Acta mathematica scientia,Series B ›› 2023, Vol. 43 ›› Issue (6): 2398-2412.doi: 10.1007/s10473-023-0605-6

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A DERIVATIVE-HILBERT OPERATOR ACTING ON HARDY SPACES*

Shanli YE, Guanghao FENG   

  1. School of Science, Zhejiang University of Science and Technology, Hangzhou 310023, China
  • Received:2022-06-29 Revised:2023-05-12 Published:2023-12-08
  • Contact: †Shanli YE, E-mail: slye@zust.edu.cn
  • About author:Guanghao FENG, E-mail: gh945917454@foxmail.com
  • Supported by:
    The research was supported by the Zhejiang Provincial Natural Science Foundation (LY23A010003) and the National Natural Science Foundation of China (11671357).

Abstract: Let μ be a positive Borel measure on the interval [0,1). The Hankel matrix Hμ=(μn,k)n,k0 with entries μn,k=μn+k, where μn=[0,1)tndμ(t), induces formally the operator as DHμ(f)(z)=n=0(k=0μn,kak)(n+1)zn,zD,where f(z)=n=0anzn is an analytic function in D.We characterize the positive Borel measures on [0,1) such that DHμ(f)(z)=[0,1)f(t)(1tz)2dμ(t) for all f in the Hardy spaces Hp(0<p<), and among these we describe those for which DHμ is a bounded (resp., compact) operator from Hp(0<p<) into Hq(q>p and q1). We also study the analogous problem in the Hardy spaces Hp(1p2).

Key words: Derivative-Hilbert operators, Hardy spaces, Carleson measures

CLC Number: 

  • 47B35
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