Papers

We introduce the generalized weighted core inverse with respect to two elements in a Banach *-algebra, extending the framework of existing generalized inverses. This new generalized inverse is characterized via the generalized weighted core decomposition, and its representations are established using the weighted g-Drazin inverse. We investigate the polar-like and principal ideal properties to reveal its structural behavior. Consequently, applications to complex matrices are derived. This finds encompasses both the core-EP and generalized core inverses, significantly broadening the theoretical framework.

We introduce an intrinsic spectral sparsity model for nonparametric density estimation on compact connected Riemannian manifolds. Instead of penalizing coefficients in an arbitrarily chosen Laplace–Beltrami eigenbasis, we group each complete eigenspace and measure the Hilbert norm of its spectral component. The resulting block-variation space is basis independent and isometry invariant. We establish its structural, atomic, and nonlinear approximation properties and clarify its relation to Sobolev, Besov, and coefficientwise spectral $\ell^1$ classes. We then construct a coordinate-free block-shrinkage estimator and prove a nonasymptotic signal-dependent $L^2$-oracle inequality that adapts to the unknown set of detectable eigenspaces. Under polynomial spectral growth, the risk theory separates the number of spectral blocks from their multiplicities and exhibits two regimes: one driven by a single high-dimensional eigenspace and the other by cumulative spectral complexity. Under matching spectral-growth and nondegeneracy assumptions, corresponding minimax lower bounds show that this multiplicity dependence is intrinsic, with sharp consequences for spheres and the rotation group $SO(3)$. Finally, we develop a positive, normalized, block-penalized exponential spectral sieve for log-densities and derive likelihood oracle inequalities together with expected Kullback–Leibler, Hellinger, and $L^2$ risk bounds. The resulting framework provides a geometry-respecting theory of sparse density estimation that remains invariant under changes of eigenbasis.

We disprove the conjecture of Gau, Wang and Wu that the numerical range of a finite-dimensional partial isometry, when circular, must be centred at the origin. More precisely, we prove that there is an \(\varepsilon>0\) such that, for every \(a\in(0,\varepsilon)\), one can find a rank-four partial isometry \(V_a\in\M_6(\R)\) satisfying $$ W(V_a)=\{\zeta\in\C:|\zeta-a|\leq 3/4\}. $$

An analog of the classical Kronecker-Weyl theorem for weak uniform distribution is obtained for an arbitrary countably infinite independent subset of a discrete abelian group using a type of interpolation sets called epsilon-Kronecker sets. Specifically, the topological size of the set of homomorphisms inducing weakly uniformly distributed sequences is established by showing that it forms a dense G-delta set.

We establish a sharp version of Grothendieck's theorem for Bessel sequences. Precisely, given a Bessel sequence $\{ x_j \}_{j\in\mathbb{N}}$ with Bessel bound $1$ in a Hilbert space, we show that there exists functions $\{ f_j \}_{j\in\mathbb{N}}$ belonging to the unit ball of $L^\infty([0,1])$ such that for all $j,k \in \mathbb{N}$ one has $$ \langle x_j,x_k\rangle = \int_0^1 f_j(x)\overline{f_k(x)}\,dx.$$ As an application, we give an affirmative answer to an extension problem of Olevskii: if $E \subset [0,1]$ is a Lebesgue measurable set such that $[0,1]\setminus E$ has positive measure, then every Bessel sequence in $L^2(E)$ with Bessel bound $1$ extends to an orthonormal system in $L^2([0,1])$ that is bounded by the (optimal) constant $\lambda([0,1]\setminus E)^{-1/2}$ on $[0,1]\setminus E$. A formalization of our main result in Lean 4 accompanies the paper.

