$\newcommand{\RR}{\mathbb R}\newcommand{\diff}{\, \mathrm d}$

We fix $T \in (0, \infty)$ and let $\mathbb T$ be the interval $[0, T]$. Let $p, q \in [1, \infty)$.

• Let $E$ be the space of all real-valued Lebesgue measurable functions on $\mathbb T \times \RR^d$ with the finite norm $$ \| g \|_E := \sup_{x \in \RR^d} \bigg ( \int_{t_0}^{t_1} \bigg ( \int_{B(x, 1)} | g(s, y) |^p \diff y \bigg )^{\frac{q}{p}} \diff s \bigg )^{\frac{1}{q}. $$

• Let $F$ be the space of all real-valued Lebesgue measurable functions on $\mathbb T \times \RR^d$ with the finite norm $$ \| g \|_F := \bigg ( \int_{t_0}^{t_1} \bigg ( \sup_{x \in \RR^d} \int_{B(x, 1)} | g(s, y) |^p \diff y \bigg )^{\frac{q}{p}} \diff s \bigg )^{\frac{1}{q}. $$

Clearly, $\| g \|_E \le \| g \|_F$ and thus $F \subset E$.

Is there a constant $c >0$ such that $\| g \|_F \le c \| g \|_E$ for all $g \in E$?

Thank you so much for your elaboration!