Difference analogues of O. A. Ladygenskaya's multiplicative inequalities for functional spaces $\mathring W{}_2^1(\Omega)$, $W_{2,0}^2(\Omega)$

Here we prove several multiplicative inequalities for the spaces $\mathring W{}_2^1(\overline\omega)$, $W_{2,0}^2(\overline\omega)$ of grid functions defined on the grid $\overline\omega\subset\overline\Omega$. The inequalities are the grid analogues of O.A. Ladygenskaya's multiplicative inequalities for functional spaces $\mathring W{}_2^1(\Omega)$, $W_{2,0}^2(\Omega)$.