Let $\mathcal{P}_{N}$ be the $(N+1)$-dimensional Hilbert space of analytic polynomials of degree at most $N$. This is the natural environment to define $SU(2)$ (Bloch) coherent states. Let $Q_{\rho }$ be the Husimi function of a density operator $\rho $ on $% \mathcal{P}_{N}$. We prove an isospectral version of Lieb-Solovej inequality: if $\rho ^{\downarrow }$ is obtained by placing the eigenvalues of $\rho $ in decreasing order along the monomial basis, then \begin{equation*} \int_{\mathbb{C}}\Phi (Q_{\rho }(z))\,dm(z)\leq \int_{\mathbb{C}}\Phi (Q_{\rho ^{\downarrow }}(z))\,dm(z) \end{equation*}% for every convex function $\Phi $ on $[0,1]$. Applying the corresponding reversed inequality to the concave function $\Phi (t)=-t\log t$ gives the Wehrl entropy. In the process it is show that the output state of $\rho $ under Lieb-Solovej's channel is majorized by the output of the state $\rho ^{\downarrow }$. As an application, among all measurable subsets of the sphere having a fixed area, spherical caps maximize every partial sum of the eigenvalues of Toeplitz operators with symbol $1_{\Omega}$. Equivalently, caps maximize all Ky Fan norms, leading to the sharp inequalities, previously known for $r=1$: \begin{equation*} \sum_{j=1}^{r}\lambda_{j}(\Omega) \leq r-\sum_{k=0}^{r-1}(r-k)\binom{N+1}{k} m(\Omega)^{k}\bigl(1-m(\Omega)\bigr)^{N+1-k}. \end{equation*} This implies isoperimetric inequalities for all Schatten sums of $T_{\Omega}$, obtained without using the spherical isoperimetric inequality.

We prove that the trigonometric system $\mathcal{E}_N = \{e_k\}_{|k| \le N}$ is not rigid in $L_p(\mathbb{T})$ for $1 \le p < 2$. That is, as $N\to\infty$, all its elements are approximated with error $o(1)$ by a subspace of dimension $o(N)$.

For the first time, we establish higher integrability of the gradient and local $L^\infty$ estimates of weak solutions to the $p(x)$-Laplacian without assuming $\log$-Hölder continuity of $p$. Instead, we establish these results for exponents satisfying a generalized Muckenhoupt condition. This admits discontinuous exponents acting as pointwise multipliers of the BMO class. Our framework bridges the gap between classical $p(x)$-regularity and weighted Muckenhoupt theory, providing a new foundation for the analysis of differential equations with highly irregular non-standard growth.

Given a countable discrete group equipped with a proper length function, one can construct a natural spectral triple on its reduced group C$^{\ast}$-algebra. A well-studied question in non-commutative metric geometry is whether the Connes pseudo-metric associated with such a triple recovers the weak$^{\ast}$-topology on the state space, thereby yielding a compact quantum metric space in the sense of Rieffel. While this metric property is known to hold for several classes of groups - including those of polynomial growth and word-hyperbolic groups - it was widely expected that not every word-length function induces a compact quantum metric space. Despite this, no explicit counterexample has been identified to date. In this note, we provide the first family of counterexamples by proving that for every integer $d \geq 2$ the canonical spectral triple of the Lamplighter group $(\mathbb{Z}/2\mathbb{Z})\wr\mathbb{F}_{d}$, equipped with the word-length function associated with a finite symmetric generating set, fails to be a spectral metric space.

In this paper, we establish variational formulas and a rearrangement inequality for principal eigenvalues of asymmetric second-order difference operators on periodic lattices. For a general irreducible nearest-neighbor operator, positive right and left eigenvectors give an explicit saddle point and hence equal minimum–maximum and maximum–minimum formulas over positive profiles and periodic logarithmic correctors. The corrector is unique and satisfies a nonlinear discrete flux-conservation law. For exponentially tilted symmetric diffusion, the formula separates the discrete Dirichlet and potential terms from an exact nonlinear periodic correction. In one dimension the flux is constant, and an elementary scalar-flux representation yields a bell-shaped cyclic rearrangement that maximizes the tilted principal eigenvalue for every tilt.

