幂平均的有限 Hardy 相理论：通过 Carleman 参数的精确解析射击族
Finite Hardy Phase Theory for Power Means: An exact analytic shooting family through the Carleman parameter
AI总结：
本文针对幂平均的有限 Hardy 常数，通过解析参数 q 建立统一射击映射，证明存在唯一实解析相，给出渐近展开系数，统一了 Carleman 与 Hardy 修正。
AI中文摘要：
对于幂平均 $P_t$（$t<1$），设 \\[ \Lambda_N(t)=\sup_{x_k>0} \frac{\sum_{n=1}^N P_t(x_1,\ldots,x_n)}{\sum_{n=1}^N x_n} \\] 为其有限 Hardy 常数。负幂、几何平均和通常的正指数 Hardy 不等式分别对应于 $t<0$、$t=0$ 和 $0<t<1$。利用解析参数 $q=t/(1-t)\in(-1,\infty)$，并记 $\lambda_N(q)=\Lambda_N(q/(1+q))$，我们为所有三种情形导出了一个精确的标量射击映射；其在 $q=0$ 处的表观奇点是可去的，并精确给出有限 Carleman 问题。该映射具有共同的终端条件，其极限向量场在 \\[ y_*(q)=1+q,\qquad \Lambda_*(q)=(1+q)^{(1+q)/q} \\] 处有一个二次临界瓶颈。我们证明存在唯一的实解析相 $\kappa(q)$，使得有限缺陷展开的三次系数为零，并且对 $(-1,\infty)$ 的紧子集中的 $q$ 一致地，对每个固定的 $L\ge2$，有 \\[ \Lambda_*(q)-\lambda_N(q) =\sum_{j=2}^{L}\frac{A_j(q)}{(\log N+\kappa(q))^j} +O\\!\left((\log N)^{-L-1}\right). \\] 在此相归一化尺度下，\\[ A_2=2\pi^2(1+q)\Lambda_*,\qquad A_3=0,\qquad A_4=-\frac{\pi^2}{3}(2q^2+5q+5)A_2. \\] 因此，经典的 Carleman 和正 Hardy 主修正是一个单一解析族的投影。在 Carleman 参数处，我们得到 $A_2(0)=2e\pi^2$ 和 $A_4(0)=-(10/3)e\pi^4$；后者也通过直接的局部留数计算恢复。相本身包含全局离散缺陷。在相归一化之后，涉及非恒定离散响应的第一个系数是 $A_5$，阶为 $(\log N+\kappa(q))^{-5}$。
英文摘要：
For the power mean $P_t$, $t<1$, let \[ Λ_N(t)=\sup_{x_k>0} \frac{\sum_{n=1}^N P_t(x_1,\ldots,x_n)}{\sum_{n=1}^N x_n} \] be its finite Hardy constant. Negative powers, the geometric mean, and the usual positive-exponent Hardy inequality correspond respectively to $t<0$, $t=0$, and $0<t<1$. With the analytic parameter $q=t/(1-t)\in(-1,\infty)$, and writing $λ_N(q)=Λ_N(q/(1+q))$, we derive one exact scalar shooting map for all three regimes; its apparent singularity at $q=0$ is removable and gives the finite Carleman problem exactly. The map has a common terminal condition and its limiting vector field has a quadratic critical bottleneck at \[ y_*(q)=1+q,\qquad Λ_*(q)=(1+q)^{(1+q)/q}. \] We prove that there is a unique real-analytic phase $κ(q)$ for which the finite-defect expansion has zero cubic coefficient and, uniformly for $q$ in compact subsets of $(-1,\infty)$, for every fixed $L\ge2$, \[ Λ_*(q)-λ_N(q) =\sum_{j=2}^{L}\frac{A_j(q)}{(\log N+κ(q))^j} +O\!\left((\log N)^{-L-1}\right). \] In this phase-normalized scale, \[ A_2=2π^2(1+q)Λ_*,\qquad A_3=0,\qquad A_4=-\frac{π^2}{3}(2q^2+5q+5)A_2. \] Thus the classical Carleman and positive-Hardy leading corrections are projections of a single analytic family. At the Carleman parameter we obtain $A_2(0)=2eπ^2$ and $A_4(0)=-(10/3)eπ^4$; the latter is also recovered by a direct local residue calculation. The phase itself contains the global discrete defect. After phase normalization, the first coefficient involving a nonconstant discrete response is $A_5$, at order $(\log N+κ(q))^{-5}$.
