Papers

In this short note, we show that some well-known filtrations of infinite dimensional groups of Fredholm operators associated to certain perturbation classes are, in fact, Fredholm $\Delta$-filtrations (see Definition 5 below.) For example, let $\E$ be a separable infinite dimensional real Hilbert space. The group $\GLK(\E)$ of all invertible operators on $\E$ which are compact perturbations of the identity is the structure group for Hilbert Fredholm manifolds and bundles modeled on $\E$ \cite{ElwTr, Ksch, Mkhr}. Using an orthonormal basis, there are canonical inclusions of general linear groups: $$\GL(1) \subset \cdots \subset \GL(n) \subset \GL(n+1) \subset \cdots \subset \GL(\infty) = \varinjlim \GL(n) \subset \GLK(\E).$$ We show this is a Fredholm $\Delta$-filtration of the Fredholm manifold $\GLK(\E)$ with dimension sequence $\Delta(n) = \dim(\GL(n)) = n^2$, which was not discussed in the classical Fredholm manifold literature because of the rigid constraint that the dimensions of a filtration increase only by one.

Let $S_c=P_{(0,c)}QP_{(0,c)}$ be the one-dimensional sinc-kernel concentration operator, let $N_a(c)=\#{n:\lambda_n(c)>a}$, and set $\bar L=\log((1-\delta)/\delta)$. We prove, uniformly for each fixed $A>0$, the tail-side quantile formula $N_\delta(c)=c+\pi^{-2}\bar L\log(4\pi^2c/\bar L)+O_A(\log c+\bar L)$ for $6\le\bar L\le A\log c$. It yields corresponding additive formulas for the lower half and full plunge, with main terms respectively $\pi^{-2}\bar L\log(4\pi^2c/\bar L)$ and twice this quantity. An exact one-tail-coordinate selection gives, for fixed $A>0$, $d\ge1$, and $q\in(1/2,1)$, the one-sided tensor-product bound $\Lambda_\delta(c;d)\ge\pi^{-2}d c^{d-1}\bar L\log(4\pi^2c/\bar L)-O_{A,d,q}(c^{d-1}(\log c+\bar L))$ for $L_{d,q}\le\bar L\le A\log c$, where $L_{d,q}=\log(q^{-(d-1)}(e^6+1)-1)$; the tensor content is nontrivial for $d\ge2$. The analytic input is a signed growing-parameter sine-kernel determinant asymptotic: uniformly for $0\le\omega\le A\log s$, $\log\det(I+(e^{2\omega}-1)K_s)=4\omega s/\pi+2\pi^{-2}\omega^2\log(4s)+2\log|G(1+i\omega/\pi)|^2+O_A((1+\omega)^4\log^2s/s)$, where $G$ is the Barnes $G$-function. We prove this negative-coupling counterpart of the Bothner–Deift–Its–Krasovsky theorem by direct IIKS steepest descent. We also retain the uniform head-side results and use a two-way determinant reduction to obtain the moving-depth lower-half bridge bound with constant $1/(32\pi^2)$; extending it to the deeper range uses Kulikov–Dam Larsen and may require a smaller constant. These counting formulas are additive. Their errors become uniformly relative when $\bar L$ tends uniformly to infinity; fixed thresholds are covered separately by Landau–Widom. A Lambert-$W_{-1}$ formula is recorded only for the continuous main term, not for individual eigenvalues.

Let \[\mathfrak O=\{(x,y)\in[0,1]^2:x+y\le \sqrt2\} \] be the first-quadrant sector of a regular octagon. For quadratic forms \(P(x,y)=ax^2+bxy+cy^2\), we study the supremum norm over \(\mathfrak O\). We obtain a complete five-region formula for the norm, according to whether the norming contact occurs at an endpoint or in the interior of one of the three radial sides. We then prove that the projection of the unit ball onto the \(ac\)-plane is exactly \([-1,1]^2\), compute both endpoints of every vertical section, and thereby parametrize the entire unit sphere. Finally, we characterize the extreme points of the unit ball as four explicit curves, their negatives, and four pairs of isolated points. The resulting description is fully explicit and reduces subsequent convex extremal problems on this polynomial space to four one-parameter families and finitely many isolated polynomials.

