发表机构
University of Texas at Austin（德克萨斯大学奥斯汀分校）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文证明了 Wigderson 矩方法中单函数估计在尺度不变区域内成立的充要条件，量化了临界线上的常数爆破与端点损失，并给出两个尖锐 Lorentz 端点及最优次要指标，确定了直接约化的端点损失。
AI 中文摘要
设 $$ M_{r,p}^{(d)}(g)=\int_{\mathbb R^d}|x|^r|g(x)|^p\\,dx, \qquad 1<p<\infty. $$ 记 $p'=p/(p-1)$ 且 $\omega_d=|B(0,1)|$。Wigderson--Wigderson 矩方法中的直接步骤要求单函数估计 $$ \lVert g\rVert_1\le K\lVert g\rVert_q^{\\,1-p\eta} \bigl(M_{r,p}^{(d)}(g)\bigr)^\eta, \qquad \eta=\frac{d(1-1/q)}{r+d-dp/q}. $$ 我们给出一个自包含的证明：在其尺度不变区域内，该估计恰好当 $r>d(p-1)$ 时成立。然后我们量化这一转变：最佳常数在临界线上以 $[r-d(p-1)]^{-1/p'}$ 的方式爆破，而在环域 $1\le |x|\le R$ 上端点损失恰好为 $(d\omega_d\log R)^{1/p'}$。然而，临界线仍有两个尖锐的 Lorentz 端点。强矩控制 $L^{1,p}$，即 $$ \lVert g\rVert_{L^{1,p}}\le \omega_d^{1/p'}M_{d(p-1),p}^{(d)}(g)^{1/p}, $$ 并且将加权范数从 $L^p$ 加强到 $L^{p,1}$ 可恢复强目标，即 $$ \lVert g\rVert_1\le \omega_d^{1/p'} \lVert |x|^{d/p'}g\rVert_{L^{p,1}}. $$ 更一般地，每个幂矩都有一个尖锐的目标 Lorentz 空间，其次要指标是最优的。这些结果确定了直接同 $q$ 约化所失去的端点。它们并不阻碍 Wigderson 框架的辅助范数版本，后者通过不同途径达到 Cowling--Price 范围。
英文摘要
Let $$ M_{r,p}^{(d)}(g)=\int_{\mathbb R^d}|x|^r|g(x)|^p\,dx, \qquad 1<p<\infty. $$ Write $p'=p/(p-1)$ and $ω_d=|B(0,1)|$. A direct step in the Wigderson--Wigderson moment method asks for a single-function estimate $$ \lVert g\rVert_1\le K\lVert g\rVert_q^{\,1-pη} \bigl(M_{r,p}^{(d)}(g)\bigr)^η, \qquad η=\frac{d(1-1/q)}{r+d-dp/q}. $$ We give a self-contained proof that, in its scale-invariant regime, this estimate holds exactly when $r>d(p-1)$. We then quantify the transition: the best constant blows up like $[r-d(p-1)]^{-1/p'}$ at the critical line, while on the annulus $1\le |x|\le R$ the endpoint loss is exactly $(dω_d\log R)^{1/p'}$. The critical line nevertheless has two sharp Lorentz endpoints. The strong moment controls $L^{1,p}$, $$ \lVert g\rVert_{L^{1,p}}\le ω_d^{1/p'}M_{d(p-1),p}^{(d)}(g)^{1/p}, $$ and strengthening the weighted norm from $L^p$ to $L^{p,1}$ restores the strong target, $$ \lVert g\rVert_1\le ω_d^{1/p'} \lVert |x|^{d/p'}g\rVert_{L^{p,1}}. $$ More generally, every power moment has a sharp target Lorentz space, and its secondary index is optimal. These results identify the endpoint lost by the direct same-$q$ reduction. They do not obstruct auxiliary-norm versions of the Wigderson framework, which reach the Cowling--Price range by a different route.
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