Papers

Let $L_v=-\Delta-2v\cdot\nabla$ on $L^2(\mathbb R^n,e^{2v\cdot x}\,dx)$, where $v\in\mathbb R^n\setminus\{0\}$ is a constant drift vector. We prove that the full vector Riesz transform $\nabla L_v^{-1/2}$ satisfies a weak-type $(1,1)$ estimate for real-valued functions with constant at most $2$, uniformly in both the dimension and the drift vector. Our proof extends the fractional obstacle method of Ouyang, Spector and Stockdale (https://arxiv.org/abs/2608.18068) to the exponentially weighted setting. Interpolation then yields strong $L^p$ bounds for $1<p\le 2$, with constants depending only on $p$.

In this paper, we establish a quantum Cheeger inequality for primitive KMS-symmetric quantum Markov semigroups in terms of projection conductance. We discuss both projection conductance and classical conductance for graph-based KMS-symmetric quantum Markov semigroups. We show that hypercontractivity and the logarithmic Sobolev inequality hold for primitive KMS-symmetric quantum Markov semigroups. We also present applications of the quantum Cheeger inequality to logarithmic Sobolev inequalities, hypercontractivity, and complete modified logarithmic Sobolev inequalities.

Let $A$ and $B$ be $C^*$-algebras, not necessarily unital, and let $T:A_+\to B_+$ be a positively homogeneous order isomorphism. We first prove that $T$ is additive and extends uniquely to a bounded real-linear order isomorphism between the self-adjoint parts of $A$ and $B$. Passing to the biduals, we then obtain a representation theorem for positively homogeneous order isomorphisms. Finally, we study surjective maps between the positive cones of $C^*$-algebras, again not necessarily unital, which preserve the norm of the arithmetic mean. This extends a previous result by removing the assumption that at least one of the algebras is unital.

Let $X_{\mathsf{KW}}$ be the Kadets–Werner modification of a closed subspace of $L_1$ constructed by Bourgain and Rosenthal in 1980. In this short note, it is shown that $X_{\mathsf{KW}}$ is asymptotically midpoint uniformly convex. Since $X_{\mathsf{KW}}$ has the Daugavet Property, it fails the Point of Continuity Property and thus does not admit an equivalent norm that is asymptotically uniformly convex. Therefore, the previously known properties of $X_{\mathsf{KW}}$ and the new geometric observation answer, in the negative, the question of Dilworth, Kutzarova, Randrianarivony, Revalski and Zhivkov whether asymptotic midpoint uniform convexity and asymptotic uniform convexity are isomorphically equivalent and a question of Perreau whether asymptotic midpoint uniform convexity implies the Point of Continuity Property.

For $N\ge2$, $0<\beta<N$ and $q>1$, we study maximizers of the critical singular anisotropic Moser–Trudinger integral \[ \int_{\mathbb{R}^N} \frac{\Phi_{N,q,\beta} (\lambda_N(1-\beta/N)|u|^{N/(N-1)})} {F^o(x)^\beta}\,\mathrm{d}x, \qquad \|F(\nabla u)\|_N^a+\|u\|_q^b\le 1. \] where $a>0$, $0<b\le N$ and $\Phi_{N,q,\beta}$ is the integrable Taylor remainder. The supremum is attained for every $a>0$ when $b<N$. For $b=N$ and $1<q<q_+$, the first retained Taylor term separates two regimes, with \[ q_-:=\frac{N^2(N-2)}{(N-1)(N-\beta)},\qquad q_+:=\frac{N^2}{N-\beta}. \] Attainment holds for every $a>0$ if $1<q<q_-$, while for $q>1$, $q_-\le q<q_+$, there is a finite threshold $a_c>N$ and attainment holds if and only if $0<a\le a_c$. An exact Euclidean reduction shows that the threshold is independent of the anisotropy. The proof combines critical–subcritical scaling with a nonlinear Green-function concentration bound. Radial flux and Pohozaev identities determine the first correction to the subcritical supremum, and refined Green tests give the strict comparison needed for attainment at the finite threshold, including in dimension two.

