq \mu, \nu, \kappa \leq 1,\qquad\\
 \left(\mu,1-\kappa,1,\nu,0,\kappa \right),  \qquad 0 \leq \mu, \nu, \kappa \leq 1.\qquad
  \end{split}$$

Hence, the computation of the set $\mathcal{E}^*$ is finished.

Global Inhomogeneous Estimates
------------------------------

The key point in the proof of the global inhomogeneous estimates is to decompose the bilinear operator $B(F,G)$ into a series of time-localized operators to which we can apply the local inhomogeneous estimates. We denote by $\Omega$ the region $\{(t,s)|s<t\}$ on the $t,s$-coordinate plane.

We call any positive integer that is a power of two a dyadic number. Furthermore, we call a square $Q$ in ${\mathbb R}^2$ dyadic if its side length is a dyadic number and the coordinates of its vertices are integer multiples of dyadic numbers.

Applying Whitney’s decomposition to partition $\Omega$, we obtain the family $\mathcal O$ of disjoint dyadic squares $Q$ (overlapping on the sides is not excluded) such that the distance between any square $Q \in \mathcal{O}$ and the boundary of $\Omega$ ($\{(t,s) | t=s \}$) is approximately proportional to the diameter of $Q$, see figure \[fig: Whitney\]. By $\Ol$ we denote the collection of all squares in $\mathcal{O}$ whose side length is $\lambda$.

Thus we obtain the representations $$\Omega = \bigcup_\lambda \bigcup_{Q \in \Ol}, \quad B(F, G) = \sum_\lambda \sum_{Q \in \Ol} B_Q(F, G),$$ where $$\label{BQ}
B_Q(F, G) = \iint_{Q} \langle U^*(s)F(s), U^*(t)G(t) \rangle ds dt.$$

The benefit of the above decomposition is that whenever $Q = J \times I$ and $Q \in \Ol$ we have $$\label{lambdaIJ}
\lambda = {\left\vertI\right\vert} = {\left\vertJ\right\vert} \sim \mathrm{dist}(\Omega, \partial\Omega)\sim \mathrm{dist}(I, J).$$ This allows us to obtain the estimate $$\label{BilStrichQ}
{\left\vertB_{Q}(F, G)\right\vert} \lesssim \lambda^{\beta (q,r,{\tilde{q}}, {\tilde{r}})} \normjt{F} \normi{G},$$ whenever $\trip$ and $\tript$ are such that we have the scaling .

\[lem: 1/p+\]Suppose $\rp p + \rp {\tilde{p}} \geq 1$. Then $$\sum_{Q \in \Ol, \ Q=J\times I}{\left\Vertf\right\Vert}_{L^{\tilde{p}}(J)}{\left\Vertg\right\Vert}_{L^{p}(I)}\leq
{\left\Vertf\right\Vert}_{L^{\tilde{p}}({\mathbb R})}{\left\Vertg\right\Vert}_{L^{p}({\mathbb R})}.$$

The lemma follows directly from the inequality $$\sum_{j} {\left\verta_j b_j\right\vert} \leq \left( \sum_j |a_j|^{\tilde p} \right)^{\frac 1 {\tilde p}}
\left( \sum_j |b_j|^{p} \right)^{\frac 1 {p}},$$ which holds in the range $\rp p + \rp {\tilde{p}} \geq 1$, and the fact that for each dyadic interval $I$ there are at most two dyadic squares in $\Ol$ with side $I$.

In the next estimate we relate the scaling argument with the dyadic decomposition.

\[cor: 1/q+\] If $\rp q + \rp {\tilde{q}} \leq 1$, then $$\label{cor1Est}
\sum_{Q \in \Ol}{\left\vertB_{Q}(F, G)\right\vert} \lesssim  \lambda^{\beta(q,r,{\tilde{q}}, {\tilde{r}})} \normrt{F}\normr{G}.$$

In view of (\[BilStrichQ\]) $$ \sum_{Q \in \Ol} {\left\vertB_Q(F, G)\right\vert} \lesssim  \lambda^{\beta (q,r,{\tilde{q}}, {\tilde{r}})} \sum_{Q \in \Ol} \normjt{F}\normi{G}.\qquad
 $$ An application of lemma \[lem: 1/p+\] concludes the proof.

We denote by $l^p_s$ the space of number sequences with a norm $$\begin{aligned}
 {\left\Vert\{a\}_{j \in {\mathbb Z}}\right\Vert}_{l^p_s}      &= \left(2^{js}{\left\verta_j\right\vert}^p \right)^{1/p}, \quad 1\leq p<\infty,\\
 {\left\Vert\{a\}_{j \in {\mathbb Z}}\right\Vert}_{l^\infty_s} & = \sup_{j \in {\mathbb Z}} 2^{js}{\left\verta_j\right\vert},  \qquad p=\infty.\end{aligned}$$

\[lem: interp l\] We have the identity $$\begin{aligned}
 \left(l^\infty_{s_0}, l^\infty_{s_1} \right)_{\theta, 1} = l^1_{s},\end{aligned}$$ where $s_0, s_1 \in {\mathbb R}$, $s_0 \neq s_1$ and $s=(1-\theta)s_0+\theta s_1$.

