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ЖУРНАЛЫ // Уфимский математический журнал // Архив

Уфимск. матем. журн., 2019, том 11, выпуск 4, страницы 114–129 (Mi ufa496)

Эта публикация цитируется в 1 статье

Realization of homogeneous Triebel–Lizorkin spaces with $p=\infty $ and characterizations via differences

M. Benallia, M. Moussai

Laboratory of Functional Analysis and Geometry of Spaces, Mohamed Boudiaf University of M'Sila, 28000 M'Sila, Algeria

Аннотация: In this paper, via the decomposition of Littlewood–Paley, the homogeneous Triebel-Lizorkin space $\dot{F}_{\infty,q}^{s}$ is defined on $\mathbb{R}^n$ by distributions modulo polynomials in the sense that $\|f\|=0$ ($\|\cdot\|$ the quasi-seminorm in $\dot F^{s}_{\infty,q}$) if and only if $f$ is a polynomial on $\mathbb{R}^n$. We consider this space as a set of “true” distributions and we are lead to examine the convergence of the Littlewood-Paley sequence of each element in $\dot F^{s}_{\infty,q}$. First we use the realizations and then we obtain the realized space $\dot{\widetilde{F}}{^{s}_{\infty,q}}$ of $\dot{F}_{\infty,q}^{s}$.
Our approach is as follows. We first study the commuting translations and dilations of realizations in $\dot{F}_{\infty,q}^{s}$, and employing distributions vanishing at infinity in the weak sense, we construct $\dot{\widetilde{F}}{^{s}_{\infty,q}}$. Then, as another possible definition of $\dot{F}_{\infty,q}^{s}$, in the case $s>0$, we make use of the differences and describe $\dot{\widetilde{F}}{^{s}_{\infty,q}}$ as $s>\max(n/q-n,0)$.

Ключевые слова: Triebel–Lizorkin spaces, Littlewood–Paley decomposition, realizations.

MSC: 46E35

Поступила в редакцию: 11.10.2018

Язык публикации: английский


 Англоязычная версия: Ufa Mathematical Journal, 2019, 11:4, 115–130

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