Iterated and Mixed Weak Norms with Applications to Geometric Inequalities

In this paper, we consider a new weak norm, iterated weak norm in Lebesgue spaces with mixed norms. We study properties of the mixed weak norm and the iterated weak norm and present the relationship between the two weak norms. Even for the ordinary Lebesgue spaces, the two weak norms are not equivalent and any one of them can not control the other one. We give some convergence and completeness results for both weak norms. We study the convergence in truncated norm, which is a substitution of the convergence in measure for mixed Lebesgue spaces. And we give a characterization of the convergence in truncated norm. We show that H\"older's inequality is not always true on mixed weak spaces and we give a complete characterization of indices which admit H\"older's inequality. As applications, we establish some geometric inequalities related to fractional integration in mixed weak spaces and in iterated weak spaces which essentially generalize the Hardy-Littlewood-Sobolev inequality.