Sharp $L^2$-Caffarelli--Kohn--Nirenberg and weighted Poincar\'e inequalities on half-spaces and orthants and their stability
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 Poincar\'e inequalities associated with measures of the form \[ e^{-\delta|x|^{\tau}}|x|^{\beta}\bigl(\prod_{i=n-k+1}^{n}x_i^{2}\bigr)\,dx, \] together with their sharp constants, extremizers and stability estimates, which substantially extend those of the classical Gaussian Poincar\'e 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.