THE CONVOLUTION IN ANISOTROPIC TRIEBEL–LIZORKIN SPACES

Abstract

In this paper, we investigate the boundedness of the norm of the convolution operator in anisotropic Triebel-Lizorkin spaces. The Triebel-Lizorkin spaces are based on the Lorentz spaces pq L . In the anisotropic case, we take the anisotropic Lorentz space pq L as the base. The main goal of the paper is to solve the following problem: let f and g be functions from some classes of the Triebel-Lizorkin space scale. It is necessary to determine which conditions on the parameters of the spaces from f and g are taken and study which space belongs to their convolution gf  . An analogue of the O'Neil theorem was obtained for the Triebel-Lizorkin space scale αq pτF , where α , τ, p , q are vector parameters. Relations of the form γξ hν βη rμ F F  ↪ αq pτF are obtained, with the corresponding ratios of vector parameters γ βα  , hrp 11 1 1   , νμτ 111  , ηξq 111  . The research method is the functional spaces theory and inequalities of functional and harmonic analysis.