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 