Zavarzina asked whether every bijective non-contractive self-map of a precompact quasi-metric space is an isometry. We answer the question in the negative, in a stronger form: on a hereditarily precompact quasi-metric space (a tolled extension of the irrational rotation over Z), the shift is a bijective non-contractive map that is not an isometry. The construction generalizes: for every metric space Y, every bijective isometry $\sigma$ of Y, and every point whose $\sigma$-orbit is aperiodic and totally bounded, the tolled Z-extension is such a counterexample; instances include the irrational rotation and the p-adic odometer. The conjugate space is not precompact, so the two-sided hypothesis of Zavarzina's positive result is sharp, and the space is not expand-contract plastic. The core results, including the general scheme and the circle instance, are machine-checked in Lean 4 over mathlib, sorry-free (no unproved steps), under a pinned toolchain. A structural section proves a monochromy theorem: for every strict pair on a hereditarily precompact space, one endpoint's backward self-returns have self-defect at least a quarter of the pair's defect, uniformly on a chain-closed displacement set. Boundary results delimit what remains open: one-sided compactness together with separation.

Unlike ordinary time, a Stieltjes clock may advance continuously, remain constant over intervals, or jump. For an ordinary continuous clock, Hölder regularity with exponent $\alpha>1$ forces constancy, whereas jumps of a Stieltjes clock may allow nonconstant behavior. We refer to Hölder regularity of exponent $\alpha>1$ measured relative to g as super-g-Hölder regularity, and show that, in finite dimensions, such functions admit an atomic representation determined entirely by their jumps, with their g-Hölder seminorms given exactly by the corresponding normalized jump sizes. This atomic representation yields a linear isometric description of the associated function space, together with its Banach and separability properties, an exact total-variation formula with an optimal bound, and finite-jump approximation results. We then apply this structure to Stieltjes differential equations. Nonzero state jumps require positive clock jumps, leading to quantitative finite-jump bounds and, for affine dynamics, exact thresholds for the number of jumps and extremal terminal distances under a prescribed total Stieltjes mass. Finally, we study recovery of the Stieltjes clock from observed state jumps. For known single-valued dynamics, clock jumps can be recovered under a natural local identifiability condition, while for nonconvex differential inclusions the super-g-Hölder bound can reduce, and in some cases remove, nonuniqueness.

An important result due to Radjabalipour asserts that if ${A\in\BB(\H)}$ is an invertible positive operator and ${T\in\BB(\H)}$ is arbitrary, then the operator sequence $\{A^{-n}TA^n\}_{n\in\ZZ}$ is norm-bounded if and only if ${TA=AT}.$ Using the concept of spectral gaps, we show that if $A$ possesses a spectral gap and the sequence $\{A^{-n}TA^n\}_{n\in\ZZ}$ is norm-bounded, then $T$ must be block diagonal with respect to a certain orthogonal decomposition of $\H$. We also provide an alternative proof of Radjabalipour's theorem. We also extend the theorem to some rich classes of operators, including invertible normal operators and invertible weighted sums of projections. Moreover, we provide simple proofs for the fact that a positive definite matrix ${A\in\MM_N}$ is a positive multiple of the identity if and only if the sequence $\{A^{-n}TA^n\}_{n\in\ZZ}$ is norm-bounded for every~${T\in\MM_N}.$