\[lem: interp T\] Suppose that $(\mathcal A_0, \mathcal A_1)$, $(\mathcal B_0, \mathcal B_1)$, $(\mathcal C_0, \mathcal C_1)$ are interpolation couples and that the bilinear operator $T$ acts as a bounded transformation as indicated below: $$\begin{aligned}
                T: \mathcal A_0 \times \mathcal B_0 \rightarrow \mathcal C_0,\\
            T: \mathcal A_0 \times \mathcal B_1 \rightarrow \mathcal C_1,\\
        T: \mathcal A_1 \times \mathcal B_0 \rightarrow \mathcal C_1.\end{aligned}$$ If $\theta_0, \theta_1 \in (0,1)$ and $p,q,r \in [1, \infty]$ such that $1/ p+ 1/q \geq 1$, then $T$ also acts as a bounded transformation in the following way: $$\begin{aligned}
                T: (\mathcal A_0, \mathcal A_1)_{\theta_0, pr} \times (\mathcal B_0, \mathcal B_1)_{\theta_1, qr} \rightarrow (\mathcal C_0, \mathcal C_1)_{\theta_0+\theta_1, r}.\end{aligned}$$

After this preparatory work, we can now begin with the crux of the main argument of this section. Consider the bilinear operator $T: \spaceLqrptd \times \spaceLqrp \rightarrow l^\infty_s$, defined by the formula $$\begin{aligned}
 T(F, G) = \left\{ b_\lambda \right\}_{\lambda \in 2^{{\mathbb Z}}} = \left\{ \sum_{Q \in \Ol} {\left\vertB_Q(F, G)\right\vert} \right\}_{\lambda \in 2^{{\mathbb Z}}}.\end{aligned}$$ Our aim is to find the range of exponent triplets $\trip$ and $\tript$ where we have the estimate $${\left\Vert\{ b_\lambda\}\right\Vert}_{l^1_0} \lesssim \normQRP{F}{\tilde q'}{\tilde r'}{\tilde p'}\normQRP{G}{q'}{r'}{p'}, \quad \forall F \in \spaceLqrptd,\; \forall G \in \spaceLqrpd.$$ Suppose that $(1/q, 1/\tilde q) \in \Delta$, where $\Delta = \{1/q>0, 1/\tilde q > 0, 1/q+1/\tilde q < 1\}$, and that $\trip,\, \tript \in \mathcal{E}^*$. Then in virtue of corollary \[cor: 1/q+\] we have the estimate $${\left\vertb_\lambda\right\vert} \lesssim \lambda^{\beta(q,r,\tilde q, \tilde r)} \normQRP{F}{\tilde q'}{\tilde r'}{\tilde p'}\normQRP{G}{q'}{r'}{p'},$$ or in other words $\{ b_\lambda\} \in l^\infty_{\beta(q,r,\tilde q, \tilde r)}$. Since $\Delta$ is an open set (triangle) on the $1/q, 1/{\tilde q}$-coordinate plane, we can always find a small enough open neighborhood of points in $\Delta$ around $(1/q, 1/\tilde q)$. Let us set $$1/q_0 = 1/q + \epsilon, \quad  1/\tilde q_0 = 1/\tilde q + \epsilon, \quad 1/q_1 = 1/q - 3\epsilon, \quad  1/\tilde q_1 = 1/\tilde q - 3\epsilon.$$ Suppose that $\epsilon >0$ is small enough so that $(1/q_0, 1/\tilde q_0), (1/q_1, 1/\tilde q_1) \in \Delta \cap \mathcal{E}^*$. Suppose also that $$\begin{aligned}
 \label{cond: scal 1/q}
 1/q + 1/ {\tilde q} = n\left(1-1/ r - 1/ {\tilde r}\right).\end{aligned}$$ Then we have that $\beta(q_0, r, \tilde q_0, \tilde r) = 2\epsilon$, and $\beta(q_1, r, \tilde q_0, \tilde r)=\beta(q_0, r, \tilde q_1, \tilde r)  = - 2\epsilon$. Thus we obtain the maps $$\begin{aligned}
                T: \SpaceLqrptd{q_0}{r}{p}  \times \SpaceLqrpd{q_0}{r}{p} &\rightarrow l_{2\epsilon}^\infty,\\
            T: \SpaceLqrptd{q_0}{r}{p}  \times \SpaceLqrpd{q_1}{r}{p} &\rightarrow l_{-2\epsilon}^\infty,\\
        T: \SpaceLqrptd{q_1}{r}{p}  \times \SpaceLqrpd{q_0}{r}{p} &\rightarrow l_{-2\epsilon}^\infty,\end{aligned}$$ are bounded. In virtue of lemma \[lem: interp T\] we have that the map $$\begin{aligned}
                T: (\SpaceLqrptd{q_0}{r}{p}, \SpaceLqrptd{q_1}{r}{p})_{1/4, \tilde q'}  \times (\SpaceLqrpd{q_0}{r}{p}, \SpaceLqrpd{q_1}{r}{p})_{1/4, q'}   \rightarrow (l_{2\epsilon}^\infty, l_{-2\epsilon}^\infty)_{1/2,1}.\end{aligned}$$ is also bounded. Finally, in view of the well-known interpolation identities of the Lorentz spaces and that of lemma \[lem: interp l\], this simplifies to $$\begin{aligned}
                T: \spaceLqrptd  \times \spaceLqrpd \rightarrow l^1,\end{aligned}$$ QED.

Now let us recapitulate all conditions that we have imposed so far on the exponents. We have the conditions of the local estimates (set $\mathcal E^*$) plus the scaling condition . Note that conditions together with are equivalent to $\trip$ and $\tript$ being KT-acceptable. Let us also note that the two locally acceptable cubes in give rise to the two globally acceptable sets (cubic sections) $\Sigma_1$ and $\Sigma_2$ in definition \[dfn: joint accept\].

