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)
Weighted composition operators from Bergman-Privalov-type spaces to weighted-type spaces on the unit ball. (English) Zbl 1250.47030

Necessary and sufficient conditions for boundedness and for compactness of weighted composition operators μC φ from the weighted Bergman-Privalov-type space AN p,α to a weighted type space or a little weighted-type space on the unit ball in the n-dimensional complex vector space n are given. The spaces and the composition operator mentioned above are defined in the following way. Let B be the unit ball in n , H(B) be the space of all holomorphic functions on B, dV(z) be the normalized Lebesgue measure on B and dV α (z)=c α (1-|z| 2 ) α dV(z), where α>-1 and c α is a normalization constant, that is, V α (B)=1. For each p1 and α>-1, the weighted Bergman-Privalov-type space AN p,α is defined in the following way:

AN p,α ={fH(B): B ln p (1+|f(z)|)dV α (z)<}·

The weighted-type space H μ is defined as the space that consists of all fH(B) such that sup zB μ(z)|f(z)|<, where μ(z) is a positive continuous function (a weight) on B. The little weighted-type space H μ,o consists of all fH(B) such that

lim |z|1 μ(z)|f(z)|=0·

For fH(B), the corresponding weighted composition operator is defined by

(μC φ )(f)(z)=μ(z)f(φ(z)),zB,

where φ is a holomorphic self-map of B and μH(B) is fixed.

MSC:
47B33Composition operators
32A37Spaces of holomorphic functions (several variables)
46E15Banach spaces of continuous, differentiable or analytic functions
47B38Operators on function spaces (general)
References:
[1]Clahane, D.; Stević, S.: Norm equivalence and composition operators between Bloch/Lipschitz spaces of the unit ball, J. inequal. Appl. 2006, 11 (2006) · Zbl 1131.47018 · doi:10.1155/JIA/2006/61018
[2]Cowen, C. C.; Maccluer, B. D.: Composition operators on spaces of analytic functions, (1995) · Zbl 0873.47017
[3]Gu, D.: Weighted composition operators from generalized weighted Bergman spaces to weighted-type spaces, J. inequal. Appl. 2008, 14 (2008) · Zbl 1160.32010 · doi:10.1155/2008/619525
[4]Li, S.; Stević, S.: Weighted composition operators between H and α-Bloch spaces in the unit ball, Taiwanese J. Math. 12, 1625-1639 (2008) · Zbl 1177.47032
[5]Matsugu, Y.; Ueki, S. I.: Isometries of weighted Bergman – Privalov spaces on the unit ball of cn, J. math. Soc. jpn. 54, 341-347 (2002) · Zbl 1027.32012 · doi:10.2969/jmsj/05420341
[6]Rudin, W.: Function theory in the unit ball of cn, (1980)
[7]Stević, S.: Composition operators between H and the α-Bloch spaces on the polydisc, Z. anal. Anwend. 25, No. 4, 457-466 (2006) · Zbl 1118.47015 · doi:10.4171/ZAA/1301
[8]Stević, S.: Essential norms of weighted composition operators from the α-Bloch space to a weighted-type space on the unit ball, Abstr. appl. Anal. 2008, 11 (2008) · Zbl 1160.32011 · doi:10.1155/2008/279691
[9]Stević, S.: Norm of weighted composition operators from Bloch space to Hμ on the unit ball, Ars. combin. 88, 125-127 (2008) · Zbl 1224.30195
[10]Stević, S.: Essential norms of weighted composition operators from the Bergman space to weighted-type spaces on the unit ball, Ars. combin. 91, 391-400 (2009) · Zbl 1216.47041
[11]Stević, S.: Weighted composition operators from weighted Bergman spaces to weighted-type spaces on the unit ball, Appl. math. Comput. 212, 499-504 (2009) · Zbl 1186.47020 · doi:10.1016/j.amc.2009.02.057
[12]Stević, S.: Norms of some operators on bounded symmetric domains, Appl. math. Comput. 216, 187191 (2010) · Zbl 1209.32010 · doi:10.1016/j.amc.2010.01.030
[13]Ueki, S. I.: Composition operators on the Privalov spaces of the unit ball of cn, J. korean math. Soc. 42, No. 1, 111-127 (2005) · Zbl 1062.32004 · doi:10.4134/JKMS.2005.42.1.111
[14]Ueki, S. I.; Luo, L.: Essential norms of weighted composition operators between weighted Bergman spaces of the ball, Acta sci. Math. (Szeged) 74, 829-843 (2008) · Zbl 1199.30274
[15]Xiao, J.: Composition operators: Nα to the Bloch space to Qβ, Stud. math. 139, 245-260 (2000) · Zbl 0963.30021
[16]Yang, W.: Weighted composition operators from Bloch-type spaces to weighted-type spaces, Ars. combin. 93, 265-274 (2009) · Zbl 1216.47042
[17]Zhu, X.: Weighted composition operators from area Nevanlinna spaces into Bloch spaces, Appl. math. Comput. 215, No. 12, 4340-4346 (2010) · Zbl 1185.30058 · doi:10.1016/j.amc.2009.12.064