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

Martin-Reyes, F. J.; Sawyer, E.

Proc. Am. Math. Soc. 106, No. 3, 727-733 (1989).

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

Keywords:

generalized Hardy operator; weight functions; Riemann-Liouville fractional integration operators