The inclusion problem for mixed norm spaces (Q2822717)

The article deals with the inclusion problem for spaces \((P_1,P_2,\dots,P_n)\) of functions \(f(x_1,x_2,\dots,x_n)\) (\(x_1 \in \Omega_1,x_2 \in \Omega_2,\dots,x_n \in \Omega_n\)) with mixed norms NEWLINE\[NEWLINE\| f\| _{\sigma(P)} = \| \dots(\| \, \| f\| _{P(\sigma(1))}\| _{P(\sigma(2))}) \cdots \| _{P(\sigma(n))}.\eqno(1)NEWLINE\]NEWLINE Here, \(P = (P_1,P_2,\dots,P_n)\) is a set of Lebesgue function spaces \(P_j = L_{\mu_j}^{p_j}\) (\(j = 1,2,\dots,n\)) of measurable functions \(f_j:\;\Omega_j \to {\mathbb R}\), \(\Omega_j\) is \(\sigma\)-finite measure space with the measure \(\mu_j\), \(\mu = \mu_1 \times \mu_2 \times \dots \times \mu_n\), \(\sigma\) is a permutation of \(\{1,2,\dots,n\}\). The main part of the article is devoted to the analysis of the inclusion NEWLINE\[NEWLINE(L_\nu^q,L_\mu^p) \subseteq (L_\kappa^r,L_\lambda^s).\eqno(2)NEWLINE\]NEWLINE In the case \(\max\, (q,s) \geq \min\, (p,r)\), the authors prove that inclusion (2) is true if and only if \(L_\mu^p \subseteq L_\kappa^r\) and \(L_\nu^q \subseteq L_\lambda^s\). In the case \(\max\, (q,s)< \min\, (p,r)\), the situation depends on the measures \(\nu,\mu,\kappa,\lambda\) and numbers \(q,p,r,s\); the authors consider separately the cases when the measures \(\nu,\mu,\kappa,\lambda\) are atomless, not purely atomic, and purely atomic. In all cases, when Inclusion (1) holds, either exact formulas for the norm of the inclusion operator or effective estimates of these norm are obtained. At the end of the article, the authors consider applications of the results about inclusion (2) to the general case of the mixed Lebesgue spaces with norms of type (1) for \(n> 2\).