Mellin transform of log-Lipschitz functions and equivalence of K-functionals and modulus of smoothness generated by the Mellin Steklov operator

In this paper, we examine the order of magnitude of the Mellin transform for complex-valued functions belonging to the log-Lipschitz class. In addition, in the intersection of the spaces Xc1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$X_{c}^{1}$$\end{document} and Xc2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$X_{c}^{2}$$\end{document}, using the Mellin Steklov operator, we construct the Mellin modulus of smoothness, and also using the Mellin derivative we define the Mellin K -functional. The second main result of our article is the proof of the equivalence between Mellin K-functionals and Mellin modulus of smoothness.