We prove that the Lipschitz-free space $\mathcal{F}(M)$ contains a complemented copy of $\mathcal{F}(\mathbb{Z}^n)$ whenever $M\subset\mathbb{R}^n$ is not porous. Consequently, if $M\subset\mathbb{R}^n$ is uniformly discrete and not porous then $\mathcal{F}(M)$ is isomorphic to $\mathcal{F}(\mathbb{Z}^n)$.
Let $D\subseteq \mathbb{C}^n$ be a bounded $\mathbb{C}$-convex domain with $C^1$ boundary whose outward unit normal admits a modulus of continuity $\omega$ satisfying $\lim_{t\to 0^+}\dfrac{\omega(t)}{\sqrt t}$. We prove that $D$ admits an increasing exhaustion by bounded $C^{\infty}$ strongly linearly convex domains....
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 $\mu$ be a finitely supported probability measure on $\mathrm{SL}_2(\mathbb{R})$ whose support generates a Zariski-dense discrete subgroup. We prove that its stationary probability measure on $\mathbb{P}^1(\mathbb{R})$ is singular with respect to the $\mathrm{SO}(2)$-invariant probability measure. The proof compare...
Let $\mathcal{M}$ be a finite von Neumann algebra with normalized center-valued trace $T_{\mathcal{M}}$. We prove that every $A\in\mathcal{M}$ with $T_{\mathcal{M}}(A)=0$ admits a representation $A=BC-CB$, where $B,C\in\mathcal{M}$ satisfy $\|B\|\|C\|\le K\|A\|$ for an absolute constant $K$ independent of $\mathcal{M}$...
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...
Let $F:\mathbb{R}^3\to\mathbb{R}$ be a polynomial that is irreducible over $\mathbb{R}$ with $\text{deg} F\geq2$. We prove that, for any finite $X\subset Z(F)$ that does not concentrate on affine lines, \[ E(X)=\#\{(a,b,c,d)\in X^4: a+b=c+d\}\ll_{\text{deg} F,\,\epsilon}(\# X)^{2+\epsilon}. \] In particular, this answe...
Yi-Fan Jing, Shukun Wu· 1 citation
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