Papers

The classical Borell–Brascamp–Lieb inequality for multiple functions is an integral inequality relating finitely many input functions to a single output function. In this paper, we give an extension that allows a distinct output function for each input function. More precisely, we establish the following inequality. For a weight $ \lambda = ( \lambda_1 , \dots , \lambda_m ) $, we set $ z_\lambda ( x ) := \sum_{ i = 1 }^m \lambda_i x_i $ and denote by $ M_p^\lambda $ the weighted power mean of power $ p $. In addition, the power transform $ Q_d $ is defined as the continuous extension of $ p \mapsto p / ( 1 - d p ) $. We consider integrable functions $ f_1 , \dots , f_m , g_1 , \dots , g_m \colon \mathbb{ R }^d \to \mathbb{ R }_{ \geq 0 } $ satisfying $ 0 < \| f_i \|_1 < \infty $ for every $ i = 1 , 2 , \dots , m $. Then one has \[ \operatorname*{ess\,inf}_{ x = ( x_1 , \dots , x_m ) } M_p^\lambda \left( \frac{ g_1 ( z_\lambda ( x ) ) }{ f_1 ( x_1 ) }, \dots , \frac{ g_m ( z_\lambda ( x ) ) }{ f_m ( x_m ) } \right) \leq M_{ Q_d ( p ) }^\lambda \left( \frac{ \| g_1 \|_1 }{ \| f_1 \|_1 }, \dots , \frac{ \| g_m \|_1 }{ \| f_m \|_1 } \right) \] for any $ - \infty \leq p \leq 1 / d $, where the essential infimum is taken over the set of points satisfying $ f_i ( x_i ) > 0 $ for every $ i = 1 , 2 , \dots , m $. When all output functions $ g_1 , \dots , g_m $ are equal, this recovers the classical Borell–Brascamp–Lieb inequality for multiple functions. When $ p = 0 $ and the hypothesis is imposed pointwise, it coincides with a special case of the multi-output Prékopa–Leindler inequality of Cordero-Erausquin–Maurey.

Let $\Omega\subset\mathbb C$ be a bounded John domain and set $\delta(z)=\text{dist}(z,\partial\Omega)$. For $1<p<\infty$ and $\alpha>\text{dim}_A(\partial\Omega)-2$, we establish a norm equivalence between $\int_\Omega |g|^p\delta^\alpha\,dA$ and $\int_\Omega |g'|^p\delta^{\alpha+p}\,dA$ for analytic functions $g$ on $\Omega$, with a point-evaluation term fixing the additive constant. The estimate of the derivative term is local and holds on every proper planar domain, whereas the converse follows from a distance-weighted Poincaré inequality on John domains. Taking $\alpha=mp-2$ yields the corresponding comparison between the $m$-th and $(m+1)$-st derivatives. For $m\ge2$ the boundary-dimension condition is automatic, so the only dimension-sensitive case is the comparison between $\int_\Omega |f'|^p\delta^{p-2}\,dA$ and $\int_\Omega |f''|^p\delta^{2p-2}\,dA$ when $1<p<2$. We show that this restriction is sharp within the class of quasidisks by using self-similar Rohde snowflakes. We also construct, for every $s>1$, an inward-cusp $s$-John domain on which the comparison fails, showing that the ordinary John condition cannot in general be weakened.

In 1974, C. D. Aliprantis posed several questions concerning topological completions of locally solid vector lattices. We answer all of them in the negative. We construct Hausdorff locally convex-solid vector lattices showing that, without metrizability, neither the $\sigma$-Lebesgue property nor property (B, i) need pass to the completion; a positive element of the completion need not be the limit of a decreasing sequence of upper elements; the generalized (A, 0) property need not make the canonical image a regular sublattice; and regularity of this image need not imply order density. All five counterexamples have been formalized in Lean 4 using the Banach lattice Lean library.

We construct a subadditive weight on the type I$_\infty$ factor that preserves directed suprema but is not $\sigma$-weakly lower semi-continuous. This provides a counterexample to Haagerup's first problem posed in 1975.