Thus, in view of the two inhomogeneous estimates that we proved in section \[sect: admiss mix Lp\], all estimates in theorem \[thm: infty case\] are now proven.

Sharpness of the Estimates {#sect: counterex}
==========================

In this section we present some counterexamples to the Strichartz estimates (\[eq: hom Strich\]), (\[eq: hom Strich mix\]) for $U(t)$ and the inhomogeneous estimates (\[eq: inhom Strich\]) for $W(t)$. Thus we show sharpness of the estimates that we stated in the three theorems of section \[sect: main res\].

Geometric interpretation of the Strichartz estimates for $U(t)$
---------------------------------------------------------------

Let us begin with the case $n=1$. We consider the velocity averages $A$ of the kinetic transport propagator $U(t)$, that is $$A[f](t,x) := \Int f(x-tv,v)dv.$$ It is not hard to see that $$A[f] = \frac 1 {\sqrt{1+t^2}}\int_{\gamma_{x,t}} f(l) dl,$$ where in the above line integral we integrate along the straight line $\gamma_{x,t}$ on the plane $Oxv$, passing through the point $X=(x,0)$ and having a gradient $-1/t$. Let $f(x,v)=\chi_{Q}$, where $\chi_{Q}$ is the characteristic of the measurable subset $Q$ of the plane $Oxv$.

Let us interpret $\normBC{U(t)\chi_{Q}}{\infty}{1}$ geometrically. By the above we have $$\normBC{U(t)\chi_{Q}}{\infty}{1} = \frac 1 {\sqrt{1+t^2}} \sup_{x\in{\mathbb R}}{\int_{\gamma_{x,t}} \chi_{Q}(l) dl},$$ or in other words $\normBC{U(t)\chi_{Q}}{\infty}{1}$ is the supremum of the (line) measure of all intersections of $Q$ with straight lines in a fixed direction (times a factor in $t$).

\[lem: Besic\] There exist an open set $Q$ on the plane of arbitrary small positive measure $\epsilon$ that contains in its interior a unit line segment in every direction. The sets $Q(\epsilon)$ can be chosen to be uniformly bounded with respect to $\epsilon$.

We repeat the construction of a Besicovitch set, see Besicovitch [@Be]. As we do not need to turn the line segment continuously, we drop from the construction the joints between the triangles. This allows us to keep $Q$ bounded regardless of $\epsilon$. We iterate a finite number of times, thus the resulting set is a polygon, a modification of the so called Perron-Schoenberg tree. For $Q$ we choose any open set that properly contains the latter so that the measure of their difference is small enough. We shall also call the set $Q$ a Besicovitch set.

It is now clear that the one-dimensional endpoint $$\begin{aligned}
 \label{est: endpoint}
 \normQRP{U(t)f}{2}{\infty}{1} \lesssim \normA{f}{2}\end{aligned}$$ fails on the characteristic functions of Besicovitch sets (thus we recover the result of [@GP] by different means).

\[lem: failed est\]

-   The Strichartz estimate $$\begin{aligned}
     \normQRP{U(t)f}{q}{\infty}{p}  \lesssim \normA{f}{a},          \tag{\ref{eq: hom Strich}'}\end{aligned}$$ where $0<a<\infty,\; 1\leq q,p \leq \infty$, fails for some $f \in L^a({\mathbb R}^{2n})$.

-   The Strichartz estimate $$\begin{aligned}
     \normQRP{U(t)f}{q}{\infty}{p}  &\lesssim \normBC{f}{b}{c}, \tag{\ref{eq: hom Strich mix}'}\end{aligned}$$ where $0< q,p, b,c\leq \infty$, fails for some $f \in L_x^bL_v^c$, apart from the trivial case $q=p=b=c=\infty$.

-   The Strichartz estimate $$\begin{aligned}
    \normQRP{W(t)F}{q}{\infty}{p} &\lesssim \normQRP{F}{\tilde q'}{\tilde r'}{\tilde p'},   \tag{\ref{eq: inhom Strich}'}\end{aligned}$$ where $1\leq q,\tilde q, p, \tilde r\leq \infty$, $1<\tilde p$, fails for some $F \in \spaceLqrptd$. Moreover, if $\tilde p=1$, estimate (\[eq: inhom Strich\]’) holds for all $F \in {L^{\tilde q'}_tL^{\tilde r'}_xL^{\infty}_v}$ if and only if $\trip$ and $\tript$ are two jointly KT-acceptable triplets, that is if and only if $\triptrip \in \Sigma_1 \cup B$, see definition \[dfn: joint accept\], when $n=1$, and the same up to an endpoint in $n>1$. The only unresolved estimate of this kind, therefore, is in $n>1$.

-   The Strichartz estimate $$\begin{aligned}
    \normQRP{W(t)F}{q}{r}{p} &\lesssim \normQRP{F}{\tilde q'}{1}{\tilde p'},   \tag{\ref{eq: inhom Strich}''}\end{aligned}$$ where $1\leq q,\tilde q, p, \tilde r\leq \infty$, $1<p$, fails for some $F \in \spaceLqrptd$. Moreover, if $p=1$, estimate (\[eq: inhom Strich\]’) holds for all $F \in {L^{\tilde q'}_tL^{1}_xL^{\tilde p'}_v}$ if and only if $\trip$ and $\tript$ are two jointly KT-acceptable triplets, that is if and only if $\triptrip \in \Sigma_2 \cup C$, see definition \[dfn: joint accept\], when $n=1$, and the same up to an endpoint in $n>1$. The only unresolved estimate of this kind, therefore, is in $n>1$.

