环面上平均值的双参数变差估计

中文摘要 AI 辅助

单参数变差不等式已得到充分发展，而其多参数对应物则远未被充分理解。我们研究了$\mathbb{R}^3$中环面上平均值的双参数变差不等式。为了捕捉潜在的双参数结构，我们引入了一个局部双参数$r$-变差范数，该范数将矩形增量与沿边界的变差相结合。由此产生的变差算子在逐点意义上支配相应的双参数局部极大函数，并且与极大函数不同，它还能捕捉跨两个参数的振荡。我们建立了该变差算子直至端点的尖锐$L^p$-$L^q$界。作为比较，我们还获得了相应的局部单参数变差算子直至端点的尖锐$L^p$界，揭示了一参数与二参数有界区域之间的本质差异。证明结合了双参数传播子的平方函数估计与通过混合范数插值的局部光滑估计。

英文摘要

One-parameter variational inequalities are well developed, whereas their multi-parameter counterparts remain much less understood. We explore two-parameter variational inequalities for averages over tori in $\mathbb{R}^3$. To capture the underlying two-parameter structure, we introduce a local two-parameter $r$-variation norm that combines rectangular increments with variations along the boundary. The resulting variation operator pointwise dominates the corresponding two-parameter local maximal function and, unlike the maximal function, also captures oscillation across the two parameters. We establish sharp $L^p$--$L^q$ bounds for this variation operator up to endpoints. For comparison, we also obtain sharp $L^p$ bounds up to endpoints for the corresponding local one-parameter variation operator, revealing a genuine difference between the one- and two-parameter boundedness regions. The proof combines square function estimates for two-parameter propagators with local smoothing estimates through mixed-norm interpolation.

发表机构

• Korea Institute for Advanced Study（韩国高等科学研究院）
• Seoul National University（首尔大学）
• Xiamen University（厦门大学）

机构由 AI 辅助整理，请以论文原文为准。