Rosenthal's classical theorem says that, for every infinite compact space $X$, the Banach space $C(X)$ admits a quotient isomorphic to either $c_0$ or $\ell_2$. The corresponding question for $C_p(X)$, the space $C(X)$ endowed with the topology of pointwise convergence, is much subtler and still open; the only compact spaces for which the existence of an infinite-dimensional metrizable quotient is not presently settled in ZFC are Efimov compacta. We prove that the Banach space case cannot be reproduced with the usual sequence spaces carrying their pointwise topologies: Let $X$ be a Tychonoff space and let $E\subseteq c_0$ be a non-trivial symmetric sequence ideal endowed with the subspace topology inherited from $\mathbb{R}^{\mathbb{N}}$. Then the existence of a continuous linear surjection $T:C_p(X)\rightarrow E_p$ implies $E=c_0$, where $E_p$ means $E$ with the topology inherited from $\mathbb{R}^{\mathbb{N}}$. Hence, no proper non-zero symmetric sequence ideal of $c_0$ can be realized as a continuous linear image of a $C_p$-space. Combining this result with the characterization of the Josefson–Nissenzweig property for $C_p(X)$ obtained by Banakh, Kąkol, and Śliwa, we derive a complete characterization of all pairs $(X,E)$ for which such a surjection exists. In particular, for every $0<q<\infty$, there is no continuous linear surjection $C_p(X)\rightarrow(\ell_q)_p$.

Let \( \Lambda=\{\lambda_j\}_{j\in\mathbb N}\subset[0,1) \) be a finite union of Carleson sequences, and let \( \Lambda^p=\{\lambda_j^p\}_{j\in\mathbb N} \) for \(p>0\). We prove that \(\Lambda\) and \(\Lambda^p\) admit compatible Hartmann-type cluster decompositions and that, under the natural coordinate identification, \( H^2(\Lambda)=H^2(\Lambda^p). \) After taking account of the change in the normalization weights, this trace-space identity gives \( \operatorname{Ran}(T_\Lambda) = \operatorname{Ran}(T_{\Lambda^p}) \) for the associated weighted evaluation operators. The characterization of finitely generated dynamical frames for normal operators then implies that, for every positive operator \(A\in\mathcal B(H)\), every finite index set \(I\), and every \(p>0\), \( \{A^nf_i\}_{n\in\mathbb N,\,i\in I} \text{ is a frame for }H\) if and only if \( \{A^{pn}f_i\}_{n\in\mathbb N,\,i\in I} \text{ is a frame for }H. \) In particular, for every \(q\in\mathbb N^+\), the subfamily \( \{A^{qn}f_i\}_{n\in\mathbb N,\,i\in I} \) is still a frame, and hence every such dynamical frame has infinite excess. An example shows that the positivity assumption cannot in general be replaced by normality.

One of the main challenges for the future in logistics lies in energy consumption. France has the ambition to reach net zero carbon emissions by 2050. The most ambitious research work envisioning the solutions to do so is ___Futurs énergétiques 2050' (October 2021) by RTE, the company in charge of the French Electricity Transportation Network. The work assesses multiple scenarios involving various energy mixes (wind, solar, nuclear etc.). It concludes that the net carbon neutrality in 2050 can only be made possible through sobriety, despite mobility switch to electricity (i.e. at least a 40% decrease final energy consumption, plummeting from 1600 TWh to 930 TWh in 2050). In the logistics field, vast progress has been made to lower the energy intensity of road freight transport. Some promising solutions and technologies have been introduced, namely the improvement of vehicle performances, the alternatives fuels, as well as the parallel evolution of physical and digital infrastructures. It resulted in a cut from 0.077 tonne of oil equivalent per thousand tonne-km in 1992 to 0.057 in 2012 (Llorca and Jamasb, 2017). But the simultaneous soar in freight flows quantity and traffic prevented a global decrease in energy consumption. This challenge is stepped up in urban logistics, as it involves managing -in already congested cities- supply channels and small vehicles that are amplifying, multiplying, and becoming more and more intricate. It seems that all the solutions and technologies that have been introduced are necessary, but insufficient, to meet the expectations in terms of energy sobriety. Significant gains will come from a radical shift in logistics shared infrastructure and from innovative organizations. In this vein, CargoCap (Stein and Schoesser, 2003) and other projects are exploring new infrastructures potential while horizontal collaborative transport as a new organization of freight systems has been largely studied and its effectiveness and efficiency has been proven in recent literature (Pan et al., 2019). In this work we will focus on the energy effectiveness of a new freight dedicated infrastructure. This work assesses the potential a large scale cooperative dedicated system, involving shared infrastructure, for the distribution of freight in the city. This is conceived as an alternative to scattered road haulage, in order to significantly cut down energy consumption. With the goal of energy sobriety, we set the twofold objective of modelling the energy consumption of such a dedicated freight supply system, and of drawing conclusions regarding the best practices by means of an impact assessment of the main design parameters. A distribution system's energy demand strongly depends on specific choices of implementation, which can involve putting back on the agenda some discontinued practices, as well as taking advantage of new technologies and innovative processes. Keeping that in mind, our methodology is devised to comprehensively assess all the available implementation choices, both in technology and in layout. The method consists in taking a step back to fundamental energy theorems, which parameters can be set to stand for any scenario of interest. The model includes transportation modes characteristics (e.g. aerodynamics, rolling resistance etc.), as well as kinetics repercussions of delivery management, handling, and infrastructure accessibility (e.g. driving cycles, congestion etc.). For our purposes of generality, but not limited to, the urban layout and the main routing steps are simplified as a general dense area accessed via a ring road. It is applied to the specific case of Paris, benefiting from previous studies and surveys, as well as government approved open data, quantifying all aspects of freight supply. As far as limits are concerned, it can be put forward that the model is overly simplified and blind on indirect energy consumption which is considerable in overall energy demand. Our work precisely spots out the most impactful parameters, and the high potential axis for refinement, factoring uncertainty. It also specifies where direct and indirect energy drives align, and, on the contrary, a judicious compromise should be investigated. These findings give the necessary tools to further the research into a refined and expanded model, a methodology and insights for urban freight distribution applied here in large cities.