We introduce a unified framework for supervised and unsupervised learning through density estimation using reproducing kernel Hilbert spaces (RKHSs). To this end, we introduce the notion of a nested RKHS, that is, an increasing sequence of RKHSs whose union is dense in an ambient $L^2$-space. This construction allows us to exploit the computational structure of RKHS methods at each finite approximation level while preserving the approximation power of the entire $L^2$-space. We first study statistical learning in a fixed RKHS. In both unsupervised and supervised settings, we establish strong convergence results for empirical averages of random elements taking values in an RKHS continuously embedded in $L^2$. We then use the nested RKHS structure to combine these statistical convergence results with the density of the union of the RKHSs in the ambient $L^2$-space. We apply this construction to probability density estimation and binary Bayesian classification. For density estimation, we construct explicit empirical estimators of the orthogonal projections of the target density onto the RKHSs and establish their almost sure convergence. A diagonal selection procedure then yields a sequence of estimators converging almost surely to the target density in $L^2$. For binary Bayesian classification, we formulate the approximation of the Bayes target as the minimization of suitable population and empirical risk functionals. We prove consistency of the empirical minimizers and use the nested structure to obtain convergence towards the Bayes target in $L^2$. We further provide a geometric interpretation in terms of hyperplanes and weighted distances in the ambient $L^2$-space. Finally, we show that the population risk admits a direct probabilistic interpretation in terms of the probability of misclassification.

We prove new small-ball inequalities for boolean matrix-series. The leading example is $\mathbb E_s[{\text{det}(I-S^2)^\beta\,\mathbf 1_{\{\|S\|<1\}}}]\ge e^{-O(\beta\tau)}$, which holds for boolean matrix-series $S=\sum_i s_iA_i$ formed using symmetric matrices $A_1,\dots,A_n$ and uniformly random signs $s\in\{\pm1\}^n$. Specifically, this inequality holds for all $\beta\ge1$ with $\tau=\sum_i\text{Tr} A_i^2$, as soon as the maximum of $(\text{Tr} A_i^2)_{i=1}^n$ and a certain variance term are bounded above by universal constants. The proof combines the Gaussian reciprocal estimate of (Akbas and Sra 2026), the directional-variation signing theorem of (Guo, Fang, and Lu 2026), and a replica argument that turns existence into a Gibbs law on good signings. Most notably, boolean small-ball delivers a new, interlacing-free proof of Kadison-Singer (most general case); it also recovers Matrix Spencer and Komlós as quick corollaries, while yielding more than six almost immediate proofs of an assortment of discrepancy theoretic problems.

We prove that hypercontractivity implies entropy contraction for a single quantum channel, without a detailed balance condition. For primitive quantum Markov semigroups that are KMS-symmetric with respect to a faithful invariant state \(\sigma\), we obtain the modified Log-Sobolev bound $\alpha_1\geq \frac{\lambda}{(2+\log\|\sigma^{-1}\|_\infty)}$ where $\lambda$ is the spectral gap. This removes the assumption of \(L_p\)-regularity for the comparison through the log-Sobolev constant. As an application, we show that the hypercontractivity of an average of two conditional expectations implies the approximate tensorization of relative entropy.

We revisit the analysis of Bayesian convergence to the truth under finite additivity in a recent paper by Nielsen (J. Philos. Logic, 2021, doi:10.1007/s10992-020-09569-2) Its principal theorem proves that the posteriors of a probability function converge to the truth almost uniformly if and only if the function has two properties: an approximation property, and so-called countable additivity on conditional hitting times. We show that the two properties are necessary but not sufficient, so that the theorem is false, and we locate the error in its published proof. We construct a merely finitely additive probability function that has both properties and whose posteriors converge to the truth almost surely but not almost uniformly. Three of the paper's four remaining theorems, and its corollary, lose their published proofs as well, two with the failed implication and two to a separate defect that we also identify. Two of the four results we reprove and one we leave undecided; the last is the corollary, which our counterexample does not refute, and which we establish for a family including the counterexample and leave open in general. We also show that almost-sure convergence of posteriors to the truth for every event does not characterize countable additivity. We close with what a repaired characterization of almost-uniform convergence would have to add.

We prove that every compact operator $T$ on a separable infinite-dimensional complex Hilbert space is a commutator of two compact operators, thereby answering a question of Pearcy and Topping. Moreover, the compact factors $A$ and $B$ can be chosen such that $[A, B] = T$ and $\max\{\Vert{}A\Vert{},\Vert{}B\Vert{}\}\leq c\Vert{}T\Vert{}^{1/2}$ for a universal constant $c$.

