Melchior Wirth
We establish a relation between the exponential decay rates of quantum Markov semigroups with respect to different inner products. More precisely, it was conjectured by Fagnola, Poletti, Sasso and Umanità that for a Gaussian quantum Markov semigroup, the exponential decay rate with respect to the KMS inner product is bounded below by the exponential decay rate for the GNS inner product. We show that this is indeed the case and not limited to Gaussian quantum Markov semigroups, but holds for quantum Markov semigroups with a faithful normal invariant state on arbitrary von Neumann algebras. Additionally, the KMS inner product can be replaced by a whole class of inner products induced by operator monotone functions.
Manzi Huang, Panu Lahti, Jiang Li, Zhuang Wang
For $0<δ,τ<1$ and $1\le s\le \frac{n}{n-δ}$, we prove that for a given $s$-John domain $Ω\subset \mathbb{R}^n$, the following Boxing inequality holds for every Lebesgue measurable set $U\subsetΩ$ with $|U|/|Ω|\leγ<1$: \[ \mathcal{H}^{s(n-δ)}_{\infty}(U\setminus\mathcal{N}_U)\le C(1-δ)\int_Ω\int_{|x-y|<τ\operatorname{dist}(y,\partialΩ)}\frac{|χ_U(x)-χ_U(y)|}{|x-y|^{n+δ}}\,dx\,dy, \] where $\mathcal{H}^{s(n-δ)}_{\infty}(U)$ denotes the $s(n-δ)$-dimensional Hausdorff content of $U$, $\mathcal{N}_U$ is a set of Lebesgue measure zero and the constant $C$ depends only on $n,τ,s,γ$, the John constant and the diameter of $Ω$. Moreover, we establish the functional formulation of the above Boxing inequality and discuss the equivalence between these two formulations. Based on the Boxing inequality, we prove the fractional Poincaré--Wirtinger trace inequality on $s$-John domains, of which the fractional Sobolev--Poincaré inequality and fractional Hardy-type inequality are special cases. Notably, we prove all of the aforementioned inequalities with the Bourgain--Brezis--Mironescu (BBM) factor $1-δ$. Furthermore, with the aid of the Bourgain--Brezis--Mironescu formula, we recover the Poincaré--Wirtinger trace inequality. Finally, by showing that, under the separation property, any domain supporting the Boxing inequality is necessarily a John domain, we conclude that the John domain condition is essentially sharp for the above inequalities. All the above inequalities with the BBM factor are new even for Lipschitz domains.
Minghui Ma, Rui Shi, Shanshan Yang
An operator $T$ in a separable factor $\mathcal{M}$ is said to be irreducible in $\mathcal{M}$ if the von Neumann subalgebra $W^*(T)$ generated by $T$ is an irreducible subfactor of $\mathcal{M}$, i.e., $W^*(T)'\cap\mathcal{M}=\mathbb{C}I$. We say that $T$ is a single generator of $\mathcal{M}$ if $W^*(T)=\mathcal{M}$. In this paper, we study generators of separable factors related to maximal abelian self-adjoint subalgebras. As an application, we obtain a complete characterization of normal operators in separable factors which are similar to irreducible operators.
Alberto Dayan, Adrián Llinares, Miguel Monsalve-López
We study properties of $A^p_α$ spaces in the Dirichlet range, recently defined by Brevig, Kulikov, Seip and Zlotnikov as the set of all holomorphic functions on the unit disc $\mathbb{D}$ such that \[ \int_{\mathbb{D}} |f(z)|^{p-2} |f'(z)|^2 (1 - |z|^2)^α \, dA(z) < \infty, \] when $0<α< 1$ and $p > 0$. We answer in the negative two questions posed by Brevig et al. by showing that, if $p\ne2$ and $p > \frac{1}{2}$, $A^p_α$ is not a vector space and that the norm is in general not increasing in $p$. This is achieved by means of an equivalent description for $A^p_α$ which is given in terms of the Poisson integral of the boundary function of its inhabitants. Such norm also leads to a description of $A^p_α$ functions in the Dirichlet range given in terms of their inner and outer factors. As a corollary, we show that $A^1_α$ is contained in the weak product of a Dirichlet-type space.