We begin with the case $n=1$. Let us first consider the homogeneous estimates (\[eq: hom Strich\]’) and (\[eq: hom Strich mix\]’). We set $f=\chi_Q$, where $Q$ is a Besicovitch set on the plane $Oxv$ of measure $\epsilon > 0$. It suffices to consider the case $c<\infty$, the case $c=\infty$ is trivial since then we must have $q=p=b=c=\infty$, see section \[sect: counterex mix\]. By Hölder, $\normBC{\chi_Q}{b}{c} \lesssim \normBC{\chi_Q}{c}{c} = \epsilon^{1/c}$, thus disproving (\[eq: hom Strich mix\]’) is reduced to disproving (\[eq: hom Strich\]’). In view of the power invariance (\[power transform\]), we can always assume that $p=1$ in (\[eq: hom Strich\]’). But then we can use the argument that we produced to disprove above.

To construct a counterexample for $W(t)$ we consider an inhomogeneous term $F(t,x,v) = \chi(t\in[0,1]; (x,v) \in Q)$, where $Q$ is a Besicovitch set on the plane $Oxv$ of measure $\epsilon > 0$. Then $F$ is a characteristic function of a set in ${\mathbb R}^{2+1}$ with measure $\epsilon$. Therefore, as above estimate (\[eq: inhom Strich\]’) fails whenever $\tilde p>1$. The case when $\tilde p=1$ has two subcases. The first is when $1/q+1/\tilde q<1$, then estimate (\[eq: inhom Strich\]’) holds in view of theorem \[thm: infty case\], set $\Sigma_1$. In the endpoint case $1/q+1/\tilde q=1$ we have that $r=\tilde r=\infty$, $p=\tilde p=1$. These estimates would fail if we can find an explicit $F \neq 0$ satisfying $$\begin{aligned}
 \normBC{W(t)F}{\infty}{1} \gtrsim \int_{0}^{t} \rp {t-s} \normBC{F(s)}{1}{\infty}ds=\infty,\end{aligned}$$ supp $\normBC{F(s)}{1}{\infty} \subseteq[0,1]$, for example. For simplicity we can choose $F$ to be of product type, i.e. $F(t,x,v)=\phi(t)\psi(x)g(v)$ and set $\phi(t) = \chi_{[0,1]}$, $g(v)\equiv 1 \in L^{\infty}({\mathbb R})$, and choose any $\psi \in L^1$, $\psi \geq 0$. We have $$\begin{split}
  \Int W(t)F dv = \inT \Int \phi(s)\psi(x-(t-s)v) dvds \\
  \geq \normP{g}{\infty}\normP{\psi}{1} \inT \rp {t-s}\phi(s)ds,
 \end{split}$$ which proves the desired inequality. Note that the change of order of integration above is crucial for this argument. Consequently, the argument works only for $p=1$.

By duality, see the duality lemma in [@O], estimate (\[eq: inhom Strich\]”) is equivalent to estimate (\[eq: inhom Strich\]’). And finally, when in $n>1$ we can reproduce the same arguments by replacing $Q$ with the product set $Q^n=Q\times Q \times ... Q$.

Note that the problem of finding all possible Strichartz estimates for the operators $U(t)$ and $W(t)$ is now completely solved in $n=1$ in view of counterexamples in the previous lemma that covered all endpoint cases in that dimension, and the counterexamples in the following subsections, which cover all other non-endpoint cases.

Let us now try finding a geometric interpretation of $\normBC{U(t)\chi_{Q}}{r}{1}$, where $1 \leq r < \infty$. It is not hard to see that $$\begin{aligned}
 \Int \int_{\gamma_{x,t}} \chi_{Q}(l) dl dx = \sqrt{1+t^2} \iint_{{\mathbb R}^2} R(\pi/2 - \theta)\chi_{Q}(l,m) dl dm,\end{aligned}$$ where by $R(\psi)$ we denote rotation by an angle $\psi$ and $\cot \theta = -t$. Therefore, if we denote $$N_r(\theta, Q) = \normBC{ R(\pi/2 - \theta)\chi_{Q}}{r}{1},$$ we obtain the representation $$\begin{aligned}
 \normQRP{U(t)\chi_Q}{q}{r}{1} &= {\left\Vert\left( \rp {\sqrt{1+t^2}} \right)^{1/r'} N_r(\theta,Q)\right\Vert}_{L^q({\mathbb R})}\\
                               &= {\left\Vert\left(\sin \theta \right)^{1/r'-2/q} N_r(\theta,Q)\right\Vert}_{L^q(0, \pi)}\end{aligned}$$

Note that when the exponent triplet $(q,r,1)$ is KT-admissible in $n=1$, that is $1/q=1/2(1/p-1/r)$ we have that $$\begin{aligned}
 \normQRP{U(t)\chi_Q}{q}{r}{1} = {\left\VertN_r(\theta,Q)\right\Vert}_{L^q(0, \pi)}.\end{aligned}$$

We can generalize the above idea to any dimension $n$ by considering the product $Q^n=Q\times Q\times \dots Q$. By repeating the argument above we obtain $$\begin{aligned}
 \normQRP{U(t)\chi_{Q^n}}{q}{r}{1} &= {\left\Vert\left( \rp {\sqrt{1+t^2}} \right)^{n/r'} N_r^n(\theta,Q)\right\Vert}_{L^q({\mathbb R})}.\end{aligned}$$

Let $Q$ be the unit circle in the plane $Ox_1v_1$. Then obviously $N_r(\theta,Q^n) = $ const for all $\theta \in [0, \pi]$. Consequently, the norm above is finite only if $r \geq 1$, $qn/r'> 1$. By the power invariance (\[power transform\]), we generalize to any admissible $\trip$ and obtain the restrictions $$\begin{aligned}
\label{cond: range r}
  r \geq a, \quad \rp q < n \left(\rp p - \rp r\right), \quad \text{or } \; q=\infty,\; 1 \leq\, p=r \,\leq \infty,\end{aligned}$$ to the range of validity of estimate . The restrictions above trivially imply that $p \leq r$.

