Papers

We completely classify the possible complexity classes of non-closed operator ranges on separable Banach spaces over a Polish non-Archimedean non-trivially valued field. These are precisely $\boldsymbol{\Pi }_{1+\lambda +n+2}^{0}$ for a countable ordinal $\lambda $ that is either zero or a limit ordinal, and a finite ordinal $n$. Considering Fréchet spaces produces the additional complexity classes $\boldsymbol{\Pi }_{\lambda }^{0}$ for a countable limit ordinal $\lambda $.

We study real multilinear forms with coefficients in $\{-1,1\}$ on finite-dimensional Hilbert spaces. Every trilinear sign form on $\ell_2^r\times\ell_2^n\times\ell_2^n$ has norm at least $\sqrt n$. Writing $K_{r,n}$ for the least norm divided by $\sqrt n$, we prove that $K_{r,n}=1$ exactly when a Hadamard matrix of order $n$ exists and $r\le\rho(n)$, where $\rho$ is the Hurwitz–Radon function. If equality fails, we obtain an explicit gap above $\sqrt n$. We also prove two asymptotic results. If $1\le m_n\le n$ and $\limsup r_n/\log_2 n<2$, there are sign forms on $\ell_2^{r_n}\times\ell_2^{m_n}\times\ell_2^n$ with norm $(1+o(1))\sqrt n$. In the square case, if $r\ge2\lceil\log_2(8n)\rceil$, then $K_{r,n}-1\ge c(1+\log_2 n)^{-4}$. We also prove a fourth-moment estimate in every fixed multilinear order, characterize equality, and give exact and asymptotic constructions.

We study O'Neil-type inequalities in Lorentz spaces for two extremal geometric configurations of translation-invariant multilinear convolution operators. For the one-parameter model, we complement the existing weak-kernel theory with estimates for kernels in the Lorentz space $L^{r,s}$, obtaining a stronger target Lorentz space when $s$ is finite. For the full-dimensional model, we prove a Lorentz refinement of Oberlin's multilinear Young inequality in the interior of its admissible range. For nonnegative full-dimensional kernels, we extend Hörmander's necessary condition to the multilinear setting in Lorentz spaces and obtain an additional restriction on the secondary indices in the critical case. We also prove the failure of the full-dimensional weak $L^1$ endpoint, use additive combinatorics to construct one-parameter restricted weak-type counterexamples for $p<1$, and obtain complementary positive results for kernels of special structure.

We investigate the relationship between the tame factorization property, denoted by $\mathfrak{TF}$ and introduced in the companion paper \cite{CDI}, and the DN-$\Omega$ type linear topological invariants of Fréchet spaces. Combining the basic properties of $\mathfrak{TF}$ with known characterizations of tameness and boundedness, we obtain several results identifying the triples of Fréchet spaces that possess $\mathfrak{TF}$. We further exhibit examples showing that tame factorization property is a strictly weaker condition than tameness, indeed, we construct triples possessing $\mathfrak{TF}$ none of whose individual pairs are tame. We then investigate triples consisting of an arbitrary Fréchet space $X$, a nuclear Fréchet space $Y$ satisfying the properties $\underline{DN}$ and $\Omega$, and a power series space of finite type $\Lambda_1(\mathcal{E})$ or infinite type $\Lambda_\infty(\mathcal{E})$. We show that requiring such a triple to possess the tame factorization property $\mathfrak{TF}$ characterizes the corresponding linear topological invariants of $X$; in some cases this holds without any restriction on $Y$, while in others it requires the coincidence of the approximate diametral dimension of $Y$ with that of $\Lambda_1(\mathcal{E})$ or $\Lambda_\infty(\mathcal{E})$.

