In the paper, we study derivative estimates of the iterated spherical averages (A_t)^N(f). We obtain the optimal range of exponents (α, N, p) to ensure the L^p 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: L^p(ℝⁿ) → L ^p_α (ℝⁿ).

中文翻译：

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迭代球平均和扩展的导数估计

在本文中，我们研究了迭代球面平均值 ( A _t ) ^N ( f ) 的导数估计。我们获得指数的最佳范围 ( α, N, p ) 以确保L ^p有界\(P\left( {{\partial \over {\partial x}}} \right){\left( {{A_1 }} \right)^N}(f)\)对于 1 ≤ p ≤ ∞，其中P是α次齐次多项式。主要定理扩展了一些已知结果。作为一个应用，我们得到最小的N使得 ( A ₁ ) ^N : L ^p (ℝ ⁿ) → L ^p _α (ℝ ⁿ )。