Let the space $L_{p,q}(\Bbb R^n_T)$ be defined as the set of all measurable $f:\Bbb R^{n}\times(0,T)\to\Bbb R$ such that $||f||_{p,q}<\infty$, where $$ ||f||_{p,q}:=\left(\int_0^T\left( \int_{\Bbb R^n}|f|^pdx \right)^{q/p}dt \right)^{1/q}. $$
Is it true that, for $p,q\in[1,\infty)$, $C^0(\Bbb R^{n}\times (0,T))$ and $C^{\infty}_c(\Bbb R^{n}\times (0,T))$ are dense in $L_{p,q}(\Bbb R^n_T)$?
For $p=q$ this is not hard:
Density of $C^0(\Bbb R^{n}\times (0,T))$ is just the fact that $C^0(\Bbb R^{n}\times (0,T))$ is dense in $L^p$ norm. As for $C^{\infty}_c(\Bbb R^{n}\times (0,T))$ we can use convolution argument and the fact that $C^0(\Bbb R^{n}\times (0,T))$ is dense.
For $p\ne q$, I don't know how to deal with the different exponents.
I have seen that there is a theorem from Bochner integral that functions of the form $$ g(t)=\sum_{i=1}^n x_i \mathbf 1_{E_i}(t) $$ are dense in $L_q((0,T);X)$ where $X$ is a Banach space, $E_i$ is a measurable set in $(0,T)$. Convolution in time direction show that the set of all $$ g(t)=\sum_{i=1}^n x_i \mathbf \phi_i(t), $$ where $\phi_i\in C^{\infty}((0,T))$, is also dense.
If we take $X=L^p(\Bbb R^n)$, I believe we can choose smooth approximation of $x_i\in L^p(\Bbb R^n)$ and prove what I want. However, I only know that this works for $L_q((0,T);X)$ which consists of Bochner-measurable functions. I am not certain if it applies to $L_{p,q}(\Bbb R^n_T)$ as I defined it or not.