Leveraging the neural architectures which we introduced in arXiv:2109.13512v4, we show a global universal approximation theorem in the topology of $L^p(\mu)$, where $1\le p<\infty$ and $\mu$ is a Radon probability measure on a suitable infinite dimensional topological space $\mathfrak X$. Namely, any function $f:\mathfrak X\to \mathbb R$ in $L^p(\mu)$ can be approximated to any degree of accuracy by suitable infinite dimensional architectures. These architectures can be in turn approximated by almost classical neural networks which are specified by a finite number of parameters only. The vectorial case (where $f=f(x)\in E$ and $E$ is a Banach space) is also considered and analogous results are obtained.

We introduce the tame factorization property for triples of Fréchet spaces: a triple $(E,G,F)$ has the tame factorization property, denoted by $(E,G,F) \in \mathfrak{TF}$, if there exists a nondecreasing function $S:\mathbb{N}\to\mathbb{N}$ such that every operator from $E$ to $F$ factoring through $G$ is $S$-tame. We give a complete characterization of $\mathfrak{TF}$ in terms of operator seminorm estimates. We specialize this characterization to triples in which one or more of the spaces are Köthe spaces, putting explicit conditions on the Köthe matrices, and we describe how $\mathfrak{TF}$ behaves under projective tensor products of Köthe spaces. Finally, we show that a triple of Köthe spaces has the tame factorization property if and only if the family of operators that factor as products of two quasi-diagonal operators is tame.

This paper reviews the subject of equivalence after extension in the context of Toeplitz operators, truncated Toeplitz operators, and further generalizations including dual and multiband truncated Toeplitz operators. Paired operators are shown to play a key role here. Properties under investigation by these methods include the invertibility and Fredholm properties of generalized Toeplitz operators, with new results showing how the kernels of such operators can often be fully described given only limited information.

Recent work produced a linearly dependent system of twelve time–frequency shifts of a Schwartz function, disproving the HRT conjecture. We show that four shifts already suffice, and hence that four is the smallest possible cardinality of a dependent finite Gabor system. More precisely, set $\alpha=\frac13+10^{-12}\sqrt2$ and $\beta=\frac13+10^{-12}\sqrt3$. We construct a nonzero complex-valued function $f\in\mathcal S(\mathbb R)$ and $\lambda\ne0$ such that $\left(I+\frac12W(1,0)+\frac12W(0,1/2)\right)W(\alpha,\beta/2)f=\lambda f$, where $W$ denotes the Weyl time–frequency shift. Since every system of at most three shifts of a nonzero $L^2(\mathbb R)$ function is linearly independent, this gives the sharp cardinality threshold. The construction uses the rank-two Zak bundle naturally associated with the covolume-$1/2$ lattice generated by $(1,0)$ and $(0,1/2)$. At the rational translation $(1/3,1/3)$, the three-step return has a uniformly dominated contracting line. A finite outward-rounded interval certificate proves that this line is topologically trivial. A quantitative perturbation argument carries the dominated line to the explicit algebraic translation above. A winding calculation and a Diophantine cohomological equation then flatten its scalar multiplier, and inverse Zak folding produces the required Schwartz function.

In this paper we prove Hölder inequalities for the Besov-Morrey spaces $ \mathcal{N}^{s}_{u,p,q}(\mathbb{R}^{d}) $ and the Triebel-Lizorkin-Morrey spaces $ \mathcal{E}^{s}_{u,p,q}(\mathbb{R}^{d}) $. We observe that these Morrey smoothness spaces are pointwise multiplier spaces if certain conditions concerning the parameters also including $ s > d \max ( 0, \frac{1}{p} - 1) $ are fulfilled. In this context we significantly generalize the Hölder inequalities for the original Besov and Triebel-Lizorkin spaces obtained by Sickel and Triebel in 1995. For the proofs we use paramultiplication and some Franke-Jawerth embeddings obtained by Haroske and Skrzypczak.