Homogeneous Estimates
---------------------

By scaling, see lemma \[lem: scaling\], estimate $$ \normQRP{U(t)f}{q}{r}{p} \lesssim \normA{f}{a}, \quad \forall f \in \spaceLa{a}, \tag{\ref{eq: hom Strich}}$$ can only hold if $$\begin{aligned}
 \rp q + \frac n r = \frac n a, \quad a=\operatorname{HM}(p,r).\end{aligned}$$ By the $TT^*$-principle, estimate (\[eq: hom Strich\]) with $a=2$ is equivalent to $$\normQRP{TT^*F}{q}{r}{r'} \lesssim \normQRP{F}{q'}{r'}{r}, \quad \forall F \in {L^{q'}_tL^{r}_xL^{r'}_v}.$$ By translation invariance in $x$, see proposition \[prop: Horm\], $r \geq r'$, or equivalently $r \geq 2$. By the power invariance (\[power transform\]) we generalize the case $a=2$ to any $0<a<\infty$, the respective condition is $r \geq a$.

We now show the upper bound on $r$, that is $r \leq r^*(a)$. To that end, we first consider again the special case $a=2$. Note that the condition $q \geq 2$ is equivalent to the one at hand. By translation invariance in $t$, see proposition \[prop: Horm\], $q \geq q'$, or equivalently $q \geq 2$. By the power invariance (\[power transform\]) we generalize the case $a=2$ to any $0<a<\infty$, the respective condition is $q \geq a$, or equivalently $r \leq r^*(a)$. Thus in view of , $p^*(a)\leq p\leq a \leq r\leq r^*(a)$.

Homogeneous estimates for mixed $L^p$-data {#sect: counterex mix}
------------------------------------------

If we consider the mixed $L^p$-data estimate $$ \normQRP{U(t)f}{q}{r}{p} \lesssim \normBC{f}{b}{c}, \tag{\ref{eq: hom Strich mix}}, \quad \forall f \in \spaceLbc{b}{c}$$ then most of the preceding section applies as well. By scaling, we have that the conditions $$\begin{aligned}
 \label{cond: range r b}
 \rp q + \frac n r = \frac n b, \quad \operatorname{HM}(p,r)=\operatorname{HM}(b,c) \stackrel{def}{=}a,\end{aligned}$$ are necessary. The conditions $p\leq a \leq r$ and $a \leq r$ carry over from the preceding section. We do not have, however, a suitable counterexample giving that exact upper bound $r^*(b,c)$ to $r$ in estimate in $n>1$. The last condition we need to verify is that $b\leq c$. Estimate is equivalent to $$ \normQRP{TT^*F}{q}{r}{p} \lesssim \normQRP{F}{1}{b}{c},  \quad \forall F \in L^1_t\spaceLbc{b}{c}.$$ By duality, it is equivalent to $$ \normQRP{TT^*F}{\infty}{b'}{c'} \lesssim \normQRP{F}{q'}{r'}{p'},  \quad \forall F \in \spaceLqrpd.$$ By the restrictions on the inhomogeneous estimates, see section \[sect: counterex inhom\], we obtain $b' \geq c'$ or equivalently $b\leq c$. Thus we have that $p < b \leq a \leq c < r$ (follows from and ) or $a=b=c=p=r$ (and $q=\infty$).

Global inhomogeneous estimates {#sect: counterex inhom}
------------------------------

In this section we present some counterexamples to the validity of the inhomogeneous estimates $$ \normQRP{W(t)F}{q}{r}{p} \lesssim \normQRP{F}{\tilde q'}{\tilde r'}{\tilde p'}. \tag{\ref{eq: inhom Strich}}$$ By scaling, see lemma \[lem: scaling\], we obtain that the conditions $$\begin{aligned}
 \rp q + \rp {\tilde q} = n \left(1 - \rp r - \rp {\tilde r} \right), \quad \operatorname{HM}(p,r)=\operatorname{HM}(\tilde p', \tilde r') \stackrel{def}{=}a,\end{aligned}$$ are necessary.

Consider $F(t,x,v) = \chi \left( 0 \leq t \leq 1, {\left\vertx\right\vert} \leq 1, {\left\vertv\right\vert} \leq 1 \right).$ When $t \gg 1$ we have that $$TT^*F \approx W(t)F \approx \chi \left( {\left\vertv- \frac x {t}\right\vert} \leq \frac 1 t, {\left\vertv\right\vert} \leq 1 \right) \approx \chi \left\{ v \sim \frac 1 t, x \sim t\right\}.$$ Hence, $$\normBC{W(t)F}{r}{p} \sim t^{\frac n r - \frac n p}, \quad t \gg 1.$$ It follows that $ \normQRP{W(t)F}{q}{r}{p} < \infty$ only if $$\label{cond: inhom accep}
 \left(\frac n r - \frac n p\right)q < -1, \quad \text{or if}\;\, q=\infty, r=p.$$ By duality, see the duality lemma of [@O], also applies to the dual exponents $\tript$. Thus we have that the conditions $p \leq  r$ and $\tilde p \leq \tilde r$ are necessary for the validity of estimate . The same conclusion applies if we replace $W(t)$ by $TT^*$ in .

