ON GENERALIZED FRACTIONAL INTEGRALS

Abstract

It is known that the fractional integral $I_\alpha (0 \lt \alpha \le n)$ is bounded from $L^p({\mathbb R}^n)$ to $L^q ({\mathbb R}^n)$ when $p \gt 1$ and $n/p - \alpha = n/q \gt 0$, from $L^p({\mathbb R}^n)$ to BMO$({\mathbb R}^n)$ when $p \gt 1$ and $n/p - \alpha = 0$, from $L^p({\mathbb R}^n)$ to $\mbox{Lip}_\beta({\mathbb R}^n)$ when $p \gt 1$ and $-1 \lt n/p - \alpha = -\beta \lt 0$, from BMO$({\mathbb R}^n)$ to $\mbox{Lip}_\alpha({\mathbb R}^n)$ when $0 \lt \alpha \lt 1$, and from $\mbox{Lip}_\beta({\mathbb R}^n)$ to $\mbox{Lip}_\gamma ({\mathbb R}^n )$ when $0 \lt \alpha + \beta = \gamma \lt 1$. We introduce generalized fractional integrals and extend the above boundedness to the Orlicz spaces and $\mbox{BMO}_\phi$.