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Mat. Sb. (N.S.), 1971, Volume 85(127), Number 3(7), Pages 403–419 (Mi msb3207)  

Transformations of multipliers for pseudodifferential operators in $L_p$

K. Tel'ner


Abstract: By a transformation of multipliers we mean the operation assigning to each pseudodifferential (ps.d.) operator $K$ with symbol $K(\xi,x)$, i.e.
$$ (Ku)(x)=\int_{\mathbf R^m}K(\xi,x)e^{i\langle\xi,x\rangle}\widehat u(\xi) d\xi, $$
a new ps.d. operator $\Phi K$ with symbol $\varphi(\xi,x)K(\xi,x)$, i.e.
$$ (\Phi Ku)(x)=\int_{\mathbf R^m}\varphi(\xi,x)K(\xi,x)e^{i\langle\xi,x\rangle}\widehat u(\xi) d\xi. $$
Here $\mathbf R^m$ is $m$-dimensional Euclidean space; $x$ and $\xi$ are points in $\mathbf R^m$; $\langle\xi,x\rangle=\xi_1x_1+…+\xi_mx_m$; $\widehat u$ is the Fourier transform of $u$. There are given two criteria for the transformation $K\to\Phi K$ to preserve the continuity of ps.d. operators in the spaces $L_p(\mathbf R^m)$. As a corollary there are obtained conditions for the boundedness of ps.d. operators (or singular integral operators) in $ L_p$.
Bibliography: 12 titles.

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English version:
Mathematics of the USSR-Sbornik, 1971, 14:3, 399–416

Bibliographic databases:

UDC: 517.43
MSC: Primary 47G05; Secondary 42A18
Received: 06.07.1970

Citation: K. Tel'ner, “Transformations of multipliers for pseudodifferential operators in $L_p$”, Mat. Sb. (N.S.), 85(127):3(7) (1971), 403–419; Math. USSR-Sb., 14:3 (1971), 399–416

Citation in format AMSBIB
\Bibitem{Tel71}
\by K.~Tel'ner
\paper Transformations of multipliers for pseudodifferential operators in~$L_p$
\jour Mat. Sb. (N.S.)
\yr 1971
\vol 85(127)
\issue 3(7)
\pages 403--419
\mathnet{http://mi.mathnet.ru/msb3207}
\mathscinet{http://www.ams.org/mathscinet-getitem?mr=290200}
\zmath{https://zbmath.org/?q=an:0247.47046}
\transl
\jour Math. USSR-Sb.
\yr 1971
\vol 14
\issue 3
\pages 399--416
\crossref{https://doi.org/10.1070/SM1971v014n03ABEH002624}


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