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Littlewood-Paley Characterizations of Hajłasz-Sobolev and Triebel-Lizorkin Spaces via Averages on Balls

Let $p\in(1,\infty)$ and $q\in[1,\infty)$. In this article, the authors characterize the Triebel-Lizorkin space ${F}^α_{p,q}(\mathbb{R}^n)$ with smoothness order $α\in(0,2)$ via the Lusin-area function and the $g_λ^*$-function in terms of difference between $f(x)$ and its average $B_tf(x):=\frac1{|B(x,t)|}\int_{B(x,t)}f(y)\,dy$ over a ball $B(x,t)$ centered at $x\in\mathbb{R}^n$ with radius $t\in(0,1)$. As an application, the authors obtain a series of characterizations of $F^α_{p,\infty}(\mathbb{R}^n)$ via pointwise inequalities, involving ball averages, in spirit close to Hajłasz gradients, here an interesting phenomena naturally appears that, in the end-point case when $α=2$, these pointwise inequalities characterize the Triebel-Lizorkin spaces $F^2_{p,2}(\mathbb{R}^n)$, while not $F^2_{p,\infty}(\mathbb{R}^n)$. 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.