# TAILIEUCHUNG - Structural properties of bilateral Grand Lebesque spaces

## In this paper we study the multiplicative, tensor, Sobolev and convolution inequalities in certain Banach spaces, the so-called bilateral Grand Lebesque spaces. We also give examples to show the sharpness of these inequalities when possible. | Turk J Math 34 (2010) , 207 – 219. ¨ ITAK ˙ c TUB doi: Structural properties of Bilateral Grand Lebesque Spaces E. Liﬂyand, E. Ostrovsky and L. Sirota Abstract In this paper we study the multiplicative, tensor, Sobolev and convolution inequalities in certain Banach spaces, the so-called Bilateral Grand Lebesque Spaces. We also give examples to show the sharpness of these inequalities when possible. Key word and phrases: Grand Lebesgue and rearrangement invariant spaces, Sobolev embedding theorem, convolution operator. 1. Introduction Let (X, Σ, μ) be a σ -ﬁnite measure space. We suppose the measure μ to be non-trivial and diﬀuse. The latter means that, for all A ∈ Σ such that μ(A) ∈ (0, ∞), there exists B ⊂ A with μ(B) = μ(A)/2. For a and b constants, 1 ≤ a 0. The Bilateral Grand Lebesgue Space (in notation, BGLS) GX (μ; ψ; a, b) = GX (ψ; a, b) = G(ψ; a, b) = G(ψ) is the space of all measurable functions h : X → R endowed with the norm def

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