We study weak openness and related geometric properties of convex combinations of slices of the unit balls of Banach spaces. We prove that, for every $m\geq 2$, properties $\mathrm{P1}^{(m)}$ and $\mathrm{CWO}^{(m)}$ are equivalent, and likewise for their closure variants. Consequently, both $\mathrm{P1}$ and $\overline{\mathrm{P1}}$ are determined by convex combinations of two slices; in particular P1 and CWO are equivalent. We construct infinite-dimensional real Banach spaces with $\mathrm{P2}^{(2)}$ but without P2, and with $\mathrm{P3}^{(2)}$ but without P3. In the latter space, every convex combination of two slices contains two points at distance 2, although the space fails the strong diameter two property. Finally, we give an equivalent norm on the real space $c_0$ for which $\overline{\mathrm{P1}}$ holds, but P1 fails. These answers several questions raised in the literature.

We establish some sharp reverse Hardy-Littlewood-Sobolev (HLS) type inequalities on \(\mathbb{R}^n\) and \(\mathbb{R}_+^n\). Using an operator representation, we overcome the difficulty that the symmetric double-integral structure is unavailable in the half-space setting. On \(\mathbb{R}^n\), for \(1 \le n < \alpha\), \(\frac{n}{\alpha} < t < 1\), and \(0 < q < 1\), there holds for nonnegative \(f \) that \[ \|E_\alpha f \|_{L^{t^\prime}(\mathbb{R}^n)} \ge \mathscr{C}(n,\alpha,q,t) \|f \|_{L^1(\mathbb{R}^n)}^{\gamma} \|f \|_{L^q(\mathbb{R}^n)}^{1-\gamma}, \quad \gamma := \frac{n - q\alpha - \frac{n}{t^\prime}q}{n(1-q)} \] for some $\mathscr{C}(n,\alpha,q,t)>0$ iff \(q>\frac{n}{\alpha}\), where \(E_\alpha\) is the extension operator with Riesz kernel and \(t^\prime\) is the conjugate of \(t\). The sharp constant is achieved when \(\frac{n t^\prime}{n + \alpha t^\prime} \le q < 1\). On \(\mathbb{R}_+^n\), with \(2 \le n < \alpha\), \(\frac{n}{\alpha} < t < 1\), and \(0 < q < 1\), we show for nonnegative \(f \) that \[ \|\widetilde{E}_\alpha f \|_{L^{t^\prime}(\mathbb{R}_+^n)} \ge\widetilde{\mathscr{C}}(n,\alpha,q,t) \|f\|_{L^1(\partial \mathbb{R}_+^n)}^{\widetilde{\gamma}} \|f\|_{L^q(\partial \mathbb{R}_+^n)}^{1-\widetilde{\gamma}}, \quad \widetilde{\gamma} := \frac{(n-1) - q(\alpha-1) - \frac{n}{t^\prime}q}{(n-1)(1-q)}, \] for some $\widetilde{\mathscr{C}}(n,\alpha,q,t)>0$ iff \(q > \frac{n-1}{\alpha-1}\), where \(\widetilde{E}_\alpha\) is the extension operator with Poisson-type kernel. The sharp constant is achieved when \(\frac{t^\prime(n-1)}{n + t^\prime(\alpha-1)} \le q < 1\). We further extend results to \(q\ge1\). The proofs use rearrangement inequalities, the sharp Carlson–Levin inequality, and refined pointwise lower bounds for the Riesz and Poisson-type potentials. Our results unify and extend the classical reverse HLS inequalities, especially on \(\mathbb{R}_+^n\).

