Some Properties Of Weak Orlicz Spaces

In some classical subjects, for example, the harmonic analysis, the martingale theory, the interpolation theory, the rearrangement invariant function space, the weighted inequality, the singular integral operator and so on, they played an important role. The concept of weak Orlicz space was introduced and its application to martingale theory was studied by Liu Peide in 2007. An interpolation theorem for weak Orlicz spaces satisfying MΔcondition was given by Jiao Yong in 2008.In this paper, some basic properties in weak Orlicz spaces are studied. The present paper is organized as follows:We first discuss the monotonicity and monotone coefficients of weak Orlicz spaces. Many people are interested in the monotonicity of Banach lattices, Orlicz spaces and Musielak-Orlicz spaces. Weak Orlicz space is the generalized space of weak L p( 0< p<∞)space. In terms of sets, weak Orlicz spaces contain Orlicz spaces. In this paper, the equivalent conditions for strict monotonicity, uniform monotonicity and upper(lower) local uniform monotonicity of weak Orlicz spaces are given. The monotone coefficients of the spaces and the upper(lower) monotone coefficients of a point of the unit sphere are computed.Second, the monotonicity of generalized weak Orlicz spaces is studied. As we know weak Orlicz spaces which are defined by different N functions are significantly generalized spaces of weak L p( 0< p<∞)spaces. In this paper, the concept of generalized weak Orlicz space is introduced, and some basic properties of this space are studied. At the same time, the criteria for strict monotonicity, uniform monotonicity, upper(lower) local uniform monotonicity of generalized weak Orlicz spaces are shown.Third, the products of weak Orlicz spaces are studied. Regardless of sets or norms are concerned, weak Orlicz spaces generated by different N functions are possibly different. We are interested in the criteria for the inclusion relations between two spaces, and the inclusion relations between their products and another space. Therefore, the criteria for the inclusion relations of weak Orlicz spaces are obtained in this paper.Finally, the convexity of weak Orlicz spaces is investigated. It is well known that the convexity is an important characteristic of Banach spaces. Studying the characteristic is good for understanding the structure of spaces. Although weak Orlicz spaces which are not Banach spaces like Orlicz spaces are quasi-Banach spaces, weak Orlicz space is p ?convex in case of 0 < p< 1which is shown. In addition, the triangle inequality in weak Orlicz spaces is also presented.