Papers

Let $p\in[1,\infty)$ and $\delta\in(0,n]$. Recently, to characterize the weighted boundedness of maximal operator on Choquet integrals based on Hausdorff content $\mathcal H_\infty^\delta$, a class of Capacity Muckenhoupt weights is introduced, denoted by $\mathcal A_{p,\delta}$, and, for any fixed $\delta\in(0,n]$, this new class of weights is proved to satisfy the self-improving property with respect to the index $p\in(1,\infty)$. In this note, we show that, for every fixed $p\in[1,\infty)$, the class of capacity Muckenhoupt weights $\mathcal A_{p,\delta}$ fails to satisfy the self-improving property with respect to the index $\delta\in(0,n]$.

We study functional inequalities along Wasserstein geodesics. If $\mu$ 0 and $\mu$ 1 are respectively $\kappa$ 0 -and $\kappa$ 1 -strongly log-concave probability measures on R n , we prove that their quadratic Wasserstein geodesic satisfies for all probability measures $\nu$ on R n . The coefficient is sharp. By linearization, this recovers the Poincaré estimate of Han and Zhu [15]. On the real line, we prove convexity of the square roots of the optimal T 1 and T 2 constants along monotone interpolation between arbitrary probability measures. The argument applies to more general transport entropy inequalities. We also establish convexity of the rescaled L p Poincaré constants for every finite p $\ge$ 1, including the square root of the Poincaré constant and the inverse Cheeger constant. Finally, we construct a planar Wasserstein geodesic whose endpoints satisfy T 2 and all finite-p L p -Poincaré inequalities, whereas every interior interpolant fails these inequalities.

We prove that equivalent locally compact, locally Hausdorff, étale groupoids with paracompact unit spaces have Morita equivalent reduced and full $L^p$-operator algebras for every $p\in[1,\infty]$. The reduced theorem requires pairing-valued approximate identities, reflecting the failure of unconditionality of the reduced norm for $1<p<\infty$, while the full theorem rests on a full-clopen reduction theorem proved by dilating spatial representations. For $1\leq p<\infty$, we also compare the concrete $p$-linking algebra of the reduced Morita equivalence with the reduced $L^p$-operator algebra of the linking groupoid: the canonical comparison is contractive, injective, and has dense range in general, and is an isometric isomorphism for $p=2$. We further study Morita cycles arising from proper étale groupoid correspondences, including the reduced case under suitable extension hypotheses. Applications include an $L^p$ version of Green's symmetric imprimitivity theorem, results for coarse groupoids and inverse semigroups, and invariance of the corresponding $K$-theory.

For $\beta \in \mathbb{R}$, we define the partition function $$ Z_\beta(X) = \sum_{i=1}^N \exp(\beta X_i) $$ of a centered Gaussian random vector $X=(X_1, \ldots, X_N)$ with $\mathbb{E}[X_i^2]= 1$. For $q \in \mathbb{R}$, we prove a complete phase transition for the generalized $L_q$-means of $Z_\beta(X)$ at $q=1$. The centered Gaussian vector with covariance matrix $\Delta_N$ obtained from a regular simplex configuration of unit vectors maximizes these means for $q<1$, and minimizes them for $q>1$. The two regimes are governed by different principles. For $q > 1$ {and $\beta\ne0$}, the moment functional is globally strongly convex on the entire set of correlation matrices, with an explicit modulus of convexity and a quantitative centroid-shape stability estimate. For $q<1$, the strategy is different. We prove a universal comparison for log-concave profiles of reverse Brascamp–Lieb type. Specializing this result to the Gumbel profile yields a Laplace-transform comparison between the partition functions.

Let $p$ be a complex polynomial of degree $n\ge2$, and let $\sigma_2(p)$ be the root mean square distance of its zeros from their centroid, including multiplicities. We prove that every zero of $p$ lies within distance $\sigma_2(p)$ of a critical point. This proves Borcea's quadratic conjecture and, in particular, gives a new proof of Sendov's conjecture. Two polynomial integration identities prove the result analytically for $n\ge2048$. For the remaining degrees, we derive explicit necessary inequalities, including a variable integration endpoint and an exact centered coefficient identity. Directed integer arithmetic certifies a finite covering of the resulting parameter domains. Degrees $4$–$26$ use $150\,845$ certified three-dimensional regions; degrees $27$–$2047$ use a further $27\,429$ regions, with degree intervals allowed. The cases of degree 2 and 3 were already covered by Khavinson–Pereira–Putinar–Saff–Shimorin.