We study Fourier extension from compact convex $C^2$ planar curves by organizing the mass $|f|^2\,\mathrm d\sigma$ through the Gauss map. The pushforward measure \[ \nu_f= N_\#\bigl(|f|^2\,\mathrm d\sigma\bigr) \] records its distribution in normal directions. The turning measure $\mathrm d\mu_\kappa=\kappa\,\mathrm d\sigma$ determines an intrinsic terminal tangential scale $r_R(\xi)$ through \[ r_R(\xi)\, \mu_\kappa\bigl(B_\Gamma(\xi,r_R(\xi))\bigr) \asymp R^{-1}, \] and hence a position-dependent angular resolution $\rho_R(\xi)=R^{-1}/r_R(\xi)$. Under a doubling hypothesis on $\mu_\kappa$, $r_R$ is, up to structural constants, the largest scale on which the curve can be linearized to precision $R^{-1}$. These scales define a normal-direction energy, and the associated terminal decomposition and transverse geometry yield local $L^4$ Fourier extension estimates and weighted variants. For the monomial curves $\gamma_k(t)=(t,t^k)$, $k\geq3$ real, the terminal scales are explicit and the energy admits a multiscale representation in terms of angular correlations of $\nu_f$. Under an $s$-dimensional Frostman condition on $\nu_f$, this yields a growth diagram with critical threshold \[ s_c(k)=\frac{k-2}{3k-4}, \] separating the flat-point and nondegenerate regimes. The resulting rates, including the critical logarithmic correction, are sharp at the energy level.

We answer Bauer and Le's question on zero products of Toeplitz operators on the Fock space $F^2(\mathbb C^n)$[JFA, 261 (2011), 9, 2617–2640]. For $n\ge2$, we construct two bounded nonradial Schwartz symbols on $\mathbb C^n$ whose Toeplitz operators are nonzero and have zero product on $F^2(\mathbb C^n)$. For $n=1$ and each $c\in(1/2,1)$, we construct two smooth nonradial symbols of growth at most $Ce^{c|z|^2}$ for some constant $C>0$. Their extended Toeplitz operators in $F^2(\mathbb C)$ are nonzero and have zero product on all holomorphic polynomials. Moreover, the second symbol is bounded when $c\geq3/4$. Our proofs use Gaussian kernel calculations, matrix identities, theta functions and Fourier transform.

We prove mixed $A_p$-$A_\infty$ estimates for sparse operators in two non-doubling settings. In the first setting, we consider sparse operators defined using stopping times in continuous time. We obtain both strong- and weak-type bounds with the same powers of the weight characteristics as in the classical setting. Some of our weak-type bounds are even new for the dyadic filtration on $\mathbb R^d$ and, in particular, imply a sharp weak-type $(2,2)$ estimate for Rubio de Francia square functions, solving a problem left open by Garg, Roncal and Shrivastava [J. Geom. Anal., 31:748-771, 2021]. In the second setting, we consider dyadic sparse forms in which distinct cubes may interact, provided their dyadic distance is bounded. We obtain strong-type bounds with the same powers of the weight characteristics as in the classical setting. As a one-dimensional application, we obtain strong-type bounds for Haar shifts over balanced non-doubling measures, answering a quantitative question posed by Conde-Alonso, Pipher, and Wagner [Math. Ann., 391:2209-2253, 2025].

We study best modular approximation in Musielak-Orlicz spaces. A geometric classification near the origin of Orlicz functions is introduced to isolate weighted $L^1$-like behavior. We establish existence results under minimal hypotheses and provide a variational characterization of minimizers via lateral derivatives. Finally, we establish uniqueness criteria, including a one-dimensional Haar-type theorem and characterizations in $0$- and $1$-spaces in higher dimensions.

