Papers

This paper introduces a new extrapolation framework within the Calculus of Variations. We develop a robust quantitative theory of the $\Gamma$-convergence approach of De Giorgi by combining it with Gagliardo's work on relative completions in interpolation theory. A central new tool is the extrapolation of \emph{variational $K$-functionals}, which associates to an arbitrary family of functionals a dynamic penalty method that simultaneously characterizes their $\Gamma$-limits and yields explicit rates of convergence for minimizers. As a technical foundation we prove compactness extrapolation theorems also involving Gagliardo completions. To illustrate the mechanism, we apply the technique to variational models arising in image processing, notably the Rudin–Osher–Fatemi (ROF) and Chan–Esedoglu models. Our main result in this setting is a contribution to the inverse regularity problem associated to discrete approximations of the ROF model: we show that the required regularity of the observed image that guarantees a prescribed optimal rate of convergence is measured in terms of the decay of variational $K$-functionals. Surprisingly, our characterization is explicitly captured by using the recently introduced spaces of Brezis, Van Schaftingen, and Yung. Moreover, our methods unify, and considerably extend, classical pointwise convergence and $\Gamma$-versions of the Bourgain–Brezis–Mironescu and Maz'ya–Shaposhnikova formulae, as well as related results in a variety of settings that involve BMO-type functionals, heat kernels, Sobolev spaces on metric measure spaces, ... By bridging De Giorgi's variational convergence with Gagliardo's completion through variational $K$-functionals and extrapolation of compactness, this work establishes a quantitative approach to the Calculus of Variations.

A classic result of Poly and Raby \cite{Poly-Raby:1984} states that, in finite-dimensional Euclidean spaces, $\mathcal{C}^k$-smoothness (with $k\geq2$) of the squared distance function near a point of a closed set characterizes smoothness of the set as a submanifold, with the same order of differentiability. In this work, we show that this characterization splits in infinite-dimensional Hilbert spaces: $\mathcal{C}^k$-smoothness of the squared distance function characterizes weakly $\mathcal{C}^k$-submanifolds, while $\mathcal{C}^k$-submanifolds are characterized by this smoothness together with an additional pointwise equicontinuity condition on the highest-order derivative of the squared distance. The diffeomorphism of Poly and Raby, its associated graph representation, and the description of its inverse remain the common geometric basis of both characterizations. The development of this work was assisted by GPT-6 Astra.

We prove that if the unit sphere of a separable infinite-dimensional real Banach space embeds uniformly into a Hilbert space, then the Banach space has Kasparov–Yu's Property (H).

A synthetic view is provided on scattered results which were never presented in this form in the literature. First we investigate precisely the oscillation length for periodic functions. Then, we establish the existence of a uniform oscillation lentgh for any scalar almost periodic function with mean-value 0. After that, we investigate upper bounds of the oscillation length for some almost periodic functions which appear in the study of continuum mechanics, and we show the existence of a finite oscillation length with respect to time in any open subset of some vibrating continuous media. The paper finally recalls a long standing open problem on the vibrations of a square membrane with fixed edge.

We prove a noncommutative maximal ergodic inequality for averages along polynomial sequences. Let $\gamma$ be a trace-preserving automorphism of a semifinite von Neumann algebra $(\mathcal N,\tau)$. We show that the associated polynomial averages \begin{equation*} A_Nf:=\frac1N\sum_{n=1}^N\gamma^{P(n)}(f), \qquad N\in\mathbb N, \end{equation*} satisfy a strong maximal inequality on $L_p(\mathcal N)$ for every $1<p<\infty$, extending the previously known restricted range of $p$. The proof follows Bourgain's major-arc strategy but requires substantially new ideas and tools for operators that may have further applications in noncommutative analysis. More precisely, we obtain a localized maximal inequality by developing a noncommutative version of Stein's extrapolation, novel combinatorial methods, and a surprising multilinear version of Doob's maximal inequality. For the required decaying $L_2$-approximation, we combine two of Bourgain's constructions in a way that avoids the multi-frequency maximal inequality used in the scalar proof.

