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Mat. Sb. (N.S.), 1983, Volume 120(162), Number 2, Pages 216–226 (Mi msb2120)  

This article is cited in 3 scientific papers (total in 3 papers)

On a problem for integral convolution operators

V. D. Stepanov


Abstract: This article considers convolution operators $T_k\colon L^2(R^N)\to L^2(R^N)$ of the form $T_kf(x)=\int_{R^N}k(x-y)f(y) dy$ which are integral operators on the whole class $L^2(R^N)$, i.e., the kernel $k(x)$ is such that $\int_{R^N}|k(x-y)f(y)| dy<\infty$ for almost all $x\in R^N$. An answer is obtained to the following question of Korotkov: if $T_k\colon L^2(R^N)\to L^2(R^N)$ is a convolution operator which is an integral operator on the whole of $L^2(R^N)$, does it follow that $\operatorname{mes}\{\xi\in R^N:|k^\wedge(\xi)|>\lambda\}<\infty$ for any $\lambda>0$? Here $k^\wedge(\xi)$ is the Fourier transform of $k(x)$. An example answering the question in the negative is given by the operator $T_{\mathscr K}\colon L^2(R^1)\to L^2(R^1)$ with kernel $\mathscr K(x)$ such that $\mathscr K^\wedge(\xi)=\sum\limits_{n\ne0}\operatorname{sign}n\chi_{[-\frac1{2|n|},\frac1{2|n|}]}(\xi-n),$ where $\chi_{[a,b]}$ is the characteristic function of $[a,b]$.
Bibliography: 4 titles.

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English version:
Mathematics of the USSR-Sbornik, 1984, 48:1, 211–221

Bibliographic databases:

UDC: 517.444
MSC: Primary 44A35, 47G05; Secondary 42B10
Received: 29.04.1982

Citation: V. D. Stepanov, “On a problem for integral convolution operators”, Mat. Sb. (N.S.), 120(162):2 (1983), 216–226; Math. USSR-Sb., 48:1 (1984), 211–221

Citation in format AMSBIB
\Bibitem{Ste83}
\by V.~D.~Stepanov
\paper On a~problem for integral convolution operators
\jour Mat. Sb. (N.S.)
\yr 1983
\vol 120(162)
\issue 2
\pages 216--226
\mathnet{http://mi.mathnet.ru/msb2120}
\mathscinet{http://www.ams.org/mathscinet-getitem?mr=687614}
\zmath{https://zbmath.org/?q=an:0542.45010|0527.45006}
\transl
\jour Math. USSR-Sb.
\yr 1984
\vol 48
\issue 1
\pages 211--221
\crossref{https://doi.org/10.1070/SM1984v048n01ABEH002671}
\isi{http://gateway.isiknowledge.com/gateway/Gateway.cgi?GWVersion=2&SrcApp=PARTNER_APP&SrcAuth=LinksAMR&DestLinkType=FullRecord&DestApp=ALL_WOS&KeyUT=A1983SV65300013}


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  • http://mi.mathnet.ru/eng/msb/v162/i2/p216

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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. V. D. Stepanov, “On the vanishing of the symbol of a convolution integral operator”, Math. USSR-Sb., 51:1 (1985), 239–253  mathnet  crossref  mathscinet  zmath
    2. Stepanov V., “A Halmos and Sander Problem”, 278, no. 2, 1984, 296–298  mathscinet  zmath  isi
    3. Stepanov V., “On Fourier Integral Multiplicators Generated by the Characteristic Set-Functions”, 299, no. 5, 1988, 1068–1070  mathscinet  zmath  isi
  • Математический сборник (новая серия) - 1964–1988 Sbornik: Mathematics (from 1967)
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