In 1940, Levin\footnotemark[1] and Milman proved that a closed linear subspace of $C[0,1]$ whose elements all have bounded Jordan variation must be finite-dimensional. We prove its analogue for variation in the sense of Young and, more generally, for every finite-valued nondecreasing gauge $\varphi:[0,\infty)\to[0,\infty)$ with $\varphi(0)=0<\varphi(t)$ for $t>0$. If $E$ is a closed linear subspace of $C[0,1]$ and every $f\in E$ satisfies $\Var_\varphi(\lambda_f f)<\infty$ at some scale $\lambda_f>0$, then $E$ is finite-dimensional. No continuity, convexity, or doubling condition is needed. The proof combines a lower-semicontinuous regularization that preserves the scaled class, Baire uniformization, Helly selection, and a quantitative nested-peaks construction for nonhomogeneous gauges. Consequently, for every infinite-dimensional closed subspace $F\subset C[0,1]$, the set $F\cap\cV_\varphi([0,1])$ is a meagre $F_\sigma$ subset of $F$. For every Young function, both the scaled and raw finite-variation families are maximal dense-lineable but not spaceable. The scaled family is itself a dense vector subspace of Hamel dimension $\mathfrak c$, whereas without local doubling the raw family need not itself be linear.
Let $A$ be an $\alpha$-H\"older cocycle over a transitive two-sided subshift of finite type, and assume that $A$ is strongly bunched. We prove that if there exist $k\in\{1,\ldots,d-1\}$ and $c>0$ such that \[ \lambda_k(p)-\lambda_{k+1}(p)\ge c \] for every periodic point $p$, then $A$ admits a dominated splitting of in...
We construct compact, exponentially stable $C_0$-semigroups $S$ on Hilbert spaces $X$ such that, for every $T>0$, the stochastic convolution $U_g(t)=\int_0^t S(t-s)g(s)\,\mathrm{d}\beta_s $ is unbounded on $[0,T]$ with positive probability for some predictable $g\in L^\infty(\Omega;L^2(0,T;X))$, where $\beta$ is a real...
In this paper we provide a positive version of Miljutin's theorem. Namely, we show that for any uncountable compact metric spaces $K$ and $L$ there exists a positive isomorphism $T:C(K)\longrightarrow C(L)$. Moreover, the inverse of $T$ is a regular operator, and $T$ may be chosen so that $\|T\|_\mathrm{r} \cdot \|T^{-...
Javier Laguna-Ricart, Gonzalo Martínez-Cervantes, Jakub Rondoš et al.· 0 citations
A classical consequence of Caffarelli's fully nonlinear regularity theory [L. A. Caffarelli, Ann. of Math. (2) 130 (1989), no. 1, 189-213] is that viscosity solutions of uniformly elliptic equations $F(D^2u)=f$, with $f\in L^p$, $p>n$, are locally $C^{1,\gamma}$ for every $\gamma<\min\{\alpha_H,\sigma_p\}$. Here $\alph...
Let $D\subset\mathbb C^n$, $n\geq 2$, be a bounded pseudoconvex domain with smooth boundary, and let $H^p_\omega(D)$ be the Hardy space defined using a weighted boundary measure $\omega\,d\sigma$, where $\omega$ is bounded above and bounded away from zero. For every $0<p<\infty$, $p\neq2$, we prove that each surjective...
Ren-Yu Chen, Song-Ying Li, Su-Juan Long et al.· 0 citations
The (B)-theorem of Cordero-Erausquin, Fradelizi and Maurey states that if $\gamma$ is the standard Gaussian in $\mathbb R^n$, $K \subset \mathbb R^n$ is an origin-symmetric convex set, and $s, t \in \mathbb R$ then $\gamma\left(e^{\frac{s + t}{2}} K\right) \ge \sqrt{\gamma(e^{s} K) \gamma(e^{t} K)}$. Herscovici, Livshy...
E. Putterman· 0 citations
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