Given a non-atomic standard probability space $(\Omega,\mu)$, we exhibit examples of non-amenable closed topological subgroups of $\mathrm{Aut}(\Omega,\mu)$ whose natural near-action on $(\Omega,\mu)$ is whirly. This answers a 2010 question by Pestov in the negative. The argument proceeds via constructing Gaussian near-actions from topologically faithful unitary representations of non-amenable whirly Polish groups.

In this paper, we introduce a pseudo-differential operator associated with the linear canonical Dunkl transform. For a particular class of symbols, we prove that this operator defines a continuous linear mapping from the Schwartz space into itself. We further establish an amplitude integral and kernel representation of the operator and investigate the smoothness and decay properties of the associated kernel. Subsequently, we establish an $L^1$-norm inequality for the pseudo-differential operator on the linear canonical Dunkl Sobolev spaces. Moreover, we extend the pseudo-differential operator to tempered distributions, proving its continuity. We then investigate the $L^2$-boundedness of the operator for the symbol class $S^m_0$. Finally, as an application, we employ the pseudo-differential operator to study non-homogeneous and nonlinear parabolic partial differential equations.

One of the basic problems in mathematical learning theory is to identify a function $f: \Omega\to {\mathbb R}$ with certain specific properties which fits a given training data $\{(x_i, \xi_i)\in \Omega\times {\mathbb R}: i=1, \ldots, n\}$, in the sense that, $f(x_i) = \xi_i$ for $i=1, \ldots, n$, where $\Omega$ is a compact subset of ${\mathbb R}^d$ for some $d\in {\mathbb N}$ and $f$ is required to have certain specified characteristics. We address this problem when $f$ belongs to an arbitrary normed linear space ${X}$, and the evaluation maps $f\mapsto f(x_i)$ are replaced by maps of the form $ f\mapsto \varphi_i(f)$ on $X$, where $\varphi_1, \ldots, \varphi_n$ are continuous linear functionals on ${X}$. Using an inner product structure on ${\mathbb R}^n$, we shall device a method of least-squares for obtaining an approximate solution for the above problem and also identify a subspace of ${X}$ in which the least-square solution is unique, which in turn is shown to be equivalent to solving a matrix equation. In the context when the matrix under consideration is ill-conditioned, a regularized equation is devised, and order optimal error estimates are derived for the exact targeted data $\xi=(\xi_1, \ldots, \xi_n)$ and also when it is noisy, by choosing the regularization parameter appropriately.

Every bounded linear operator $T$ on a complex Banach space $E$ is known to extend to a lattice homomorphism $\overline{T}$ that acts on the so-called complex free Banach lattice over $E$. We show the following three results about the spectrum $\sigma(\overline{T})$ of $\overline{T}$: (i) $\sigma(\overline{T})$ always contains the spectrum $\sigma(T)$; this answers a recent question of de Hevia and Tradacete. (ii) It can happen that $\sigma(T)$ is a singleton while $\sigma(\overline{T})$ is the entire unit circle; this shows that $\sigma(\overline{T})$ is not the closure of the cyclic hull of $\sigma(T)$ in general. (iii) Finally, we give a full characterization of $\sigma(\overline{T})$ in terms of the spectral properties of $T$. Some of our arguments also give new results about the spectral properties of general lattice homomorphisms. As a main tool we make extensive use of the Banach lattice functional calculus for continuous positively homogeneous functions.