This paper addresses the Turán extremal problem on the symmetric three-interval set $\Omega_\lambda = (-1,1) \cup (\lambda-1,\lambda+1) \cup (-\lambda-1,-\lambda+1)$, $\lambda \geq 0$. We establish a discrete approximation principle relating the continuous Turán problem on $\Omega_\lambda$ to the discrete Turán problem on an associated finite discrete set. This enables the continuous problem to be exactly expressed as a limit of finite nonnegative trigonometric polynomial problems, which are equivalent to finite semidefinite programs. Using this framework, we compute the Turán constant for integer and rational parameters $\lambda$. We further analyze the dependence of the Turán constant on $\lambda$ and derive upper bounds for all $\lambda > 5$.

For $m\geq 2$, let $c_p(m)$ be the all-dimensional best constant in $$ \left\|\sum_{k=1}^m A_k\right\|_p \leq c_p(m)\left\|\sum_{k=1}^m |A_k|\right\|_p. $$ Tang and Zhang conjectured an explicit formula for every finite $p>1$. We disprove the conjecture with two explicit real $2\times 2$ rank-one matrices at $p=3/2$. The comparison is certified by seven strict rational inequalities and, in particular, places the attained ratio above $207/200$, while the conjectured constant lies below $207/200$. On the positive side, we prove the conjectured sharp bound for every family of rank-at-most-one summands when $2\leq p<\infty$, and classify all equality cases. We also prove the corresponding endpoint statement for $p=\infty$. Finally, for arbitrary complex matrices, we establish the conjectured sharp constant in the case $m=2$, $p=4$.

The goal of this paper is to characterize the boundedness property of the uncentered Hardy–Littlewood maximal operator on ball Banach function spaces and their Köthe associate spaces in terms of the vector-valued maximal inequality. The main result of this paper supplements the recent results by Nieraeth.

It is well known the local integral inequality which is called Poincaré–Sobolev inequality on each domain cube (or ball). After that many authors investigated the generalization for this inequality with Muckenhoupt-type weights or global weights which are independent of the domain cube. In this paper, we investigated similar weighted generalization for this inequality with the local weights which are depending on the domain cube without assuming the Muckenhoupt condition. Moreover, we considered the weighted generalization for the homogeneous and inhomogeneous-type Sobolev embedding theorems as some applications.

We study lower tails of support functions of isotropic convex bodies. An endpoint negative moment estimate, combined with a spherical cap argument, gives polynomial deviation estimates for the inradius of random projections in every dimension; in the unconditional class this yields the optimal order of the median inradius and shows that the cube is extremal. The same argument gives estimates for convex hulls of independent rotations and, after an additional projection, a mixed rotation-projection theorem. We also prove weighted lower and upper estimates for Minkowski averages of rotated polars and their random projections, as well as an inradius dependent estimate for the geometric distance of global averages from the Euclidean ball. A family of isotropic product cylinders shows that the latter estimate is sharp, up to absolute constants, throughout the possible range of the inradius. For the cube, an explicit negative moment computation gives a quantitative projected $(m,k,n)$-profile and, in full dimension, recovers the classical order $n/\ln(en)$ for bounded geometric distance. Finally, we determine the sharp volume profile of intersections of independent rotations in the unconditional isotropic class and show that the cube is extremal.

For a real Banach space $X$. The Dehghan–Rooin magnitude compares the angular distance $\alpha[x,y]=\|x/\|x\|-y/\|y\|\|$ together with the skew angular distance $\beta[x,y]=\|x/\|y\|-y/\|x\|\|$. Motivated by the recent use of Birkhoff orthogonality in the Massera–Schaffer inequality, we introduce and study the Birkhoff-restricted Dehghan–Rooin constant \[ \operatorname{DR}_{B}(X)=\sup\left\{\frac{\alpha[x,y]}{\beta[x,y]}:x,y\in X\setminus\{0\},\ x\perp_{B} y,\ \beta[x,y]\ne0\right\}. \] The restriction $x\perp_{B} y$ turns the global comparison of $\alpha$ and $\beta$ into a directional invariant which detects the interaction between supporting functionals and the radial normalization map. We obtain a scale-profile formula, sharp elementary bounds, stability under subspaces and ultrapowers, a variational characterization of the extremal case $\operatorname{DR}_{B}(X)=1$, estimates in terms of the classical Dehghan–Rooin constant, and consequences for uniformly convex and uniformly smooth spaces. In smooth two-dimensional spaces we derive an explicit formula through the normalized duality mapping, and in Radon planes we reduce the constant to a one-dimensional optimization on an arc of the unit circle. Several model computations are included, including Hilbert spaces, polyhedral planes, and two-dimensional $p$-norm spaces. The final section gives normal-structure type consequences through a companion Birkhoff angular modulus and lists several problems suggested by the new constant.

