The uncertainty principle for operators commuting with translations. I

Let $X$ be a class of distributions (in $\mathbb R$), $K$ a distribution, $E\subset\mathbb R$. The set $E$ is said to be a $(K,X)$-set if there is no non-zero $f\in X$ such that. The $f\in X$, $f|E=(k\ast f)|E$ article begins with some examples of kernels $K$ for which every non-empty interval is a $(K,L^2)$-set (among them are the M. Btiesz kernels). Some connections with the Cauchy problem for the laplace equation and with approximations by linear combinations of translations of the kernel are discussed. The principal results treat kernels $K$ with the “semirational” Fourier transforms (symbols) $\hat K$. This means that $\hat K$ coincides with a rational function $\tau$ on a ray $(C,+\infty)$ and $\operatorname{mes}\{\xi\in(-\infty,b]:\hat{K}(\xi)=r(\xi)\}=0$ for a $b\le c$ is proved that every Carleson set $E$ with $\operatorname{mes}E>0$ is a $(K,X)$-set if $K$ has a semiratioaal symbol and $X$ is the domain (in $L^2$) of the operator $f\to K*f$ (a compact set $E$ of real numbers is said to be a Carleson set if $\sum|\ell|\log|\ell|>-\infty$, the sum being taken over the family of all bounded complementary intervals $l$ of $E$). This result implies some uniqueness theorems for weakly perturbed Hilbert transforms.