This article is cited in 1 paper

Mathematics

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

L. N. Lyakhov^ab, N. I. Trusova<sup>b

a</sup> Voronezh State University, Voronezh, Russia
^b Lipetsk State Pedagogical University named after P. P. Semenov-Tyan-Shanskiy, Lipetsk, Russia

Abstract: Two types of linear integral operators with partial integrals are considered, which are defined on functions given in a finite rectangle $D=D_1\times D_2$ of the Euclidean point space $\mathbb{R}_2$. Operators of the first type are constructed according to the type of Romanovsky integrals and are studied in the space $C(D_1;L_{p}(D_2))$ norms, space of continuous functions on $\overline{D_1}$ with values in the Lebesgue class $L_p(D_2)$. For general operators, the authors prove that they belong to the class of linear bounded operators from the anisotropic class of functions $L_{p,p^2}$ for $p>1$ to the class of functions with a mixed norm $C (D_1;L_{p}(D_2))$.

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

UDC: 517.983.23

Received: 07.02.2020
Revised: 02.03.2020

DOI: 10.24411/2500-0101-2020-15102