intervals of arbitrary lengths. Optimizing over $\epsilon$ yields $$\| D^{\sigma} \nabla u\|_{L^p L^q}^2 \lesssim \| u\|_{H^1}^2 +
 \| u\|_{H^1} \|\Box_g u\|_{L^2}$$ We want to apply this result to the functions $S_{\lambda}u_{{\lambda},d}$ in Definition \[xst\]. By we obtain

a\) Let $(\sigma,p,q)$ verifying $$\sigma = -\frac{n}2 + \frac{1}p +\frac{n}q, \qquad 2 \leq p \leq
\infty, \ \ 2 \leq q < \infty$$ Then for $(\sigma,p,q)$ as in we have $$\| S_{\lambda}\nabla u\|_{L^p L^q}  \lesssim \lambda^{1-s-\sigma}
d^{\frac12-\theta} \|u\|_{X^{s,\theta}_{{\lambda},d}}$$ If additionally $\theta > \frac12$ then $$\| S_{\lambda}\nabla u\|_{L^p L^q}  \lesssim \lambda^{1-s-\sigma}
\|u\|_{X^{s,\theta}}$$ b) If instead $$\frac{2}p +\frac{n-1}q \geq \frac{n-1}2$$ then $$\| S_{\lambda}\nabla u\|_{L^p L^q}  \lesssim  \lambda^{1-
s-\sigma+\frac12(\frac{2}p +\frac{n-1}q - \frac{n-1}2)}
d^{\frac12-\theta-\frac12(\frac{2}p +\frac{n-1}q - \frac{n-1}2)}
\|u\|_{X^{s,\theta}_{{\lambda},d}}$$

\[sob\]

The interesting triplets of indices for $(\sigma,p,q)$ in $4+1$ dimensions are $$(0,\infty,2) \text{(energy)} \qquad (-\frac12,\frac{10}3,\frac{10}3)
\text{(Strichartz)}  \qquad (-\frac56,2,6) \text{(Pecher)}$$ In addition, we can also use the index $q=\infty$. Thus we obtain the triplets $$(-2,\infty,\infty), \qquad (-\frac32,2,\infty)$$ For the case when $\theta<\frac12$, we rely on the additional triplets $$(-\frac16,2,3),\qquad (\frac14,4,2)$$

For convenience we summarize the bounds we need for $\tilde
X^{s,\theta}_{{\lambda},d}$:

For $0 < \theta < 1$ we have $${\lambda}^{s-1} \|S_{\lambda}\nabla u\|_{L^\infty L^2} + {\lambda}^{s-\frac52} \|S_{\lambda}\nabla u\|_{L^2
  L^\infty} + {\lambda}^{s-3} \| S_{\lambda}\nabla u\|_{L^\infty} \lesssim
d^{\frac12-\theta} \|u\|_{\tilde X^{s,\theta}_{{\lambda},d}}$$ $${\lambda}^{s-\frac{17}{12}} \|S_{\lambda}\nabla 