Complex Analysis and Operator Theory

, Volume 6, Issue 2, pp 373–396 | Cite as

Hardy–Littlewood Inequalities for Fractional Derivatives of Invariant Harmonic Functions

  • Guangbin Ren
  • Uwe KählerEmail author
  • Jihuai Shi
  • Congwen Liu


We establish Hardy–Littlewood inequalities for fractional derivatives of Möbius invariant harmonic functions over the unit ball of \({\mathbb R^n}\) in mixed-norm spaces. In doing so we introduce a new criteria for the boundedness of operators in mixed-norm L p -spaces in terms of hyperbolic geometry of the real unit ball.


Hardy–Littlewood inequalities Invariant harmonic functions Mixed-norm spaces Normal functions 

Mathematics Subject Classification (2000)

Primary 46E35 Secondary 47B38 


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Copyright information

© Springer Basel AG 2010

Authors and Affiliations

  • Guangbin Ren
    • 1
  • Uwe Kähler
    • 2
    Email author
  • Jihuai Shi
    • 1
  • Congwen Liu
    • 1
  1. 1.Department of MathematicsUniversity of Science and Technology of ChinaHefeiChina
  2. 2.Department of MathematicsUniversity of AveiroAveiroPortugal

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