Let $Hf(x)=x^{-1}\int_0^x f(t)\,dt$ be the Hardy operator on $\mathbb R_+$. Korenovskii proved that $$ \Vert Hf\Vert_{\mathrm{BMO}}\geq \frac{e\alpha_0}{4}\Vert f\Vert_{\mathrm{BMO}}, $$ for every nonincreasing and locally integrable $f$, where $\alpha_0$ is defined by the relation $\Vert H\chi_{(0,1)}\Vert_{\mathrm{BMO}} = \alpha_0 \Vert \chi_{(0,1)}\Vert_{\mathrm{BMO}}$, and conjectured that the factor $e/4$ could be removed. We prove this conjecture by showing that every nonincreasing locally integrable $f$ satisfies $$ \Vert Hf\Vert_{\mathrm{BMO}}\geq \alpha_0\Vert f\Vert_{\mathrm{BMO}}. $$ The constant $\alpha_0$ is optimal, with equality for the one-jump functions $\chi_{(0,a)}$.

Let $n\geq 2$, $1\leq \ell\leq n-1$, and $1<p<\infty$. We prove that every $v\in \dot W^{n/p,p}(\mathbb{R}^n;\Lambda^\ell)$ has a representative $u\in \dot W^{n/p,p}\cap L^\infty$ with $du=dv$ and $\max{|u|{\dot W^{n/p,p}},|u|{L^\infty}}\lesssim |v|{\dot W^{n/p,p}}$. Equivalently, $d[\dot W^{n/p,p}\Lambda^\ell]=d[(\dot W^{n/p,p}\cap L^\infty)\Lambda^\ell]$ with equivalent quotient norms. The proof reduces the selection problem to an endpoint graph estimate for a Riesz potential and the exact Hodge projection. Its main analytic input is a finite-dimensional-input Maz'ya–$\Phi$ inequality for operator-valued homogeneous kernels. For the Riesz/Hodge pair, a nonlinear spherical profile built from the projected kernel has exact atomic cancellation and is coercive by the identity $\int{S^{n-1}}P(\theta),d\bar{\sigma}=(\ell/n)\operatorname{Id}$. Frequency-localized graph closure and Hahn–Banach duality then return a bounded representative.

We disprove the conjectural converse to Kato's positivity criterion for commutators of functions of the canonical position and momentum operators $Q$ and $P$ by showing that the operator \[ i\,[\,\arctan(P),\,\arctan(Q)\,] \] is nonnegative and nonzero.

In this paper, we first establish a generalized Cauchy–Kovalevskaya (GCK) extension for axially polyanalytic functions of order $2$. We prove that the extension can be written as a power series involving differential operators acting on two initial functions. We also study the decomposition of the GCK extension in terms of integrals over the sphere 2-sphere $\mathbb{S}$ involving plane-wave type functions. We further establish a connection between the GCK extension for polyanalytic functions of order $2$ and the Fueter theorem. This is one of the most important results in hypercomplex analysis and can be described in two steps. In the first step, starting from holomorphic functions of one complex variable, the application of a suitable operator yields slice hyperholomorphic functions. In the second step, applying the Laplace operator in four real variables (called Fueter map) one obtains axially monogenic functions, i.e. functions in the kernel of the Fueter operator. A suitable factorization of the Fueter map in terms of the Fueter operator and its conjugate gives rise to two intermediate classes of functions between slice hyperholomorphic functions and axially monogenic functions: axially harmonic functions and axially polyanalytic functions of order $2$. Another goal of this paper is to study the invertibility of the factorizations of the Fueter map for harmonic and polyanalytic functions of order 2 on suitable open sets, and to derive integral representation formulas for the inverse of the factorized Fueter map. These integral representations are based on the Cauchy formula for polyanalytic functions of order $2$ and the Poisson integral formula for harmonic functions.

We prove a Bauer-Poulsen dichotomy theorem for simplices of invariant measures associated with permutation groups. More precisely, let $G$ be a transitive group of permutations of a countable set $\mathcal{S}$, and let $H$ be the stabilizer of a point of $\mathcal{S}$. Let $\overline{G}$ and $\overline{H}$ denote their closures in the topology of pointwise convergence. Assume the Polish group $\overline{H}$ has relative Property (T) in $\overline{G}$. Then the simplex $\mathcal{M}_\mathrm{inv}(2^\mathcal{S})$ of invariant probability measures for the induced action $G\curvearrowright 2^\mathcal{S}$ is Bauer if and only if $\overline{G}$ has Property (T), and is Poulsen otherwise. This addresses some examples and questions considered by Austin. We deduce this result from a more general model-theoretic statement of independent interest. To this end, we initiate the study of existentially closed models in affine logic.

