In this paper, 2-microlocal Herz type Besov and Triebel-Lizorkin spaces with
variable exponents are introduced for the first time. Then, we give characterizations of
these spaces by so-called Peetre's maximal functions. Further, the atomic and molecular decompositions of these spaces are obtained. Finally, using the characterizations of
the spaces by local means and molecular decomposition we obtain the wavelet characterizations.
In this note, we investigate the properties of Gaussian BV functions and give
a heat semigroup characterization of BV functions in Gauss space. In particular, the latter is the nontrivial generalization of classical De Giorgi's heat kernel characterization
of function of bounded variation on Euclidean space to the case of Gauss space.
The targets of this article are threefold. The first one is to give a survey on the
recent developments of function spaces with mixed norms, including mixed Lebesgue
spaces, iterated weak Lebesgue spaces, weak mixed-norm Lebesgue spaces and mixed
Morrey spaces as well as anisotropic mixed-norm Hardy spaces. The second one is
to provide a detailed proof for a useful inequality about mixed Lebesgue norms and
the Hardy–Littlewood maximal operator and also to improve some known results on
the maximal function characterizations of anisotropic mixed-norm Hardy spaces and
the boundedness of Calderón–Zygmund operators from these anisotropic mixed-norm Hardy spaces to themselves or to mixed Lebesgue spaces. The last one is to correct
some errors and seal some gaps existing in the known articles.
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