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Izv. Akad. Nauk SSSR Ser. Mat., 1990, Volume 54, Issue 3, Pages 645–656 (Mi izv1090)  

This article is cited in 16 scientific papers (total in 16 papers)

Two-weighted estimates of Riemann–Liouville integrals

V. D. Stepanov

Institute for Applied Mathematics, Khabarovsk Division, Far-Eastern Branch of the Russian Academy of Sciences

Abstract: Weighted estimates
\begin{equation} (\int\limits_0^\infty|I_rf(x)u(x)|^q dx)^{1/q}\leqslant C(\int\limits_0^\infty|f(x)v(x)|^p dx)^{1/p} \end{equation}
are considered, where the constant $C$ does not depend on $f$, for fractional Riemann– Liouville integrals
$$ I_r(f(x)=\frac {1}{\Gamma (r)}\int\limits_0^x(x-t)^{r-1}f(t) dt,\quad r>0, $$
and the following problem is examined: find necessary and sufficient conditions on weight functions $u$ and $v$ under which estimate (1) is valid for all functions for which the right-hand side of (1) is finite. The problem is solved for $1\leqslant p\leqslant q\leqslant\infty$ and $r>1$. This result is definitive, and it generalizes known results for integral operators when $r=1$.

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English version:
Mathematics of the USSR-Izvestiya, 1991, 36:3, 669–681

Bibliographic databases:

UDC: 517.51
MSC: 26D10
Received: 06.09.1988

Citation: V. D. Stepanov, “Two-weighted estimates of Riemann–Liouville integrals”, Izv. Akad. Nauk SSSR Ser. Mat., 54:3 (1990), 645–656; Math. USSR-Izv., 36:3 (1991), 669–681

Citation in format AMSBIB
\by V.~D.~Stepanov
\paper Two-weighted estimates of Riemann--Liouville integrals
\jour Izv. Akad. Nauk SSSR Ser. Mat.
\yr 1990
\vol 54
\issue 3
\pages 645--656
\jour Math. USSR-Izv.
\yr 1991
\vol 36
\issue 3
\pages 669--681

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    This publication is cited in the following articles:
    1. R. C. Brown, D. B. Hinton, “A WEIGHTED HARDY'S INEQUALITY AND NONOSCILLATORY DIFFERENTIAL EQUATIONS”, Quaestiones Mathematicae, 15:2 (1992), 197  crossref
    2. D. A. Labutin, “Embedding of Sobolev Spaces on Hölder Domains”, Proc. Steklov Inst. Math., 227 (1999), 163–172  mathnet  mathscinet  zmath
    3. A. A. Kalybai, “A generalization of the weighted Hardy inequality for a class of integral operators”, Siberian Math. J., 45:1 (2004), 100–111  mathnet  crossref  mathscinet  zmath  isi  elib
    4. V. G. Maz'ya, S. V. Poborchi, “Imbedding theorems for Sobolev spaces on domains with peak and on Hölder domains”, St. Petersburg Math. J., 18:4 (2007), 583–605  mathnet  crossref  mathscinet  zmath  elib
    5. R. Oinarov, “Boundedness and compactness of Volterra type integral operators”, Siberian Math. J., 48:5 (2007), 884–896  mathnet  crossref  mathscinet  zmath  isi  elib  elib
    6. A. A. Vasil'eva, “Kolmogorov Widths of Weighted Sobolev Classes on Closed Intervals”, Math. Notes, 84:5 (2008), 631–635  mathnet  crossref  crossref  mathscinet  isi  elib  elib
    7. Meskhi A., “Measure of Non-Compactness For Integral Operators in Weighted Lebesgue Spaces”, Measure of Non-Compactness For Integral Operators in Weighted Lebesgue Spaces, Nova Science Publishers, Inc, 2009, 1–121  isi
    8. A. A. Vasil'eva, “Estimates for the widths of weighted Sobolev classes”, Sb. Math., 201:7 (2010), 947–984  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib  elib
    9. B. Florkiewicz, K. Wojteczek, “Some second-order integral inequalities of generalized Hardy type”, Proceedings of the Royal Society of Edinburgh: Section A Mathematics, 129:05 (2011), 947  crossref
    10. Vasil'eva A.A., “Kolmogorov Widths of Weighted Sobolev Classes on a Domain For a Special Class of Weights. II”, Russ. J. Math. Phys., 18:4 (2011), 465–504  crossref  isi
    11. Vasil'eva A.A., “Kolmogorov Widths of Weighted Sobolev Classes on a Domain For a Special Class of Weights”, Russ. J. Math. Phys., 18:3 (2011), 353–385  crossref  isi
    12. A. A. Vasilyeva, “Kolmogorov widths and approximation numbers of Sobolev classes with singular weights”, St. Petersburg Math. J., 24:1 (2013), 1–27  mathnet  crossref  mathscinet  zmath  isi  elib  elib
    13. A. Gogatishvili, V. D. Stepanov, “Reduction theorems for weighted integral inequalities on the cone of monotone functions”, Russian Math. Surveys, 68:4 (2013), 597–664  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib  elib
    14. Alois Kufner, Komil Kuliev, Ryskul Oinarov, “Some criteria for boundedness and compactness of the Hardy operator with some special kernels”, J Inequal Appl, 2013:1 (2013), 310  crossref
    15. A. A. Vasil'eva, “Widths of Sobolev weight classes on a domain with cusp”, Sb. Math., 206:10 (2015), 1375–1409  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib
    16. V. D. Stepanov, G. E. Shambilova, “On the boundedness of quasilinear integral operators of iterated type with Oinarov's kernels on the cone of monotone functions”, Eurasian Math. J., 8:2 (2017), 47–73  mathnet  mathscinet
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