In this paper, we establish the weak type $(1,1)$ boundedness of noncommutative singular integral operators with rough kernels. To deal with the rough kernels, we develop some orthogonality of the kernels themselves in the pseudo-localization arguments for the good functions and use the microlocal decomposition and $TT^*$ arguments for the bad functions.

We show that every UCP extension problem for a finite dimensional operator system can be reduced to three dimensions after finite matrix amplification. The set of extensions is preserved by an affine bijection. The matrix size is of order $\sqrt{d}$, where $d$ is the dimension of the original system. This also gives a three dimensional (smallest) counterexample to Arveson's hyperrigidity conjecture.

We construct a separable reflexive Banach lattice $X$ and a closed sublattice $E\subseteq X$ such that every bounded linear operator from $X$ into $\ell_2$ is compact, whereas there exists a positive operator from $E$ into $\ell_2$ that does not attain its norm and admits a positive maximizing sequence converging weakly to a nonzero vector. This answers negatively the question posed by Dantas, Jung and Martínez-Cervantes in [4] concerning the inheritance of the weak maximizing property by closed subspaces of the domain. It also answers negatively the corresponding question for the positive weak maximizing property and closed sublattices of reflexive Banach lattices. Canonical lattice complexification yields a counterexample to the original question over the complex field as well.

We study whether every infinite-dimensional weakly compactly generated (WCG) Banach space fails the Lotz property. The answer is affirmative in the nonseparable case: a classical decomposition theorem for WCG spaces provides a Schauder decomposition, and an explicit diagonal $C_0$-semigroup has an unbounded generator. The unrestricted assertion is false. Our main abstract result is a Calkin-algebra criterion: if $X$ fails the bounded compact approximation property and, for some fixed $m$, every noninvertible element of $\Cal(X)=\Lop(X)/\Kop(X)$ has $m$th power zero, then $X$ has the Lotz property. In particular, this applies when every operator is scalar-plus-strictly-singular and one fixed power of every strictly singular operator is compact. Combining this criterion with the Maurey–Pisier–Szankowski extraction theorem and constructions of Argyros and Motakis yields separable hereditarily indecomposable Lotz spaces over both scalar fields: reflexive examples, and, over the complex field, an example containing no infinite-dimensional reflexive subspace. Moreover, the ambient Argyros–Motakis spaces considered here are Lotz-saturated, although they themselves are non-Lotz. Thus every nonseparable WCG space is non-Lotz, while the separable WCG class contains both Lotz and non-Lotz spaces.

We prove that the intersection of $2d$ halfspaces containing the Euclidean unit ball in $\mathbb{R}^d$ contains a point of norm at least $\sqrt{d}$. Equality characterizes the cube. The proof gives a lower bound for the average squared norm of the vertices, weighted by the Gaussian measures of their normal cones. The main step is an inequality for Gaussian orthant probabilities, proved by a one-dimensional comparison of mass, centroid, and boundary density. We also discuss a quadratic extension and give an independent spectral estimate.

Let $I=(a,b)$ be an open interval with finite or infinite endpoints. For $1<p_1,p_2<\infty$ and $0<q<\infty$, we study the weighted bilinear Hardy operator $ \mathcal H_2(f,g)(x)=(\int_a^x f)(\int_a^x g) $ from $L^{p_1}(v_1;I)\times L^{p_2}(v_2;I)$ to $L^q(u;I)$, including the quasi-Banach target range $0<q<1$. With $\vartheta=\min\{1,q\}$, the target and bounded-bilinear-map spaces are treated through their complete powered metrics. Under explicit local integrability assumptions on the weights, compactness is equivalent to boundedness and vanishing endpoint-tail quasi-norms. Whenever the full operator is bounded, the $\vartheta$-power of the directed combined-tail limit equals exactly the powered essential distance, finite-rank distance, and Hausdorff measure of noncompactness in the target metric. Equivalently, the unpowered quantities agree under the stated quasi-norm-radius normalisation. Piecewise-constant middle-window approximants give quantitative bounds for the bilinear approximation numbers. The finite-rank-distance identity then implies that these numbers converge to the essential (quasi-)norm. Applying published norm-equivalent boundedness characteristics to endpoint-truncated output weights yields explicit compactness criteria in the convex, mixed, moderately subcritical, and deeply subcritical regimes. Algebraic companion exponents are used to make the negative powers arising for $q<1$ explicit. Four worked examples include an asymmetric mixed-range threshold and a nonlocally convex $L^{1/2}(0,1)$ example for which $a_{m+1}^{(2)}(\mathcal H_2)=O(m^{-1/2})$.

