Abstract
The classical inequalities of Bochkarev play a very important role in harmonic analysis. The meaning of these inequalities lies in the connection between the metric characteristics of functions and the summability of their Fourier coefficients. One of the most important directions of harmonic analysis is the theory of Fourier series. His interest in this direction is explained by his applications in various departments of modern mathematics and applied sciences, as well as the availability of many unsolved problems. One of these problems is the study of the interrelationships of the integral properties of functions and the properties of the sum of its coefficients. The solution of these problems was dedicated to the efforts of many mathematicians. And further research in this area are important and interesting problems and can give new, unexpected effects. In the article we receive a two-dimensional analog of the Bochkarev type theorem for the Fourier transform.