Given an ideal $\mathcal I$ on $\omega$, a subgroup $H$ of the unit circle $\mathbb T$ is said to be $\mathcal I$-characterized if there exists a sequence of integers $(a_n:n\in\omega)$ such that $$ H= \left\{ x\in\mathbb T: \mathcal I\text{-}\lim_{n\to\infty}a_nx=0 \right\}. $$ We investigate the descriptive complexity and Polishability of these subgroups in terms of the structural and topological properties of the ideal $\mathcal I$. Our main structural result shows that $\mathcal{I}$ is an analytic $P$-ideal if and only if all $\mathcal I$-characterized subgroups are Polishable. In such case, we explicitly describe a compatible finer Polish group topology. Using results on Polishable subgroups, we obtain a trichotomy for their possible Borel complexities. If $\mathcal{I}$ is a generalized density ideal, we show the sharper dichotomy that every proper $\mathcal I$-characterized subgroup is either countable or $F_{\sigma\delta}$-complete. We also prove that this fails for general analytic $P$-ideals by constructing a subgroup characterized by a summable ideal which is neither $F_\sigma$ nor $F_{\sigma\delta}$-complete. Finally, we give explicit descriptions of the subgroups associated with the sequences of powers, the Fibonacci sequence, and the sequence of factorials. We conclude with several open questions.

Let $\omega$ be a radial $\widehat{\mathcal D}$-weight and $u$ be a bounded function on the unit disk $\mathbb D$. We prove that the Toeplitz operator \(T_{\omega,u}\) is compact on \(A_\omega^2\) if and only if its Berezin transform vanishes at the boundary. Our approach is based on a polynomial frame for $A_\omega^2$ and a detailed localization analysis of the resulting infinite matrix representation of $T_{\omega,u}$. Even in the unweighted Bergman space \(A^2\), our argument is new and does not rely on the classical translation operators. We further show that this Axler–Zheng compactness characterization does not extend, in general, to products of Toeplitz operators on \(\widehat{\mathcal D}\)-weighted Bergman spaces, and hence to the corresponding Toeplitz algebra generated by bounded symbols. More precisely, we construct a radial log-subharmonic \(\widehat{\mathcal D}\)-weight \(\omega\) and bounded symbols \(u,v\) such that the product \(T_{\omega,v}T_{\omega,u}\) is noncompact, whereas its Berezin transform vanishes at the boundary.

In 2006 Neeb asked whether the Hausdorff completion of a continuous inverse algebra must again be a continuous inverse algebra. The noncommutative case remained open, while the commutative case was known to be true. We give a negative answer after twenty years. We construct a Hausdorff metrizable locally m-convex complex continuous inverse algebra whose completion is a Fréchet locally m-convex algebra, where inversion stays continuous but the set of invertible elements is not open. The construction uses finite-support sequences in a dense nil subalgebra of a Jacobson-semisimple Banach algebra. Finite support makes every element nilpotent, giving continuous inversion without a locally uniform bound on nilpotence indices. In the completion, shifting a fixed noninvertible element to later and later coordinates produces noninvertible elements converging to the identity. Thus completion destroys exactly the local spectral stability at the identity. The counterexample is necessarily noncommutative and marks the precise boundary of the commutative completion theorem.

We prove that the heat semigroup on the circle has optimal complete modified logarithmic Sobolev constant \(1\). The proof is based on a matrix-valued Wirtinger inequality and yields the stronger Bogoliubov–Kubo–Mori Fisher information contraction with rate \(e^{-2t}\). As a consequence, the complete tensorization and transference principle yields the same sharp constant for the heat semigroups on classical and quantum tori.

