Let $X$ and $Y$ be compact regions of $n$- and $m$-dimensional Euclidean spaces respectively.
For any $p,q \in [1,\infty)$, define $L^{p,q}(X \times Y)$ be the space of real valued functions $f :X \times Y \to \mathbb{R}$ such that $\int_X \Bigl(\int_Y \lvert f(x,y) \rvert^q dy\Bigr)^{p/q} dx < \infty$.
Let $\phi^n : \mathbb{R}^n \to \mathbb{R}$ be the standard mollifier defined by \begin{equation} \phi_n(x):=C\exp\Bigl(\frac{1}{\lvert x \rvert^2-1}\Bigr) \chi_{\lvert x \rvert \leq 1}(x) \end{equation} where $C>0$ is chosen so that $\int_{\mathbb{R}^n} \phi^n=1$ Note that $\phi^n$ is a smooth function compactly supported on the closed unit ball.
Let $\phi^n_{\epsilon}(x):=\frac{1}{\epsilon^n}\phi_n(x/\epsilon)$ for any small $\epsilon>0$.
Similarly, we think of $\phi^m_{\epsilon}(y) : \mathbb{R}^m \to \mathbb{R}$.
For any $f \in L^{p,q}(X \times Y)$, define the convolution with $\phi^n_\epsilon \otimes \phi^m_\epsilon$ by \begin{equation} f*[\phi^n_\epsilon \otimes \phi^m_\epsilon](x,y):=\int_{X \times Y} f(x',y')\phi^n_\epsilon(x-x')\phi^m_\epsilon(y-y')dx'dy' \end{equation}
I wondder if $f * [\phi^n_\epsilon \otimes \phi^m_\epsilon]$ converges to $f$ as $\epsilon \to 0^+$ in the above $L^{p,q}$. That is, \begin{equation} \int_X \Bigl(\int_Y \bigl\lvert f*[\phi^n_\epsilon \otimes \phi^m_\epsilon](x,y) -f(x,y)\bigr\rvert^q dy\Bigr)^{p/q} dx \to 0^+ \end{equation} holds as $\epsilon \to 0^+$?
Could anyone please clarify for me?
