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

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.

References in corpus (4)

• Riesz's and Bessel's Operators in Bilateral Grand Lebesgue Spaces
• Nikol'skii-type inequalities for rearrangement invariant spaces
• Quadratic $A_1$ bounds for commutators of singular integrals with BMO functions
• Some Fractional Functional Inequalities and Applications to Some Constrained Minimization Problems Involving a Local Non-linearity

Cited by in corpus (10)

• Necessary Conditions for Fractional Hardy-Sobolev's Inequalities
• Exact norm estimates for multivariate dilation operators between two Bilateral Weight Grand Lebesgue Spaces
• Sharp estimates for module of continuity of fractional integrals and derivatives
• Lebesgue spaces norm estimates for fractional integrals and derivatives
• Grand Lebesgue Spaces are really Banach algebras relative to the convolution on unimodular locally compact groups
• Exact constant in Sobolev's and Sobolev's trace inequalities for Grand Lebesgue Spaces with monomial weight
• Poincare type inequalities for two different Bilateral Grand Lebesgue Spaces
• Lebesgue-Riesz norm estimates for fractional Laplace transform
• Quasi Grand Lebesgue Spaces
• Multidimensional Lusin-type inequalities for Grand Lebesgue Spaces