We establish a dynamic generalization of Nevanlinna-Pick interpolation for functions invariant under the action of a finite Blaschke product $f$, reducing global bounded holomorphic extension on orbit spaces to structured block-kernel positivity. Furthermore, we demonstrate that membership in $H^\infty(\mathbb{D})$ is universally detectable via deformations by any finite Blaschke product. These results are proved via an underlying operator-theoretic lifting framework for covariant operator pairs in the spirit of Sarason. Finally, in the setting of non-commutative function theory, we show that despite the universal validity of the matrix-valued von Neumann inequality over the free polydisk, Arveson-type complete spectral set representations break down for non-commutative inner varieties.

The purpose of this paper is twofold. First, we investigate the Hardy-Littlewood type phenomena for Dirichlet solutions to the Möbius invariant Laplace equation on the unit ball in $\mathbb{R}^n$. Our work extends and improves several key results due to Pavlovć [Rev. Mat. Iberoam. 23: 831-845, 2007] and Chen et al. [J. Geom. Anal. 34: 23 pp, 2024]. In particular, we give a complete answer to a question raised by Makoto Masumoto. Second, motivated by Aikawa's work, we study the boundedness of the operator norm of $P_{\alpha}$, where $P_{\alpha}[\varphi]$ is the Dirichlet solution of such equation for the boundary data $\varphi$. By using alternative proof techniques, we obtain an equivalent characterization of the boundedness of the operator norm of $P_{\alpha}$. Finally, we show that the Girela-Peláez conjecture holds positively for more general classes of functions induced by the Möbius invariant Laplacian operator.

In this paper, we completely characterize when the semi-commutator of two Cauchy singular integral operators on $L^2$ is compact. Our main idea is to study Cauchy singular integral operators via Hankel operators, Toeplitz operators and function algebras. Moreover, we obtain a necessary and sufficient condition for Cauchy singular integral operators to be essentially normal.

We construct two examples clarifying the position of enriched $P$-contractions among related classes of mappings on normed spaces. Our first example is a $P$-contraction operator on a finite subset of a normed linear space that is not an enriched contraction. In the second example we have a continuous operator on the whole of a normed linear space that is a $P$-contraction but not an enriched contraction. These examples answer in negative a question raised in \cite{AltunHancerAtes2024} asking whether every enriched $P$-contraction is an enriched contraction.

This paper explores the algebraic invariants and ergodic behavior of the finite-index inclusions forming commuting squares of $\text{II}_1$ factors. We begin by establishing how regularity, intermediate-subfactor lattices, and Weyl groups transfer across the commuting square under relative commutant and irreducibility conditions. Furthermore, we provide a complete characterization, via a novel factorization theorem, of the unitary normalizers of the upper inclusion when the lower inclusion is regular. Applying these findings to crossed-product inclusions induced by discrete groups, we introduce a relative eigenbasis property that preserves regular inclusions. Finally, we contrast this by proving that a relative weak mixing condition on the group action forces the crossed-product inclusion to be singular.

Factorization for positive semidefinite matrix-valued polynomials over a nonsingular compact affine real surface is established. Corollaries include Fejér-Riesz factorization for bivariate matrix polynomials that take positive semidefinite values and resolutions to questions posed by Mehta-Slofstra-Zhao and Savchuk-Schmüdgen. An explicit example shows a conclusion of [Dri25, Theorem, p. 519] that arises organically from its proof need not hold. The difficulty is traced to [Dri25, Theorem 3.7].

Texture is commonly described through statistical descriptors, filter-bank responses, or multiscale representations, yet its functional-analytic structure remains largely unexplored. In this work we investigate texture from the viewpoint of Gaussian scale-space evolution, interpreting it as the persistence of local structures across scales. This perspective naturally leads to a multiscale texture operator that quantifies the evolution of local image structures under Gaussian diffusion. We establish the fundamental analytical properties of this operator, proving its boundedness on $L^p(\mathbb{R}^n)$, its infinite-order smoothing effect in the Hilbert setting, and an explicit spectral characterization showing that it acts as a localized Gaussian band-pass filter. Motivated by the logarithmic organization of Gaussian scales, we introduce an $r$-adic Difference-of-Gaussians discretization and prove Littlewood–Paley-type stability estimates together with an explicit reconstruction formula. Building on this multiscale representation, we define multiscale texture norms that quantify the persistence of local structures across scales. These norms induce a natural family of Banach spaces, with a distinguished Hilbertian case, and we prove that they provide an equivalent characterization of the classical Besov and Sobolev scales. Consequently, classical functional regularity can be described intrinsically through the persistence of texture under Gaussian scale evolution. These results establish the first functional-analytic characterization of texture persistence under Gaussian scale evolution, revealing that classical Besov and Sobolev regularity admit an equivalent description in terms of multiscale texture persistence. This provides a rigorous bridge between Gaussian scale-space theory, harmonic analysis, and variational models based on texture.

