zbMATH — the first resource for mathematics

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.

a & b logic and
a | b logic or
!ab logic not
abc* right wildcard
"ab c" phrase
(ab c) parentheses
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)
Composition operators on small spaces. (English) Zbl 1114.47028
Let $A^p_{\alpha,s}$ denote the space of holomorphic functions $f$ on the unit disk such that $\int_{\mathbb D}\vert {\mathcal R}^s f(z)\vert ^p(1-\vert z\vert ^2)^\alpha\,dA(z)<\infty$, that is to say that its $s$-fractional derivative ${\mathcal R}^sf(z)=\sum_{n=0}^\infty (n+1)^\alpha a_nz^n$ belongs to $A_\alpha^p $. The authors analyze the boundedness and compactness of $C_\phi$ on the spaces $A^p_{\alpha,s}$ in some particular situations. Motivated by the notion of “suitably small” Banach space, they denote by $\Cal R$ the set of parameters $(s,p,\alpha)$ such that either $sp>\alpha+2$ or $sp=\alpha+2,\ 0<p\le 1$. Their main results establish that if $(s-1,p,\alpha)\in{\Cal R}$ (where $s$ is assumed to be a positive integer for $\alpha=-1$) and $\phi\in A^p_{\alpha,s}$, then the composition operator $C_\phi$ is bounded on $A^p_{\alpha,s-k}$ for all $k\in {\mathbb Z}^+,\ k\le s$. In particular, they obtain in such a case that $C_\phi$ is bounded on $A^p_{\alpha,s}$ if and only if $\phi\in A^p_{\alpha,s}$. As a byproduct, they get that $A^p_{\alpha,s}$ is an algebra under multiplication for $(s,p,\alpha)\in {\Cal R}$. Among other interesting results in the paper, the authors also give a complete description of symbols for the boundedness of $C_\phi$ on the Zygmund class $\Lambda_1$ of functions such that $\vert f''(z)\vert =O\left(\frac{1}{1-\vert z\vert }\right)$. Finally, some examples concerning the boundedness of $C_\phi$ on $A^p_{\alpha,s}$ if $\alpha+1\le sp\le \alpha+2+p$ are provided.

47B33Composition operators
46E15Banach spaces of continuous, differentiable or analytic functions
30D55H (sup p)-classes (MSC2000)
Full Text: DOI