On generalized fractional integrals

Abstract

It is known that the fractional integral Iα (0 < α < n) is bounded from Lp(ℝn) to Lq(ℝn) when p > 1 and n/p - α = n/q > 0, from Lp(ℝn) to BMO(ℝn) when p > 1 and n/p - α = 0, from Lp(ℝn) to Lipβ(ℝn) when p > 1 and -1 < 0, from BMO(ℝn) to Lipα(ℝn) when 0 < α < 1, and from Lipβ(ℝn) to Lipγ(ℝn) when 0 < α + β = γ < 1. We introduce generalized fractional integrals and extend the above boundedness to the Orlicz spaces and BMOφ.