We now show that conditions $$\begin{aligned}
   \frac 1 q + \frac 1 {\tilde q} \leq 1,   \quad \frac 1 r + \frac 1 {\tilde r} \leq 1,
 \end{aligned}$$ are necessary for the validity of estimate . Indeed, that follows from the translation invariance of $W(t)$ in $t$ and $x$ and proposition \[prop: Horm\]. We check this fact only for $t$. Consider $F_{\tau}(t) = F(t-\tau)$ and $W(t)F_{\tau}$. We have $$\begin{aligned}
    \intI U(t-s)F(s-\tau) ds = \int_{\infty}^{t-\tau} U(t-\tau-\sigma)F(\sigma) d\sigma,
 \end{aligned}$$ or in other words $W(t)F_{\tau}=W(t-\tau)F$, QED.

Thus we have fully verified the necessity of the condition that $\trip$ and $\tript$ be two jointly KT-acceptable exponent triplets (apart from some boundary cases, e.g. $q=\infty$, etc.)

We do not have a suitable counterexample showing the necessity of condition $$\frac {n -1}{p'} < \frac {n}{\tilde{r}}, \quad \frac {n-1}{\tilde{p}'} < \frac {n}{r}. \tag{\ref{cond: glob inhom}}$$ However, we can show that the similar condition $$\label{cond: glob inhom a}
        \frac n {p'} < \rp {\tilde q} + \frac {n}{\tilde{r}}, \quad \frac {n}{\tilde{p}'} < \rp {q} + \frac {n}{r}, \tag{\ref{cond: glob inhom}a}$$ is sharp. Indeed, is a direct consequence of the two conditions , . Condition implies whenever $p' \leq \tilde q$ and $\tilde p' \leq q$. Thus, if there are some other global inhomogeneous estimates for $W(t)$ not included in theorem \[thm: infty case\], they must belong to the range $\tilde q < p'$ or $q < \tilde p'$.

Local inhomogeneous estimates {#sect: counterex loc inhom}
-----------------------------

In this section we show that the condition $\operatorname{HM}(p,r)=\operatorname{HM}(\tilde p', \tilde r')$ is necessary for the validity of the local inhomogeneous estimates, e.g. $$\label{eq: inhom Strich loc}
 {\left\VertW(t)F\right\Vert}_{L^{q}_t \left([2,3]; L^{r}_x L^{p}_v\right)} \lesssim {\left\VertF\right\Vert}_{L^{\tilde q'}_t([1,2]; L^{\tilde r'}_xL^{\tilde p'}_v)}. $$ This implies that we cannot perturbate the exponents $p,r,\tilde p, \tilde r$ freely. Consequently, the method of Keel and Tao [@KT] that was successful in treating the endpoints for the wave and the equations does not apply in the present context.

Let us set $F(t,x,v)=\chi \left(t \in [0, 1],\; (x,v) \in Q_R \right)$, where by $Q_R$ we denote the cube of side length $2R$ with center at the origin of ${\mathbb R}^{2n}$, that is $$\begin{aligned}
 Q_R = \left\{ (x,v) : \normInfty{x} \leq R,\; \normInfty{v} \leq R \right\},\end{aligned}$$ where $\normInfty{x} = \sup_{1 \leq i \leq n} {\left\vertx_i\right\vert}, \text{for } x = (x_1,..,x_n)$. Hence, $${\left\VertF\right\Vert}_{L^{\tilde q'}_t([1,2]; L^{\tilde r'}_xL^{\tilde p'}_v)} \sim R^{\frac n {\tilde p'} + \frac n {\tilde r'}}.$$ We set $\tau=t-s$, and consider the set $Q_R(\tau)$ given by $$\begin{aligned}
 \normInfty{x - (t-s)v} \leq R,\; \normInfty{v} \leq R.\end{aligned}$$ Then, for $t \in [2,3]$, $s \in [0,1]$, and therefore $\tau \in [1,3]$, we have the inclusions $$Q_{R/4} \subset Q_R(\tau) \subset Q_{4R}.$$ Hence, $${\left\VertW(t)F\right\Vert}_{L^{q}_t \left([2,3]; L^{r}_x L^{p}_v\right)} \sim R^{\frac n p + \frac n r}.$$ We conclude that condition $$\frac 1 r + \frac 1 p = \frac 1 {\tilde r'} + \frac 1 {\tilde p'}$$ is necessary for the validity of the local estimates .

Some Open Questions
===================

The following estimates/questions remain unresolved.

-   In $n>1$, the endpoint $$\begin{aligned}
    \label{est: endpoint r}
     \normQRP{U(t)f}{2}{r*}{{r^*}'} \lesssim \normA{f}{2}.\end{aligned}$$

-   In $n>1$, inhomogeneous endpoints of the type $$\begin{aligned}
     \label{est: endpoint infty}
     \normQRP{W(t)F}{q}{\infty}{p} \lesssim \normQRP{F}{\tilde q}{p}{\infty}, \quad 1/q+1/{\tilde q}=1, \; p=n.\end{aligned}$$

-   In $n>1$, inhomogeneous endpoints of the type $$\begin{aligned}
     \label{est: endpoint 1}
     \normQRP{W(t)F}{q}{p}{1} \lesssim \normQRP{F}{\tilde q}{1}{p}, \quad 1/q+1/{\tilde q}=1, \; p=n.\end{aligned}$$

-   In $n>1$, the full range of acceptability for the estimate $$\begin{aligned}
      \normQRP{W(t)F}{q}{r}{p} \lesssim \normQRP{F}{\tilde q'}{\tilde r'}{\tilde p'}.\end{aligned}$$

-   In $n>1$, the upper bound $r^*(b,c)$ to $r$ in the mixed $L^p$-data estimate $$\begin{aligned}
      \normQRP{U(t)f}{q}{r}{p} \lesssim \normBC{f}{b}{c}.\end{aligned}$$ Is $r^*(b,c)=r^*(a)$ or is $r^*(b,c) > r^*(a)$?

