Estimates of the norm of the convolution operator in anisotropic Besov spaces with the dominated mixed derivative
In this paper, we investigate the boundedness of the norm of the convolution operator in Sobolev spaces with the dominated mixed derivative and anisotropic Nikolsky-Besov spaces. For Sobolev spaces with the dominated mixed derivatives, an analogue of Young’s inequality is obtained, namely, relations of the form W<sup>γ</sup><sub>p</sub>*W<sup>β</sup><sub>r</sub>→W<sup>α</sup><sub>q</sub> (1) are proved when the corresponding conditions on the parameters are satisfied. The main goal of the paper is to solve the following problems. Let f and g be functions from some classes of the Nikolsky-Besov space scale. We would like to find the Nikolsky-Besov space such that the convolution f*g belongs to this space. Using relation (1) and the Nursultanov interpolation theorem for anisotropic spaces, an analogue of the O’Neil theorem was obtained for the Nikolsky-Besov space scale B<sub>pq</sub><sup>α</sup>, where α, p, q are vector parameters. Relations of the form B<sub>ps1</sub><sub>γ</sub> B<sub>rs2</sub><sub>β</sub>→B<sub>qs</sub><sup>α</sup> are obtained, with the corresponding ratios of vector parameters. The theorems obtained in this paper complement the results of Batyrov and Burenkov, where similar problems were considered in isotropic Nikolsky-Besov spaces, that is, in spaces where the parameters are scalars.