离散极大函数的高阶正则性
Sharp higher order regularity of discrete maximal functions
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中文总结 AI 辅助
研究离散非中心极大算子应用于特征函数时\(k = 0,1,2\)阶导数在\(\ell^p(\mathbb{Z})\)中的精确界，\(k = 1,2\)时为Hardy-Littlewood极大函数导数首获精确界，还给出\(k\geq3\)及一般函数的下界。
中文摘要 AI 辅助
我们推导了离散非中心极大算子应用于特征函数\(f:\mathbb{Z}\to\{0,1\}\)时，\(k = 0,1,2\)阶导数在\(\ell^p(\mathbb{Z})\)中的精确界。当\(k = 1,2\)且\(1 < p < \infty\)时，这是连续或离散情形下Hardy-Littlewood极大函数导数的首个精确界。我们还建立了\(k\geq3\)及一般函数\(f:\mathbb{Z}\to\mathbb{R}\)的几个下界。
英文摘要
We derive sharp $\ell^p(\mathbb{Z})$ bounds for the $k$th derivative of the discrete uncentered maximal operator applied to characteristic functions $f:\mathbb{Z}\to\{0,1\}$ in the cases $k=0,1,2$. When $k=1,2$ these are the first sharp bounds for derivatives of a Hardy-Littlewood maximal function in continuous or discrete settings when $1<p<\infty$. We also establish several lower bounds for $k\geq 3$ and for general functions $f:\mathbb{Z}\to\mathbb{R}$.
