zbMATH — the first resource for mathematics

Examples
Geometry Search for the term Geometry in any field. Queries are case-independent.
Funct* Wildcard queries are specified by * (e.g. functions, functorial, etc.). Otherwise the search is exact.
"Topological group" Phrases (multi-words) should be set in "straight quotation marks".
au: Bourbaki & ti: Algebra Search for author and title. The and-operator & is default and can be omitted.
Chebyshev | Tschebyscheff The or-operator | allows to search for Chebyshev or Tschebyscheff.
"Quasi* map*" py: 1989 The resulting documents have publication year 1989.
so: Eur* J* Mat* Soc* cc: 14 Search for publications in a particular source with a Mathematics Subject Classification code (cc) in 14.
"Partial diff* eq*" ! elliptic The not-operator ! eliminates all results containing the word elliptic.
dt: b & au: Hilbert The document type is set to books; alternatively: j for journal articles, a for book articles.
py: 2000-2015 cc: (94A | 11T) Number ranges are accepted. Terms can be grouped within (parentheses).
la: chinese Find documents in a given language. ISO 639-1 language codes can also be used.

Operators
a & b logic and
a | b logic or
!ab logic not
abc* right wildcard
"ab c" phrase
(ab c) parentheses
Fields
any anywhere an internal document identifier
au author, editor ai internal author identifier
ti title la language
so source ab review, abstract
py publication year rv reviewer
cc MSC code ut uncontrolled term
dt document type (j: journal article; b: book; a: book article)
Characterization of the Besov spaces via the commutator operator of Coifman, Rochberg and Weiss. (English) Zbl 0838.42006
{\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 $f$. 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.

MSC:
42B20Singular and oscillatory integrals, several variables
46E35Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems
WorldCat.org
Full Text: DOI