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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