Vignon Oussa
We record a Lean-certified theorem package for the four-point Heil--Ramanathan--Topiwala configuration \[ Λ=\{0,a,b,ν\}\subset \R^2, \qquad \Lzero=\Z a+\Z b, \qquad ν=r a+s b, \] with $a$ and $b$ linearly independent. The principal certified theorem states that if $|\symp(a,b)|>1$ and $1,r,s$ are linearly independent over $\Q$, then for every nonzero $f\in L^2(\R)$ the four vectors \[ f,\qquad π(a)f,\qquad π(b)f,\qquad π(ν)f \] are linearly independent. A second certified theorem treats the rational-coordinate case $r,s\in \Q$, where the configuration lies in a finer full-rank lattice and linear independence follows from Linnell's theorem. The paper is written in standard mathematical prose. An appendix records the precise Lean certification ledger and the explicit analytic inputs used by the formal development and a download link is provided.
Leandro Candido, Pedro L. Kaufmann
We continue the study on Kurzweil--Stieltjes integration on compact lines initiated in [doi:10.1007/s11117-025-01161-9]. Given a real valued function $G$ on a compact line, the presented integral is called the Kurzweil--Stieltjes integral with respect to $G$, or simply the $G$-integral. %Given a compact line $K$ and a right-continuous function $G:K\to\mathbb{R}$ of bounded variation, we consider the Radon measure $μ_G$ naturally induced by $G$. Our main results concern the relationship between $G$-integrability and measurability. We prove that, whenever $G$ is nondecreasing, every $G$-integrable function is $μ_G$-measurable, where $μ_G$ is the natural Radon measure induced by $G$. We also show that, for an arbitrary $G$ of bounded variation, every bounded $G$-integrable function is $μ_G$-measurable. %, where $|μ_G|$ denotes the total variation measure of $μ_G$. As an application, we provide a full characterization of Lebesgue integrablility with respect to Radon measures in terms of the $G$-integral, and demonstrate that the $G$-integral represents an extension of the Lebesgue integral with respect to $μ_G$ for suitable $G$. In addition, we establish a version of Hake's theorem for the $G$-integral in this setting.
Andrea Kubin, Enrico Pasqualetto
We develop a measure and integration theory for random normed modules. Given a probability space $({\rm X},Σ,\mathfrak m)$, we introduce and study measures taking values into the space $L^0(\mathfrak m)$ of $\mathfrak m$-measurable functions quotiented up to $\mathfrak m$-a.e. equality. Moreover, we develop a Bochner-type integration theory with respect to an $L^0(\mathfrak m)$-valued measure $μ$, for maps whose target ${\rm M}$ is a complete random normed module with base $({\rm X},Σ,\mathfrak m)$, or equivalently an $L^0(\mathfrak m)$-Banach $L^0(\mathfrak m)$-module. Inter alia, we prove versions of the Radon-Nikodým theorem and of the Riesz-Markov-Kakutani representation theorem for $L^0(\mathfrak m)$-valued measures. We also outline several applications of our integration theory: we introduce a notion of martingale with values in a complete random normed module, we propose a definition of random Radon-Nikodým property and we discuss random sets of finite perimeter.
Santiago Chaves, Andreas Debrouwere, Alberto Hernández Alvarado, Jasson Vindas, Rafael Zamora
We establish new strong factorization properties for the smooth vectors of representations of exponential solvable Lie groups on Fréchet spaces. In particular, our results improve upon the Dixmier-Malliavin factorization theorem for simply connected nilpotent Lie groups.
Christos P. Tantalakis, Michał Wojciechowski
We demonstrate a construction of products on the quantum torus $\mathbb{T}_θ^2$ that generalises the usual construction of finite Riesz products on the commutative torus $\mathbb{T}^2$. We explain why the former constitutes a natural analogue of the latter in the non-commutative setting and, based on this construction, as well as on previous results by K. Kazaniecki and the second author, we prove a non-commutative version of an Ornstein non-inequality.
