ON THE BOUNDEDNESS AND THE NORM OF A CLASS
OF INTEGRAL OPERATORS??
Lifang ZHOU(周立芳)
【摘 要】Abstract：The boundedness and the norm of a class of integral operators Ta,b,conspaces are studied in this paper.The author not only gives the sufcient and necessary condition for the boundedness of Ta,b,con,but also obtains its accurate norm onfor some range under the condition of c=n+a+b.【期刊名称】数学物理学报（英文版）【年(卷),期】2015(035)006【总页数】8
【关键词】Key words：integral operators;sufcient condition;necessary condition;operator norm; hypergeometric functions
2010 MR Subject Classifcation47B38;47G10
1 Introduction
Let Bndenote the open unit ball in Rn,and dv be the Lebesgue measure on Bn,such that v(Bn)=1.For λ∈R,the weighted measure dvλis given by dvλ(x):=(1-|x|2)λdv(x).It is easy to know that the weighted measure dvλis fnite if and only if λ＞-1.Suppose 1≤p＜∞; Lp:=Lp(Bn,dv)denotes the Lpspace under the measure v,and:=Lp(Bn,dvλ)stands for the weighted Lpspace under the weighted measure vλ.
It is well known that the weighted harmonic Bergman projection
is bounded on for 1＜p＜∞and λ＞-1,where Rλ(x,y)is the weighted harmonic Bergman reproducing kernel;see[1-3]for details.And the boundedness of Pλoncomes from the boundedness of the integral operator,defned by
onaccording to the estimation of Bergman kernel;see[4].For the optimal norm estimate and the exact norm of Sλ,we refer the reader to[5,6].Here and subsequently,[x,y]denotes [x,y]2=1-2x·y+|x|2|y|2.However,the variant of higher-dimension analogue of
Hilbert’sinequality also used the boundedness and the norm of the integral operator Sλwith λ=0;see [7].The Berezin-type transform,which is similar to the integral operator Sλ,was discussed while studying the Toeplitz operator on harmonic Bergman
ON THE BOUNDEDNESS AND THE NORM OF A CLASS OF INTEGRAL OPERATORS