For $\alpha>1$, we study the weighted multiplicative Hilbert operator \[ (T_\alpha f)(s) = \int_{1/2}^{\infty} f(t)\bigl(\zeta(s+t)^\alpha-1\bigr) \left(t-\frac12\right)^{\alpha-1}\,dt \] on the Hilbertian Bergman space $A_{\alpha,0}^2$ of Dirichlet series with zero constant term. In the natural orthonormal basis, $T_\alpha$ is represented by the weighted multiplicative Hilbert matrix \[ \left( \frac{\Gamma(\alpha)\sqrt{d_\alpha(m)d_\alpha(n)}} {\sqrt{mn}\,(\log(mn))^\alpha} \right)_{m,n\ge2}, \] where $d_\alpha$ is the generalized divisor function defined by $\zeta(s)^\alpha=\sum_{n\ge1}d_\alpha(n)n^{-s}$ for $\Re s>1$. Set $\Lambda_\alpha=\Gamma(\alpha/2)^2/\Gamma(\alpha)$. We prove that the essential spectrum of \(T_\alpha\) is \([0,\Lambda_\alpha]\) and that the spectrum in this interval is purely absolutely continuous with multiplicity one. Above \(\Lambda_\alpha\), there are at most finitely many eigenvalues, all of which are simple. Moreover, there is a unique \(\alpha_*\in(1,2)\) such that \(T_\alpha\) has eigenvalues if and only if \(\alpha>\alpha_*\). Consequently, the associated sharp embedding inequality admits a non-zero extremizer precisely when $\alpha>\alpha_*$, and this extremizer is unique up to scalar multiplication. We also establish the asymptotic formula $\|T_\alpha\|\sim\Gamma(\alpha+1)/(2(2\log 2)^\alpha)$ as $\alpha\to\infty$. A key step in the proof combines Mellin-transform asymptotics, strict total positivity, and eigenvalue counting to exclude embedded eigenvalues. The proof includes a general result on the essential and singular continuous spectra of weighted integral Hankel operators.

We prove that the Toeplitz operator $T_{\bar{z}+h}$ is invertible on the Bergman space provided that $|\bar{z}+h|$ is bounded below by some positive constant on the open unit disk, where $h$ is in the disk algebra. This provides a general class of harmonic functions for which the answer to the Douglas question on the Bergman space is affirmative.

We prove a common generalization of two structure theorems: the dimension-free decomposition theorem for idempotent Schur multipliers and the idempotent theorem in harmonic analysis. Roughly speaking, our result shows that an invariant integer-valued kernel with Hilbert-space factorization norm $\gamma$ admits a signed decomposition into at most $2^{O(\gamma^4)}$ elementary pieces. In the matrix setting these pieces are blocky matrices, while in the group setting they are indicators of cosets. In the locally compact abelian setting this quantitatively strengthens the theorem of Green and Sanders, while in the non-abelian setting it gives a quantitative strengthening of Host's idempotent theorem and, for finite groups, of Sanders's quantitative result. It also improves the exponent in the dimension-free matrix decomposition from $\gamma^6$ to $\gamma^4$.

