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Zbl 0838.42006
Paluszyński, M.
Characterization of the Besov spaces via the commutator operator of Coifman, Rochberg and Weiss. (English)
[J] Indiana Univ. Math. J. 44, No.1, 1-17 (1995). ISSN 0022-2518

{\it R. R. Coifman}, {\it R. Rochberg} and {\it G. Weiss} [Ann. Math., II. Ser. 103, 611-635 (1976; Zbl 0326.32011)]\ proved that the commutator operator, $C_f (g)= T(f\cdot g) -f\cdot T(g)$, where $T$ is a Calderón-Zygmund singular integral operator is bounded on some $L^p (\bbfR^n)$, $1< p<\infty$, 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 $f$. \par 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 $C_f^\alpha (g)= I^\alpha (f\cdot g)-f\cdot I^\alpha (g)$. The function space $\dot \Lambda_\beta$ is the homogeneous Lipschitz space defined in terms of the $k$th-difference operator $\Delta_h^{k+1} f(x)= e^k_h f(x+ h)- \Delta^k_h f(x)$, where $\Delta^1_h f(x)= f(x+h)- f(x)$. We say that $f\in \dot \Lambda_\beta$ if $|f|_{\dot \Lambda_\beta}= \sum_{x,h\in \bbfR^n, h\ne 0} {{|\Delta_h^{[\beta]+1} f(x)|}/{|h|^\beta}} <\infty$. The homogeneous Triebel-Lizorkin space is denoted by $\dot F_p^{\beta, \infty}$. The characterization of $\dot F_p^{\beta, \infty}$, of fundamental importance to this paper, is that for $0< \beta< 1$ and $1<p <\infty$, $|h|_{\dot F_p^{\beta, \infty}} \approx |\sup_{Q\ni\cdot} {1\over {|Q|^{1+ \beta/n}}} \int_Q|h-h_Q |\ |_p$. In the first theorem when $0< \beta< 1$, $1< p< \infty$ the author establishes the equivalence of the three conditions (i) $f\in \dot \Lambda_\beta$, (ii) $C_f$ is a bounded operator from $L^p (\bbfR^n)$ to $\dot F_p^{\beta, \infty}$ and (iii) $C_f$ is a bounded operator from $L^p (\bbfR^n)$ to $L^q (\bbfR^n)$, $1/p- 1/q= \beta/n$ if $1/p> \beta/n$. The second theorem concerns the operator $C_f^\alpha$ and the setting $1< p< q< \infty$, $0< \beta <1$, $1/p- 1/q= \alpha/n$. In that scenario, he proves that conditions (i) $f\in \dot \Lambda_\beta$, (ii) $C_f^\alpha$ is a bounded operator from $L^p (\bbfR^n)$ to $\dot F_q^{\beta, \infty}$ and (iii) $C_f^\alpha$ is a bounded operator from $L^p (\bbfR^n)$ to $L^r (\bbfR^n)$, $1/p- 1/r= (\alpha+ \beta)/n$ if $1/p> (\alpha+ \beta)/n$ are equivalent.
[S.Staples (Fort Worth)]

MSC 2000:: *42B20 Singular integrals, several variables
46E35 Sobolev spaces and generalizations

Keywords: Besov space; BMO; commutator operator; Calderón-Zygmund singular integral operator; Riesz potential; homogeneous Lipschitz space; homogeneous Triebel-Lizorkin space

Citations: Zbl 0326.32011

Cited in: Zbl 1209.42009 Zbl 1084.42012

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