Spectral Barron spaces are Fourier-Lebesgue-type spaces that arise in the approximation of functions by neural networks. Conventional spectral Barron spaces are mostly considered on the whole space or on a bounded domain with certain smoothness assumptions on the boundary. In this work, we investigate spectral Barron spaces for periodic functions, thereby extending the study of spectral Barron spaces to high-dimensional tori and, more generally, to locally compact Abelian groups. These extensions are carried out by considering Fourier series expansions that are isometrically isomorphic to weighted sequence spaces built upon the $\ell^1$ sequence space. We discuss the well-posedness of PDEs in these newly introduced spaces. Two specific inverse problems for evolution equations are investigated, yielding conditional stability within the framework of spectral Barron spaces on the torus.

We prove that every subspace $V\subset\ell_\infty^N$ of codimension one or two with absolute projection constant $\lambda(V)<4/3$ satisfies $$ d(V,\ell_\infty^{\dim V}) \leq\frac{\lambda(V)}{4-3\lambda(V)}. $$ We conjecture that the codimension assumption can be removed and that the same estimate holds for every subspace $V\subset\ell_\infty^N$ with $\lambda(V)<4/3$.

In this paper, we give the complete characterization of the boundedness of the discrete fractional maximal operator $M_\gamma $, defined on $\mathbb{Z}^n$, $n\in\mathbb{N}$, $\gamma\in [0,n)$, between the classical Lorentz sequence spaces $\lambda_p(v)$ and $\lambda_q(w)$, where $0<p,q<\infty$, $\{v(m)\}$ and $\{w(m)\}$ are non-negative sequences defined on $\mathbb{N}$. We first obtain a sharp upper estimate for the nonincreasing rearrangement of the discrete fractional maximal operator $M_\gamma x$, with a supremal operator involving a discrete Hardy operator and the nonincreasing rearrangement of the sequence $\{x(k)\}$, and we show that this estimate is sharp by showing a lower estimate for special sequences. Using this result, we can reduce the characterization of the boundedness of the discrete fractional maximal operator $ M_\gamma$ between the discrete weighted Lorentz sequence spaces to the boundedness of the discrete supremum operator between weighted Lebesgue sequence spaces, restricted to nonincreasing sequences. This problem was investigated in the authors' recent paper, which enables us to obtain the final characterization.

Let $w \colon (0,1) \to \mathbb{R}_{+}$ be a weight. We prove that for an arbitrary Schauder basis $\{e^{i \lambda_n t}\}_{n \in \mathbb{Z}}$ in $L^2(0,1)$ and an arbitrary lacunary defect set $A \subset \mathbb{Z}$ the system $\{w(t) r_n(t)\}_{n \in \mathbb{Z} \setminus A}$ is always complete and never minimal for any weight $w$ satisfying a natural decay condition, which is sharp on the exponential scale. Moreover, we establish a simple combinatorial criterion for systems of the form $ \{ w(t) e^{2\pi i n t} \}_{n \in \mathbb{Z} \setminus A} $ to be complete and minimal in $L^2(0,1)$ for an arbitrary weight $w$ and a finite defect set $A \subset \mathbb{Z}$.

The variational derivation of two-dimensional von Kármán shell energies from three-dimensional nonlinear elasticity is investigated in the presence of constant pressure loads. On the one hand, the paper extends the existing analysis of pressure-loaded plates to the curved setting of shells. On the other hand, it complements the classical variational derivation of shell theories by incorporating a nonlinear pressure force.

We consider Walsh's process on a star graph perturbed by the fact that its paths need to additionally pass through semi-permeable soft membranes, each located on a different edge at a small distance from the center, and each having its own permeability coefficient. We describe the limit processes as the distances converge to $0$, in terms of the characteristics of the approximating ones. The main finding is that the limit process is still of Walsh type, but its parameters need to be appropriately modified. The paper is a close companion to A. Bobrowski and A. Pilipenko “On Portenko's approximation of skew Brownian motion” (arXiv:2601.20440), and to A. Pilipenko and G. Vilmart, “Limit theorems for one-dimensional diffusions: reflected, skew, sticky, skew-sticky, and snapping-out limits.”