We investigate Banach spaces generated by uniformly bounded families of functions taking values in a fixed finite set and establish a rigidity principle connecting the cardinality of the generating family with the geometry of its closed linear span. We prove that such a space is superreflexive precisely when the generating family is finite, or equivalently, when the resulting Banach space is finite-dimensional. The main ingredient is a finite-range spreading-model obstruction showing that an infinite family of finite-valued functions cannot generate a superreflexive space under the supremum norm. This principle is applied to Banach spaces generated by the characteristic functions of the left derivatives of formal languages. It yields geometric characterizations of regular languages in terms of finite dimensionality, superreflexivity, and the existence of an equivalent uniformly convex norm. We also obtain a canonical representation of the associated language space as a coordinate-function subspace of a space of continuous functions on a compact shift-orbit closure. The corresponding language Hankel operator is shown to be compact exactly for regular languages. In the nonregular case, we determine its exact distance from both the compact and finite-rank operators and compute all its nontrivial approximation numbers. Finally, we construct a single binary language whose associated Banach space contains an isometric copy of every separable real Banach space. These results reveal a sharp contrast between the geometric rigidity associated with regular languages and the universality that may occur in the nonregular setting.

We establish an abstract storage theorem for projectively $C_0$ unitary representations admitting a pair of one-parameter subgroups with conjugation escape. It produces, for every $0<\varepsilon<1$, a single orbit whose sampling set is relatively separated modulo the projective kernel and whose frame bounds are $(1-\varepsilon)^2$ and $(1+\varepsilon)^2$. Two applications are obtained from the same mechanism. First, every nonabelian exponential Lie group admits a near-Parseval frame of left translates for its left regular representation. Consequently, a positive-dimensional exponential Lie group is an FT group if and only if it is nonabelian; in particular, the three-dimensional Heisenberg group admits such a frame. Second, every infinite-dimensional irreducible unitary representation of an exponential Lie group admits a near-Parseval discrete orbit frame. When the effective projective quotient is abelian, the orbit may be chosen to be an orthonormal basis arising from a transported Weyl lattice.

We broaden Buzano-type inequalities to provide novel numerical radius bounds for operators of the type $AXB$, thereby generalizing the results obtained by Sababheh et al. For the $\lambda$-mean transform $M_\lambda(T)$, we provide a counterexample demonstrating that $r_\sigma(M_\lambda(T)) \le r_\sigma(T)$ fails to hold in general for $\lambda \in (0, 1)$, establish that $(r_\omega(M_\lambda(T)))^n$ and $r_\omega(T^n)$ are typically incomparable for $n \ge 2$, and confirm that $M_\lambda(T^*) = (M_\lambda(T))^*$ is valid for $\lambda \in [0, 1)$ if and only if $T$ is a member of a newly established $\sigma$-class. Furthermore, we examine the transformation characteristics of $T$ and the tensor products $T \otimes S$, refine Zamani's inequalities, and unify operator modulus bounds $|\widetilde{T}| \le |\widehat{T}| \le |T|$. In this application, we demonstrate that the conditions for norm preservation, $\|\widetilde{T}\| = \|T\|$ and $\|M_\lambda(T)\| = \|T\|$, are equivalent to the statement $\|T^2\| = \|T\|^2$, and we offer precise norm estimates for $2 \times 2$ off-diagonal block operator matrices under $\lambda$-mean transformation.

As is known, the $A_2$-continued representation of numbers is not topologically equivalent to the classical binary representation; therefore, the digit projector of such representations is a discontinuous function. In this paper, we introduce a continuous function that serves as an analogue of the digit projector of the classical binary representation of numbers into the digits of the $A_2$-continued representation with zero redundancy, namely a function of the form \[f(\Delta^2_{\alpha_1\alpha_2...\alpha_{2n-1}\alpha_{2n}...})= \Delta^{A_2}_{(\frac{1}{2})^{1-\alpha_1}(\frac{1}{2})^{\alpha_2}... (\frac{1}{2})^{1-\alpha_{2n-1}}(\frac{1}{2})^{\alpha_{2n}}...}, \alpha_n\in \{0,1\}.\] It is proved that the function $f$ is well-defined, continuous, and monotone. Using the normal properties of numbers with respect to their binary representation and Lebesgue's theorem asserting the existence of a finite derivative for a continuous monotone function almost everywhere, the singularity of the function $f$ is established. The paper also establishes a relationship between the considered function, the right-shift operator on digits, and the inversor of the continued representation of numbers. This relationship is then used to establish the singularity of the inversor.

