Derivative estimates of averaging operators and extension

Abstract

We study the derivative operator of the generalized spherical mean {{\displaystyle S}}_t^{\gamma }. By considering a more general multiplier {{\displaystyle m}}_{\gamma ,b}^{\mathrm{\Omega }}={{\displaystyle V}}_{\frac{n-\mathrm{2}}{\mathrm{2}}+\gamma }(\left|\xi \right|){\left|\xi \right|}^b\mathrm{\Omega }(\xi \text{'}) and finding the smallest \gamma such that {{\displaystyle m}}_{\gamma ,b}^{\Omega } is an H^p multiplier, we obtain the optimal range of exponents (\gamma ,\beta ,p) to ensure the H^p({ℝ}^n) boundedness of {\partial }^{\beta }{{\displaystyle S}}_{\mathrm{1}}^{\gamma }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.

Key words： Generalized spherical mean; Bessel function; H^p multiplier; wave equation; oscillatory integrals

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