On the spaces introduced by N. K. Karapetyants and B. S. Rubin, and their connection with Lorentz–Zygmund spaces

Abstract

We consider spaces introduced by N. K. Karapetyants and B. S. Rubin in 1982, to characterize, in particular, the image of the fractional integral Riemann–Liouville operator. These spaces lie near L∞\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${L}^{\\infty }$$\\end{document}. We show that they coincide with well-known Lorentz–Zygmund spaces. This allows us to reformulate one result from N. K. Karapetyants and B. S. Rubin dealing with Riemann–Liouville fractional integral operator J0+α\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${J}_{0+}^{\\alpha }$$\\end{document} defined on Lp0,1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${L}^{p}\\left(\ ext{0,1}\\right)$$\\end{document} (1<p<∞\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$1<p<\\infty$$\\end{document}) in the borderline case α=1/p\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\alpha =1/p$$\\end{document}. Using of the well-developed theory of Lorentz–Zygmund spaces leads to new results on the fractional integral Riemann–Liouville operator.