We study a variable-exponent double-phase Neumann problem on a domain in a metric measure space, where the boundary measure satisfies an upper codimension-\(\theta\) bound for \(0\le\theta<1\). Under a local comparability condition on the double-phase growth function, we establish a modular estimate for a fractional maximal operator and use it to prove the existence and boundedness of the associated trace operator. We then establish coercivity of the associated energy functional on the normalized double-phase Newtonian space and, under reflexivity, prove the existence of minimizers. We further show that the set of minimizers is closed and convex, and that any two minimizers have the same minimal weak upper gradient almost everywhere. Finally, we prove the stability of minimizers under perturbations of the Neumann data.

For the Dirichlet-type space $\mathcal D(\rho^{(1)},\rho^{(2)})$ on the unit bidisc $\mathbb D^2,$ where $\rho^{(1)}$ and $\rho^{(2)}$ are finite positive Borel measures on the closed unit disc $\overline{\mathbb D},$ we show that the Lévy measure $\nu_{(\mathscr M_z,1)}$ associated with the completely alternating multisequence $\left\{\|z^\alpha\|_{\mathcal D(\rho^{(1)},\rho^{(2)})}^2 \right\}_{\alpha\in\mathbb Z_+^2}$ admits the explicit representation \begin{equation*} d\nu_{(\mathscr M_z,1)}(x)=\frac{1}{1-x_1}d((S_{*}\rho^{(1)})\times\delta_1)(x)+\frac{1}{1-x_2}d(\delta_1\times (S_{*}\rho^{(2)}))(x) \end{equation*} on $[0,1]^2\backslash\{(1,1)\},$ where $S_{*}\rho^{(i)},$ $i=1,2$, denotes the pushforward measure of $\rho^{(i)}$ by the map $S:\overline{\mathbb D}\rightarrow [0,1]$ defined by $S(z)=|z|^2,$ $z\in \overline{\mathbb D},$ and $\delta_1$ denotes the Dirac measure at 1.

We prove that every traceless real or complex matrix $A$ can be written as $A=BC-CB$, with factors of the same size and over the same field satisfying $\|B\|\,\|C\|\le K\|A\|$, where $K$ is an absolute constant. The proof combines an approximate-rank dichotomy with stable commutator representations obtained from quantum expansion.

We solve a problem posed by Audenaert and Kittaneh in 2012 by proving that the Aujla–Bourin subadditivity inequality can be strengthened to an optimal form. We also answer questions posed by Bourin and the second author: (1) we significantly extend both the reciprocal Lie–Trotter theorem of Audenaert and Hiai and Kato's limit theorem; (2) we determine the best constants in inequalities for sums of contractions and positive block matrices, including the sharp triangle inequality $$ |A+B+C| \leq \frac{3}{4} I + |A|+|B|+|C| $$ for three contractions $A,B,C$; and (3) we prove a remarkable eigenvalue inequality for the symmetric modulus, thereby determining the best dimension-free lower bound in a question studied by Bourin, Lee, and Zhang.

We give a computer-assisted proof that the universal real Grothendieck constant satisfies $K_G^{\mathbb R}\le 1.7813$. The construction combines an explicit odd Hermite threshold of degree $11$ with a signed correlation polynomial of degree $51$. A sufficient inverse-majorant inequality is certified by enclosing the scalar coefficient head through degree $301$ and bounding the entire remaining tail using a weighted Gaussian trace estimate. The finite integral is bounded on a complete interval partition; its spatial exterior is controlled analytically. Exact parameters and source code that regenerate all accepted numerical inputs accompany the paper. The bound improves both the explicit bound $1.7818666069360661$ in the recent literature and the subsequently reported, system-tested value $1.7813319810625639$.

