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

Research paper by Der-;Chen Chang, Jun Liu; Dachun Yang; Wen Yuan

Indexed on: 12 Aug '16Published on: 10 Aug '16Published in: Potential Analysis

#### Abstract

Abstract Let $$p\in (1,\infty )$$ and $$q\in [1,\infty )$$ . In this article, the authors characterize the Triebel-Lizorkin space $${F}^{\alpha }_{p,q}(\mathbb {R}^{n})$$ with smoothness order α ∈ (0, 2) via the Lusin-area function and the $$g_{\lambda }^{*}$$ -function in terms of difference between f(x) and its ball average $$B_{t}f(x):=\frac 1{ B(x,t) }{\int }_{B(x,t)}f(y)\,dy$$ over the ball B(x, t) centered at $$x\in \mathbb {R}^{n}$$ with radius t ∈ (0, 1). As an application, the authors obtain a series of characterizations of $$F^{\alpha }_{p,\infty }(\mathbb {R}^{n})$$ 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 $$F^{2}_{p,2}(\mathbb {R}^{n})$$ , while not $$F^{2}_{p,\infty }(\mathbb {R}^{n})$$ , and that some of other obtained pointwise characterizations are only known to hold true for $$F^{\alpha }_{p,\infty }(\mathbb {R}^{n})$$ with $$p\in (1,\infty )$$ , 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.AbstractLet $$p\in (1,\infty )$$ and $$q\in [1,\infty )$$ . In this article, the authors characterize the Triebel-Lizorkin space $${F}^{\alpha }_{p,q}(\mathbb {R}^{n})$$ with smoothness order α ∈ (0, 2) via the Lusin-area function and the $$g_{\lambda }^{*}$$ -function in terms of difference between f(x) and its ball average $$B_{t}f(x):=\frac 1{ B(x,t) }{\int }_{B(x,t)}f(y)\,dy$$ over the ball B(x, t) centered at $$x\in \mathbb {R}^{n}$$ with radius t ∈ (0, 1). As an application, the authors obtain a series of characterizations of $$F^{\alpha }_{p,\infty }(\mathbb {R}^{n})$$ 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 $$F^{2}_{p,2}(\mathbb {R}^{n})$$ , while not $$F^{2}_{p,\infty }(\mathbb {R}^{n})$$ , and that some of other obtained pointwise characterizations are only known to hold true for $$F^{\alpha }_{p,\infty }(\mathbb {R}^{n})$$ with $$p\in (1,\infty )$$ , 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. $$p\in (1,\infty )$$ $$p\in (1,\infty )$$ $$q\in [1,\infty )$$ $$q\in [1,\infty )$$ $${F}^{\alpha }_{p,q}(\mathbb {R}^{n})$$ $${F}^{\alpha }_{p,q}(\mathbb {R}^{n})$$α $$g_{\lambda }^{*}$$ $$g_{\lambda }^{*}$$fx $$B_{t}f(x):=\frac 1{ B(x,t) }{\int }_{B(x,t)}f(y)\,dy$$ $$B_{t}f(x):=\frac 1{ B(x,t) }{\int }_{B(x,t)}f(y)\,dy$$Bxt $$x\in \mathbb {R}^{n}$$ $$x\in \mathbb {R}^{n}$$t $$F^{\alpha }_{p,\infty }(\mathbb {R}^{n})$$ $$F^{\alpha }_{p,\infty }(\mathbb {R}^{n})$$α $$F^{2}_{p,2}(\mathbb {R}^{n})$$ $$F^{2}_{p,2}(\mathbb {R}^{n})$$ $$F^{2}_{p,\infty }(\mathbb {R}^{n})$$ $$F^{2}_{p,\infty }(\mathbb {R}^{n})$$ $$F^{\alpha }_{p,\infty }(\mathbb {R}^{n})$$ $$F^{\alpha }_{p,\infty }(\mathbb {R}^{n})$$ $$p\in (1,\infty )$$ $$p\in (1,\infty )$$ααnp