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The $L^{p,\infty}$ quasi-norm of functions on a measure space can be characterized in terms of their pairing with normalized characteristic functions. We generalize this result to the case of the outer $L^{p,\infty}$ quasi-norms for appropriate sizes. This characterization provides an explicit form for certain implicitly defined quasi-norms appearing in an article of Di Plinio and Fragkos., Comment: 21 pages, no figures

We establish the sharp conditions for the embedding between Wiener amalgam spaces $W_{p,q}^s$ and some classical spaces, including Sobolev spaces $L^{s,r}$, local Hardy spaces $h_{r}$, Besov spaces $B_{p,q}^s$, which partially improve and extend the main result obtained by Guo et al. in J. Funct. Anal., 273(1):404-443, 2017. In addition, we give the full characterization of inclusion between Wiener amalgam spaces $W_{p,q}^s$ and $\alpha$-modulation spaces $M_{p,q}^{s,\alpha}$. Especially, in the case of $\alpha=0$ with $M_{p,q}^{s,\alpha} = M_{p,q}^s$, we give the sharp conditions of the most general case of these embedding. When $0, Comment: 33 pages,6 figures, all comments are welcome

We study the double iterated outer $L^p$ spaces, namely the outer $L^p$ spaces associated with three exponents and defined on sets endowed with a measure and two outer measures. We prove that in the case of finite sets, under certain conditions between the outer measures, the double iterated outer $L^p$ spaces are isomorphic to Banach spaces uniformly in the cardinality of the set. We achieve this by showing the expected duality properties between them. We also provide counterexamples demonstrating that the uniformity does not hold in any arbitrary setting on finite sets, at least in a certain range of exponents. We prove the isomorphism to Banach spaces and the duality properties between the double iterated outer $L^p$ spaces also in the upper half $3$-space infinite setting described by Uraltsev, going beyond the case of finite sets., Comment: 44 pages, no figures

We prove that the outer $L^p_��(\ell^r)$ spaces, introduced by Do and Thiele, are isomorphic to Banach spaces, and we show the expected duality properties between them for $1 < p \leq \infty, 1 \leq r < \infty$ or $p=r \in \{ 1, \infty \}$ uniformly in the finite setting. In the case $p=1, 1 < r \leq \infty$, we exhibit a counterexample to uniformity. We show that in the upper half space setting these properties hold true in the full range $1 \leq p,r \leq \infty$. These results are obtained via greedy decompositions of functions in $L^p_��(\ell^r)$. As a consequence, we establish the equivalence between the classical tent spaces $T^p_r$ and the outer $L^p_��(\ell^r)$ spaces in the upper half space. Finally, we give a full classification of weak and strong type estimates for a class of embedding maps to the upper half space with a fractional scale factor for functions on $\mathbb{R}^d$., 32 pages, 1 figure

We refine the definition of generalized Stummel classes and study inclusion properties of these classes. We also study the inclusion relation between Stummel classes and other function spaces such as generalized Morrey spaces, weak Morrey spaces, and Lorentz spaces. In addition, we show that these inclusions are proper. Our results extend some previous results in \cite{CRR, RZ}., 14 pages

We consider functions of vanishing mean oscillation on a bounded domain $��$ and prove a $\rm{VMO}$ analogue of the extension theorem of P. Jones for $\rm{BMO}(��)$. We show that if $��$ satisfies the same condition imposed by Jones (i.e.\ is a uniform domain), there is a linear extension map from $\rm{VMO}(��)$ to $\rm{VMO}(\mathbb{R}^n)$ which is bounded in the $\rm{BMO}$ norm. Moreover, if such an extension map exists from $\rm{VMO}(��)$ to $\rm{BMO}(\mathbb{R}^n)$, then the domain is uniform., 26 pages

In this paper, we prove that the inclusions between Morrey spaces, between weak Morrey spaces, and between a Morrey space and a weak Morrey space are all proper. The proper inclusion between a Morrey space and a weak Morrey space is established via the unboundedness of the Hardy-Littlewood maximal operator on Morrey spaces of exponent 1. In addition, we also give a necessary condition for each inclusion. Our results refine previous inclusion properties studied in [Gunawan et al, \emph{Math. Nachr.} {\bf 290} (2017), 332--340]., 8 pages

In this paper, we introduce a new scale of tent spaces which covers, the (weighted) tent spaces of Coifman-Meyer-Stein and of Hofmann-Mayboroda-McIntosh, and some other tent spaces considered by Dahlberg, Kenig-Pipher and Auscher-Axelsson in elliptic equations. The strong factorizations within our tent spaces, with applications to quasi-Banach complex interpolation and to multiplier-duality theory, are established. This way, we unify and extend the corresponding results obtained by Coifman-Meyer-Stein, Cohn-Verbitsky and Hyt\"onen-Ros\'en., Comment: 20 pages. Comments are welcome

In this paper we establish the following estimate \[ w\left(\left\{ x\in\mathbb{R}^{n}\,:\,\left|[b,T]f(x)\right| > \lambda\right\} \right)\leq \frac{c_{T}}{\varepsilon^{2}}\int_{\mathbb{R}^{n}}\Phi\left(\|b\|_{BMO}\frac{|f(x)|}{\lambda}\right)M_{L(\log L)^{1+\varepsilon}}w(x)dx \] where $w\geq0, \, 0\lambda\}\right)\leq c_T\,[w]_{A_{\infty}}\left(1+\log^{+}[w]_{A_{\infty}}\right)^{2}\int_{\mathbb{R}^{n}} \Phi\left(\|b\|_{BMO}\frac{|f(x)|}{\lambda}\right)Mw(x)dx\] We also obtain the analogue estimates for symbol-multilinear commutators for a wider class of symbols., Comment: 31 pages. Final version, accepted for publication in Israel J. Math

Let $T$ be an arbitrary operator bounded from $L^{p_0}(w)$ into $L^{p_0, \infty}(w)$ for every weight $w$ in the Muckenhoupt class $A_{p_0}$. It is proved in this article that the distribution function of $Tf$ with respect to any weight $u$ can be essentially majorized by the distribution function of $Mf$ with respect to $u$ (plus an integral term easy to control). As a consequence, well-known extrapolation results, including results in a multilinear setting, can be obtained with very simple proofs. New applications in extrapolation for two-weight problems and estimates on rearrangement invariant spaces are established too., Comment: 29 pages