Let $\Omega\subset\mathbb R^n$ be a bounded Lipschitz domain. We prove and widely generalize a conjecture of A.\,I.~Nazarov \cite{Naz21}: for $s\in(1,\frac 32)$ the quadratic form $Q^{\rm SP}_s[u]$ of the spectral fractional Dirichlet Laplacian strictly increases under the map $u\mapsto|u|$ provided $u\in\tilde H^s(\Omega)$ changes sign in $\Omega$.
In this paper, it is proved that the commutator $\mathcal{H}_{β,b}$ which is generated by the $n$-dimensional fractional Hardy operator $\mathcal{H}_β$ and $b\in \dot{Λ}_α(\mathbb{R}^n)$ is bounded from $L^P(\mathbb{R}^n)$ to $L^q(\mathbb{R}^n)$, where $0
Qin Zheng, Zun-Wei Fu· Communications in Mathematic...· 1 citation
Let $F \in \mathbb{Z}[x_1, \ldots, x_n]$ be a homogeneous form of degree $d \geq 4$ which defines a smooth hypersurface in $\mathbb{P}^{n-1}_{\mathbb{C}}$. For $n \geq 24 d^4 2^d$, we prove an asymptotic formula for the number of prime solutions to the equation $F(x_1, \ldots, x_n) = 0$, provided $F$ satisfies suitable...
We construct a bounded pseudoconvex complete Reinhardt domain $\Omega$ with smooth boundary in $\mathbb{C}^3$ such that the $\bar\partial$-Neumann operator $N_1$ is compact although $b\Omega$ contains an analytic disc and thus also fails Catlin's Property $(P)$ and McNeal's Property $(\tilde P)$. This example solves an...
We study spectral and variational properties of the (possibly) fractional Dirichlet Laplacian $\left(-\Delta_{n+k}\right)^s\!$, $s>0$, on bounded subsets of $\mathbb R^n\times\mathbb R^k$ whose extent in one or more directions becomes much larger than in the remaining ones. We first investigate the asymptotic behaviour...
Let $\Gamma$ be a finitely generated hyperbolic group, and let $Q$ denote the conformal dimension of its Gromov boundary $\partial\Gamma$. We prove that for every $p<\frac{Q}{Q-1}$, the group $\Gamma$ admits a proper uniformly Lipschitz affine action on $\ell^p$ with compression exponent $1/p$. The $\ell^p$-space hosti...
This paper is concerned with the following semilinear elliptic equation involving the fractional Laplacian: $$(-\Delta)^s u+ g|u|^{p-1}u= \lambda \frac{u}{|x|^{2s}}+f(x),$$ in a bounded domain $\Omega$ of $\mathbb{R}^N\,(N>2s)$, subject to the zero Dirichlet condition in $\mathbb{R}^N\setminus \Omega$, where $01$ and $...
