Boundedness of Operators in Bilateral Grand Bebesgue Spaces with Exact and Weakly Exact Constant Calculation

TL;DR

This paper studies the boundedness of various operators in Bilateral Grand Lebesgue Spaces, providing exact or near-exact constants, interpolation theorems, and extending the theory to anisotropic spaces with applications to Fourier inequalities.

Contribution

It introduces methods for calculating exact operator bounds in GLS, develops interpolation theorems, and extends the framework to anisotropic spaces with new estimates.

Findings

Exact constants for operator boundedness in GLS obtained

Interpolation theorems for Bilateral Grand Lebesgue Spaces established

Boyd's indices calculated for anisotropic GLS

Abstract

In this article we investigate an action of some operators (not necessary to be linear or sublinear) in the so-called (Bilateral) Grand Lebesgue Spaces (GLS), in particular, double weight Fourier operators, maximal operators, imbedding operators etc. We intend to calculate an exact or at least weak exact values for correspondent imbedding constant. We obtain also interpolation theorems for GLS spaces.We construct several examples to show the exactness of offered estimations. In two last sections we introduce anisotropic Grand Lebesgue Spaces, obtain some estimates for Fourier two-weight inequalities and calculate Boyd's multidimensional indices for this spaces.