Weighted conditional type operators between different Orlicz spaces

1. Introduction and Preliminaries

The continuous convex function Φ :R→R is called a Young’s function whenever:
(1)Φ(x) = 0 if and only ifx= 0.
(2) Φ(x) = Φ(−x).
(3) limx→∞ Φ(x)/x =∞, limx→∞Φ(x) =∞.

With each Young’s function Φ one can associate another convex function Φ*, having similar properties, which is defined by Φ*(y) = sup{x|y| −Φ(x) :x≥0}, y∈R.

A Young’s function Φ is said to satisfy the△2condition (globally) if Φ(2x)≤ kΦ(x), x≥x0≥0 (x0= 0) for some constantk >0. Also, Φ is said to satisfy the △′ (▽′) condition, if∃c >0 (b >0) such that Φ(xy)≤cΦ(x)Φ(y), x, y≥x0≥0.

2. Bounded weighted conditional type operators

We begin by defining the weighted conditional type operator (WCT). Let (Ω,Σ, μ) be aσ-finite measure space and letAbe aσ- sub-algebra of Σ such that (Ω,A,A) is alsoσ-finite. LetEbe the corresponding conditional expectation operator relative toA. Ifu∈L 0 (Σ) (the spaces of Σ- measurable functions on Ω) such thatufis conditionable andE(uf)∈L Ψ (Σ) for allf∈ D ⊆L Φ (Σ), whereDis a linear subspace, then the corresponding weighted conditional type operator (or WCT operator) is the linear transformation Ru:D →L Ψ (Σ) defined byf→E(uf).

In theorem 2.2, we propose that let WCT operatorRu:D ⊆L Φ (Σ)→L Ψ (Σ) be well defined, then certain conditions hold. If ΦˉΨ and some sufficient conditions, when ΦΨ, and E(u) should be essentially bounded.

3. Conclusions

If Φ∗ ◦Ψ∗ −1 is a Young’s function, Φ∗ ∈ ▽′ globally andRuis bounded from L Φ (Σ) intoL Ψ (Σ), then certain conditions must hold. This paper investigates the boundedness of conditional type operators in different Orlicz spaces by considering GCH-inequality. The results of section 2 generalizes earlier results and we also find upper and lower bounds for weighted conditional type operators on Orlicz spaces.