# Isometries of weighted Bergman-Privalov spaces on the unit ball of C^n

## Abstract

Let B denote the unit ball in \bm{C}<SUP>n</SUP>, and v the normalized Lebesgue measure on B. For α>-1, define dv<SUB>α</SUB>(z)=Γ(n+α+1)/{Γ(n+1)Γ(α+1)}(1-|z|<SUP>2</SUP>)<SUP>α</SUP>dv(z), z∈ B. Let H(B) denote the space of holomorphic functions in B. For p≥q 1, define<br>(\displaystyle AN)^{\bm{p}}(v<SUB>α</SUB>)={f∈ H(B):\left//f\right//≡[∈t<SUB>B</SUB>{log(1+|f|)}<SUP>p</SUP>dv<SUB>α</SUB>]<SUP>1/p</SUP><∞}.<br>(AN)<SUP>p</SUP>(v<SUB>α</SUB>) is an F-space with respect to the metric ρ(f, g)≡\left//f-g\right//. In this paper we prove that every linear isometry T of (AN)<SUP>p</SUP>(v<SUB>α</SUB>) into itself is of the form Tf=c(f\circψ) for all f∈(AN)<SUP>p</SUP>(v<SUB>α</SUB>), where c is a complex number with |c|=1 and ψ is a holomorphic self-map of B which is measure-preserving with respect to the measure v<SUB>α</SUB>.

## Journal

• Journal of the Mathematical Society of Japan

Journal of the Mathematical Society of Japan 54(2), 341-347, 2002-04

The Mathematical Society of Japan

## Codes

• NII Article ID (NAID)
10008204974
• NII NACSIS-CAT ID (NCID)
AA0070177X
• Text Lang
ENG
• Article Type
ART
• ISSN
00255645
• NDL Article ID
6153492
• NDL Source Classification
ZM31(科学技術--数学)
• NDL Call No.
Z53-A209
• Data Source
CJP  NDL  J-STAGE

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