\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.
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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, 