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 necessarily origin-symmetric. These identities contain explicit nonnegative $R_p$-remainders and yield sharp anisotropic radial $L^p$-Hardy and $L^p$-Caffarelli-Kohn-Nirenberg inequalities. In the noncritical parameter regions, we characterize the full family of extremal functions in the natural weighted completion space, while in the critical case the sharp constant is not attained by any nonzero function in that space. As applications, we establish sharp anisotropic $L^p$-Heisenberg uncertainty principles and obtain explicit anisotropic Gaussian-type optimizers. Finally, letting $K^*$ denote the polar body of $K$, we derive norm-based anisotropic gradient inequalities associated with the origin-symmetric body $K^*\cap(-K^*)$. When $K$ is origin-symmetric, the corresponding gradient constants are sharp, including those in the critical 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 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 study nonnegative entire spacelike graphs in Lorentz--Minkowski space satisfying $\sigma_k(A[u])=u^p$. For $2\le k<n$ and $p\ge k$, pointwise strict spacelikeness and closed $k$-admissibility yield universal bounds for the height and Lorentz factor. Their proof combines block coercivity in the full G\r{a}rding cone,...
We study sparse Kahane--Salem--Zygmund constructions and weighted Hardy--Littlewood inequalities for homogeneous polynomials. For supports of cardinality $n^{d+o(1)}$, we determine the sharp power of $n$ governing the smallest norm of a unimodular $m$-linear form on $\ell_{p_1}^n\times\cdots\times\ell_{p_m}^n$; in the...
A. Barbosa, D. Núñez-Alarcón, A. Raposo et al.· 0 citations
We prove a Liouville theorem for nonnegative entire strictly spacelike solutions of \[ \operatorname{div}\left(\frac{\nabla u}{\sqrt{1-|\nabla u|^2}}\right)+u^p=0 \qquad\text{in }\mathbb R^n. \] If $n=2$ and $p\geqslant1$, or if $n\geqslant3$ and $1\leqslant p\leqslant\frac{n+2}{n-2}$, every nonnegative $C^2$ solution...
Xi-nan Ma, Tian Wu, Wang-Zhe Wu et al.· 0 citations
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...
Yi-Tian Zhang· 0 citations
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