Multidimensional inequalities between distinct metrics in spaces with an asymmetric norm

Jackson–Nikol'skii inequalities in the spaces $L_{p_1,p_2}(\mathbb T^d)$ and $L_{p_1,p_2}(\mathbb R^d)$ endowed with asymmetric norms are studied for trigonometric polynomials and entire functions of exponential type, respectively. It is shown that for any $d\in {\mathbb N}$, $\mathbf n\in {\mathbb N}^d$ and $p_1,p_2,q_1,q_2\in (0,\infty]$ a trigonometric polynomial $T_{\mathbf n}$ of degree $n_j$ in $x_j$ satisfies the inequality
$$ \|T_{\mathbf n}\|_{L_{q_1,q_2}(\mathbb T^d)} \leqslant C_{p_1,p_2,q_1,q_2,d}\biggl (\prod ^d_{j=1}n_j\biggr ) ^{\psi (p_1,p_2,q_1,q_2,d)}\|T_{\mathbf n}\|_{L_{p_1,p_2}(\mathbb T^d)}, $$
where $C_{p_1,p_2,q_1,q_2,d}$ is a constant independent of $\mathbf n$ and $\psi$ is an explicitly indicated function. Examples of polynomials show that this estimate is sharp in order. A similar result is obtained for functions of exponential type.