The ability of large language models to assist professional mathematicians has been progressing rapidly. Earlier this year, a group of researchers in Banach lattice theory and phase retrieval began incorporating this technology into their research workflows. Facing challenges about the reliability of these models, they also decided to couple the discovery process with Lean verification. Here, we present a case study of how this has led to a more united community and a deeper understanding of our field.

We present a Lean 4 library for the theory of Banach lattices. Its purpose is to support the systematic formalization of contemporary research in Banach lattices and related areas. As evidence of this, we describe three research-level formalizations built using the library. Writing the library at scale was made possible by the use of LLMs with careful human supervision and planning. Unlike autoformalization, this approach allows for an actual understanding of the code. This, in turn, led to new mathematical insights that are also discussed. Judging by the interest expressed by other researchers, we expect the library to become a communal effort in the near future. For this reason, we also describe several parts of the theory that could be added next.

We present new formulae providing equivalent quasi-norms on rearrangement-invariant quasi-Banach function spaces.

In this paper, we establish the noncommutative maximal weak type $(1,1)$ and strong type $(p,p)$ estimates for the family of operators $(T_N)_N$, defined by $$T_Nf=\sum_{k=N_1}^{N_2}\nu_{k}(M_{k}-\mathsf{E}_k)f,$$ where $M_k$ denotes the dyadic Hardy–Littlewood average operator, $\mathsf{E}_{k}$ is the conditional expectation with respect to the dyadic cubes of side-length $2^{-k}$, $N=(N_1,N_2)$ with $N_1<N_2$ and $(\nu_{k})\in\ell_{\infty}$. The main novelty of our approach is the development of a noncommutative Cotlar-type inequality for non-smooth kernels, a result that is new even in classical harmonic analysis. As an application, we obtain the boundedness theory of the noncommutative maximal differential transforms for averaging operators.

Maps $\Phi:D\to D$ of the unit disk preserve frame generator sets for single orbits of normal diagonal operators exactly when they preserve Carleson interpolating sequences bidirectionally and satisfy $1-|\Phi(z)|^2\asymp1-|z|^2$ uniformly. Denote this class by $\mathcal P$. Equality of frame generator sets forces the boundary estimate, since every frame generator $f=(f_j)$ satisfies $|f_j|^2\asymp 1-|\lambda_j|^2$. No continuity, measurability, or analyticity is assumed; Carleson preservation alone remains flexible and, to our knowledge, unclassified in general. Equivalently, normalized Szego kernel Riesz sequences are preserved bidirectionally, with uniformly comparable unnormalized kernel norms at $z$ and $\Phi(z)$. Equivalently, we require uniform two-sided Carleson norm bounds for ordinary pushforwards of finite positive discrete measures and bidirectional preservation of pairwise pseudohyperbolic separation. Every $\Phi\in\mathcal P$ has a canonical bilipschitz radial trace $h_\Phi(\zeta)=\lim_{r\to1^-}\Phi(r\zeta)$ on $T=\partial D$, with uniform convergence. We give an intrinsic geometric characterization of $\mathcal K=\{\Psi\in\mathcal P:h_\Psi=\mathrm{id}\}$ and prove the unique factorization $\Phi=\Psi\circ E_{h_\Phi}$, $\Psi\in\mathcal K$, where $E_h(r\zeta)=rh(\zeta)$ and $E_h(0)=0$; thus $\mathcal P\cong\mathcal K\rtimes\mathrm{BiLip}(T)$ as monoids. For $m$ orbits, $\mathcal P_m=\mathcal P$ for every finite $m\ge1$, whereas $\mathrm{Aut}(D)\subsetneq\mathcal P_\omega\subsetneq\mathcal P$. Countable-orbit preservers in $\mathcal P_\omega$ are uniform pseudohyperbolic homeomorphisms. If $\varphi\in\mathrm{Aut}(D)$ and $|\Phi(z)-\varphi(z)|=o(1-|z|)$ uniformly as $|z|\to1$, then $\Phi\in\mathcal P_\omega$ exactly when $\Phi$ is a disk homeomorphism. Two-sided tests characterize $\mathcal P_\omega$. Holomorphic members of these classes are exactly disk automorphisms.

