Weighted norm inequalities for operators of Hardy type. (English) Zbl 0753.42010

Let the kernel \(\varphi(x,y)\) satisfy the assumptions: \(\varphi(x,y)>0\) if \(x>y\); \(\varphi(x,y)\) is nondecreasing in \(x\) and in \(y\); \(\varphi(x,y)\approx\varphi(x,z)+\varphi(z,y)\) if \(y<z<x\). The authors consider the operator \[ T_ rf(x)=\int^ x_ 0[\varphi(x,y)]^ rf(y)dy,\quad x>0, \] and its adjoint \(T^*_ r\). Their main result is the following: Let \(1<p\leq q<\infty\) and let \(u\) and \(v\) be nonnegative, measurable functions on \((0,\infty)\) with \(0<u\),\(v<\infty\) a.e. Then \[ \| uTf\|_ q\leq C\| vf\|_ p \] for all \(f\geq 0\) if and only if \[ I^*[(v^{-1}I^*u^ q)^{p'}]\leq C(I^*u^ q)^{p'/q'}<\infty \] and \[ I^*[(v^{-1}T^ x_ qu^ q)^{p'}]\leq C(T^*_ qu^ q)^{p'/q'}<\infty \] a.e. on \((0,\infty)\), where \(I=T_ 0\). The proof is easy in the sense that it uses no other means except Hölder and Minkowski inequalities. A comparison with some results of V. D. Stepanov and T. Martin-Rayes and E. Sawyer is given. We note that the recent papers by V. D. Stepaov [Izv. Akad. Nauk SSSR, Ser. Mat. 54, No. 3, 645-656 (1990; Zbl 0705.26015); Sib. Mat. Zh. 31, No. 3(181), 186-197 (1990; Zbl 0727.42007)] are also relevant.


42B25 Maximal functions, Littlewood-Paley theory
26A33 Fractional derivatives and integrals
26D15 Inequalities for sums, series and integrals
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[1] G. Hardy, J. Littlewood, and G. Polya, Inequalities, Cambridge Univ. Press, 1967.
[2] F. Martin-Reyes and E. Sawyer, Weighted norm inequalities for the Riemann-Liouville fractional integral operators, preprint. · Zbl 0704.42018
[3] V. D. Stepanov, Two-weight estimates for Riemann-Liouville integrals, Izv. Akad. Nauk SSSR Ser. Mat. 54 (1990), no. 3, 645 – 656 (Russian); English transl., Math. USSR-Izv. 36 (1991), no. 3, 669 – 681. · Zbl 0705.26015
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