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In this article we investigate an action of some operators (not necessary tobe 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 exactvalues for correspondent imbedding constant. We obtain also interpolationtheorems for GLS spaces.We construct several examples to show the exactness ofoffered estimations. In two last sections we introduce anisotropic GrandLebesgue Spaces, obtain some estimates for Fourier two-weight inequalities andcalculate Boyd's multidimensional indices for this spaces.

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