Restriction of Families of Conversion Operators in <i>L<sup>p</sup></i> Spaces
We study a one-parameter family of convolutional operators acting in Lebesgue L<sup>p</sup> spaces. The case of integral kernels given by the Fourier coefficients is considered. It is established that the condition of the coefficients being quasiconvex ensures the boundedness of the corresponding maximal operators. The limiting behavior of families in the metrics of spaces of continuous functions and L<sup>p</sup> , p ≥ 1, classes is studied, and their convergence is obtained almost everywhere. The ways of possible generalizations and distributions are indicated.