Bochkarev's inequalities in the Anisotropic grand Lorentz Spaces

TL;DR

This paper extends Bochkarev's inequalities to anisotropic grand Lorentz spaces, providing new bounds on Fourier coefficients and summability properties in this generalized setting.

Contribution

It introduces Bochkarev-type inequalities for anisotropic grand Lorentz spaces using extrapolation methods, generalizing classical results to broader function spaces.

Findings

Derived new Bochkarev-type inequalities for anisotropic grand Lorentz spaces.

Established summability properties of Fourier coefficients in these spaces.

Extended classical inequalities to arbitrary orthonormal systems.

Abstract

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 $2 < r \le \infty$. 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.