Let \[ A_{\min}(x)=\bigl(x_{\min(i,j)}\bigr)_{i,j=1}^n, \qquad A_{\max}(x)=\bigl(x_{\max(i,j)}\bigr)_{i,j=1}^n \] be the Min and Max matrices generated by a real sequence \(x=(x_1,\ldots,x_n)\). Using their classical cone parametrizations, we give exact characterizations of entrywise maps preserving positive semidefiniteness, total nonnegativity, Loewner order, and Loewner convexity. Our main results concern the Loewner structure.Without assuming continuity or differentiability, entrywise Loewner-order preservation is equivalent to \(f\) being nondecreasing and convex. On the Min and Max cones, requiring the Loewner-convexity inequality on arbitrary pairs is rigid and forces \(f\) to be affine. Under the standard convention of restricting the inequality to Loewner-comparable pairs, the condition automatically forces \(f\in C^1([0,\infty))\) and is equivalent to convexity of both \(f\) and \(f'\). The analogous statement holds for Loewner concavity. In each case, the condition is already detected in dimension two. We also show that \(f:[0,\infty)\to\mathbb R\) preserves positive semidefiniteness entrywise on all positive semidefinite Min or Max matrices if and only if \(f\) is nonnegative and nondecreasing; the same condition characterizes total-nonnegativity preservation. Finally, we determine the power-function ranges and Loewner-order automorphisms, characterize the entrywise preservers of the strict Min and Max classes, and relate these strict cones to inverse \(M\)-matrices and oscillatory matrices.

The theory of quantum channels is traditionally studied either on finite-dimensional state spaces or within the Heisenberg picture as completely positive maps on C^*-algebras. In this paper, we consider quantum channels as completely positive maps on the duals of general von Neumann algebras in the Schrodinger picture. We investigate the construction of such channels through Pettis integrals using representations of topological groups.

By employing the Moore$-$Penrose inverse of a bounded linear operator, we derive several bounds for the numerical radius and operator norms of the sum of operators in semi$-$Hilbertian space that generalize and improve the classical bounds. We establish novel inequalities pertaining to the $\mathbb{A}$$-$Davis$-$Wielandt radius for $n \times n$ operator matrices and further explore their ramifications, particularly concerning $\mathbb{A}$$-$Davis$-$Wielandt radius bounds for $2 \times 2$ operator matrices, where diagonal operator matrix $\mathbb{A}$ contains positive bounded operator $A.$ Ultimately, we get an improved upper bound for the $A$$-$numerical radius inequalities relating to the commutators of operators.

We answer a question raised by Vogt by proving that the growth bound of an individually eventually positive $C_0$-semigroup on an $L^p$-space ($1 < p < \infty$) coincides with the spectral bound of its generator. Our proof follows the general strategy of Vogt's argument for the uniformly eventually positive case. The key new ingredient is a variant of an operator-range uniformisation principle of Arora and Glück, adapted here to time-dependent operator ranges. When applied to principal ideals, this principle turns individual eventual positivity into the uniform domination estimate required for Vogt's stability argument.

We extend the classical Taibleson-Weiss molecular theory to arbitrary radii, reaching the critical decay |x|^{-n/p}, with the resulting embedding shown sharp. This molecular theory governs the largest class of functions certifying membership in h^p(R^n), 0 < p <= 1, through Goldberg's convolution splitting: a critical Hölder modulus and an \ell^p-Dini decay majorant, sharp on the regularity, decay, and cancellation axes.

