On generalized fractional integrals

It is known that the fractional integral I-alpha (0 < <alpha> < n) is bound ed from L-p(R-n) to L-q (R-n) when p > 1 and n/p - alpha = n/q > 0, from L- p(R-n) to BMO(Rn) when p > 1 and n/p - ce = 0, from LP(Rn) to Lipo (Rn) whe n p > 1 and -1 < n/p - <alpha> = -beta < 0, from BMO(R-n) to Lip(<beta>)(R- n) when 0 < <alpha> < 1, and from Lip(<beta>)(R-n) to Lip(gamma)(R-n) when 0 < <alpha> + beta = gamma < 1. We introduce generalized fractional integra ls and extend the above boundedness to the Orlicz spaces and BMO<phi>.