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A function space, Lθ,∞)(Ω), 0≤θ<∞, is defined. It is proved that Lθ,∞)(Ω) is a Banach space which is a generalization of exponential class. An alternative definition of Lθ,∞)(Ω) space is given. As an application, we obtain weak monotonicity property for very weak solutions of 𝒜-harmonic equation with variable coefficients under some suitable conditions related to Lθ,∞)(Ω), which provides a generalization of a known result due to Moscariello. A weighted space Lwθ,∞)(Ω) is also defined, and the boundedness for the Hardy-Littlewood maximal operator Mw and a Calderón-Zygmund operator T with respect to Lwθ,∞)(Ω) is obtained.