Naor proved that $L_p$ satisfies the metric $X_p$ inequality with the sharp scaling parameter and asked for the optimal dependence of its constant on $p$. The best previously known upper bound was $O(p^4/\log p)$, while first-degree chaos gives a lower bound of order $p/\log p$. We prove the matching upper bound. More generally, if $h$ is a mean-zero scalar function on the Hamming cube, $p\geq2$, and $S$ is uniformly distributed over the $k$-subsets of $[n]$, then \[ \bigl(\E_S\|\E_{[n]\setminus S}h\|_p^p\bigr)^{1/p} \lesssim \frac p{\log p} \left[\frac{k}{n}\sum_{j=1}^n\|\partial_jh\|_p^p+ \left(\frac{k}{n}\right)^{p/2}\|h\|_p^p\right]^{1/p}. \] The proof combines a sharp fixed-cardinality Rosenthal comparison with the exact reconstruction $h=\sum_jD_j\Delta^{-1}h$. A dimension-free inverse-gradient square-function estimate follows by integrating a pointwise reverse Poincaré inequality for the Walsh heat semigroup. Naor's transference argument then yields the optimal metric bound $O(p/\log p)$.

Banach asked whether a normed space must be Hilbert if, for one fixed dimension greater than one, all subspaces of that dimension are linearly isometric. We prove the complex case. By the codimension-one reduction, the essential finite-dimensional problem is to characterize a balanced convex body in a complex $(n+1)$-space whose complex hyperplane sections are all complex-linearly equivalent. We show that such a body is a Hermitian ellipsoid. The proof adapts the recent bundle–degree method of Lu and Yang for the real problem, with two features specific to the complex setting. After covariance normalization, write $G=\operatorname{Aut}{\mathbb C}(S)$ for the complex-linear symmetry group of a model section $S$. The exact section maps form a principal $G$-bundle over $S^{2n+1}$; reduction to $G^{\circ}$ places its obstruction in $\pi{2n}(G^{\circ})$, which is finite. Pulling back by a suitable positive-degree self-map trivializes this bundle and yields a global Lipschitz family of exact complex-linear section maps. Brouwer degree and a signed degree formula then imply that $p_S^{2n+2}$ and $p_S^{2n+4}$ are real homogeneous polynomials, where $p_S$ is the norm of a model section. Unique factorization forces $p_S^2$ to be quadratic, and phase invariance makes the resulting quadratic form Hermitian. The parallelogram identity then yields the general complex Banach-space statement.

Let $X$ be a complex Banach space with an unconditional Schauder basis. We prove that the monomials associated with this basis, endowed in each homogeneous degree with the square order and globally with any compatible ordering, form a Schauder basis for the space $(\calH(X),\tau_0)$ of entire holomorphic functions on $X$ with the compact-open topology. The proof relies on two main ideas. First, coordinate-tail conditions yield a fundamental system of compact solid subsets of $X$. Second, Fourier projections in the coordinates give the uniform estimate $c_{K,n}\leq n+1$ for the basis constant of the degree-$n$ monomials with respect to the supremum seminorm on each such compact set $K$. We derive applications to Banach sequence lattices with dense $c_{00}$, including classical, Lorentz, Orlicz-heart and Schreier-type sequence spaces, and to general vector-valued $E$-sums. In particular, the result covers mixed $c_0$- and $\ell_r$-sums of finite-dimensional $\ell_p$ spaces and the Lorentz predual $d_*(w,1)$.

We study rigidity phenomena for maps on vector and normed lattices arising from isometric, order-theoretic, and modulus-preserving conditions. We first investigate the role of strict convexity in Baker's nonsurjective version of the Mazur-Ulam theorem. In the process, we develop a theory of midpoint injective functions and investigate their connections with convexity and monotonicity. Using this framework, we prove a generalization of Baker's theorem. As our first main application, we show that every modulus isometry, i.e., a map $T:X\to Y$, where $X$ is a sublattice of a vector lattice $Y$, satisfying $$|T(x)-T(y)|=|x-y|, \qquad x,y\in X,$$ is affine and has a disjointness-preserving linear part. When $X$ is an ideal, the linear part is moreover a bijection of $X$ onto itself and an involution. In the normed lattice setting, we also show that every positive norm isometry into a strictly convex normed lattice is not only affine, but has a linear part that is a lattice homomorphism. Finally, we prove a nonlinear order-isometric rigidity theorem. After shifting to the origin, every order-preserving norm isometry between normed lattices whose codomain has a strictly monotone norm, preserves suprema and infima and is both disjointness preserving and disjointly additive. These conclusions do not force linearity, as shown by an explicit family of nonlinear isometries on $L_1[0,1]$.

