## Weighted inequalities for Riemann-Liouville fractional integrals of order one and greater.(English)Zbl 0704.42018

Let $$\phi$$ : (0,1)$$\to (0,\infty)$$ be nonincreasing and such that $$\phi (ab)\leq D[\phi (a)+\phi (b)]$$ for a, b in (0,1), and let $$T_{\phi}$$ be the generalized Hardy operator: $$T_{\phi}f(x)=\int^{x}_{0}\phi (\frac{t}{x})f(t)dt,$$ for a function f on (0,$$\infty)$$. Necessary and sufficient conditions on the weight functions v and w: (0,$$\infty)\to (0,\infty)$$ are given so that $\int^{\infty}_{0}| T_{\phi}f(x)|^ pw(x)dx\leq C\int^{\infty}_{0}| f(x)|^ pv(x)dx,$ for some p in (1,$$\infty)$$, and “all” functions f. In particular, taking $$\phi (s)=\Gamma (\alpha)^{-1}(1- s)^{\alpha -1}$$ for $$\alpha\geq 1$$, weighted estimates for the Riemann- Liouville fractional integration operators are obtained.
Reviewer: M.Cowling

### MSC:

 42B25 Maximal functions, Littlewood-Paley theory 42A99 Harmonic analysis in one variable
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