Littlewood-Paley Characterizations of Haj{\l}asz-Sobolev and Triebel-Lizorkin Spaces via Averages on Balls

摘要

Let $p\in(1,\infty)$ and $q\in[1,\infty)$. In this article, the authorscharacterize the Triebel-Lizorkin space ${F}^\alpha_{p,q}(\mathbb{R}^n)$ withsmoothness order $\alpha\in(0,2)$ via the Lusin-area function and the$g_\lambda^*$-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 $B(x,t)$ centeredat $x\in\mathbb{R}^n$ with radius $t\in(0,1)$. As an application, the authorsobtain a series of characterizations of $F^\alpha_{p,\infty}(\mathbb{R}^n)$ viapointwise inequalities, involving ball averages, in spirit close to Haj{\l}aszgradients, here an interesting phenomena naturally appears that, in theend-point case when $\alpha =2$, these pointwise inequalities characterize theTriebel-Lizorkin spaces $F^2_{p,2}(\mathbb{R}^n)$, while not$F^2_{p,\infty}(\mathbb{R}^n)$. In particular, some new pointwisecharacterizations of Haj{\l}asz-Sobolev spaces via ball averages are obtained.Since these new characterizations only use ball averages, they can be used asstarting points for developing a theory of Triebel-Lizorkin spaces withsmoothness orders not less than $1$ on spaces of homogeneous type.