For each $N\geq1$, consider the normalized König bilinear form $B_{\mathrm K}:L_\infty(\mathbb R^N)\times L_\infty(\mathbb R^N)\to\mathbb R$ given by \[ B_{\mathrm K}(f,g):=\frac{1}{(\sqrt{2}\pi)^N} \iint_{\mathbb R^N\times\mathbb R^N} f(x)g(y)e^{-(\lVert x\rVert^2+\lVert y\rVert^2)/2} \sin\langle x,y\rangle\,\mathrm d x\,\mathrm d y, \] We define the König constant by \[ \mathfrak K_{\mathrm K}:=\sup_{N\geq1}\sup_{\substack{f,g:\mathbb R^N\to\{\pm1\}\\ f,g\ \mathrm{measurable}}}B_{\mathrm K}(f,g). \] The study of this bilinear form arose from efforts to determine the exact value of the Grothendieck constant. König~\cite{KONIG} conjectured that the sharp value should instead be given by the one-dimensional half-spaces $B_{\mathrm K}(\operatorname{sgn}(x_1),\operatorname{sgn}(x_1))=\frac{2}{\pi}\log(1+\sqrt{2})$. A positive answer to this conjecture, together with a classical upper bound of Krivine \cite{KRIVINE}, would determine the exact value of the Grothendieck constant. In a breakthrough~\cite{BMMN}, Braverman, Makarychev, Makarychev, and Naor disproved König's conjecture already in dimension two and used their counterexamples to obtain the first strict improvement over Krivine's bound. One question in \cite{BMMN} attempts to determine the Grothendieck constant through alternating Krivine rounding schemes arising from König's bilinear form in high dimension. More recently, Li et al.~\cite{LISK} constructed high-dimensional examples showing that $\mathfrak K_{\mathrm K}\ge 0.59357$. An elementary Fourier argument gives $\mathfrak K_{\mathrm K}\le 1$ and excludes equality for every finite-dimension. In this paper, we prove that $\mathfrak K_{\mathrm K}=1$ by constructing a family of Boolean pairs in high dimensions. In particular, this gives a negative answer to the high-dimensional aspect of the question in \cite{BMMN}.

We show that an inductive AH-system with diagonal connecting maps describes an action of the canonical AF-groupoid on the spectrum of the canonical C$^*$-diagonal, and that the canonical groupoid model is given by the transformation groupoid associated to this action. This divides the groupoid structure into two distinct aspects: the acting AF-groupoid (which is well-studied) and the unit space on which it acts. We describe the unit space by enriching the Bratteli diagram with extra topological information, and apply these descriptions to examples of interest, including Villadsen algebras of the first kind.

We give a deterministic proof of Bourgain's slicing theorem. The proof uses neither stochastic localization nor stochastic calculus, does not rely on Guan's covariance bound, and avoids $M$-ellipsoids, Pisier's regularization machinery, and complex interpolation. Instead, the argument is based on a deterministic one-parameter quadratic perturbation of the measure, a covariance-survival principle, and a geometric analysis of $L_p$-centroid bodies.

We develop an abstract Banach principle in vector lattices and apply it to obtain lattice-theoretic versions of the individual and maximal ergodic theorems, without recourse to any measure representation. Considering a sequence of bounded operators with values in a Dedekind $\sigma$-complete vector lattice endowed with a locally solid topology satisfying the $\sigma$-Lebesgue property, we prove that the set of points at which the sequence is order convergent is a closed subspace and coincides with the whole space whenever convergence holds on a dense subset. Investigating positive, power-bounded, mean ergodic operators on order continuous Banach lattices, we construct a topology on the universal completion induced by a strictly positive order continuous functional and prove that the Cesàro means converge in order in the universal completion and, in particular, uo-converge in the original lattice. Moreover, we introduce the notion of a superinvariant pair and derive a lattice-theoretic Hopf inequality together with a weak type estimate via band projections, which yields an abstract maximal ergodic theorem. A spectral-theoretic version of the theorem follows from classical Perron–Frobenius theory. Finally, we specialise the abstract framework to the model space $L^0(\Omega)$ and revisit the classical Banach principle, the Hopf–Dunford–Schwartz theorem and Doob's martingale convergence theorem.

