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Weighted norm inequalities for the Riemann-Liouville and Weyl fractional integral operators. (English) Zbl 0664.26002

This paper studies weighted norm inequalities \[ [\int^{\infty}_{0}| (Tf)(x)u(x)|^ q dx]^{1/q}\leq C[\int^{\infty}_{0}| f(x)v(x)|^ p dx]^{1/p}, \] where u and v are nonnegative weight functions, \(\alpha >0,\quad 1<p<1/\alpha,\quad 1/q=1/p-\alpha,\) and C is a constant depending on p, q, \(\alpha\), u, v but independent of f and T is one of Riemann-Liouville or Weyl fractional integral from which inequalities for other fractional integral operators T, such as Erdélyi-Kober, can be deduced. Some questions raised by B. Muckenhoupt [Proc. Symp. Pure Math. 35, No.1, 69-83 (1979; Zbl 0428.26009)] have been answered through Theorems 2 and 3.
Reviewer: R.N.Kalia

MSC:

26A33 Fractional derivatives and integrals
26D10 Inequalities involving derivatives and differential and integral operators
42B25 Maximal functions, Littlewood-Paley theory

Citations:

Zbl 0428.26009
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References:

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