We study the derivative operator of the generalized spherical mean S
 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.

Derivative estimates of averaging operators and extension

In the paper, we study derivative estimates of the iterated spherical averages (At)N(f). We obtain the optimal range of exponents (α, N, p) to ensure the Lp boundedness of $$P\left( {{\partial \over {\partial x}}} \right){\left( {{A_1}} \right)^N}(f)$$
 for 1 ≤ p ≤ ∞, where P is a homogeneous polynomial of degree α. The main theorem extends some known results. As an application, we obtain the smallest N such that (A1)N: Lp(ℝn) → L
 (ℝn).

Derivative Estimates of Iterated Spherical Average and Extension

Abstract In this article, we study the boundedness properties of the averaging operator S t γ on Triebel-Lizorkin spaces $$\dot F_{p,q}^\alpha \left( {{\mathbb{R}^n}} \right)$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:msubsup><mml:mover><mml:mi>F</mml:mi><mml:mo>˙</mml:mo></mml:mover><mml:mrow><mml:mi>p</mml:mi><mml:mo>,</mml:mo><mml:mi>q</mml:mi></mml:mrow><mml:mi>α</mml:mi></mml:msubsup><mml:mrow><mml:mo>(</mml:mo><mml:mrow><mml:msup><mml:mi>ℝ</mml:mi><mml:mi>n</mml:mi></mml:msup></mml:mrow><mml:mo>)</mml:mo></mml:mrow></mml:mrow></mml:math> for various p,q . As an application, we obtain the norm convergence rate for S t γ ( f ) on Triebel-Lizorkin spaces and the relation between the smoothness imposed on functions and the rate of norm convergence of S t γ is given. 

/pdf/certain-averaging-operators-on-triebel-lizorkin-spaces-2imq632j.pdf

Certain averaging operators on Triebel-Lizorkin spaces

In the paper, we study some multiplier operator $$\mu _{\gamma ,\alpha }$$ raised from studying the $$L^{p}$$-approximation of the spherical mean $$S_{t}^{\gamma }.$$ We obtain the optimal range of exponents $$ (\alpha ,\gamma ,p)$$ such that $$\mu _{\gamma ,\alpha }$$ is an $$L^{p}$$ multiplier. As an application, we obtain the convergence rate for $$ S_{t}^{\gamma }\left( f\right) \ $$in the $$L^{p}$$ spaces.

Certain averaging operators on Lebesgue spaces

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Certain averaging operators on Lebesgue spaces