We consider a family of non-local and non-convex functionals depending on three real parameters $(\gamma,p,\lambda)$, which was introduced by H. Brezis, A. Seeger, J. Van Schaftingen and P. L. Yung. In the case $\gamma<-p$ we prove a lower bound for the Gamma-liminf of these functionals as $\lambda\to 0^+$ and a Poincaré-type inequality. We also show that such inequality cannot hold if $\gamma>0$, and we establish a modified version in this case. These results answer some open questions posed by these authors and by H. M. Nguyen.

We study continuous Brascamp–Lieb inequalities associated with isotropic measures on the sphere $S^{n-1}$ and on the Grassmannian $G(n,k)$. We give two proofs of the continuous Grassmannian inequality. The first proof approximates the continuous isotropic measure by discrete isotropic measures and then applies the equal-rank geometric Brascamp–Lieb inequality. In the rank-one case, the construction of the discrete isotropic measures follows from Barthe's work, which is standard, whereas in the higher-rank case the construction of finite approximate isotropic measures is given by a Tchakaloff-type theorem. Weak convergence, geometric Brascamp–Lieb inequality and Fatou's lemma then yield the continuous inequalities. The second proof generalizes the heat-semigroup preservation method of Barthe and Huet to isotropic measures on $G(n,k)$. We prove a heat-semigroup preservation theorem in the Grassmannian setting and show that this theorem yields a Grassmannian Brascamp–Lieb inequality. At the end, we also give a sketch for the mixed-rank case of the continuous Brascamp–Lieb inequality.

A Banach space has the ball-covering property (BCP) if its unit sphere can be covered by countably many balls that miss the origin. A Banach space has the renorming-stable ball-covering property (RBCP) if it has the BCP with respect to every equivalent norm. We investigate the RBCP from the perspective of $C(K)$-spaces. In particular, we show that there are many non-separable $C(K)$-spaces having the RBCP. This yields an emphatic negative answer to a question of Cheng, Kato and Zhang from 2020.

In this paper, we introduce a class of function-preserving neural network operators designed to preserve a prescribed strictly positive function. We examine the approximation behaviour of the proposed operators on the space of continuous functions and show that they are positive, linear, well-defined, and bounded. Specifically, we demonstrate uniform convergence on compact intervals and get quantitative estimates of the approximation error in terms of the modulus of continuity. To support the theoretical analysis, we present several numerical experiments focusing on the preservation of exponential and logarithmic functions. These examples show how well the proposed operators perform and compare them with existing neural network operators. Also, an application to denoising modulated signals is provided, demonstrating the practical relevance of the proposed approach.

We study the large time asymptotics of the Feynman-Kac semigroups of the symmetric Lévy process on unbounded open sets. Our main result proves the exact exponential asymptotic decay rate for the survival probability given in terms of the bottom of the spectrum of the associated nonlocal Schrödinger operator. We also prove quantitative upper bounds with an explicit polynomial correction. The proof is probabilistic and is done by decomposing the Lévy process into a finite-range jump process collecting the small jumps and an independent compound Poisson process describing the large jumps. Our approach gives a direct link between the spectral properties of nonlocal Schrödinger operators and pointwise decay of survival probabilities, which was previously unknown for jump processes.

Together with Bourgain's slicing problem and the variance conjecture, the Kannan-Lovász-Simonovits (KLS) conjecture has been a driving force for mathematical developments in high dimensional geometry in recent decades. We present here a proof of this conjecture. The crucial step in the proof is the very recent criterion from Song and Zhang involving high-order derivatives of tilt averages.

For $d\ge3$, let $\lambda_d$ denote the sharp constant in the nearest-neighbor Hardy inequality on $\mathbb Z^d$ with Euclidean inverse-square weight for functions vanishing at the origin. Recent work established that the continuous Hardy coefficient \[ C_d=\frac{(d-2)^2}{4} \] is sharp in dimensions three and four, while it is not sharp for $d\ge9$. We prove $\lambda_d=C_d$ for the remaining cases $5\le d\le8$. The proof uses reciprocal edge fields and pointwise estimates for the associated vertex weights. The exterior estimates follow from polynomial inequalities, and the weights near the origin are treated by explicit estimates and local corrections.

