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