Transformations of multipliers for pseudodifferential operators in~$L_p$

K. Tel'ner

Sbornik. Mathematics, Tome 14 (1971) no. 3, pp. 399-416

Résumé

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 $\mathbf R^m$; $\langle\xi,x\rangle=\xi_1x_1+\dots+\xi_mx_m$; $\widehat u$ 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.