We consider a general, nonsymmetric Ornstein–Uhlenbeck semigroup $(\mathcal H_t)_{t>0}$. We prove an $L^p$ bound for the jump quasi-seminorms for $1 < p < \infty$ and a weak type (1,1) oscillation inequality, both with respect to the invariant measure. These results are established for the order $\varrho=2$. To do so, we analyze specific components of $(\mathcal H_t)_{t>0}$, by distinguishing between small and large values of $t$, and between local and global spatial zones. This decomposition allows us to explicitly identify which parts of the semigroup remain bounded and which are responsible for the failure of boundedness, both in a weak and in a strong sense, even with respect to Lebesgue measure.

We study the reciprocal problem for weighted Bergman spaces: if $f\in A_\alpha^p(\Bn)$ and $\inf_{\Bn}|f|>0$, does it follow that $1/f\in A_\alpha^p(\Bn)$? We determine the range of parameters for which the answer is affirmative. In particular, we prove that functions in $A_{\alpha}^p(\D)$ have the reciprocal property for all $\alpha \in \R$ and $p\geq 1,$ where $\D$ denotes the unit disk in $\C.$ Moreover, functions in $A_{\alpha}^p(\Bn)$ have the reciprocal property for all $\alpha \in \R$ when $n=2$ and $p=2.$ In addition, we resolve the reciprocal problem in the three-dimensional Drury–Arveson space $H_3^2$ and obtain an equivalent condition in the four-dimensional space $H_4^2$. We also settle the reciprocal problem for general $A_{\alpha}^p(\Bn)$ under some additional conditions.

Talagrand asked whether the Rademacher cotype of an operator is universally controlled by the maximum of its Gaussian cotype and its $(q,1)$-summing norm. We give a negative answer already for $q=2$. The counterexample was discovered by ChatGPT (GPT-5.6).

Let $\mu$ be a finite positive Borel measure on $[0,1)$ and let $\gamma>0$. We establish sharp mapping criteria for the generalized Cesàro operator \begin{equation*} \mathcal C_{\mu,\gamma}f(z) =\sum_{n=0}^\infty \mu_n \left(\sum_{k=0}^n \frac{\Gamma(n-k+\gamma)}{\Gamma(\gamma)(n-k)!}a_k\right)z^n,\qquad z\in \mathbb{D},\end{equation*} between Hardy spaces, including the source and target $H^\infty$ endpoints. For $0<p<q<\infty$, for $0<p=q<1$, and for $0<p\le1$ with $q=\infty$, the boundedness of $ \mathcal C_{\mu,\gamma}: H^p \to H^q$ is equivalent to $\mu$ being a $(\gamma+1/p-1/q)$-Carleson measure, without further restrictions on $\gamma$. For $1\le q<p\le\infty$, set $1/r=1/q-1/p$. In this range, boundedness and compactness are equivalent to \[ \int_0^1 \left(\frac{\mu([t,1))}{(1-t)^{\gamma-1/r}}\right)^r \frac{dt}{1-t}<\infty. \] This condition is also equivalent to $F_{\mu,\gamma}\in H^r$ and to $\sum_{n\ge0}(n+1)^{r\gamma-2}\mu_n^r<\infty$, where $F_{\mu,\alpha}=\mathcal C_{\mu,\alpha}(1)$ and $\mu_n=\int_{[0,1)}t^n\,d\mu(t)$. For $1<p<\infty$, boundedness and compactness from $H^p$ to $H^\infty$ are characterized by the shifted condition $F_{\mu,\gamma+1}\in H^{p'}$, where $p'=p/(p-1)$. We therefore obtain a complete boundedness classification of the generalized Cesàro operators $\mathcal C_{\mu,\gamma}$ between Hardy spaces $H^p$ and $H^q$ for the full range $0<p,q\le\infty$.