We develop a structural framework for irregularity in linear dynamics centered on the space $\Upsilon$ of observation schemes. This approach separates the underlying dynamical behavior from the observation mechanism and provides a unified setting in which classical notions such as Li–Yorke chaos, mean Li–Yorke chaos, and distributional chaos arise as particular cases corresponding to Dirac measures and Cesàro averages. We establish two abstract criteria ensuring the existence of large linear structures, leading to dense-lineability and spaceability results for both absolutely $(\mu_m)$-irregular and distributionally $(\mu_m)$-irregular vectors. Furthermore, under a natural density assumption, we obtain a trichotomy describing the global behavior of irregularity across $\Upsilon$, together with rigidity phenomena for the classes of observation schemes generating each type of chaotic behavior.

We investigate the behavior of the singular spectrum of self-adjoint operators under families of Hermitian perturbations, even non-compact ones. Under mild assumptions we have the “shift” of the singular spectrum for almost all values of the parameter. Moreover, if we consider trace class perturbations, we observe the “shift” outside a countable set of the values of the parameter. While similar results were known for finite rank perturbations, the extensions to trace class perturbations are far from easy, and the proofs require significant new ideas. In addition, some of our results hold (surprisingly enough!) even for all positive and bounded perturbations. This is a consequence of our generalized Aleksandrov disintegration theorem established in this paper. We use some advanced techniques, involving Sz.-Nagy–Foia\c s theory and the operator ${\bf A}_2$ condition.

In this note, we show that invertible composition operators on S(R) are never generalized hyperbolic and, when the symbol has a fixed point, the corresponding operator fails to have the positive shadowing property. We then establish sufficient conditions on the symbol under which the associated composition operator is positively topologically expansive. In particular, we obtain a complete characterization for affine symbols and prove expansivity for broad classes of odd-degree polynomial symbols, while showing that polynomial symbols of even degree cannot generate topologically expansive operators. Our results reveal a strong connection between the dynamics of the symbol and the linear dynamics of the induced composition operator, and provide new examples and obstructions for topological expansivity in locally convex spaces.

Many nonlinear and variational problems are posed fibrewise over a probability space, whereas the dynamically relevant objects are measurable sections coupled by measure-preserving transformations. We study structural obstructions created by this fibre-to-section lifting. First, for a global continuously differentiable potential on a real Hilbert space, absence of influence between closed subspaces is reciprocal; for finite orthogonal decompositions this yields an additive decomposition over dependency components. Second, finite Koopman memory channels are identifiable exactly under a sharp orbit-separation condition. Resolving the classical Hilbert-space potentiality criterion channel by channel then gives a shifted-adjoint balance between forward and reverse memories. At the global finite-difference level, if a section map with arbitrary finite dynamical memory is the gradient of a global continuously differentiable potential, its genuinely active lags are invariant under time reversal. Thus globally variational memory is reversal-complete, and one-sided memory collapses to present-state locality. Continuous proximity operators inherit the same rigidity through their convex-potential representation; a discontinuous proximal selection shows that continuity cannot generally be removed. Finally, under pointwise nondegeneracy of two coherent branches, vanishing branchpasting residual is equivalent to asymptotic invariance of the branch labels. For finitely generated ergodic probability-preserving actions, strong ergodicity is therefore the exact threshold for relative compactness of all branch-pasting approximate-zero sequences. We also record a strong-compactness criterion for full measurable-selection lifts and a nonconvex Wasserstein minimizing-step illustration.

In this paper we focus on the stability of positive semidefinite matrix sensing maps $\Phi_{\mathcal{A}}(X)=(\langle A_i,X\rangle)_{i=1}^m$ where $A_i\succeq 0$, $X\succeq0$ and $\operatorname{rank}(X)\le r$. We introduce the bi-Lipschitz constants of $\Phi_{\mathcal{A}}(X)$ and define the global condition numbers as the ratio of upper and lower Lipschitz constants. We give deterministic universal lower bounds for these condition numbers, that depend only on the rank $r$ and on the underlying field. We then investigate the random rank-one Gaussian measurements and show that our lower bounds on condition numbers are asymptotically sharp, and therefore the random rank-one Gaussian measurements are asymptotically optimal. As an application, we derive the stability guarantees for an $\ell_1$-residual PhaseLift-type estimator at the optimal sampling scale.

