Assume $p,q\in (1,\infty)$, $r\in [p,\infty)$, $s\in [q,\infty)$ and $$ 1<\frac{d}{p}+\frac{2}{q}=1+\frac{d}{r}+\frac{2}{s}. $$ Let $u\in C_c^\infty((0,1)\times B_1)$, where $B_1=\{x\in \mathbb{R}^d|\ |x|<1\}$.

I was wondering whether the following inequality is correct? \begin{equation} \|\nabla u\|_{L^s_tL^r_x} \leq C(d, p, q, r,s) \left( \|\partial_t u\|_{L^q_tL^p_x}+ \|\nabla^2 u\|_{L^q_tL^p_x}\right). \end{equation}

Remark: I know the above estimate is true if $1<p=q<d+2$ and $r=s=\frac{(d+2)p}{d+2-p}$. I guess the general cases can be obtained by some results in interpolation theory, but I am not familiar with that.