Let p∈(1,∞)\documentclass[12pt]{minimal}
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\begin{document}$p\in (1,\infty )$\end{document} and q∈[1,∞)\documentclass[12pt]{minimal}
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\begin{document}$q\in [1,\infty )$\end{document}. In this article, the authors characterize the Triebel-Lizorkin space Fp,qα(ℝn)\documentclass[12pt]{minimal}
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\begin{document}${F}^{\alpha }_{p,q}(\mathbb {R}^{n})$\end{document} with smoothness order α ∈ (0, 2) via the Lusin-area function and the gλ∗\documentclass[12pt]{minimal}
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\begin{document}$g_{\lambda }^{*}$\end{document}-function in terms of difference between f(x) and its ball average Btf(x):=1|B(x,t)|∫B(x,t)f(y)dy\documentclass[12pt]{minimal}
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\begin{document}$B_{t}f(x):=\frac 1{|B(x,t)|}{\int }_{B(x,t)}f(y)\,dy$\end{document} over the ball B(x, t) centered at x∈ℝn\documentclass[12pt]{minimal}
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\begin{document}$x\in \mathbb {R}^{n}$\end{document} with radius t ∈ (0, 1). As an application, the authors obtain a series of characterizations of Fp,∞α(ℝn)\documentclass[12pt]{minimal}
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\begin{document}$F^{\alpha }_{p,\infty }(\mathbb {R}^{n})$\end{document} via pointwise inequalities, involving ball averages, in spirit close to Hajłasz gradients, here some interesting phenomena naturally appear that, in the end-point case when α = 2, some of these pointwise inequalities characterize the Triebel-Lizorkin spaces Fp,22(ℝn)\documentclass[12pt]{minimal}
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\begin{document}$F^{2}_{p,2}(\mathbb {R}^{n})$\end{document}, while not Fp,∞2(ℝn)\documentclass[12pt]{minimal}
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\begin{document}$F^{2}_{p,\infty }(\mathbb {R}^{n})$\end{document}, and that some of other obtained pointwise characterizations are only known to hold true for Fp,∞α(ℝn)\documentclass[12pt]{minimal}
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\begin{document}$F^{\alpha }_{p,\infty }(\mathbb {R}^{n})$\end{document} with p∈(1,∞)\documentclass[12pt]{minimal}
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\begin{document}$p\in (1,\infty )$\end{document}, α ∈ (0, 2) or α ∈ (n/p, 2). In particular, some new pointwise characterizations of Hajłasz-Sobolev spaces via ball averages are obtained. Since these new characterizations only use ball averages, they can be used as starting points for developing a theory of Triebel-Lizorkin spaces with smoothness orders not less than 1 on spaces of homogeneous type.