We prove that if $f_1,\ldots,f_N$ is an uniformly bounded orthonormal system, then for all $p\in[1,2)$ there is an estimate for its Kolmogorov widths: $d_n(\{f_1,\dots,f_N\},L_p) \gtrsim \min\{1,n^{-1/p}N^{1/2}\}$, $n\le N/2$. In the regime $n\asymp N$ this provides the lower bound $N^{-\alpha_p}$ with a sharp exponent $\alpha_p:=1/p-1/2$. Besides that, it follows that a good approximation of such ONS requires the dimension $n\gtrsim N^{p/2}$. Our second result is an approximation theorem for Fourier matrices of finite abelian groups (this includes the usual DFT matrices). For every $\eta>0$ there is $a=a(\eta)>0$ such that any Fourier matrix $F$ of sufficiently large order has an approximation of rank $N^{1-a}$ with the row-wise $\ell_1$-error at most $N^{1/2+\eta}$; the exponent $1/2$ is sharp. Consequences include the approximation of the same kind for circulant matrices; the approximation of the trigonometric functions $\exp(2\pi i\langle \lambda,x\rangle)$, $\lambda\in \Lambda=K\cap\mathbb{Z}^d$, $K$ is symmetric convex, with dimension $|\Lambda|^{1-a}$ and an optimal error $|\Lambda|^{-\alpha_p+\eta}$; bounds for widths of weighted Wiener classes.

Hilbert C*-modules whose coefficient C*-algebra is equipped with a finite faithful trace generate Banach Gelfand triples. This provides a natural setting for a function space interpretation of Hilbert C*-modules. In particular, when the modules are finitely generated and projective, their external tensor products interact well with projective tensor products, allowing us to derive abstract kernel theorems for Hilbert C*-modules. As an application, we apply our results to Heisenberg modules and obtain new kernel theorems for time-frequency analysis.

We determine the cotangent operator space structure at the origin of the tetrablock (denoted by $\mathbb E$). In particular we prove that $\operatorname{COT}_0(\overline{\mathbb E}) =\operatorname{MIN}(\ell_1^2\oplus_\infty\mathbb C)$ completely isometrically. This shows the existence of commuting tuple of operators having $\overline{\mathbb E}$ as a spectral set but not as a complete spectral set.

For a bounded linear operator on an infinite-dimensional Hilbert space, we consider the class $\beta(H)$ of operators whose restrictions to all nonzero reducing subspaces attain their norms. We prove that $\beta(H)$ is dense in $\mathcal{B}(H)$ in the operator norm. Since $\beta(H)$ is not stable under arbitrary compact perturbations, we introduce a natural subclass $\beta_0(H)$ and investigate its perturbation properties. We show that $\beta_0(H)$ is stable under finite-rank perturbations and obtain a complete characterization of those compact perturbations that preserve membership in $\beta_0(H)$. We also determine the maximal compactly perturbation-invariant subset of $\beta_0(H)$. These results clarify the spectral structure and stability of norm-attaining operators on reducing subspaces.

A complete classification is established for continuous, SL($n$) contravariant, and translation invariant tensor valuations defined on the Sobolev space $W^{1,p}(\mathbb R^n)$. When these valuations are further assumed to be homogeneous, the classification reveals that they are precisely the Fisher information tensors, which constitute a higher-order generalization of the Fisher information matrix.

For any Tychonoff space $X$, let $D(X)$ denote either the space $C(X)$ of all continuous real-valued functions on $X$ or the space $C^*(X)$ of all bounded continuous real-valued functions on $X$. We write $D_p(X)$ when $D(X)$ is endowed with the topology of pointwise convergence. In our recently published paper [8, Theorem 1.6], we obtained the following result: if $T: D_{p}(X) \to D_{p}(Y)$ is a linear continuous surjection, where $X$ is a metrizable space and $Y$ is a perfectly normal space, then $Y$ inherits a given topological property $\mathcal{P}$ from $X$. A linear continuous surjection $T: E_{p}(X)\to E_{p}(Y)$ is said to be densely defined if $E(X)$ and $E(Y)$ are dense linear subspaces of $D_{p}(X)$ and $D_{p}(Y)$, respectively, and $E(X)$ is correct (see Definition 1.5(a)). In the present paper, we establish sufficient conditions under which the above statement remains valid for a densely defined linear continuous surjection $T: E_{p}(X) \to E_{p}(Y)$. In particular, $\mathcal{P}$ can be zero-dimensionality, $\sigma$-compactness or strong countable-dimensionality. Additionally, for arbitrary Tychonoff spaces $X$ and $Y$, assuming that $T: E_p(X)\to E_p(Y)$ is a densely defined linear continuous operator, we show that $X\in\mathcal P$ implies $Y\in\mathcal P$, where $\mathcal P$ is the property $(\kappa)$, the strong $\sigma$-scatteredness, or the property of being a $\Delta_1$-space.

