For the classes of functions of two variables $W_2(\Omega_{m,\gamma},\Psi)=\{ f \in L_{2,\gamma}(\mathbb{R}^2) : \Omega_{m,\gamma}(f,t) \leqslant \Psi(t) \, \forall t \in (0,1)\}$, $m \in \mathbb{N}$, where $\Omega_{m,\gamma}$ is a generalized modulus of continuity of the $m$-th order, and $\Psi$ is a majorant, the upper and lower bounds for the ortho-, Kolmogorov, Bernstein, projective, Gel'fand, and linear widths in the metric of the space $L_{2,\gamma}(\mathbb{R}^2)$ are found. The condition for a majorant under which it is possible to calculate the exact values of the listed extreme characteristics of the optimization content is indicated. We consider the similar problem for the classes
$W^{r,0}_2(\Omega_{m,\gamma},\Psi)=L^{r,0}_{2,\gamma}(D,\mathbb{R}^2) \cap W^r_2(\Omega_{m,\gamma},\Psi)$, $r,m \in \mathbb{N}$, $\big(D=\frac{\displaystyle \partial^2}{\displaystyle \partial x^2} + \frac{\displaystyle \partial^2}{\displaystyle \partial y^2} -2x\frac{\displaystyle \partial}{\displaystyle \partial x} -2y\frac{\displaystyle \partial}{\displaystyle \partial y}$ being the differential operator$\big)$. Those classes consist of functions $f \in L^{r,0}_{2,\gamma}(\mathbb{R}^2)$ whose Fourier--Hermite coefficients are $c_{i0}(f) = c_{0j}(f)=c_{00}(f)=0$ $\forall i, j \in \mathbb{N}$. The $r$-th iterations $D^rf = D(D^{r-1}f)$ $(D^0f \equiv f)$ belong to the space $L_{2,\gamma}(\mathbb{R}^2)$ and satisfy the inequality $\Omega_{m,\gamma}(D^rf,t) \leqslant \Psi(t)$ $\forall t \in (0,1)$. On the indicated classes, we have determined the upper bounds (including the exact ones) for the Fourier--Hermite coefficients. The exact results obtained are specified, and a number of comments regarding them are given.