In this paper, we prove the $L^p(p>1)$-uncertainty principles for any $1<p<n$, including the classical Heisenberg-Pauli-Weyl inequality, Caffarelli-Kohn-Nirenberg interpolation inequality and Hardy inequality in $\mathbb R^n$ as special cases, on $n(\geq 2)$-dimensional forward complete and noncompact Finsler measure spaces $(M, F, \mathfrak{m})$ with curvatures bounded from above or below by constants. Further, we characterize the sharpness of $L^p$-uncertainty principles in terms of the reversibility of $F$ and the bounds of flag (or Ricci) curvature and S-curvature induced by the measure $\mathfrak m$ and obtained some rigidity results, which generalize the related ones in [HKZ] in Finslerian case and [KKPZ] in Riemannian case.
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...
L. Nguyen, Yukta Lodha, Guo-Zhen Lu et al.· 4 citations
We study complete
n
-dimensional submanifolds immersed in the Gaussian space
$$\mathbb {G}^{n+p},$$
that is, the Euclidean space
$$\mathbb {R}^{n+p}$$
endowed with its standard metric and the Gaussian weight
$$e^{-|x|^2/4}$$
. More precisely, in codimension
$$p\ge 2,$$
we consider submanif...
W. Barboza, H. D. de Lima, Wallace Gomes· Annali di Matematica Pura ed...· 0 citations
Let $2\leq k<n/2$ and let $(M^n,[g])$ be a smooth, closed, connected, and locally conformally flat Riemannian manifold with a $k$-admissible metric in the conformal class $[g]$. We prove a stability result of the $\sigma_k$-curvature inequality on $M$, in the sense that if equality is almost satisfied for some conforma...
Let $1<p<n$ and let $v(x)=(1+|x|^{p/(p-1)})^{-(n-p)/p}$ be the standard radial extremal for the sharp Sobolev inequality. We determine all eigenvalues and eigenspaces of the linearized $p$-Laplacian at $v$, defined by its closed quadratic form in $L^2(\mathbb{R}^n,v^{p^*-2} dx)$. After decomposition into spherical harm...
We prove a H\"ormander multiplier theorem for the Dunkl transform associated with an arbitrary finite reflection group. Uniform $H^\sigma(\mathbb{R}^N)$ bounds for the normalised dyadic pieces of a measurable symbol, with $\sigma>\mathbf N/2$ and $\mathbf N$ the homogeneous dimension, imply $L^p(d\omega)$ boundedness f...
Der-Chen Chang, Ji Li, Chao-Jie Wen et al.· 2 citations· ⚡1
We establish exact anisotropic $L^p$-Hardy and Caffarelli-Kohn-Nirenberg identities associated with the Minkowski functional $\|\cdot\|_K$ and the anisotropic radial derivative $\mathcal R_K$, where $1<p<\infty$ and $K\subset\RN$ is a smooth and strictly convex body containing the origin in its interior, not necessaril...
Zhen-Zhen Wei· 0 citations
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