Let $M({\mathbf x}, {\mathbf y}), {\mathbf x}, {\mathbf y}\in {\mathbb R}^2$ be such a bounded function that the operator:
$O[M](f) = \int_{{\mathbb R}^2} M({\mathbf x}, {\mathbf y}) f({\mathbf y}) d {\mathbf y}$
is a self-adjoint linear operator from $L_{2}({\mathbb R}^2)$ to $L_{2}({\mathbb R}^2)$. I.e., we know that $O[M](f)\in L_{2}({\mathbb R}^2)$ whenever $f\in L_{2}({\mathbb R}^2)$.
Let $G_\epsilon(x) = \frac{1}{\sqrt{2\pi\epsilon^2}}e^{-\frac{x^2}{2\epsilon^2}}$ be the gaussian. Let us define:
$M_{\epsilon} (x_1, y_1) = \int_{{\mathbb R}^2} M(x_1, x_2, y_1, y_2) G_\epsilon(x_2) G_\epsilon(y_2) d x_2 y_2$
If we additionally require that $M$ is Lipschitz, then $M_{\epsilon} (x_1, y_1) \mathop\rightarrow\limits^{\epsilon\rightarrow +0} M(x_1, 0, y_1, 0)$ uniformly over $x_1, y_1$, because:
$|M_{\epsilon} (x_1, y_1) - M(x_1, 0, y_1, 0)| = |\int_{{\mathbb R}^2} (M(x_1, x_2, y_1, y_2) - M(x_1, 0, y_1, 0)) G_\epsilon(x_2) G_\epsilon(y_2) d x_2 y_2| \leq C \int_{{\mathbb R}^2} (|x_2|+|y_2|) G_\epsilon(x_2) G_\epsilon(y_2) d x_2 y_2 = O(\epsilon)$
My question is the following. If we omit any additional requirements, i.e. we only have that $|M({\mathbf x}, {\mathbf y})|\leq C, O[M]: L_{2}({\mathbb R}^2)\rightarrow L_{2}({\mathbb R}^2)$, is it enough to prove that
$M_{\epsilon} (x_1, y_1) \mathop\rightarrow\limits^{\epsilon\rightarrow +0} \tilde{M}(x_1, y_1)$ uniformly over $x_1, y_1$,
where $\tilde{M}(x_1, y_1) = \lim_{\epsilon\rightarrow +0}M_{\epsilon} (x_1, y_1)$ is a pointwise limit?