A Generalization Of Exponential Class And Its Applications

This paper is divided into two parts:In the first part, we define three spaces:grand Lebesgue space Lθ,∞)(Ω); weak Lebesgue space Lweak θ,∞(Ω) and weighted Lebesgue space Lω θ,∞), respectively. We prove that L∞(Ω) is a proper subspace of Lθ,∞)(Ω), which is equivalent to Lweak θ,∞(Ω). We obtain Lθ,∞)(Ω) is a Banach Spaces, which is a generalization of Exponential class.In the second part, we give some applications of the above spaces. We obtain weak monotonicity property for very weak solutions of A-harmonic equation in grand Lebesgue space Lθ,∞)(Ω). In a weighted space Lω θ,∞(Ω), the boundedness for the Hardy-Littlewood maximal operator Mω and Calderon-Zygmund operator T are obtained.