AI agents are increasingly used in mathematics research, but it is often unclear how to use them effectively. Towards this, we present an extensive case study of how AI was used to improve bounds on the Grothendieck constant $K_G$, which captures the hardness between combinatorial problems and their continuous relaxations. Specifically, while the precise value of $K_G$ is not known, we recently tightened the best known bounds to \[ \frac{6\pi}{11} \;\le\; K_G \;\le\; \frac{\pi}{2\log(1+\sqrt2)} - 10^{-4}. \] Crucially, these improvements were achieved using an AI research system that could arrive at insights deemed novel by domain experts. We give a detailed discussion of our experience using AI for mathematics research, particularly touching upon its strengths and weaknesses, as well as our experience with creating ideal conditions for AI to arrive at breakthrough insights.

In this work we give a counterexample to maximal $\mathrm{L}^2$-regularity in Lions' problem for divergence-form differential operators. On a bounded interval, we construct a bounded, uniformly elliptic, real scalar diffusion coefficient that is $\frac{1}{2}$-Hölder continuous in time with values in spatial $\mathrm{L}^\infty$. It can be chosen arbitrarily close to the constant coefficient of the heat equation. For zero initial data and a forcing term that is continuous in time with square-integrable spatial values, the unique Lions variational solution has a time derivative that is not square integrable in space-time. Thus $\frac{1}{2}$-Hölder continuity alone does not imply maximal $\mathrm{L}^2$-regularity, even for arbitrarily small scalar perturbations of the heat equation. The construction is based on a lacunary family of oscillatory trigonometric modes localised on shrinking time intervals. The spatial profile and the oscillatory modes, together with their first spatial derivatives, vanish at both endpoints. This permits zero extension of the counterexample to the real line. Tensorisation and localisation by parabolic rescaling then yield real symmetric isotropic counterexamples on $\mathbb{R}^d$ and on every bounded domain $\Omega\subset\mathbb{R}^d$, for all $d\ge1$.

A discrete group $\Gamma$ has the \emph{$C^*$–ISR property} if every unital $\Gamma$–invariant $C^*$–subalgebra $A\subseteq C_r^*(\Gamma)$ is of the form $C_r^*(N)$ for a normal subgroup $N\triangleleft\Gamma$. We show that $\text{PSL}_n(\mathbb{Z})$ satisfies the $C^*$–ISR property for every $n\geq3$.

We will provide an example of a Banach space with the Daugavet property which does not possess the weak operator Daugavet property. We will also show that every separable Banach space with the Daugavet property contains a non-poor subspace isomorphic to $\ell^1$ while noticing that every separable subspace of $L^\infty[0,1]$ is poor.

We study funnel control for linear infinite-dimensional passive systems whose controlled output is obtained by integrating the passive output. In mechanical applications, this corresponds to position control based on a measured passive velocity output. Within the system-node framework, we propose a nonlinear recursive funnel controller with a dynamic gain supervisor for systems with distributed or boundary control and observation. The supervisor automatically increases a damping gain whenever the normalized recursive error approaches a prescribed monitoring level. The controller uses the tracking error and its derivative, with the latter obtained directly from the passive output rather than by numerical differentiation. We prove global existence and uniqueness of the closed-loop trajectory, prescribed funnel performance, and global boundedness of the derivative tracking error and the passive output. Under an additional position-compatible strict dissipativity condition, the supervised gain is increased only finitely many times and eventually becomes constant. Consequently, the gain, the internal state, and the control input are globally bounded, and the recursive-error funnel margin is uniform in time. The results are illustrated by a boundary-actuated Euler-Bernoulli beam with uniformly positive distributed damping.

We determine the sharp constants in the one-sided John-Nirenberg inequality for functions of bounded lower oscillation on an interval. More precisely, if $f\in\BLO(I_0)$, then $$ \sup_{I\subseteq I_0}\frac{1}{|I|} \left|\left\{x\in I: f(x)-\essinf_I f>\lambda\right\}\right| \le c_1\exp\left(-\frac{c_2\lambda}{\norm{f}_{\BLO(I_0)}}\right), \qquad \lambda>0, $$ with the usual interpretation when $\|f\|_{\mathrm{BLO}(I_0)}=0$. The sharp constants are $c_1^*=e$ and $c_2^*=1$. We give a proof based on the Riesz rising sun lemma and an independent Bellman function proof. Finally, we present several consequences of the sharp estimate.