We study optimal polynomial approximants in spaces of Dirichlet-type over the bidisc. Given a nonzero function f, the linear optimal polynomial approximant (opa) to its reciprocal is the affine polynomial p for which pf is closest to the constant function 1 in the space norm. We prove that, for every positive value of the Dirichlet-type parameter, there exists a symmetric polynomial f with no zeros on the closed bidisc, while the corresponding linear optimal approximant has a zero inside the bidisc. This provides polynomial counterexamples to the version of the Weak Shanks Conjecture on these spaces. The construction also extends to higher-dimensional polydiscs and to anisotropic Dirichlet-type spaces.

We prove dimension-independent Hyers–Ulam stability for surjective approximate isometries of the positive trace-class cone in both the trace and Bures metrics. Here $\varepsilon\geq0$ is a uniform upper bound on the additive error in preserving distances. Such an approximate isometry is uniformly approximated by a Wigner symmetry (a unitary or antiunitary conjugation), with error at most $9\varepsilon/2$ in the trace metric and $2\sqrt2\varepsilon$ in the Bures metric, even if it does not fix zero. For maps fixing zero, the trace bound improves to $3\varepsilon$ even under the hypothesis of $\delta$-surjectivity, independently of $\delta$. In infinite dimensions, we construct spectrum-preserving homeomorphisms of the full state space, including mixed states, whose distortions tend to zero in both metrics but which stay a fixed uniform distance from every Wigner symmetry. For state spaces, stability holds in each fixed finite dimension with a dimension-dependent modulus tending to zero as $\varepsilon\to0$, but no such modulus can be chosen independently of dimension.

Let $\operatorname{UT}_n$ denote the group of $n\times n$ real upper triangular matrices with positive diagonal entries. We derive a closed-form analytic expression for the Karcher mean on $\operatorname{UT}_2$ and obtain a sharp convergence threshold for the projection mean iteration with fixed initial diagonal, together with a sufficient convergence condition for arbitrary initial points. In addition, we obtain explicit formulas for the log-Euclidean mean and establish a Lie–Trotter type formula for the Karcher mean on $\operatorname{UT}_2$. For $\operatorname{UT}_n$ we derive the diagonal and first superdiagonal entries of every solution of the Karcher equation and transfer nonconvergence examples from $\operatorname{UT}_2$.

For every $\alpha>1$ we construct a finite set $A\subset\mathbb{D}$ such that the zero-based invariant subspace $I_A$ fails the wandering subspace property. Consequently, the Beurling-type theorem holds on the weighted Bergman space $A^2_\alpha$ if and only if $-1<\alpha\le1$, confirming a conjecture of Shimorin. As a further application, we extend the prescribed-curvature construction of Hedenmalm and Perdomo to all $\alpha>1$, thereby replacing the constant $\alpha_0\approx1.04$ in their result by $1$.

Let \[ M_{k,\ell}=[z^k-w^\ell]\subset H^2(\mathbb D^2), \qquad k,\ell\in\mathbb N,\quad k\ne\ell . \] We study the large-index asymptotics of Yang's numerical invariants and the boundary structure of their generating function. Starting from the exact staircase formula obtained in our preceding work, we prove that \[ \Sigma_j(M_{k,\ell}) = \frac{C_{k,\ell}}{j} + O(j^{-2}), \] where \[ C_{k,\ell} = \int_0^\infty \frac{x^2} {(x+1/k)^2(x+1/\ell)^2}\,dx . \] The first strict descent together with $C_{k,\ell}$ recovers the unordered pair $\{k,\ell\}$, yielding an asymptotic rigidity principle. At the next order we obtain a periodic correction \[ \Sigma_j(M_{k,\ell}) = \frac{C_{k,\ell}}{j} + \frac{\Psi_{k,\ell}(j)}{j^2} + O(j^{-3}), \] whose least period is \(\operatorname{lcm}(k,\ell)\), and we determine the leading amplitudes of the strict drops. For Yang's generating function \[ \mathcal P_{k,\ell}(t) = \sum_{j=0}^{\infty}\Sigma_j(M_{k,\ell})t^j, \] we prove that it has radius of convergence one and a logarithmic singularity at $t=1$; hence it is never a polynomial for $k\ne\ell$, giving a negative answer to Yang's polynomiality question within this quasi-homogeneous family. The periodic higher-order corrections generate root-of-unity polylogarithmic boundary modes. The leading logarithmic coefficient together with the second-order boundary support determines the unordered pair $\{k,\ell\}$, while successive renormalized boundary limits recover the Fourier coefficients of every finite-order periodic asymptotic term.

