Boundedness of operators with partial integrals with the mixed norm. II

L. N. Lyakhov ; N. I. Trusova

Čelâbinskij fiziko-matematičeskij žurnal, Tome 5 (2020) no. 3, pp. 293-305

Résumé

The general form of a linear integral operator with partial integrals in $\mathbb{R}_3$ is considered as the sum of eight integral expressions, including partial integrals for one and two variables. The action of the specified operator is studied within the space $C(\Omega_1;L_{p}(\Omega_2))$ of continuous functions on $\overline{\Omega_1}$ with values in the Lebesgue class $L_p (\Omega_2)$, $1$, where $\Omega_1\times\Omega_2=D$ is the the finite parallelepiped in $\mathbb{R}_3$. We prove that the considered operators belong to the class of linear bounded operators from the anisotropic class of Lebesgue functions $L_{p,p^2}$ to the class of functions with the mixed norm $C (\Omega_1;L_{p}(\Omega_2))$.

Détail

Keywords: function with values in a Banach space, partial integral, linear operator with partial integrals, anisotropic classes of Lebesgue functions.