Let $X$ be an isotropic log-concave random vector in $\mathbb{R}^n$ and let $G$ be standard Gaussian. Starting from the first moment comparison for gauges, we derive corresponding estimates at arbitrary moment orders. For every gauge $\phi$ and every $q\geqslant 1$, $$ \|\phi(G)\|_q\leqslant C\sqrt{\ln(en)+q}\,\|\phi(X)\|_q,\qquad \|\phi(X)\|_q\leqslant C\left(\sqrt{\ln(en)}+\psi(X)\sqrt q\right)\|\phi(G)\|_q. $$ Applied to support functions, this gives the sharp worst case order $C\sqrt{\ln(en)}$ for the $L_2$-Sudakov constant and quantitative $L_p$-Sudakov estimates. Combining, at each level of Latala's dyadic chain, the strongest of the three quantitative $L_p$-Sudakov estimates used here yields $$ \left(\mathbb{E}\|Y\|^p\right)^{1/p}\leqslant C\left(n^{1/4}\sqrt{\ln(en)}\,\ln(e+\ln(en))\,\mathbb{E}\|X\|+\sigma_p(Y)\right) $$ whenever the weak moments of $Y$ are dominated by those of $X$. In a second direction, we study the generalized dual Sudakov problem for the self generated metrics associated with $Z_r(X)^\circ$ and prove dimension free packing estimates for $Z_p(X)$. We also obtain mean norm estimates for centroid bodies, factorization through arbitrary symmetric convex bodies and an affine dimensional refinement.

In this paper, we give an explicit construction of surface measures on a class of surfaces with arbitrary, possibly infinite, codimension in $\ell^2$. These measures are constructed from local representations associated with a fixed Gaussian product measure. We then establish local and global Gauss–Green-type formulas, introduce a notion of top-degree differential form, and derive an associated Stokes-type identity. We also determine the orientation of the boundary induced by the orientation of the surface. Moreover, therelationship between $\mathcal F$-continuity and Borel measurability is examined,which reveals a phenomenon specific to the infinite-dimensional setting.

We give a detailed proof of interior $W^{2,p}$ regularity for uniformly elliptic equations in nondivergence form \[ a^{ij}(x)D_{ij}u+b^i(x)D_i u=f. \] The leading matrix is assumed to have sufficiently small mean oscillation on the balls under consideration. The proof uses a modern real-variable argument: a constant-coefficient harmonic replacement yields a sharp-function estimate for the Hessian, and the Fefferman–Stein and Hardy–Littlewood theorems permit the coefficient error to be absorbed. A parameter estimate, obtained by Agmon's auxiliary-variable argument, supplies a consistent resolvent on all $L^p$ spaces and makes the subsequent gain of integrability non-circular. The first-order term is retained throughout and is controlled by the scale-invariant quantity $R^{1-n/q}\|b\|_{L^q(B_R)}$, with $q>\max\{n,p\}$. As a consequence, coefficients in $\VMO_{\rm loc}$ give the usual local $W^{2,p}$ regularity for every finite $p$.

Many approximation methods enlarge a trial space by adjoining function blocks generated by different operators. Exact redundancy and strong cross-level interaction can make coefficients nonunique and render pairwise or diagonal-dominance tests needlessly pessimistic. For \(V_m=\sum_{\ell\leq m}S_\ell(E_\ell)\) in a Hilbert space $\mathcal H$, we quotient coefficients representing the same function and control successive orthogonal innovations to obtain Riesz bounds independent of \(m\). The setting includes factored operators \(S_\ell=T_\ell\circ\cdots\circ T_1:E_\ell\to\mathcal H\), with compatible intermediate spaces, but the theorem allows arbitrary bounded \(S_\ell\). The same constants control approximation, truncation, perturbation, and levelwise error. A block Schur complement identifies the intrinsic new dimension and gives the exact reduction in squared best-approximation error, leading to a constructive enrichment procedure. Nonstationary and lifted examples give positive intrinsic bounds where diagonal-dominance estimates are negative or labelled Gram matrices are singular; adaptive and recycled-subspace calculations illustrate the distinct roles of representation stability and application-specific utility.

We characterize the extreme points of the parametric space of quadratic forms on ${\mathbb R}^2$ endowed with the sup norm on the squares $\square_h := [h,h+1] \times [0,1]$ with $h>0$.