Relationship Between an Atomic Decomposition of Double and Unary Systems in Grand-Lebesgue Spaces

The grand-Lebesgue space is defined. Based on the shift operator, a separable subspace is determined in which continuous functions are dense. The concepts of frame and atomic decomposition are defined. An atomic decomposition of double and unary systems of functions in grand-Lebesgue spaces is considered. Relationship between atomic decomposition of these systems in grand-Lebesgue spaces is established

Approximation by trigonometric polynomials in the framework of variable exponent grand Lebesgue spaces

In this paper we establish direct and inverse theorems on approximation by trigonometric polynomials for the functions of the closure of the variable exponent Lebesgue space in the variable exponent grand Lebesgue space.

Variable exponents and grand Lebesgue spaces: Some optimal results

Consider p : Ω → [1, +∞[, a measurable bounded function on a bounded set Ø with decreasing rearrangement p* : [0, |Ω|] → [1, +∞[. We construct a rearrangement invariant space with variable exponent p* denoted by [Formula: see text]. According to the growth of p*, we compare this space to the Lebesgue spaces or grand Lebesgue spaces. In particular, if p*(⋅) satisfies the log-Hölder continuity at zero, then it is contained in the grand Lebesgue space Lp*(0))(Ω). This inclusion fails to be true if we impose a slower growth as [Formula: see text] at zero. Some other results are discussed.

The Distance toL∞in the Grand Orlicz Spaces

We establish a formula for the distance toL∞from the grand Orlicz spaceLΦ)Ωintroduced in Capone et al. (2008). A new formula for the distance toL∞from the grand Lebesgue spaceLn)Ωintroduced in Iwaniec and Sbordone (1992) is also provided.