Frontiers of Mathematics in China

, Volume 14, Issue 2, pp 475–491 | Cite as

Derivative estimates of averaging operators and extension

  • Junyan ZhaoEmail author
  • Dashan Fan
Research Article


We study the derivative operator of the generalized spherical mean S t γ . By considering a more general multiplier \(\mathfrak{m}_{\gamma,b}^\Omega\;=\;V_{\frac{n-2}{2}+\gamma}(\mid\xi\mid)\mid\xi\mid^b\Omega(\xi\prime)\) and finding the smallest γ such that \(\mathfrak{m}_{\gamma,b}^\Omega\) is an Hp multiplier, we obtain the optimal range of exponents (γ, β, p) to ensure the Hp(ℝn) boundedness of ∂βS 1 γ f(x). As an application, we obtain the derivative estimates for the solution for the Cauchy problem of the wave equation on Hp(ℝn) spaces.


Generalized spherical mean Bessel function Hp multiplier wave equation oscillatory integrals 


42B15 42B20 47B38 42B30 


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The authors would like to thank the referees for carefully reading the manuscript and for making several helpful suggestions. This work was supported by the National Natural Science Foundation of China (Grant Nos. 11771388, 11371316, 11471288, 11601456).


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Copyright information

© Higher Education Press and Springer-Verlag GmbH Germany, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Department of MathematicsZhejiang UniversityHangzhouChina
  2. 2.Department of Mathematical SciencesUniversity of Wisconsin-MilwaukeeMilwaukeeUSA

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