Weighted estimates of the Fourier transformation

The Fourier transformation is regarded as an operator from $\mathcal L_2(-\pi,\pi)$ to $\mathcal L_2(\mathbb R,\mu)$, where $\mu$ is a measure on the real axis $\mathbb R$. Some criteria are obtained for this operator to be bounded or compact, or to belong to some symmetrically normed ideal with the domination property. These results can be viewed as a description of the Carleson measures for the Paley–Wiener space of entire functions. Bibliography: 15 titles.