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Sibirsk. Mat. Zh., 1993, Volume 34, Number 4, Pages 184–196 (Mi smj1643)  

This article is cited in 4 scientific papers (total in 4 papers)

On weighted estimates for a class of integral operators

V. D. Stepanov


Abstract: The weighted estimates of the form
\begin{equation} (\int_0^\infty|T_\varphi f(x)u(x)|^q dx)^{1/q}\le C(\int_0^\infty|f(x)v(x)|^p dx)^{1/p}, \tag{1} \end{equation}
are considered with
$$ T_\varphi f(x)=\int_0^x\varphi(t/x)f(t) dt, $$
where the measurable function $\varphi$ satisfies the conditions:
a) $\varphi(t)\geqslant0$ and $\varphi(t)$ is nonincreasing for $t\in[0,1]$,
b) $\varphi(t_1,t_2)\leqslant D(\varphi(t_1)+\varphi(t_2))$, $0<t_1$, $t_2<1$ and $D$ independent of $t_1$$t_2$.
We state necessary and/or sufficient conditions for (1) to hold if $1<p$, $q<\infty$ or $0<q<1<p<\infty$.

Full text: PDF file (1050 kB)

English version:
Siberian Mathematical Journal, 1993, 34:4, 755–766

Bibliographic databases:

UDC: 517.51
Received: 24.02.1992

Citation: V. D. Stepanov, “On weighted estimates for a class of integral operators”, Sibirsk. Mat. Zh., 34:4 (1993), 184–196; Siberian Math. J., 34:4 (1993), 755–766

Citation in format AMSBIB
\Bibitem{Ste93}
\by V.~D.~Stepanov
\paper On weighted estimates for a class of integral operators
\jour Sibirsk. Mat. Zh.
\yr 1993
\vol 34
\issue 4
\pages 184--196
\mathnet{http://mi.mathnet.ru/smj1643}
\mathscinet{http://www.ams.org/mathscinet-getitem?mr=1248804}
\zmath{https://zbmath.org/?q=an:0815.47064}
\transl
\jour Siberian Math. J.
\yr 1993
\vol 34
\issue 4
\pages 755--766
\crossref{https://doi.org/10.1007/BF00975180}
\isi{http://gateway.isiknowledge.com/gateway/Gateway.cgi?GWVersion=2&SrcApp=PARTNER_APP&SrcAuth=LinksAMR&DestLinkType=FullRecord&DestApp=ALL_WOS&KeyUT=A1993MA84100023}


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    Citing articles on Google Scholar: Russian citations, English citations
    Related articles on Google Scholar: Russian articles, English articles

    This publication is cited in the following articles:
    1. Goldman M.L., Stepanov V.D., Heinig H.P., “The principle of duality in Lorents spaces”, Doklady Akademii Nauk, 344:6 (1995), 740–744  mathnet  mathscinet  zmath  isi
    2. GolDman M.L., Heinig H.P., Stepanov V.D., “On the principle of duality in Lorentz spaces”, Canadian Journal of Mathematics–Journal Canadien de Mathematiques, 48:5 (1996), 959–979  crossref  mathscinet  zmath  isi
    3. Jain P., Jain P.K., Gupta B., “On a conjecture of Kufner and Persson”, Rocky Mountain Journal of Mathematics, 37:6 (2007), 1941–1951  crossref  mathscinet  zmath  isi
    4. 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
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