Iterated Weak and Weak Mixed-Norm Spaces with Applications to Geometric Inequalities

In this paper, we consider two types of weak norms, the weak mixed-norm and the iterated weak norm, in Lebesgue spaces with mixed norms. We study properties of two weak norms and present their relationship. 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 the two weak norms, respectively. We study the convergence in the truncated norm, which is a substitution of the convergence in measure for mixed-norm Lebesgue spaces. And we give a characterization of the convergence in the truncated norm. We show that Hölder’s inequality is not always true on weak mixed-norm Lebesgue spaces and we give a complete characterization of indices which admit Hölder’s inequality. As applications, we establish some geometric inequalities related to fractional integrals in weak mixed-norm spaces and in iterated weak spaces, respectively, which essentially generalize the Hardy–Littlewood–Sobolev inequality.