We would also like to ask the following questions

-   Is there any improvement to the above estimates for spherically symmetric $f$ or $F$?

-   Is there any improvement to the above estimates in the case when is defined over a finite velocity space, (the only physically relevant setting), that is $v \in V$ for some compact subset $V$ of ${\mathbb R}^n$?

-   What happens with the estimates of lemma \[lem: failed est\] that we showed to fail, if one replaces the $L^\infty$-norm by the BMO-norm?

-   What is the full range of acceptability of the local estimates, that is the set $\mathcal E$? The answer to this question is unknown even in the context of the equation, see Foschi [@F sect. Open questions].

Applications {#sect: applic}
============

Below we make an application of the estimates that we have derived above to a nonlinear kinetic model of bacterial chemotaxis. The model describes a population of bacteria in motion in a field of a chemoattractant, see [@B], [@BC], and the references therein. So we consider the following system $$\begin{aligned}
 &{\partial_t u(t,x,v) + v\cdot\nabla_x u(t,x,v)} = \int_{v' \in V} T[S](t,x,v,v')u(t,x,v')dv', \label{eq: kin mod 1}\\
 &\qquad -\int_{v' \in V} T[S](t,x,v',v)u(t,x,v)dv', \quad t>0,\, x \in {\mathbb R}^n, v \in V \notag\\
 &-\Delta S + S =\rho(t,x) \stackrel{def}{=} \int_{v\in V}u(t,x,v)dv,\label{eq: kin mod 2}\\
 &u(0,x,v)=f(x,v) \geq 0, \label{eq: kin mod 3}\end{aligned}$$ where by $u(t,x,v)$ we denote the cell density in phase space and we assume that the space of admissible velocities $V \in {\mathbb R}^n$ is bounded. The cell density in physical space is denoted by $\rho(t,x)$, where $t$ and $x$ are a time and a space coordinate respectively. The free transport operator ${\partial_t u(t,x,v) + v\cdot\nabla_x u(t,x,v)}$ describes the free runs of the bacteria which have velocity $v \in V$. The nonlinear terms in the right hand side of denote a scattering operator that expresses the reorientation process occurring during bacterial migration towards regions of high concentration in chemoattractant $S(t,x)$. In this model the second integral term is always positive and will be omitted in our estimates below. Based on biologically realistic assumptions, the turning kernel $T[S](t,x,v,v')$ is assumed to satisfy the following bound $$\label{cond: turn ker Lp}
 \normXVV{T[S](t,\cdot, \cdot, \cdot)}{r}{p_1}{p_2} \lesssim_{{\left\vertV\right\vert}, p_1, p_2} {\left\VertS(t,\cdot)\right\Vert}_{L^{r}} + {\left\Vert\nabla S(t, \cdot)\right\Vert}_{L^{r}},$$ whenever $r \geq p_1,\, p_2$.

Following the bootstrap argument in theorem 3 of [@B], we shall extend their result to dimensions $n \geq 3$.

\[thm: kin mod\] Suppose that the initial data $f \in L^1({\mathbb R}^{2n}) \cap L^{n/2}({\mathbb R}^{2n})$, where $n \geq 3$ and ${\left\Vertf\right\Vert}_{L^{n/2}({\mathbb R}^{2n})}$ is sufficiently small and the turning kernel $T$ satisfies condition . Then the IVP - has a global weak solution $u \in L^{q}([0, \infty]; L^{r}({\mathbb R}^{2n};$\
$L^{p}(V)))$, where $$q=n,\quad  r=\frac {n^2} {2n-1},\quad  p = \frac {n^2} {2n+1}.$$

