## 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.

### MSC:

 42B25 Maximal functions, Littlewood-Paley theory 26A33 Fractional derivatives and integrals 26D15 Inequalities for sums, series and integrals

### Citations:

Zbl 0705.26015; Zbl 0727.42007
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### References:

 [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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