AI 中文总结
该研究针对半空间与卦限，显式计算尖锐$L^2$-CKN不等式及加权Poincaré不等式的最优常数、极值函数，建立稳定性估计，推广了经典高斯Poincaré不等式的结果。
AI 中文摘要
尽管尖锐$L^2$-Caffarelli--Kohn--Nirenberg（CKN）不等式在整个欧氏空间中已被广泛研究，但边界包含原点的区域上的对应问题仍在很大程度上未被探索。我们通过显式计算最优常数、确定所有可能的极值函数并建立亏量的精确恒等式，研究半空间和卦限$\boldsymbol R^{n}_{k,+}$上的尖锐$L^2$-CKN不等式。由于奇异权重$|x|^{-2b}$排除了适用于更简单海森堡不确定性原理的提升论证，我们开发了一种基于变换$u(x)=|x|^{m}v(x)$（其中$m$为适当选取的参数）结合球谐函数分解与加权恒等式的方法。此外，我们建立了与形如$e^{-\tau|x|^{\tau}}|x|^{\beta}\bigl(\boldsymbol \times_{i=n-k+1}^{n}x_i^{2}\bigr)\text{d}x$的测度相关的加权Poincaré不等式，连同其尖锐常数、极值器和稳定性估计，这极大地推广了经典高斯Poincaré不等式的相关结果。在全卦限上，线性模式不再是可容许的竞争者，因为所有奇次球谐函数都会被提升算子湮灭；此时第一个非径向模式为二次阶，尖锐常数和最优解流形也随之改变。最后，我们在半空间和卦限的全参数范围内建立了CKN不等式的若干稳定性估计及二阶稳定性估计。
英文摘要
Though the sharp $L^{2}$-Caffarelli--Kohn--Nirenberg (CKN) inequalities have been extensively studied in the entire Euclidean spaces, the corresponding problem on domains whose boundary contains the origin remains largely unexplored. We investigate the sharp $L^{2}$-CKN inequalities on half-spaces and orthants $\mathbb R^{n}_{k,+}$ by computing explicitly the optimal constants, determining all possible extremal functions, and establishing exact identities for the deficits. Since the singular weights $|x|^{-2b}$ rule out the lifting argument that is available for the simpler Heisenberg Uncertainty Principle, we develop an approach based on the transformations $u(x)=|x|^{m}v(x)$ for an appropriately chosen $m$ combined with spherical harmonic decompositions and weighted identities. Moreover, we establish weighted Poincaré inequalities associated with measures of the form \[ e^{-δ|x|^τ}|x|^β\bigl(\prod_{i=n-k+1}^{n}x_i^{2}\bigr)\,dx, \] together with their sharp constants, extremizers and stability estimates, which substantially extend those of the classical Gaussian Poincaré inequality. On the full orthant, the linear modes cease to be admissible competitors, since all odd spherical harmonics are annihilated by the lifting; the first non-radial mode is then of degree two, and both the sharp constant and the manifold of optimizers change accordingly. Finally, we establish several stability estimates, and second-order stability estimates, of the CKN inequalities on the half-spaces and orthants throughout the full parameter range.