We study three affine variational quantities built on the affine $L^p$ energy of Lutwak, Yang and Zhang: the affine Poincaré–Sobolev constants $\Lambda^{\mathcal{A}}_{p,q}(\Omega)$ of a bounded open set, the associated affine torsional rigidity, and the affine $p$-capacity of a compact set. We show that ellipsoids are the only extremal sets in their corresponding isoperimetric inequalities, and that the affine capacity of a compact set of positive volume is attained by a potential solving the affine $p$-Laplace equation. We also prove an affine Kohler-Jobin inequality: among sets with prescribed affine torsional rigidity, only ellipsoids minimize $\Lambda^{\mathcal{A}}_{p,q}$. Following the approach of Haddad, Jiménez and Montenegro to affine Sobolev inequalities, our methods rest on the $L_p$ Busemann–Petty centroid inequality. Proving a stability version of it, we obtain quantitative forms of the isoperimetric inequalities for $\Lambda^{\mathcal{A}}_{p,q}$, the affine torsional rigidity and the affine $p$-capacity, together with a stability result for the affine Kohler-Jobin inequality on convex sets.

We construct Anderson-type operators in the full subcritical regime via a SPDE argument. Our work encompasses perturbations by a Gaussian noise of the fractional Laplacian on the torus or on the full space, and of the harmonic oscillator on the full space. In every case, we are able to derive an optimal estimate on the difference between the solution and the deterministic heat flow, which allows us to get a Weyl law for the fractional case on the torus and for the harmonic oscillator case, and so in the full subcritical regime. For the non-translation-invariant operator, the main novelty is that we develop a general argument in order to prove how to renormalise the operator with some constants. The SPDE argument uses the flow approach introduced by Duch. To construct the operators, we use the Hille-Yosida theorem for the fractional Laplacian on the torus and the harmonic oscillator, and the Klein-Landau theorem for the fractional Laplacian in the full space.

We study a log-Hardy-Littlewood-Sobolev (log-HLS) inequality under constraints and a naturally induced Wasserstein flow. The constrained log-HLS inequality is dual, in a variational sense, to the constrained Trudinger-Moser-Onofri-Aubin inequality. The corresponding gradient-flow formulation leads to a parabolic-elliptic Patlak-Keller-Segel (PKS) system whose global dynamics exhibit a dichotomy between global existence and finite-time blow-up depending on the initial mass. Remarkably, the constraint raises the critical mass threshold of the flow by a factor of two compared with the unconstrained setting. This provides a distinct mechanism for suppressing blow-up from those arising in related formulations of the PKS-system, where blow-up is mitigated through additional effects such as a forcing term, coupling with a fluid equation, or the ambient flow generated by harmonic perturbations of the Newtonian potential.

We determine the sharp uniform constant in the quartic stability estimate for the logarithmic Sobolev inequality on finite cycles. We equip $C_{n}=\mathbb{Z}/n\mathbb{Z}$ with the uniform probability measure, and we let $\mathcal{E}_{n}$ be the Dirichlet form of the simple random walk on $C_{n}$, normalized so that its spectral gap is one. Frank and Ivanisvili recently proved the sharp logarithmic Sobolev inequality $2\mathcal{E}_{n}(u)\geq \mathrm{Ent}(u^{2})$ for all $n\geq 4$. We prove that $2\mathcal{E}_{n}(u)-\mathrm{Ent}(u^{2})\geq \frac{1}{3}\Vert u-1\Vert _{2}^{4}$ for all $n\geq 4$ and all nonnegative $u$ with $\Vert u\Vert _{2}=1$. This is the sharp form, uniformly in $n$, of the quartic stability estimate of Xie and Zhang, who obtained the constant $\frac{1}{12}$. The same estimate holds on the circle and for the word-length energy on $C_{n}$. Among these models, the four-cycle is the only extremal one: on $C_{4}$ the constant $\frac{1}{3}$ is sharp but not attained, while on $C_{n}$ with $n\geq 5$ and on the circle the sharp constant is at least $\frac{1}{3}+c$, where $c>0$ does not depend on $n$. We also give explicit upper bounds for these sharp constants. Our second main result is a stability estimate for the cubic Sobolev inequality of Frank and Ivanisvili, which is the main step in their proof. It gives the quantitative version of this inequality expected by Frank and Ivanisvili, with the optimal powers of $\Vert u-1\Vert _{2}$: four on $C_{n}$ with $n\geq 5$ and on the circle, and six on $C_{4}$. Two key new ingredients of the proofs are a bound for the first Fourier coefficient of $u$ in terms of its mean, which holds because $u$ is nonnegative, and a sharp stability estimate for Gross's two-point logarithmic Sobolev inequality, with sharp constants in both the fourth and the sixth powers of the distance.

