Derivative estimates of averaging operators and extension

doi:10.1007/s11464-019-0755-y

Front. Math. China

2019, Vol. 14

Issue (2) : 475-491 https://doi.org/10.1007/s11464-019-0755-y

RESEARCH ARTICLE

Derivative estimates of averaging operators and extension

Junyan ZHAO¹(

), Dashan FAN²

¹. Department of Mathematics, Zhejiang University, Hangzhou 310027, China
². Department of Mathematical Sciences, University of Wisconsin-Milwaukee, Milwaukee, WI 53201, USA

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Abstract

We study the derivative operator of the generalized spherical mean $S t γ$ . By considering a more general multiplier $m γ, b Ω = V n − 2 2 + γ (| ξ |) | ξ | b Ω (ξ')$ and finding the smallest $γ$ such that $m γ, b Ω$ is an H^p multiplier, we obtain the optimal range of exponents $(γ, β, p)$ to ensure the $H p (ℝ n)$ boundedness of $∂ β S 1 γ f (x)$ . As an application, we obtain the derivative estimates for the solution for the Cauchy problem of the wave equation on $H p (ℝ n)$ spaces.

Keywords Generalized spherical mean Bessel function H^p multiplier wave equation oscillatory integrals

Corresponding Author(s): Junyan ZHAO

Issue Date: 14 May 2019

Cite this article:

Junyan ZHAO,Dashan FAN. Derivative estimates of averaging operators and extension[J]. Front. Math. China, 2019, 14(2): 475-491.

URL:

https://academic.hep.com.cn/fmc/EN/10.1007/s11464-019-0755-y
https://academic.hep.com.cn/fmc/EN/Y2019/V14/I2/475

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