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In this article, the authors give a survey on the recent developments of both the John--Nirenberg space $JN_p$ and the space BMO as well as their vanishing subspaces such as VMO, XMO, CMO, $VJN_p$, and $CJN_p$ on $mathbb{R}^n$ or a given cube $Q_0subsetmathbb{R}^n$ with finite side length. In addition, some related open questions are also presented.
With a view towards Riemannian or sub-Riemannian manifolds, RCD metric spaces and specially fractals, this paper makes a step further in the development of a theory of heat semigroup based $(1,p)$ Sobolev spaces in the general framework of Dirichlet
Interpolation inequalities in Triebel-Lizorkin-Lorentz spaces and Besov-Lorentz spaces are studied for both inhomogeneous and homogeneous cases. First we establish interpolation inequalities under quite general assumptions on the parameters of the fu
The main result of this paper is an intersection representation for a class of anisotropic vector-valued function spaces in an axiomatic setting `a la Hedberg$&$Netrusov, which includes weighted anisotropic mixed-norm Besov and Lizorkin-Triebel space
We prove that for $alpha in (d-1,d]$, one has the trace inequality begin{align*} int_{mathbb{R}^d} |I_alpha F| ;d u leq C |F|(mathbb{R}^d)| u|_{mathcal{M}^{d-alpha}(mathbb{R}^d)} end{align*} for all solenoidal vector measures $F$, i.e., $Fin M_b(math
The classical theorems of Banach and Stone, Gelfand and Kolmogorov, and Kaplansky show that a compact Hausdorff space $X$ is uniquely determined by the linear isometric structure, the algebraic structure, and the lattice structure, respectively, of t