On Function Spaces with Mixed Norms — A Survey (Journal of Mathematical Study, 2021)
The targets of this article are threefold. The first one is to give a survey on therecent developments of function spaces with mixed norms, including mixed Lebesguespaces, iterated weak Lebesgue spaces, weak mixed-norm Lebesgue spaces and mixedMorrey spaces as well as anisotropic mixed-norm Hardy spaces. The second one isto provide a detailed proof for a useful inequality about mixed Lebesgue norms andthe Hardy–Littlewood maximal operator and also to improve some known results onthe maximal function characterizations of anisotropic mixed-norm Hardy spaces andthe boundedness of Calderón–Zygmund operators from these anisotropic mixed-norm Hardy spaces to themselves or to mixed Lebesgue spaces. The last one is to correctsome errors and seal some gaps existing in the known articles.
Keywords: Mixed norm; (weak) Lebesgue space; Morrey space; Hardy space; maximal function; Littlewood–Paley function; Calderón–Zygmund operator.