Applying estimate to we have $$\begin{aligned}
 \label{eq: append 1}
 \normQRP{u}{q}{r}{p} \lesssim \normA{f}{a} + \normQRP{\int_{V} T[S]u'dv'}{\tilde q'}{\tilde r'}{\tilde p'},\end{aligned}$$ where we have set $u'=u(t,x,v')$ and assumed that the Lebesgue exponents above are subject to the conditions of theorem \[thm: Strich admiss\]. To the integral term above we first apply Hölder’s inequality to get $$\begin{aligned}
 \int_{V} T[S](t,x,v,v')u(t,x,v')dv' \leq {\left\VertT[S](t,x,v,\cdot)\right\Vert}_{L^{p'}_{v'}}{\left\Vertu(t,x, \cdot)\right\Vert}_{L^p_{v'}}.\end{aligned}$$ We next take the $L^{\tilde p'}$-norm in $v$ and then take the $L^{\tilde r'}$-norm in $x$ using Hölder’s inequality with $\frac 1 {\tilde r'} = \rp r + \rp w$ to get $$\begin{aligned}
 {\left\Vert\int_{V} T[S]u(t,x,v')dv'\right\Vert}_{L^{\tilde r'}_xL^{\tilde p'}_v} \leq \normXVV{T[S](t,\cdot,\cdot,\cdot)}{w}{\tilde p'}{p'}{\left\Vertu(t,\cdot, \cdot)\right\Vert}_{L^r_xL^p_{v}}.\end{aligned}$$ Assuming that $w \geq \tilde p',\, p'$, we can use condition to get $$\begin{aligned}
 \normXVV{T[S](t,\cdot,\cdot,\cdot)}{w}{\tilde p'}{p'} \lesssim {\left\VertG*\rho\right\Vert}_{L^w_x} + {\left\Vert\nabla G*\rho\right\Vert}_{L^w_x},\end{aligned}$$ where $G$ is the Bessel potential $$G(x)= \int_{0}^{\infty} e^{-\pi \frac {{\left\vertx\right\vert}^2} {4s}} s^{\frac {-n+2} 2} \frac {ds} s,$$ and we have used the representation formula $S=G*\rho$. We recall the following two estimates on $G$, see [@Stein], $G \in L^{b}({\mathbb R}^n)$, for $1\leq b < \frac n {n-2}$, and $\nabla G(x) \lesssim \rp {{\left\vertx\right\vert}^{n-1}}$ for all $x$. We use Young’s inequality to estimate the $G$-term and the HLS inequality to estimate the $\nabla G$-term to get $$\begin{aligned}
 {\left\VertG*\rho\right\Vert}_{L^w_x} \lesssim {\left\VertG\right\Vert}_{L^{n/(n-1)}} {\left\Vert\rho\right\Vert}_{L^{r}},\\
 {\left\Vert\nabla G*\rho\right\Vert}_{L^w_x} \lesssim {\left\Vert\nabla G\right\Vert}_{L^{n/(n-1)}} {\left\Vert\rho\right\Vert}_{L^{r}},\end{aligned}$$ where $1+\rp w = \frac {n-1} n + \rp r$ and $n \geq 2$. Noting that ${\left\Vert\rho\right\Vert}_{L^{r}} \lesssim_{{\left\vertV\right\vert}} {\left\Vertu(t,\cdot, \cdot)\right\Vert}_{L^{r}_xL^p_v}$ and setting $\tilde q'=q/2$ we obtain $$\begin{aligned}
 \normQRP{\int_{V} T[S]u'dv'}{q}{r}{p} \lesssim {\left\Vert{\left\Vertu(t,\cdot, \cdot)\right\Vert}_{L^{r}_xL^p_v}^2\right\Vert}_{L^{q/2}_t} \lesssim \normQRP{u}{q}{r}{p}^2.\end{aligned}$$ Hence, we obtain the following a-priori estimate for $u$ $$\label{eq: append 2}
 \normQRP{u}{q}{r}{p} \lesssim \normA{f}{a} + \normQRP{u}{q}{r}{p}^2,$$ whenever the following system of conditions $$\begin{aligned}
 \operatorname{HM}(r,p)=\operatorname{HM}(\tilde r', \tilde p') =a,\\
 \rp q = \frac n 2 \left(\rp p - \rp r\right), \quad \rp {\tilde q} = \frac n 2 \left(\rp {\tilde p} - \rp {\tilde r}\right),\\
 \tilde q' = q / 2, \quad w \geq \tilde p', p', \quad n \geq 2\\
 p^*(a) < p \leq a, \quad a \leq r < r^*(a),\\
 p^*(a') \leq \tilde p \leq a', \quad a' \leq \tilde r \leq r^*(a'),\\
 1+\rp w = \frac {n-1} n + \rp r,\end{aligned}$$ admits a solution. One could check that the condition $w \geq p'$ is equivalent to $a \geq n/2$. Therefore, we shall look for a solution with the mildest regularity assumption $a=n/2$. Thus, $a'=n/(n-2)$ and $p^*(a) = n^2/2(n+1)$, $r^*(a)=n^2/2(n-1)$, $p^*(a')=n^2/(n+1)(n-2)$, $r^*(a')=n^2/(n-1)(n-2)$ in $n \geq 3$. Since this section serves for illustrative purposes only, we shall try to guess a simple solution without making any attempt to find out the full (two parameter) solution space. An educated guess can be $q = {n}$, $r = {n^2}/(2n-1)$, $p = {n^2}/(2n+1)$, and $a=n/2$. This implies $\tilde q = n/(n-2)$, $\tilde r = n^2/(n-1)(n-2)$, $\tilde p = n^2/(n+1)(n-2)$, and $n \geq 3$. Note that $\trip$ is a non-endpoint KT-admissible triplet while $\tript$ is an endpoint KT-admissible triplet. The condition $w \geq \tilde p'$ holds as we have that $w=n^2/(n-1))$ and $\tilde p'=n^2/(n+2)$. Apparently, all other conditions in the system above are satisfied. Hence estimate holds with the chosen exponents due to theorem \[thm: Strich admiss\], see also remark \[rem: admiss endpoint\].

As a final comment we mention that the a-priori estimate is used in a standard bootstrap argument to finish the proof of the theorem, see remark 4.2 in [@B].

[^1]: The author would like to thank to Damiano Foschi and Francisco Villarroya for fruitful discussions related to this work

[^2]: That is equation with $F(t) = 0$.

[^3]: With $L^{q,c}_t$ replaced by $L^q_t$.

[^4]: However, the exact upper bound on $r$ is not known in the case $n>1$, $b \neq c$, see section \[sect: counterex mix\], and we cannot discard the possibility of omissions of some estimates in that case.