We prove that every surjective Fischer–Muszély map between positive cones of arbitrary JB-algebras is additive and positively homogeneous, and extends uniquely to a bounded positive linear surjection. In particular, this gives a complete affirmative answer to Molnár's problem for arbitrary $C^*$-algebras. The theorem includes nonunital and exceptional JB-algebras and assumes neither injectivity nor continuity. The proof connects the norm-valued functional equation to positive-cone rigidity through translation invariance of an induced pseudometric. For a surjective map satisfying the Fischer–Muszély identity with uniform error $\varepsilon$, we also construct a unique bounded positive linear map $L$ at uniform distance at most $3\varepsilon$. The set $L(A_+)$ is dense in $B_+$, although $L$ need not be surjective, even when the original map is continuous and bijective. For bijective FM maps, we obtain weighted Jordan representations on the biduals. We also extend the characterization of surjective norm-sum preservers to JB-algebras and characterize norm arithmetic-midpoint and harmonic-mean identities on positive invertible cones of unital JB-algebras.

We study the Friedrichs Laplacian associated with the pullback of the round metric on \(\mb P^1\) by a nonconstant meromorphic function \(\varphi:X\to\mb P^1\) on a compact Riemann surface. For arbitrary ramification profiles, including several ramification points over the same branch value, we prove the local formula \[ \operatorname{Det}_{\zeta}(\Delta_{[\varphi],\mc F}) =C\,\det\operatorname{Im}B\,|\tau_B|^2 \prod_{k=1}^N\rho(z_k,\overline{z_k})^{c_k}. \] The zero eigenvalue is omitted. Here \(B\) is the period matrix, \(\tau_B\) is the local Bergman tau-function, \(z_k\) are the branch-value coordinates, \(\rho(z,\overline z)=4(1+|z|^2)^{-2}\), and \(c_k=\frac1{12}\sum_j(n_{kj}-n_{kj}^{-1})\), where \(n_{kj}\) are the ramification indices over the \(k\)-th branch value. The constant \(C>0\) is independent of the Hurwitz coordinates, and \(\det\operatorname{Im}B\) is taken to be \(1\) in genus zero. The proof combines smooth trivializations and trace-norm variation of resolvent powers with matrix comparison to spherical conic models. The Davies–Gaffney estimate provides the required uniform high-energy control, while the zero-energy terms are identified through the Schiffer bidifferential and Rauch variational formulas.

In this article, we give various real-variable properties of local Muckenhoupt matrix weights $W$ and establish the quantitative boundedness of several operators on local matrix-weighted Lebesgue spaces $L^p(W)$, including local (fractional) maximal operators, local fractional integral operators, local Haar square functions, and Calderón–Zygmund operators with exponential decay. For the local maximal operators, we obtain the sharp quantitative bounds when $p\in(1,2]$, while, for local fractional integral operators, we obtain the quantitative bounds matching the global best known exponents, whose scalar case is known to be sharp. The key used strategies include giving a new extension property (which can clarify their relationships with global ones) and an optimal scale lifting property (which can balance the locality of weights and operators under consideration) of local matrix weights. As an application, we establish the quantitative boundedness on $L^p(W)$ of the Riesz transform associated with Schrödinger operators $-\Delta+m^2I$ with $m\in(0,\infty)$ being large enough, whose quantitative bound when $p=2$ is precisely $[W]_{\mathscr{A}^{\operatorname{loc}}_{2}(r)}^{\frac 32}$.

Wasserstein barycenters provide a notion of weighted mean for probability measures on a metric space. We prove that the barycenter curvature-dimension condition ${\rm BCD}(K,N)$ is equivalent to ${\rm RCD}(K,N)$ for $K\in \mathbb R$ and $1<N<\infty$. This gives a new characterization of Riemannian curvature-dimension spaces by entropy inequalities at barycenters of finite families of measures. As a byproduct, we introduce an almost ${\rm BCD}$ condition that allows an additive error in the entropy inequality. It is stable under measured Gromov-Hausdorff convergence and implies essential non-branching when the error is sufficiently small. The resulting class contains non-Riemannian Finsler spaces. Within the ${\rm BCD}$ framework, this answers an open problem posed by Ambrosio in his 2018 ICM survey. We further bound the failure of the parallelogram identity for cotangent norms in terms of the entropy error.