Weicong Su, Zhuang Wang, Yi Ru-Ya Zhang
We investigate the geometric behavior of $τ(E)$ for bounded finite-perimeter sets $E \subset \mathbb R^n$, where $τ(E)$ is the trace constant introduced by Figalli--Maggi--Pratelli [Invent. Math. 2010]. This quantity is a key ingredient in proving a quantitative isoperimetric inequality with the optimal exponent. We first show that for every $ε>0$ one can find a bounded open set $Ω\subset \mathbb R^n$ that is very close to the unit ball $\mathbb B^n$ in the sense that $$ τ(\mathbb B^n)>τ(Ω)>τ(\mathbb B^n)-ε\quad \text{and} \quad P(ΩΔ\mathbb B^n)\le C(n)ε, $$ while at the same time the complement of $Ω$ has infinitely many connected components. Thus, $τ(Ω)$ can be made arbitrarily close to $τ(\mathbb B^n)$ even when $Ω$ has highly intricate geometry. We then establish, under a mild additional hypothesis, the equivalence between a condition formulated in terms of $τ$ and two classical criteria from the literature for open sets that admit trace inequalities. As a consequence, we obtain the John-type characterization of domains that support a trace inequality, assuming the ball separation property.
Thomas P. Wihler
Fixed point iterations are a fundamental tool in numerical analysis and scientific computing for the approximation of solutions to nonlinear problems. Their convergence is often established via the Banach fixed point theorem, provided that a suitable contraction property can be verified. However, such conditions are typically too restrictive for more complex nonlinear equations that lack key structural features such as monotonicity or convexity. In this paper, we develop a general framework for the weak convergence of fixed point iterations based on asymptotic bounds. In particular, we introduce and exploit a weak sequential non-expansiveness property, which is significantly weaker than the global Lipschitz assumptions commonly employed in this context. This approach permits to extend classical convergence results to a broader class of mappings in general (reflexive) Opial spaces, without relying on additional geometric assumptions such as uniform convexity.
Luciano Abadías, Alejandro Mahillo, Pedro J. Miana
In this paper we study boundedness and detailed spectral properties for the Cesàro-Hardy operator and some generalizations in $L^p[0,1]$. The study employs $C_0$-semigroup theory, expressing the Cesàro-Hardy operators and their dual operators through subordination with $C_0$-semigroups $T(t)$ and $S(t)$ respectively. The spectral properties of the semigroup's infinitesimal generators are transferred to the Cesàro-Hardy operators using functional calculus methods. Furthermore, some implications for the Invariant Subspace Problem are explored by demonstrating the universality of certain translations related to the semigroup $T(t)$, and providing results on the invariant subspaces of these operators.
Sergio Albeverio, Suji Kawasaki, Yumi Yahagi, Minoru W. Yoshida
This paper introduces partial results, in the current situation, of ongoing considerations corresponding to the above title. A construction on exact relativistic quantum field model with the space time dimension $d \in {\mathbb N}$, including the case where $d \geq 4$, is going to be discussed. Firstly, Hermitian scalar quantum fields $<{\cal H}, U, ψ, D>$, within a relaxed framework of the Gårding-Wightman Axioms, is constructed by making use of the stochastic calculus arguments with respect to the {\it{stationary additive random fields }} on ${\mathbb R}^d$, i.e., the {\it{L{é}vy random fields}} on ${\mathbb R}^d$. The first constructed $<{\cal H}, U, ψ, D>$, here, satisfy all the requirements of the the Gårding-Wightman Axioms, except that the field operators $ψ(f)$ with $f \in {\cal S}({\mathbb R}^d \to {\mathbb R})$ are symmetric operators on the physical Hilbert space ${\cal H}$, which situation is denoted here as {\it{a relaxed framework}} of the Gårding-Wightman Axioms. Secondly, by taking the adequate subspaces of ${\cal H}$, non trivial exact Wightman quantum fields, which satisfy all the requirements of the Gårding-Wightman Axioms, are constructed actually. keywords: Axiomatic quantum field theory, Gårding-Wightman axioms, Bochner-Minlos theorem, L{é}vy fields on ${\mathbb R}^d$.
David Muñoz-Lahoz
We characterize the Archimedean lattice-ordered algebras with identity that admit a polynomial growth continuous function calculus. More precisely, for an $n$-tuple $\mathbf{x}=(x_1,\dots,x_n)$ in an Archimedean lattice-ordered algebra $X$ with identity $1_X$, we prove that the existence of a lattice-algebra homomorphism from the algebra $PG_n$ of continuous functions on $\mathbb{R}^n$ of polynomial growth, sending the coordinate projections to $x_1,\dots,x_n$ and the constant function to $1_X$, is equivalent to the existence of $f\ge 1_X\vee |x_1|\vee \cdots \vee |x_n|$ and an $f\!$-subalgebra $Y$ of $X$ such that $1_X,x_1,\ldots ,x_n \in Y$ and, for every $m \in \mathbb{N}$, the norm $\|{\cdot }\|_{f^{m}}$ is complete on $Y\cap I_{f^{m}}$. This result may be viewed as an analogue, for lattice-ordered algebras, of the characterization of positively homogeneous continuous function calculus for Archimedean vector lattices due to Laustsen and Troitsky. As a by-product, we describe the finitely generated free objects in the category of uniformly complete Archimedean $f\!$-algebras and also show that the existence of a nontrivial polynomial growth continuous function calculus on a vector space forces it to be a commutative $f\!$-algebra.