In his study of Cantor-type intersection theorems and fixed points of almost affine mappings, Jachymski \cite{Jachymski} posed four open questions. In this note, we provide an affirmative answer to the fourth question.

In this paper we establish a criterion on weak type $(1,\,1)$ bound of the following rough singular integral $$T_{\Omega,K,a}f(x)={\rm p.v.}\int_{\mathbb{R}^n}\Omega(x-y)K(x,y)m_{x,y}a f(y)dy,$$ where $m_{x,y}a=\int_0^1a(sx+(1-s)y)ds$ with $a\in L^1(\mathbb{R}^n)$ and $\hat{a}\in L^1(\mathbb{R}^n)$, $\Omega$ is homogeneous of degree zero, integrable in $\mathbb{S}^{n-1}$ and satisfies the cancellation condition $\int_{\mathbb{S}^{n-1}}\Omega(\theta)d\sigma(\theta)=0$ and $K$ is a measurable function defined on $\mathbb{R}^n\times\mathbb{R}^n\setminus \{(x,x):x\in\mathbb{R}^n\}$ and satisfies a Hölder condition. By assuming that $\Omega\in L\log L(\mathbb{S}^{n-1})$ and the operator $T_{\Omega, K}f(x)={\rm p.v.}\int_{\mathbb{R}^n}\Omega(x-y)K(x,y)f(y)dy$ is bounded on $L^2(\mathbb{R}^n)$, we prove the weak type (1,1) bound of $T_{\Omega,K,a}$. As several applications, we obtain a large class of singular integral operators which possess weak type $(1,\,1)$ bound. The main results of this paper essentially extend and generalize some known ones.

In this paper, by implementing the generalized inverse, known as the Moore-Penrose inverse, for Hilbert space operators, several inner product bounds that are of Cauchy-Schwarz type are presented. As a standard application, some new refined and generalized versions of numerical radius-norm bounds are also stated, in a generalized form that involves different combinations of operators.

We study the problem of finding the norm of the point evaluation operator in the Paley–Wiener space $PW^{1}(\r^d)$, consisting of $d$-variable functions of spherical exponential type that are integrable on $\r^d$. The extremal functions can be taken radial, which naturally leads us to consider a related extremal problem in a weighted Paley–Wiener space of single-variable functions. We establish that the radial extremal function must satisfy a third-order linear ODE with polynomial coefficients for every $d \geq 1$, extending Gorbachev's recent odd-dimensional result. Along the way, we prove interpolation and reciprocal formulas involving the zeros of the extremizer.

We provide a detailed study of equivalent reformulations of the classical condition derivation closedness for weight sequences. This property is crucial in many applications since it ensures that the corresponding weighted spaces satisfy the desired property of stability under taking derivatives. Moreover, it yields technical estimates which are required in various directions and proofs and so the knowledge that different conditions are equivalent to this crucial requirement is important and useful when dealing with weighted spaces. Finally, we construct an explicit (counter-)example which shows that without the standard condition log-convexity the full characterization fails in general and comment on the mixed setting as well.

The purpose of this paper is to study the existence and convergence of $\phi$-best proximity points in Banach spaces. A strong convergence theorem is established by employing a shrinking projection algorithm designed to compute common $\phi$-best proximity points of a proximally weakly suppressive mapping. Finally, we address a projection scheme to solve split $\phi$-best proximity point and variational problem in a Banach space. These contributions provide new tools for solving nonlinear problems involving non-self mappings.

We establish the orthonormal Strichartz estimates in the abstract Keel–Tao framework, under precisely the same hypotheses as in their original theorem. As an important consequence, we resolve the endpoint conjecture for the Schrödinger equation in dimensions $d\ge 2$, which was first raised by Frank, Lewin, Lieb, and Seiringer. This consequence also yields an affirmative answer to the conjecture for the free transport equation, which was stated in dual form by Bennett, Bez, Gutiérrez, and Lee. As a further application, we also establish a refinement of the Strichartz estimate for a single function.