Chollet's permanent conjecture asks whether per(A o B) <= per(A) per(B), where o denotes the entrywise (Hadamard) product, for complex Hermitian positive semidefinite matrices. We present computer-assisted proofs of two restricted forms with explicit constants strictly smaller than one. For every integer n >= 10 and every such matrix A of rank at most four, the self-conjugate ratio per(A o conjugate(A)) / per(A)^2 is bounded by 999991742359 / 10^12 when the denominator is nonzero. For order nine, we obtain the bound 999815240367 / 10^12 for correlation matrices whose four nonzero eigenvalues all equal 9/4. The first argument combines complex-sphere integrals, spectral subspace tilts, projection bounds, exact finite covers, and an analytic infinite tail. The second uses a quadratic relation among nine Gram vectors, a positive operator on the ten-dimensional space of quadratic forms, and rigorously bounded entropy. All decisive finite calculations use rational arithmetic and full closed-domain certificates. The unrestricted conjecture, including general non-tight order-nine rank-four matrices, is outside these results.

Many learning tasks require stability to small transformations while retaining sensitivity to larger ones. We introduce transversal pooling neural networks, which generalize spatial max pooling to affine group actions. We establish equivariance to a chosen subgroup and derive explicit stability bounds for individual pooled wavelet coefficients under affine perturbations of the input. Experimentally, we demonstrate the utility of our networks in low-data environments and for predicting tropical cyclone intensification.

We give a short proof of the following theorem of K. Ball: If a centrally symmetric convex body $K\subset \mathbb R^d$ is covered by finitely many planks, then the sum of their relative widths with respect to $K$ is at least $1$.

We prove that Rademacher type $s$ coincides with Talagrand type $(s,\psi_{s,s/2})$ for every $1<s<2$.

Let $R$ be a compact discrete valuation ring and $G = R^d \rtimes \operatorname{GL}_d(R)$. We show that the central Fourier algebra $\operatorname{ZA}(G)$ is not amenable, reaffirming a conjecture of Alaghmandan and Spronk. Our methods reduce to studying the Fourier algebra of the commutative orbit hypergroup $H = R^d/\operatorname{GL}_d(R)$. Along the way, we also show that dual of any commutative orbit hypergroup $H$ is the hypergroup of the corresponding dual action, and this in turn shows that $\operatorname{A}(H) \cong \operatorname{L}^1(\widehat{H})$, which aligns with the classical setting.

We introduce power variations of critical Gaussian multiplicative chaos (GMC) along refining partitions in arbitrary dimension. Under suitable renormalisation, we prove uniform fractional moment bounds via Laplace transform estimates as well as stable convergence to supercritical GMCs. Together, these results provide a unified approach to regularity and spectral questions for critical chaos, yielding: (i) the sharp exponent for the logarithmic modulus of continuity, (ii) the existence of non-empty essential spectrum for critical Liouville quantum gravity surfaces; and (iii) an almost-sure quantitative Fourier decay.

We prove a formula for the total variation of SBV functions in sub-Riemannian Heisenberg groups arising as the limit of certain BMO-type functionals.

Let $\mathscr{H}^2$ denote the Hilbert space of Dirichlet series with square-summable coefficients. It follows from a theorem of Bohr that if $f$ in $\mathscr{H}^2$ has a bounded analytic continuation to the right half-plane, then its norm can be computed from the mean values \[\|f\|_{\mathscr{H}^2}^2 = \lim_{\sigma\to 0^+} \lim_{T\to\infty} \frac{1}{2T} \int_{-T}^T \lvert f(\sigma+it) \rvert^2\,dt.\] We investigate to what extent the assumption that the analytic continuation be bounded can be relaxed, and obtain optimal results in two different directions. The first direction concerns the order of the analytic continuation, and yields as a corollary a version of the classical Landau–Schnee convergence theorem for $\mathscr{H}^2$. The second direction concerns the abscissa of convergence, and resolves a problem posed by McCarthy.