Let $X$ and $Y$ be locally compact Hausdorff spaces and let $\mathbb{K}\in \{\mathbb{R},\mathbb{C}\}$. We say that a bijection $T\colon C_0(X,\mathbb{K})\to C_0(Y,\mathbb{K})$ is a ring isomorphism in norm if \[ \|T(f+g)\|=\|T(f)+T(g)\|,\qquad \|T(fg)\|=\|T(f)T(g)\| \] for every $f,g\in C_0(X,\mathbb{K})$. We determine the form of such maps. When $\mathbb{K}=\mathbb{C}$, under the additional assumption that $\|T(\overline f)\|=\|T(f)\|$ for every $f\in C_0(X,\mathbb{C})$, there exist a continuous function $w\colon Y\to\{\lambda\in\mathbb{C}:|\lambda|=1\}$, a homeomorphism $\varphi\colon Y\to X$, and a closed and open subset $Y_0\subset Y$ such that \[ T(f)(y)= \begin{cases} w(y)f(\varphi(y)),&y\in Y_0,\\ w(y)\overline{f(\varphi(y))},&y\in Y\setminus Y_0, \end{cases} \] for every $f\in C_0(X,\mathbb{C})$ and $y\in Y$. When $\mathbb{K}=\mathbb{R}$, there exist a continuous function $w\colon Y\to\{\pm1\}$ and a homeomorphism $\varphi\colon Y\to X$ such that \[ T(f)(y)=w(y)f(\varphi(y)) \] for every $f\in C_0(X,\mathbb{R})$ and $y\in Y$. In particular, the real case extends the norm version of the Gelfand--Kolmogoroff theorem to the locally compact setting.
Let $d\geq 1$. Let $K\subset\mathbb{R}^d$ be a non-singleton self-similar set generated by a finite strongly irreducible iterated function system satisfying the open set condition, and let $\delta=\dim_{\mathrm H} K$. For $\tau>1/d$, set \[ W_d(\tau) = \left\{ \mathbf{x}\in\mathbb{R}^d: |q\mathbf{x}-\mathbf{p}|<q^{-\ta...
In this paper we consider area-preserving twist maps $f:\mathbf{T}^2\to\mathbf{T}^2$ with vertical rotation sets reduced to a single irrational number $\alpha$. Our first result states that under these hypotheses, there exists a unique $f$-invariant foliation $\mathcal{F}_G=\{\operatorname{Graph}(\phi_t)\}_{t\in\mathbf...
Let $\mathcal{X}$ be a complex manifold and $X$ a compact complex manifold with boundary in $\mathcal{X}$. For a complex Lie group $G$ and a regularity class $$\mathfrak{r}\in \big\{\mathcal{C}^k|\ k\in\mathbb{N}\cup\{\infty\}\big\}\cup\big\{\Lambda^r_{\rm loc}|\ r\in (0,\infty)\big\} $$ we define the sheaf of groups $...
Let $\mathcal{A}$ be the normalized class of analytic functions $f$ in the unit disc $\mathbb{D} := \{z \in \mathbb{C} : \vert{}z\vert{}<1\}$. For $\beta>1$, let $\mathcal{N}(\beta)$ denote the subclass of $\mathcal{A}$ satisfying $\text{Re}\{1 + z f''(z)/f'(z)\}<\beta$ for $z \in \mathbb{D}$. The main purpose of this...
We study the generalized Hilbert operator \[ \mathcal{H}_b f(z)=\int_0^1 f(t)\,\frac{(1-t)^b}{(1-tz)^{b+1}}\,dt, \qquad b>0, \] acting on the Hardy spaces $H^p$ for $1\leq p\leq \infty$. We establish the precise operator norm \[ \|\mathcal{H}_b\|_{H^p\to H^p}=B\!\left(\frac1p,b+1-\frac1p\right) \] for every $1<p<\infty...
Let $X\subset \mathbb C^n$ be a closed pure $d$-dimensional complex analytic set. We associate to $X$ its set of projective asymptotic directions $$ \Sigma_\infty(X) :=\{\ell \in \mathbb P^{n-1}(\mathbb C):\ell\cap C_\infty(X)\ne\{0\}\}, $$ where $ C_\infty(X)$ is the total tangent cone at infinity. We prove the metric...
Nhan D. V. Nguyen· 1 citation· ⚡1
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