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Register a new userFefferman-Stein decomposition for $Q$-spaces and micro-local quantities
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In this paper, we consider the Fefferman-Stein decomposition of $Q_{alpha}(mathbb{R}^{n})$ and give an affirmative answer to an open problem posed by M. Essen, S. Janson, L. Peng and J. Xiao in 2000. One of our main methods is to study the structure of the predual space of $Q_{alpha}(mathbb{R}^{n})$ by the micro-local quantities. This result indicates that the norm of the predual space of $Q_{alpha}(mathbb{R}^{n})$ depends on the micro-local structure in a self-correlation way.
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Let $Dinmathbb{N}$, $qin[2,infty)$ and $(mathbb{R}^D,|cdot|,dx)$ be the Euclidean space equipped with the $D$-dimensional Lebesgue measure. In this article, via an auxiliary function space $mathrm{WE}^{1,,q}(mathbb R^D)$ defined via wavelet expansions, the authors establish the Riesz transform characterization of Triebel-Lizorkin spaces $dot{F}^0_{1,,q}(mathbb{R}^D)$. As a consequence, the authors obtain the Fefferman-Stein decomposition of Triebel-Lizorkin spaces $dot{F}^0_{infty,,q}(mathbb{R}^D)$. Finally, the authors give an explicit example to show that $dot{F}^0_{1,,q}(mathbb{R}^D)$ is strictly contained in $mathrm{WE}^{1,,q}(mathbb{R}^D)$ and, by duality, $mathrm{WE}^{infty,,q}(mathbb{R}^D)$ is strictly contained in $dot{F}^0_{infty,,q}(mathbb{R}^D)$. Although all results when $D=1$ were obtained by C.-C. Lin et al. [Michigan Math. J. 62 (2013), 691-703], as was pointed out by C.-C. Lin et al., the approach used in the case $D=1$ can not be applied to the case $Dge2$, which needs some new skills.
Let $Dinmathbb{N}$, $qin[2,infty)$ and $(mathbb{R}^D,|cdot|,dx)$ be the Euclidean space equipped with the $D$-dimensional Lebesgue measure. In this article, the authors establish the Fefferman-Stein decomposition of Triebel-Lizorkin spaces $dot{F}^0_{infty,,q}(mathbb{R}^D)$ on basis of the dual on function set which has special topological structure. The function in Triebel-Lizorkin spaces $dot{F}^0_{infty,,q}(mathbb{R}^D)$ can be written as the certain combination of $D+1$ functions in $dot{F}^0_{infty,,q}(mathbb{R}^D) bigcap L^{infty}(mathbb{R}^D)$. To get such decomposition, {bf (i),} The authors introduce some auxiliary function space $mathrm{WE}^{1,,q}(mathbb R^D)$ and $mathrm{WE}^{infty,,q}(mathbb{R}^D)$ defined via wavelet expansions. The authors proved $dot{F}^{0}_{1,q} subsetneqq L^{1} bigcup dot{F}^{0}_{1,q}subset {rm WE}^{1,,q}subset L^{1} + dot{F}^{0}_{1,q}$ and $mathrm{WE}^{infty,,q}(mathbb{R}^D)$ is strictly contained in $dot{F}^0_{infty,,q}(mathbb{R}^D)$. {bf (ii),} The authors establish the Riesz transform characterization of Triebel-Lizorkin spaces $dot{F}^0_{1,,q}(mathbb{R}^D)$ by function set $mathrm{WE}^{1,,q}(mathbb R^D)$. {bf (iii),} We also consider the dual of $mathrm{WE}^{1,,q}(mathbb R^D)$. As a consequence of the above results, the authors get also Riesz transform characterization of Triebel-Lizorkin spaces $dot{F}^0_{1,,q}(mathbb{R}^D)$ by Banach space $L^{1} + dot{F}^{0}_{1,q}$. Although Fefferman-Stein type decomposition when $D=1$ was obtained by C.-C. Lin et al. [Michigan Math. J. 62 (2013), 691-703], as was pointed out by C.-C. Lin et al., the approach used in the case $D=1$ can not be applied to the cases $Dge2$, which needs some new methodology.
In this article, we establish the Fefferman-Stein inequalities for the Dunkl maximal operator associated with a finite reflection group generated by the sign changes. Similar results are also given for a large class of operators related to Dunkls analysis.
For $mathbb B^n$ the unit ball of $mathbb C^n$, we consider Bergman-Orlicz spaces of holomorphic functions in $L^Phi_alpha(mathbb B^n)$, which are generalizations of classical Bergman spaces. We obtain atomic decomposition for functions in the Bergman-Orlicz space $mathcal A^Phi_alpha (mathbb B^n)$ where $Phi$ is either convex or concave growth function. We then prove weak factorization theorems involving the Bloch space and a Bergman-Orlicz space and also weak factorization theorems involving two Bergman-Orlicz spaces.
We use the method of atomic decomposition to build new families of function spaces, similar to Besov spaces, in measure spaces with grids, a very mild assumption. Besov spaces with low regularity are considered in measure spaces with good grids, and results on multipliers and left compositions are obtained.
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