Equi-isoclinic (EI) and equichordal (EC) tight fusion frames (TFFs) are types of minimally coherent (maximally orthogonal) subspace packings that arise in several applications, such as compressed sensing. They seem challenging to construct, and the problem of characterizing their existence remains mostly open. In this paper, we characterize those special complex EITFFs and ECTFFs that arise from isometries whose cross-Gram matrices commute, as is the case with filter banks. In particular, we show that every filter bank EITFF is equivalent to a direct sum of equiangular tight frames (ETFs). More generally, we show that every filter bank ECTFF is analogously equivalent to a novel generalization of ETFs that we dub stratified unit norm tight frames (SUNTFs). We further establish a general theory of SUNTFs, providing various explicit constructions of them, and some necessary conditions on their existence. All of our techniques are explicit, yielding direct constructions of incoherent filter banks.

The discretization of continuous frames in Hilbert spaces is well established, while the corresponding problem for Hilbert $C^*$-modules presents additional difficulties arising from the $C^*$-algebra-valued inner product and the structure of compact operators on Hilbert modules. In this paper, we develop an operator-selection framework for the discretization of continuous frames in Hilbert $C^*$-modules over finite-dimensional $C^*$-algebras. Our main result is a trace-controlled selection theorem for positive compact operators acting on countably generated Hilbert $C^*$-modules over finite-dimensional $C^*$-algebras. In particular, we establish a module version of the binary selection principle for positive trace-controlled compact operators. The proof combines finite-rank approximation, the matrix representation of finite-dimensional $C^*$-algebras, and operator estimates adapted to the $C^*$-module setting. We then apply these selection results to the discretization of continuous Hilbert $C^*$-module frames. Under suitable assumptions on the underlying metric measure space, with the coefficient algebra $\mathcal A$ finite-dimensional, and assuming that every closed submodule of $\mathcal H$ is orthogonally complemented. The resulting sampling family is a Hilbert $C^*$-module frame with explicit frame bounds and controlled separation.

We construct a quasi-Banach space with a semi-greedy Markushevich basis which is not quasi-greedy, and hence not almost greedy. This gives a negative answer to the question of whether the equivalence between semi-greediness and almost greediness for Markushevich bases in Banach spaces extends to the general quasi-Banach setting.

Let $Q_0:=(0,1)^n$. We define $\mathcal{B}_q(Q_0)$ by replacing the local $L^1$ mean oscillation in the Bourgain–Brezis–Mironescu space $\mathcal{B}(Q_0)$ with its $L^q$ counterpart, which coincides with $\mathrm{BMO}(Q_0)$ when $n=1$. For $n\geq2$, our results reveal different endpoint behaviors for general functions, characteristic functions and the exact perimeter formula. (i) For general functions, $\mathcal{B}_q(Q_0)=\mathcal{B}(Q_0)$ precisely when $q\in(0,\frac{n}{n-1})$. We also establish the corresponding sharp self-improvement and duality results for Hardy-type preduals. (ii) For characteristic functions, the endpoint is attained: the equivalence $\|\mathbf{1}_E\|_{\mathcal{B}_q(Q_0)}\sim\|\mathbf{1}_E\|_{\mathcal{B}(Q_0)}\sim\min\{1,P(E,Q_0)\}$ holds uniformly over measurable $E\subset Q_0$ precisely when $q\in(0,\frac{n}{n-1}]$, where $P(E,Q_0)$ denotes relative perimeter. (iii) The isotropic functional $I_{\varepsilon,q}^{*}(E)$, defined using arbitrarily oriented cubes, converges to $\frac{1}{2}\min\{1,P(E,Q_0)\}$ as $\varepsilon\to0^+$ throughout $q\in(0,\frac{n}{n-1}]$. In dimension two, the formula extends to $q\in(0,q_*]$ for some universal $q_*>\frac{19}{9}$, whereas it fails in every dimension for every $q>q_0$, where $q_0>2$ is defined variationally and numerical computation suggests that $q_0\approx2.15391$. The proofs of the three sharp self-improvements use sparse median domination, atomic block decompositions, and relative isoperimetry, respectively. For the perimeter formula, a uniform De Giorgi blow-up reduces the problem to a geometric cube-section inequality; geometric averaging and an explicit packing yield the planar extension and the obstruction, respectively.