We introduce matrix-weighted anisotropic Besov-type and Triebel-Lizorkin-type spaces associated with an expansive matrix $A$. Inspired by the $\A_p$-dimensions of matrix weight of Bu et al. (2025), we study the properties of matrix weight associated with $A$. Using the nice properties of matrix weight, we obtain that these spaces are equivalent with their corresponding averaging spaces and establish the discrete $\varphi$-transform of these spaces. Finally, we introduce matrix-weighted anisotropic Triebel-Lizorkin spaces for the limiting case $p=\infty$ and obtain their $\varphi$-transform. The relation between matrix-weighted anisotropic Triebel-Lizorkin spaces and matrix-weighted anisotropic Besov-type and Triebel-Lizorkin-type spaces is also studied.

We complete the construction of tracial spectral nonassociative $\mathrm{L}^p$-spaces for general $\mathrm{JBW}^*$-algebras. More precisely, if $\mathcal{M}$ is a $\mathrm{JBW}^*$-algebra equipped with a normal finite faithful trace $\tau$ and $1 \leq p < \infty$, we prove that $\|x\|_{\mathrm{L}^p(\mathcal{M})} \overset{\mathrm{def}}{=} (\tau[(x^* \circ x)^{\frac p2}])^{\frac1p}$, where $x \in \mathcal{M}$, defines a norm on $\mathcal{M}$. This resolves the remaining exceptional case left open by the corresponding result for $\mathrm{JW}^*$-algebras. The main difficulty is the complexified Albert algebra $\mathrm{H}_3(\mathbb{O}_{\mathbb{C}})$, which admits no embedding into an associative operator algebra. To treat this case, we establish a Jordan analogue of the joint convexity of the Kiefer map $\mathrm{M}_n \times \mathrm{H}_n^{++} \to \mathrm{H}_n^{+}$, $(a,h) \mapsto a^*h^{-1}a$, where $\mathrm{H}_n$ is the space of Hermitian matrices, and a Jordan version of a variational formula of Carlen and Lieb. This provides a complex Banach space framework for Jordan-algebraic models arising in some generalized probabilistic theories.

For $d \geq 2$, $p \geq 1$ and $\epsilon > 0$, let $N_p(d,\epsilon)$ be the smallest integer $N$ such that for every integer $n$ and every $A\in\mathbb{R}^{n\times d}$, there exists a matrix $\Phi\in\mathbb{R}^{N\times n}$ satisfying $(1-\epsilon)\lVert Ax\rVert_p\leq \lVert\Phi A x\rVert_p\leq (1+\epsilon)\lVert Ax\rVert_p$ for all $x\in\mathbb{R}^d$. For every constant $p\geq 1$ with $p\not\in 2\mathbb{Z}$, when $d\gtrsim_p \log(1/\epsilon)$, the bound \[ N_p(d,\epsilon) \gtrsim_{p} \frac{d}{\epsilon^2 \operatorname{polylog}(d/\epsilon)} \] is established. This improves the previous lower bound $\Omega(1/(\epsilon^2\operatorname{polylog}(1/\epsilon)))$ due to Li et al. (SICOMP 2021) and is optimal up to logarithmic factors for $1\leq p<2$. The central technical idea originated from ChatGPT 5.6 Sol.

We investigate proper actions of arbitrary semigroups on separable locally compact metric spaces, where point orbits are allowed to escape to infinity. An action is pointwise uniquely ergodic when every compact orbit closure supports exactly one invariant probability measure and non-compact orbit closures support none. The associated ergodic map therefore assigns the selected probability measure to non-escaping points and the zero subprobability to escaping ones. Under the hypothesis that compact orbit closures admit at least one invariant measure, we establish that the weak* continuity of this ergodic map together with a vanishing at infinity condition is equivalent to the mean ergodicity of the Koopman representation on the space of continuous functions vanishing at infinity. In consequence, every such function and every finite signed measure split uniquely into invariant components and limits of coboundaries. The corresponding projections are obtained by integration against the ergodic map. Because this operator-theoretic characterization avoids explicit averaging schemes, it remains applicable even to semigroups without Følner sequences. When restricted to countable, discrete, bicancellative, and left amenable semigroups, these properties are shown to be equivalent to the uniform convergence of Følner averages and the weak-star continuity of their dual limits, extending classical results for group actions on compact spaces. Furthermore, we identify the space of ergodic measures with a compactified ergodic quotient, prove that the invariant measure simplex is Bauer, and show that these structural properties descend through proper factor maps. The theoretical framework is complemented by dynamical examples, including a continuously pointwise ergodic subshift that exhibits discontinuous entropy along the ergodic map.