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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:10.3906/mat-0812-8 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 ||h||G(ψ) = sup |h|p /ψ(p), |h|p = p∈(a,b) 1/p |h(x)| dμ(x) . p X The G(ψ) spaces with μ(X) = 1 appeared in [12]; it was proved that in this case each G(ψ) space coincides with certain exponential Orlicz space, up to norm equivalence. Partial cases of these spaces have been intensively studied, in particular, their associate spaces, fundamental functions φ(G(ψ; a, b); δ), Fourier and singular operators, conditions for convergence and compactness, reflexivity and separability, martingales in these spaces, etc.; see, e.g., [1],[4]–[9],[11], [13], [16], and a recent paper [3]. These spaces are also Banach function spaces and, moreover, rearrangement invariant (r.i.) (see [1, Ch. 1, §1]). The BGLS norm estimates, in particular, Orlicz norm estimates for measurable functions, e.g., for random variables are used in PDE [5]–[8], probability in Banach spaces [14], in the modern non-parametrical statistics, for example, in the so-called regression problem [15, 16]. 2000 AMS Mathematics Subject .
