Bochkarev's inequalities in the Anisotropic grand Lorentz Spaces

The main aim of this paper is to obtain Bochkarev-type inequalities for the anisotropic grand Lorentz spaces. In the classical setting, Bochkarev obtained inequalities of the Hardy–Littlewood type, which reveal the connection between the integral properties of functions and the summability of their Fourier coefficients. His results describe the behavior of trigonometric series in the Lorentz spaces $L_{2,r}$ for $2<r\le \mathrm{\infty }$. In this work, we extend these ideas to the framework of anisotropic grand Lorentz spaces. Using an approach based on the extrapolation of linear operators, we derive new Bochkarev-type inequalities that generalize the classical results to the case of anisotropic grand Lorentz spaces and arbitrary orthonormal systems. We investigate the summability properties of the Fourier coefficients of functions from anisotropic grand Lorentz spaces.