For integers \(n,\ell\geq1\), set \(Q_\ell=2(n+\ell)+2\) and \(q_\ell=2Q_\ell/(Q_\ell-2)\). On the Siegel domain \(\Ucal=\Heis^n\times(0,\infty)\), we study \[ -\Delta_{\Heis^n}v -4\rho\bigl(v_{\rho\rho}+T^2v\bigr)-4\ell v_\rho =|v|^{q_\ell-2}v. \] For its Dirichlet form \(E_\ell\), we determine the sharp Sobolev constant and all extremals, prove \[ S_\ell\|v\|_{q_\ell}^2\leq E_\ell(v),\qquad E_\ell(v)-S_\ell\|v\|_{q_\ell}^2 \geq\kappa_{n,\ell} \operatorname{dist}_{\dot S^1_\ell} \bigl(v,\mathfrak M_\ell^{\R}\bigr)^2, \] where \(\mathfrak M_\ell^{\R}\) is the extremal cone. We classify nonnegative finite-energy solutions and establish the linearized kernel, profile decomposition, and attainment of the optimal stability quotient. Cayley conjugation yields ground-state symmetry and nondegeneracy, global Palais–Smale compactness, and perturbative existence for critical equations on \(\CHyp^{n+1}\). The mechanism is the radial lift \(v^\uparrow(z,w,t)=v(z,t,|w|^2)\), which reverses the interior-to-boundary construction by realizing the Siegel domain as a symmetry-reduced slice of a larger Heisenberg group. The completed-space reduction resolves the degenerate axis, hidden auxiliary concentration, and the mismatch of extremal cones. Together with the Cayley transform, this supplies the missing nonlinear layer—classification, stability, bubbling, and variational compactness—on complex hyperbolic space and provides a blueprint for other rank-one symmetric spaces.

In this paper, we introduce a new family of geometric constants CtB(X) of Birkhoff-orthogonal Takahashi-von Neumann-Jordan type for real Banach spaces. We discuss their basic properties and give a characterization of uniformly non-square spaces via CtB(X). Some sufficient conditions for super-reflexivity and fixed point property are presented. We establish upper bounds of CtB(X) related to orthogonal geometric moduli and constants, and obtain the sharp value for Radon planes.

We study the behavior of strongly facially symmetric spaces under completion and passage to norm-dense subspaces. We introduce a natural face-density condition guaranteeing that the completion of a normed SFS-space remains strongly facially symmetric. We prove that a norm-dense subspace of a neutral strongly facially symmetric space inherits the neutral SFS-structure whenever it is invariant under the ambient generalized Peirce projections. Several examples and counterexamples are presented, including intermediate subspaces of the trace class and the dense subspace $C[0,1]\subset L_1[0,1]$.

We prove that the Fremlin projective tensor product of Banach lattices preserves unbounded norm convergence and unbounded absolute weak convergence of elementary tensors without any additional assumptions. The main tool is a tensor-lattice estimate controlling expressions of the form \[ \lvert x_{\alpha}\otimes y_{\beta}-x\otimes y\rvert \wedge w, \] where $w$ is positive in the tensor product. This provides a direct and simplified proof of permanence results previously obtained under extra hypotheses.

We establish a spectral representation theorem for compact operators that are self-adjoint with respect to the duality mapping on smooth Banach spaces. The result is a natural analogue of the classical spectral theorem for compact self-adjoint operators on Hilbert spaces.

