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

Abstract

Let $p\in (1,\infty )$ and $q\in [1,\infty )$. In this article, the authors characterize the Triebel-Lizorkin space $F^{\hat{I}}{\pm }_{p,q}({\mathbb{R}}^n)$ with smoothness order $\hat{I}\pm \in (0,2)$ via the Lusin-area function and the $g_{\hat{I}}{»}^{\ast }$-function in terms of difference between $f(x)$ and its average $B_{t}f(x):=\frac{1}{|B(x,t)|}{\int }_{B(x,t)}f(y)dy$ 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 $F^{\hat{I}}{\pm }_{p,\infty }({\mathbb{R}}^n)$ 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 $\hat{I}\pm =2$, these pointwise inequalities characterize the Triebel-Lizorkin spaces $F_{p,2}^2({\mathbb{R}}^n)$, while not $F_{p,\infty }^2({\mathbb{R}}^n)$. 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.

28 pages; Submitted for its publication on September 28, 2015