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

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∞. 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 defined on Lp(0,1) (1<p<∞) in the borderline case α=1/p. Using of the well-developed theory of Lorentz-Zygmund spaces leads to new results on the fractional integral Riemann-Liouville operator