Fundamental function for Grand Lebesgue Spaces

Abstract

We investigate in this short article the fundamental function for the so-called Grand Lebesgue Spaces (GLS) and show in particular a one-to-one and mutually continuous accordance between its fundamental and generating function.

Analog of modulus of convexity for Grand Lebesgue Spaces

“…in [10], [12], [13], [23], [24], [25], [26], [28], [29] - [33] etc. They are applied for example in the theory of Partial Differential Equations [12], [13], in the theory of Probability [7], [31] - [33], in Statistics [29], chapter 5, theory of random fields, [25], [26], [29], [32], in the Functional Analysis [29], [30], [32] and so one.…”

“…These spaces are rearrangement invariant (r.i.) Banach functional spaces; its fundamental function is considered in [32]. They do not coincides in general case with the classical rearrangement invariant spaces: Orlicz, Lorentz, Marcinkiewicz etc., see [28], [30].…”

Grand Lebesgue Spaces are really Banach algebras relative to the convolution on unimodular locally compact groups

“…These spaces are rearrangement invariant (r.i.) Banach functional spaces; its fundamental function has been studied in [36]. They not coincides, in the general case, with the classical rearrangement invariant spaces: Orlicz, Lorentz, Marcinkiewicz, etc., see [30], [34].…”

Exponential tail estimates in the Law of Ordinary Logarithm (LOL) for arrays of random variables

“…The Grand Lebesue spaces and several generalizations and variants of them have been widely investigated, see e.g. [21], [10], [28], [38], [8], [1], [16]. These spaces are of great interest for their applications not only in statistics, in theory of random fields, Monte-Carlo methods but also in the theory of Partial Differential Equations (PDEs) (see e.g.…”