Structural properties of Bilateral Grand Lebesque Spaces

Averaging of random variables and fields

“…and this estimate is also non -improvable still in the case when the set X consists in a single point [18].…”

“…These spaces are complete Banach functional rearrangement invariant; they are investigated in many works, see e.g. [7], [8], [9], [10], [11], [12], [13], [14], [15], [16], [17], [18], [19]. It is important for us in particular to note that there is exact of course up to finite multiplicative constant interrelations under certain natural conditions on the generating function between belonging the r.v.…”

Modified Bernstein's inequality for sums of independent random variables

“…The minimal non-negative value τ satisfying ( 22) for any λ ∈ (−λ 0 , λ 0 ) is named B(φ)-norm of the variable ξ and we write…”

“…These spaces are investigated in many works, e.g. in [9], [11], [12], [16], [17], [19], [22], [27] - [30] etc. They are applied for example in the theory of Partial Differential Equations [11], [12], in the theory of Probability [7], [29] - [30], in Statistics [27], chapter 5, theory of random fields [19], [30], in the Functional Analysis [27], [28], [30] and so one.…”

“…These spaces are rearrangement invariant (r.i.) Banach functional spaces; its fundamental function is considered in [30]. They not coincides in general case with the classical spaces: Orlicz, Lorentz, Marcinkiewicz etc., see [22], [28].…”