Recently, the weak Triebel-Lizorkin space was introduced by Grafakos and He, which includes the standard Triebel-Lizorkin space as a subset. The latter has a wide applications in aspects of analysis. In this paper, the authors firstly give equivalent quasi-norms of weak Triebel-Lizorkin spaces in terms of Peetre's maximal functions. As an application of those equivalent quasi-norms, an atomic decomposition of weak Triebel-Lizorkin spaces is given., Wenchang Li, Jingshi Xu., and Seznam literatury
For any $x\in (0,1]$, let $$ x=\frac {1}{d_1}+\frac {1}{d_1(d_1-1)d_2}+\dots +\frac {1}{d_1(d_1-1) \dots d_{n-1}(d_{n-1}-1)d_{n}}+\dots $$ be its Lüroth expansion. Denote by ${P_n(x)}/{Q_n(x)}$ the partial sum of the first $n$ terms in the above series and call it the $n$th convergent of $x$ in the Lüroth expansion. This paper is concerned with the efficiency of approximating real numbers by their convergents $\{{P_n(x)}/{Q_n(x)}\}_{n\ge 1}$ in the Lüroth expansion. It is shown that almost no points can have convergents as the optimal approximation for infinitely many times in the Lüroth expansion. Consequently, Hausdorff dimension is introduced to quantify the set of real numbers which can be well approximated by their convergents in the Lüroth expansion, namely the following Jarník-like set: $\{x\in (0,1]\colon |x-{P_n(x)}/{Q_n(x)}|<{1}/{Q_n(x)^{\nu +1}} \text{infinitely often}\}$ for any $\nu \ge 1$.