Fractional integrals on weighted $H^ p$ and $L^ p$ spaces

Abstract:

We study the two weight function problem $\parallel {I_\alpha }f{\parallel _{H_u^q}} \leqslant c\parallel f{\parallel _{H_v^p}},0 < p \leqslant q < \infty$ , for fractional integrals on Hardy spaces. If $u$ and $v$ satisfy the doubling condition and $0 < p \leqslant 1$, we obtain a necessary and sufficient condition for the norm inequality to hold. If $1 < p < \infty$ we obtain a necessary condition and a sufficient condition, and show these are the same under various additional conditions on $u$ and $v$. We also consider the corresponding problem for $L_u^q$ and $L_v^p$, and obtain a necessary and sufficient condition in some cases.