Let \(\mathcal{H}\) be a complex Hilbert space, and let \(T=H+iK\in\mathcal{B}(\mathcal{H})\) be the Cartesian decomposition of \(T\), where \(H=H^{*}\) and \(K=K^{*}\). We introduce the \(L^2\)-Cartesian numerical radius \[ \omega_2(T)=\left(\frac{1}{2\pi}\int_0^{2\pi}\|\cos\theta H+\sin\theta K\|^2\,d\theta\right)^{1/2}. \] We study its basic properties and compare it with the classical numerical radius \(\omega(T)\). In particular, we establish the sharp estimate \[ \frac{1}{\sqrt{2}}\omega(T)\le \omega_2(T)\le \omega(T), \] and establish lower and upper bounds in terms of the Cartesian parts \(H,K\). As a product, we obtain \[ \frac{1}{4}\left\|T^{*}T+TT^{*}\right\|\le \omega_2(T)^2\le \omega(T)^2. \]

Sharp inequalities between the Riesz and logarithmic capacities of compact sets in $\mathbb{R}^n$ are established, and the equality conditions are fully characterized. These results naturally extend Pólya and Szegő's conjecture on the ratio of Newtonian to logarithmic capacities in $\mathbb{R}^2$, which was posed in the 1940s.

We prove lower bounds for Fourier transforms of surface measures $$|\widehat{\psi\,d\sigma}(\xi)| \gtrsim_{\psi, K, \sigma, R_0} \sigma(B(p, |\xi|^{-1})), \text{ given }p\in K\Subset\{\psi>0\} ,\ \vec{n}_{p}\parallel \xi,\ |\xi|\ge R_0,$$ on convex hypersurfaces of finite type with at most one vanishing principal curvature at each point. These bounds are uniform in the choice of $p\in K\Subset \{\psi>0\}$ and match the classical upper bound of Bruna-Nagel-Wainger. The key new analytic ingredient is a uniform lower bound for parameter-dependent oscillatory integrals on $\mathbb{R}$ with convex phases. We also give convex examples of finite type with two vanishing principal curvatures for which $$\inf_{p\in K}|\widehat{\psi\,d\sigma}(R \vec{n}_p)|\lesssim_N R^{-N},\ \text{ for arbitrary fixed } \{0\}\Subset K\Subset\{\psi>0\}.$$ Though the estimate above has its own interest, our original motivation is to study Fourier frames for surface measures. In particular, we show that, if $\Gamma$ is a closed convex smooth surface in $\mathbb{R}^d, d\geq 2$, of finite type and at most one principal curvature vanishes at each point, then the surface measure on $\Gamma$ does not admit any Fourier frame. This weakens the non-vanishing curvature condition in a previous result of Iosevich, Lai, the second author, and Wyman. On centrally symmetric surfaces of non-vanishing Gaussian curvature, we obtain more domains without Fourier frames. For example, we show that $\{x\in S^{d-1}:0<x_d<\delta\}, d\geq 3$, does not admit any Fourier frame for all $\delta>0$, which generalizes the result of Chen and the second author on the hemisphere.

In this paper, we present an operator representation for Herglotz-Nevanlinna functions in several variables using self-adjoint linear relations. Moreover, a description of the reproducing kernel Hilbert space corresponding to a Herglotz-Nevanlinna function of several variables is also presented, along with a comparison to some existing result concerning the class of Loewner functions.