Projection method for solving equations for multidimensional operators with anisotropically homogeneous kernels of compact type

V. M. Deundyak ; A. V. Lukin

Vestnik Udmurtskogo universiteta. Matematika, mehanika, kompʹûternye nauki, Tome 29 (2019) no. 2, pp. 153-165

Résumé

We consider the Banach algebra $\mathfrak{V}_{\mathbf{n}; p}$ of operators with anisotropically homogeneous kernels of compact type in $L_p$-space on the $\mathbb{R}^n$-group. Interest in the operators from $\mathfrak{V}_{\mathbf{n}; p}$ is motivated by their natural connection with the Mellin convolution operators and multidimensional multiplicative convolution operators on the $\mathbb{R}^n$-group, as well as by their applicability to the solution of problems with complex singularities. We describe the relationship of this algebra with the algebra of multidimensional convolution operators with compact coefficients using the similarity isomorphism. For the operators from the $\mathfrak{V}_{\mathbf{n}; p}$-algebra we obtain the criterion of applicability of the projection method for solving operator equations in terms of invertibility of some set of operators in cones. We prove the criterion of applicability using the reduction of the original equation to an equation for convolution operators with compact coefficients. The proof of the applicability of the projection method is sufficiently based on the new operator version of the local principle by A.V. Kozak in the theory of projection methods, which is a modification of the well-known local principle by I.B. Simonenko in the theory of the Fredholm property. In this paper, we give illustrative examples of the equations for operators with anisotropically homogeneous kernels of compact type, where we calculate the symbol and apply the developed projection method for these operators.

Détail

Keywords: integral operator, homogeneous kernels, convolution operator, projection method
Mots-clés : compact coefficients.