Motivated by developments in metric regularity theory for set-valued mappings, this paper introduces several directional regularity notions that are weaker than the corresponding classical regularity properties. We investigate the relationships among these concepts and establish a classification framework based on directional minimal time functions. Particular attention is devoted to directional orbital regularity and directional orbital Aubin continuity, which are shown to provide a natural setting for the analysis of fixed point phenomena. Using these notions, we derive approximate and exact fixed point theorems for set-valued mappings under directional assumptions. The obtained results are expressed in terms of minimal time functions and orbital approximation procedures, allowing for anisotropic and asymmetric behaviors that are not captured by the classical nondirectional framework. We further apply the developed theory to directional coincidence point results, coupled fixed point theorems, and Milyutin-type perturbation stability properties. The proposed approach provides a unified directional perspective on fixed point theory and variational analysis. In particular, it recovers several known results from the literature as special cases and yields new extensions under substantially weaker assumptions than classical contraction or global regularity conditions.

The existence and construction of Schwartz-class windows for lattice Gabor frames is a central problem in time–frequency analysis. For general time–frequency lattices, we characterize exactly those lattices that admit Schwartz-class windows and, additionally, windows in the Feichtinger algebra. Thereby we arrive at sharp classical and amalgam Balian–Low theorems for lattices in $\mathbb{R}^d$. For separable lattices and specified block-zero forms, we construct smooth windows that are compactly supported either in time or in frequency. These constructions are based on a characterization of pairs of lattices for which there exists a single set that tiles with one lattice and whose $\varepsilon$-neighborhood packs with the other.

In this paper we investigate the spectral properties of certain self-adjoint quadratic pencils, occurring at the separation of the variables in the problem of the oscillation of a semi-infinite cylinder with a pseudo isotropic elastic structure.

For a convex body $K \subset \mathbb R^d$ let $\Delta(K)$ be the expected distance between two independent uniform points of $K$, and let $\theta(K)$ be the corresponding expectation for normalized surface measure on $\partial K$. The Zaporozhets-Tarasov conjecture asserts $\Delta(K) \le \theta(K)$. We prove this conjecture in case $d=2$. In addition, we give a six-vertex convex polytope in $\mathbb R^3$ for which the reverse strict inequality holds, and obtain counterexamples in every dimension $d \ge 3$ by taking products with segments. Finally, we show that $\theta(K)> \frac{\operatorname{per}K}{6}$ for every planar convex body.

Let $A$ and $B$ be unbounded self-adjoint operators on a Hilbert space. Suppose that there exists a dense linear subspace $D\subseteq D(AB)\cap D(BA)$ such that $ABx=BAx$ for $x \in D$. We give some characterizations of the strong commutativity of $A$ and $B$ in terms of domain equalities.

We give a negative solution to the inverse generator problem on Hilbert spaces. More precisely, we construct a bounded operator $A$ with dense range on a Hilbert space $H$ that generates a bounded, strongly stable $C_0$-semigroup, while $A^{-1}$ does not generate a $C_0$-semigroup. We also construct an exponentially stable generator $A$ with $0 \in \rho(A)$ such that the inverse semigroup is unbounded and grows at least double logarithmically. For the latter generator, every Cayley transform satisfies the ordinary Kreiss resolvent condition but is neither strongly Kreiss bounded nor power bounded. Moreover, its powers satisfy a doubly logarithmic lower bound. Therefore, the Crank–Nicolson scheme is unstable in operator norm both for every fixed step size over long times and under mesh refinement at any fixed final time. Our counterexamples are deduced from a common finite-dimensional construction. For $\alpha\in(0,1)$, we use explicit bases of $\mathbb C^{2n}$ whose partial-sum projections are uniformly bounded and whose unconditionality constants are comparable to $n^\alpha$. The matrices underlying the counterexamples are then obtained as Schauder multipliers with respect to these bases, using a sequence of eigenvalues whose moduli decay doubly exponentially.