In the present paper, the characterization of strong-type modular inequality ∫ 0 ∞φ(Sf (t))w(t)dt≤∫0∞ φ(Cf (t))w(t)dt, f↓ is given, where φ∈Δ' and S is a Hardy operator. Furthermore, the equivalent conditions of modular inequalities and norm inequalities related to weak Orlicz-Lorentz spaces are researched. We also explore the conditions for Orlicz-Lorentz spaces and weak Orlicz-Lorentz spaces to be normable. Finally, the weak boundedness of certain Hardy-type operators on Orlicz-Lorentz spaces is studied.
Let X be a rearrangement invariant space is R^n and Wxr1…,rn be an anisotropicSobolev space which is a generalization of Wpri,…,rn,The main subject of this paper is to prove the embedding theorem fot Wpri,…,rn.