id	author	title	date	pages	extension	mime	words	sentence	flesch	summary	cache	txt
ijassa-1092	Almohammad, Kh.	The Modular Inequalities for Hardy-type Operators on Monotone Functions in Orlicz Space	2021	9	.pdf	application/pdf	2750	181	78	(2.6) Here E ′ = E ′(Rn) is the associated RIS, i.e. RIS with the norm: ‖g‖E′ = sup {∫ Rn |fg| dµ : f ∈ E, ‖f‖E ≤ 1 } . Then there exists a constant A > 0 such that Φ−1 2 ( +∞∫ 0 Φ2(aKf)ω dx ) ≤ Φ−1 1 ( +∞∫ 0 Φ1(Afb dx)v ) for all positive, measurable functions f if and only there exists a constant C such that Φ−1 2 ( +∞∫ r Φ2 ( a(x) C ∥∥∥∥k(r; ·)χ(0,r)(·) εvb ∥∥∥∥ Ψ1(εv) ) ω(x) dx ) ≤ Φ−1 1 (1 ε ) and Φ−1 2 ( +∞∫ r Φ2 ( a(x)	cache/ijassa-1092.pdf	txt/ijassa-1092.txt