Pierre-Cyril Aubin-Frankowski, Virginie Ehrlacher, Gabriele Todeschi
We study the notion of debiasability for cost functions arising in optimal transport. We call a symmetric cost function $c:\mathscr{X}\times\mathscr{X}\to\mathbb{R}\cup\{+\infty\}$ debiasable if it satisfies $c(x,y)\ge \tfrac{1}{2}c(x,x)+\tfrac{1}{2}c(y,y)$ for all $x,y\in\mathscr{X}$. Building on an equivalent characterization by an inf-representation $c(x,y)=\inf_{z\in\mathscr{Z}}ψ(x,z)+ψ(y,z)$ for some set $\mathscr{Z}$ and some function $ψ: \mathscr{X}\times \mathscr{Z} \to \mathbb{R} \cup \{+\infty\}$, interpreted as a generalization of the midpoint identity for squared geodesic distances, we investigate the debiasability of costs defined on spaces of probability measures. Our primary focus is the entropic regularization of optimal transport across different regimes of the regularization parameter $\varepsilon \in [0,+\infty]$, encompassing classical optimal transport ($\varepsilon=0$), entropic optimal transport ($\varepsilon>0$), and the Maximum Mean Discrepancy ($\varepsilon=+\infty$). For $\varepsilon \in (0,+\infty]$, we investigate sufficient conditions, such as negative definiteness of the ground cost or continuity and positive definiteness of the induced kernel, handled then via a convex-nonconcave minimax argument. All our results extend naturally to unbalanced optimal transport settings and we generalize in this way the findings of \cite{feydy2019interpolating} and \cite{sejourne2019sinkhorn}. As a byproduct, we derive novel decomposition formulas for entropic optimal transport, which may be of independent interest.
Lyonell Boulton, Connor Evans
We establish a framework to determine the linear completeness of families of non-linear trajectories in Hilbert spaces, which relies on an infinite analytic block Toeplitz operator formulation. By means of this approach, we show the linear completeness in Sobolev spaces of two families of classical functions. One is the moving family of dilated Weierstrass functions. The other is the family of eigenfunctions of the Gross-Pitaevskii equation with trapping potential in an infinite square well. Our results provide a new insight on the formulation of general methods to examine this intriguing concept, bridging classical non-linear analysis and linear approximation theory.
Ziemowit M. Wójcicki
We study the local Lipschitz one subsets of a finite dimensional space, that is, sets for which there exists a continuous function whose local Lipschitz derivative is the characteristic function of said set. We give a characterization of a local Lipschitz one set on the real line in terms of a certain measure-theoretic density condition, which we call quasi-density. We show that any local Lipschitz one set needs to be quasi-dense, but the converse does not hold. Finally, we show that any regular closed subset of a normed space is a local Lipschitz one set, but there exist local Lipschitz one sets that are not regular closed.
Anil Belli, Ugur Gul, William T. Ross, Aristomenis G. Siskakis
This paper explores a version of the classical Ces`aro integral operator for the Lebesgue space L2(0, 1) where we discuss its norm, adjoint, spectral properties, and invariant subspaces. An important tool will be semigroups of weighted composition operators on L2(0, 1).
Luigi Rodino, Patrik Wahlberg
We define a filter of time-frequency anisotropic global singularities of phase space for tempered distributions. The filter contains information from the corresponding anisotropic Gabor wave front set and admits propagation results for the Cauchy problem for certain linear evolution equations of Schrödinger type that generalize the harmonic oscillator.
Nuno J. Alves
We prove a compactness criterion for asymptotic $L_p$ spaces over arbitrary measure spaces. Total boundedness is characterized by almost equiboundedness together with total boundedness in $L_p$ of all truncations. This gives a measure-theoretic counterpart to the Kolmogorov-Riesz theorem for asymptotic $L_p$ spaces on $\mathbb{R}^n$.