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Characterization of the Besov spaces via the commutator operator of Coifman, Rochberg and Weiss. (English) Zbl 0838.42006

R. R. Coifman, R. Rochberg and G. Weiss [Ann. Math., II. Ser. 103, 611-635 (1976; Zbl 0326.32011)] proved that the commutator operator, ${C}_{f}\left(g\right)=T\left(f·g\right)-f·T\left(g\right)$, where $T$ is a Calderón-Zygmund singular integral operator is bounded on some ${L}^{p}\left({ℝ}^{n}\right)$, $1, if and only if $f\in \text{BMO}$. Various generalizations of this type of boundedness result for commutator operators have been studied where one varies either the operator class of $T$ or the class of functions $f$.

In this paper, the author presents two very nice theorems of this nature and then follows with generalizations of his own results. Here we describe the notation involved and summarize the first two results alone. Let ${I}^{\alpha }$ be the Riesz potential of order $\alpha$. We define the commutator ${C}_{f}^{\alpha }\left(g\right)={I}^{\alpha }\left(f·g\right)-f·{I}^{\alpha }\left(g\right)$. The function space ${\stackrel{˙}{{\Lambda }}}_{\beta }$ is the homogeneous Lipschitz space defined in terms of the $k$th-difference operator ${{\Delta }}_{h}^{k+1}f\left(x\right)={e}_{h}^{k}f\left(x+h\right)-{{\Delta }}_{h}^{k}f\left(x\right)$, where ${{\Delta }}_{h}^{1}f\left(x\right)=f\left(x+h\right)-f\left(x\right)$. We say that $f\in {\stackrel{˙}{{\Lambda }}}_{\beta }$ if ${|f|}_{{\stackrel{˙}{{\Lambda }}}_{\beta }}={\sum }_{x,h\in {ℝ}^{n},h\ne 0}|{{\Delta }}_{h}^{\left[\beta \right]+1}f\left(x\right)|/{|h|}^{\beta }<\infty$. The homogeneous Triebel-Lizorkin space is denoted by ${\stackrel{˙}{F}}_{p}^{\beta ,\infty }$. The characterization of ${\stackrel{˙}{F}}_{p}^{\beta ,\infty }$, of fundamental importance to this paper, is that for $0<\beta <1$ and $1, ${|h|}_{{\stackrel{˙}{F}}_{p}^{\beta ,\infty }}\approx |{sup}_{Q\ni ·}\frac{1}{{|Q|}^{1+\beta /n}}{\int }_{Q}|h-{h}_{Q}{|\phantom{\rule{4pt}{0ex}}|}_{p}$. In the first theorem when $0<\beta <1$, $1 the author establishes the equivalence of the three conditions (i) $f\in {\stackrel{˙}{{\Lambda }}}_{\beta }$, (ii) ${C}_{f}$ is a bounded operator from ${L}^{p}\left({ℝ}^{n}\right)$ to ${\stackrel{˙}{F}}_{p}^{\beta ,\infty }$ and (iii) ${C}_{f}$ is a bounded operator from ${L}^{p}\left({ℝ}^{n}\right)$ to ${L}^{q}\left({ℝ}^{n}\right)$, $1/p-1/q=\beta /n$ if $1/p>\beta /n$. The second theorem concerns the operator ${C}_{f}^{\alpha }$ and the setting $1, $0<\beta <1$, $1/p-1/q=\alpha /n$. In that scenario, he proves that conditions (i) $f\in {\stackrel{˙}{{\Lambda }}}_{\beta }$, (ii) ${C}_{f}^{\alpha }$ is a bounded operator from ${L}^{p}\left({ℝ}^{n}\right)$ to ${\stackrel{˙}{F}}_{q}^{\beta ,\infty }$ and (iii) ${C}_{f}^{\alpha }$ is a bounded operator from ${L}^{p}\left({ℝ}^{n}\right)$ to ${L}^{r}\left({ℝ}^{n}\right)$, $1/p-1/r=\left(\alpha +\beta \right)/n$ if $1/p>\left(\alpha +\beta \right)/n$ are equivalent.

##### MSC:
 42B20 Singular and oscillatory integrals, several variables 46E35 Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems