Papers

Let $\|\cdot\|_p$ denote the Schatten $p$-norm and let $|A|=(A^*A)^{1/2}$. For $2\leq p<\infty$, let $x_{p,m}>1$ be the unique solution of $x_{p,m}^p=2x_{p,m}+m-1$, and set \[ C_{p,m}=\frac{\sqrt{x_{p,m}(x_{p,m}+m-1)}}{(x_{p,m}^p+m-1)^{1/p}}. \] We prove the sharp inequality \[ \|A_1+\cdots+A_m\|_p\leq C_{p,m}\bigl\||A_1|+\cdots+|A_m|\bigr\|_p \] for arbitrary complex matrices of arbitrary size. Equivalently, if $q=p/(p-1)$ and $R,X_1,\cdots,+X_m$ are positive semidefinite, then \[ \|RX_1\|_1+\cdots\|RX_m\|_1 \leq C_{p,m}\|R\|_q\|X_1+\cdots X_m\|_p. \] For $1<p<2$, we also show that the formula proposed for the optimal constant fails. We give both a numerical counterexample and a systematic analytic construction.oposed for the optimal constant fails. We give both a numerical counterexample and a systematic analytic construction.

Let $K$ be a compact Hausdorff space and let $E$ be an infinite-dimensional real Banach space. We prove that there is no continuous bijection $h\colon C_p(K)\to E_w$ whose inverse is continuous at $h(0)$. Consequently, $C_p(K)$ and $C_w(L)$ are not homeomorphic for any infinite compact Hausdorff spaces $K$ and $L$. This settles Krupski's problem and its two-space version due to Krupski and Marciszewski, and answers a question of Kąkol, Leiderman, and Michalak concerning $C_p([0,1])$ and weak Banach spaces.

Inverse learning within a statistical framework has a wide range of applications. It has garnered significant attention in machine learning, artificial intelligence, and related fields, where the goal is to infer unknown parameters from indirect and noisy observations. This work investigates the stable approximation of $u^{\dagger}$ which solves the equation $Au=g$, with $A$ being a linear operator between appropriate vector spaces. We will consider the domain to be a non-reflexive Banach Space and the co-domain to be a space of real-valued functions on a metric space $X$. The function $g$ is characterized by a finite number of independently and identically distributed data points, which are assumed to follow some unknown probability measure $\rho$. We employ Tikhonov regularization with an arbitrary convex functional to obtain the regularized solution corresponding to the given data point. The convergence analysis is carried out with respect to the Bregman distance, and an upper bound for the error is derived in probability terms. The theoretical findings are then supported by numerical experiments.

In this paper, we derive some bounds on numerical radius of the weighted shift operator $T$ with weights $(1,sq,q^2,tq^3,q^4,sq^5,q^6,tq^7,\ldots)$ where $s,t >0$ and $0<q<1$. Furthermore, we provide an entire function $F_T(z)$. The reciprocal of the minimal positive root of $F_T(z)=0$ gives the numerical radius of $T$. These results generalize several previously known results on the numerical radius of weighted shift operators discussed in \cite{chakraborty2025numerical}.

We determine the exact spectral gaps for both the finite-particle and infinite-particle unlabelled Ginibre interacting Brownian motions. In every finite-particle system, the spectral gap is equal to one, whereas for the infinite-particle system it is equal to two. The infinite-particle dynamics therefore has a strictly larger spectral gap than any of its finite-particle counterparts, revealing an enhancement of spectral gaps in the passage from finite to infinite particle systems.

Let $V\geq 0$ be a locally integrable function on $\mathbb R$. Consider the Schrödinger operator $L=-\frac{d^2}{dx^2} +V$. We prove that for all $0<a\leq 1$, there is a constant $C_a$, independent of $V$, such that the Riesz transform type operator $V^aL^{-a}$ is bounded on $L^1(\mathbb R)$ and $\| V^aL^{-a}f\|_{L^1(\mathbb R)}\leq C_a\|f\|_{L^1(\mathbb R)}$.

We study the sine-kernel time–band limiting operator $S_c$ on $L^2(0,c)$ through the entropy-type trace $\operatorname{Tr}\varphi_u(S_c)$, where $\varphi_u(x)=\log(1+(e^u-1)x)-ux$. A Fourier factorization gives $S_c=A_c^*A_c$ and $\operatorname{Tr}S_c=c$. We derive an exact identity for $\operatorname{Tr}(S_c-S_c^2)$ and the lower bound $\operatorname{Tr}(S_c-S_c^2)\ge\pi^{-2}\log c-\pi^{-3}$, which in turn yields $\operatorname{Tr}\varphi_u(S_c)\ge u\log c/(4\pi^2)$. We also obtain a positive moment expansion $\operatorname{Tr}\varphi_u(S_c)=\sum_{n\ge1}A_n(u)T_n(c)$ and, for each fixed $n$, the Landau–Widom asymptotic $T_n(c)=\log c/(\pi^2n)+o_n(\log c)$. Three natural uniformity hypotheses, denoted (LC), (MT), and (QM), lead to a quadratic lower bound of order $u^2\log c$ on logarithmically growing windows. We prove the implications among these hypotheses and the resulting bounds, with explicit constants, and explain why the fixed-index asymptotic alone does not provide the required uniformity. The hypotheses themselves remain open. Finally, a signed growing-parameter sine-kernel determinant theorem from a companion paper gives the quadratic lower bound independently of (LC), (MT), and (QM). This separates the conclusions available from elementary operator methods from those that use Riemann–Hilbert asymptotics.

Let $K$ be an origin-symmetric full-dimensional convex polytope in $\mathbb R^n$, $n\ge 3$. We show that, for $-n+3<p<1$, the normed space $(\mathbb R^n,\|\cdot\|_K)$ embeds in $L_p$ if and only if it embeds in $L_1$. The case of $p\le 0$ is understood in the generalized sense. In particular, an origin-symmetric full-dimensional convex polytope $K\subset\mathbb R^n$, $n\ge 5$, is an intersection body if and only if $K$ is the polar of a zonotope.

Let $B_1^n$ be the standard cross-polytope in $\mathbb R^n$, let $g_1,\ldots,g_m$ be independent standard Gaussian vectors in $\mathbb R^n$, and set $G_m=\operatorname{conv}{\pm g_1,\ldots,\pm g_m}$. For $m=n^3$ it is proved that $$ \mathbb P\left\{d_{\mathrm{BM}}(G_m,B_1^n)\geqslant c n^{5/8}(\ln n)^{-5/8}\right\}\geqslant 1-\frac2n $$ for a suitable absolute constant $c>0$. This independently improves the polynomial exponent $4/7$ in Friedland's preceding work. Independent concurrent work of Friedland, which appeared after completion of the present manuscript, obtains the same polynomial exponent with the stronger logarithmic factor $(\ln n)^{-1/4}$ by a different argument. The proof uses Friedland's discretization and conditioning argument together with the $K/U$ decomposition. A selected family of $K$ vectors is suppressed and the remaining $K$ vectors are quotiented out. In the resulting quotient simultaneous bounds are proved for every top-dimensional exterior product formed from the suppressed $K$ vectors and the $U$ vectors. A Dvoretzky-Rogers selection after L"owner normalization converts these determinant estimates into a bound for the minimum volume ellipsoid of the whole projected polytope and Maurey's empirical method then gives the required Gaussian measure estimate.

We develop a Hamilton–Jacobi theory for finite-horizon optimal control problems governed by state-dependent delay equations. In this setting, the delayed-time map depends on the controlled trajectory itself, so that both the state and the point at which the past state is evaluated vary simultaneously. The natural state variable is therefore the entire history, and the value functional is defined on a space of Lipschitz histories. Under suitable growth, Lipschitz, and monotonicity assumptions, we establish the dynamic programming principle and characterize the value functional as the unique viscosity solution of the associated Hamilton–Jacobi equation with co-invariant derivatives. To accommodate Lipschitz histories, we introduce a notion of viscosity solution based on finite-dimensional projections generated by polygonal extensions, and prove a comparison principle in the resulting class of functionals. Under additional regularity assumptions, we derive a Pontryagin minimum principle whose adjoint equation contains advanced terms induced by the state dependence of the delay. We also obtain a generalized transversality relation formulated through the delay superdifferential of the value functional. Finally, we establish semiconcavity in the history variable and a joint semiconcavity estimate on an appropriate solution manifold.

In a remarkable breakthrough, Faulhuber, Petersen, van Velthoven, and Voigtlaender disproved the Heil–Ramanathan–Topiwala conjecture by constructing twelve linearly dependent time–frequency shifts of a Schwartz function. The following day, the author compressed their vector-Zak and cohomological architecture to an explicit intrinsically subcritical four-point configuration, thereby attaining the minimum possible cardinality. Soon thereafter, Jasper and Mixon gave a transparent analytic refinement of the breakthrough mechanism and proved, for every Diophantine pair, that HRT counterexamples may be realized with windows in the Roumieu Gelfand–Shilov class $\mathcal S_1^1(\mathbb R)$; their theorem establishes the endpoint regularity phenomenon in broad arithmetic generality while leaving support cardinality unrestricted. The purpose of the present paper is to show that these two sharp advances can be attained simultaneously. We prove that the same explicit four points, coefficients, and matrix cocycle admit a nonzero window in $\mathcal S_1^1(\mathbb R)$. Moreover, if $0\ne f\in\mathcal S_s^\sigma(\mathbb R)$ and $s<1$ or $\sigma<1$, then every finite system of distinct time–frequency shifts of $f$ is linearly independent. Thus four is the least possible cardinality and $(1,1)$ is the sharp coordinatewise threshold within the Roumieu Gelfand–Shilov scale.

Let \(\varphi\) be an admissible concave function on \((0,1)\) and let \(X\) be a rearrangement-invariant space. We study the classes determined by the oscillation functional \[ \mathcal N_{\varphi,X}(f) = \left\| \frac{f^{**}-f^*}{\varphi} \right\|_X+\|f\|_1. \] We develop an interpolation method, based on the Aronszajn–Gagliardo extremal construction, which allows the normability problem and the determination of the optimal rearrangement-invariant Banach exterior to be treated in a unified way. A recovery principle shows that the oscillation construction reflects the inclusion order of the underlying rearrangement-invariant spaces. This makes it possible to transfer the corresponding Aronszajn–Gagliardo extremal structure to the oscillation classes. In particular, the upper extremal generates the least rearrangement-invariant Banach space containing the class, while normability occurs precisely when the lower and upper extremals collapse. At the critical fundamental scale, normability is rigid and forces the underlying space to be the corresponding Lorentz endpoint. Applications to Lorentz, limiting Lorentz, and Orlicz scales illustrate the scope of the method, including limiting normable examples for which the classical Copson absorption mechanism fails.

This work studies the relations between shadowing, topological stability, Lipschitz structural stability, and pseudo-hyperbolicity for linear dynamical systems on Banach spaces. The main result establishes that pseudo-hyperbolicity implies both topological stability and strong Lipschitz structural stability, whereas strong Lipschitz structural stability implies the shadowing property. In addition, a spectral characterization of pseudo-hyperbolicity is obtained, yielding an equivalence between pseudo-hyperbolicity, topological stability, strong Lipschitz structural stability, and the shadowing property for invertible operators whose eigenspaces associated with unimodular eigenvalues admit closed complements. In particular, these four notions are equivalent on Hilbert spaces. Combined with a recent result of Dragičević and Pituk, these results yield, for a large class of Banach spaces, an invertible operator that has the shadowing property but is not generalized hyperbolic, thereby answering an open problem in linear dynamics.

In this paper, we consider methods for the diagonal multi-omics integration of heterogeneous datasets. Several approaches to the nature of biological heterogeneity are analyzed and developed to comprehend more clearly the generated differences. Specifically, the extremal trace problems for the coupled Laplacian on sets homeomorphic to the Stiefel manifold embedded in the complex Euclidean space are investigated. The gradient ascent method for the maximization problem is elaborated in the classical terms of functional analysis, which is of significant interest in itself. On this basis, we introduce a novel characteristic of dataset heterogeneity by employing the norm of the difference between the maximum and minimum points.

We study the quantum drift-diffusion, or Derrida-Lebowitz-Speer-Spohn (DLSS), equation for a nonnegative density $\varrho$ on a bounded convex domain with Neumann boundary conditions, in the square-root variable $u=\sqrt\varrho$. We show that the DLSS operator, defined and monotone on smooth strictly positive functions, admits a unique maximal monotone extension in $L^2(\Omega)$, explicitly given by the minimal (defect-free) operator plus the normal cone of the positivity constraint. The generated semigroup, which contracts the Hellinger distance between the densities, thus yields a canonical solution - existing, unique, and stable for every nonnegative $L^2$ initial datum and in every space dimension - independent of any approximation scheme: it is in fact the unique contraction semigroup extending the classical evolutions that emanate from smooth, uniformly positive data. The implicit Euler scheme converges to it, and $\sqrt u\in L^2_{\rm loc}(H^2)$ along the flow. When the datum belongs to the domain of the operator, the solution is strong and satisfies the equation pointwise, with no reaction term created on the vacuum $\{u=0\}$. We characterize the trajectories in several equivalent ways - as Bénilan integral solutions and through one-sided weak formulations - prove the maximality of the operator also in the $H^2$-$H^{-2}$ duality and, in dimension $d\le3$, identify the flow with the weak solutions in the uniqueness class of Fischer. A second-order estimate of independent interest underlies the construction: on a convex domain with Neumann conditions the dissipation $\int_\Omega(\Delta u)^2/u\,\mathrm{d} x$ is finite exactly when $\sqrt u\in H^2(\Omega)$, and it then controls the full Hessian of $\sqrt u$, in every dimension.

We study properties of the calculus $\varphi\mapsto\varphi(A,B)$ for self-adjoint operators with commutator $[A,B]$ in the Schaten–von Neumann class $\boldsymbol{S}_p$. It turns out that this calculus defined on the Besov class $B_{\infty,1}^1({\Bbb R}^2)$ in the case $p\le2$ admits a commutator Lipschitz estimate and is multiplicative modulo $\boldsymbol{S}_p$. On the other hand in the case $p>2$ there is no commutator Lipschitz estimate. There is no commutator Lipschitz estimate in the operator norm as well.

For complex \(m\)-homogeneous polynomials, let \(D_m\) denote the optimal dimension-free constant in the polynomial Bohnenblust–Hille inequality. We prove that, for every \(B>1/2\), there exists \(K_B>0\) such that \[ D_m\le K_B m^B,\qquad m\ge1. \] More precisely, we obtain the square-root-scale estimate \[ D_m\le \sqrt m\,\exp\!\bigl(C\sqrt{\log m}\,\log\log m\bigr). \] The proof combines coefficient-preserving degree reduction with a weighted bootstrap, separating balanced and dominant degree profiles. We also determine the critical linear-dimensional asymptotics: \[ D_{m,n_m}\longrightarrow 2 \qquad\text{whenever}\qquad \frac{n_m}{m}\longrightarrow1, \] and consequently \(\liminf_{m\to\infty}D_m\ge2\). Further applications give two-sided bounds for homogeneous Sidon constants and a quantitative logarithmic remainder in the multidimensional Bohr-radius asymptotic.

Torus computed tomography (Torus CT) reconstructs the Fourier coefficients of an image from X-ray data along closed geodesics on the flat torus. We extend Torus CT to experimental X-ray tomographic data by converting the data to a parallel-beam geometry and mapping the measured projection angles to the geodesic directions on the torus. In addition, we introduce two extensions of the Torus CT framework: the Star TCT method, which recovers additional Fourier coefficients beyond the standard frequency box using the same data, and torus backprojection (TBP), which sums the data over the measured directions. We prove that filtered TBP converges to the Tikhonov-regularized solution for general data up to an explicit error term that vanishes for data which is invariant along fixed geodesics. We also show that the direction-wise averaging of the zero mode is an unbiased estimator and its mean squared error decays like $N^{-2}$. We evaluate the methods on simulated phantoms and experimental X-ray data of a walnut using an updated implementation. The results show that torus-based reconstruction methods are applicable to experimental data, avoid the streak artifacts of filtered backprojection (FBP), and yield competitive reconstruction errors on both simulated and experimental data.

We determine the exact strong converse exponent of quantum soft covering under the sandwiched Rényi divergence for all orders $\alpha\in[\frac{1}{2},\infty)$. For $\alpha\in[\frac{1}{2},1)$, the exponent is characterized by the two-parameter club-sandwiched mutual information, whereas for $\alpha\in[1,\infty)$, it is characterized by the order-$\alpha$ sandwiched Rényi mutual information. We also determine the exact strong converse exponent of privacy amplification against quantum side information under the sandwiched Rényi divergence for $\alpha\in(2,\infty)$, expressed in terms of the corresponding order-$\alpha$ sandwiched Rényi conditional entropy. To the best of our knowledge, these results provide the first exact characterization of the strong converse exponent of quantum soft covering and the first precise operational interpretation of the two-parameter club-sandwiched mutual information in the quantum setting. The key ingredient is that we establish the exponential rate of the $K$-functional, which is instrumental in deriving the strong converse exponent of quantum soft covering for $\alpha\in[\frac{1}{2},1)$.

Statistical inverse problems have garnered significant attention in recent years due to the growing importance of statistical learning theory and functional analytic approaches in the fields of machine learning and artificial intelligence. In this paper, we investigate the stable approximation of the element $u^{\dagger}$ that satisfies the equation $Au = g$, where $A$ is a linear operator that maps a Banach space into an appropriate function space. The function $g$ is observed only through independently and identically distributed data points that are corrupted by noise and assumed to follow an unknown distribution $\rho$. We employ the Tikhonov regularization scheme, leveraging statistical learning techniques and the framework of reproducing kernel Banach spaces to estimate the solution. We establish convergence and derive the convergence rate of the estimated solution with respect to the true solution as the number of data points increases, with the rate expressed in probabilistic terms. The theoretical findings are further supported by numerical experiments that demonstrate the effectiveness of the proposed approach.

We disprove the conjecture that every tree with t leaves embeds isometrically into $\ell_\infty^{\lceil \log_2 t\rceil}$. We construct a 32-leaf tree whose least isometric $\ell_\infty$-dimension is six rather than five, and prove that every tree with at most 31 leaves attains the conjectured bound; Brigham et al. had recorded equality through 21 leaves. Thus 32 is the first failure, and the example answers affirmatively a question of Fitzpatrick and Nowakowski from 2000. The same topology has dimension six under every assignment of positive edge lengths, and therefore also disproves the later sharp leaf-threshold conjecture for weighted metric trees.

In this note we establish a unified growth rate for the operator norm of $C_0$-semigroups on Hilbert spaces whose generators satisfy the generalized Kreiss resolvent condition. Our bound contains and improves several known estimates in the literature. In particular, it captures the transition between different super-linear growth behaviors.

Let $\alpha\in\mathbb{R}$, $p\in(0,\infty)$, and $q\in(0,\infty]$. In this article, we develop a theory of matrix-weighted anisotropic Besov spaces associated with an expansive matrix $A$ and an $\mathcal A_{p,\infty}$-matrix weight $W$. We first introduce the homogeneous spaces $\dot B_{p,q}^{\alpha}(A,W)$ and establish their $\varphi$-transform characterization. Then we construct counterexamples to show that the assumption $W\in\mathcal A_{p,\infty}$ in this characterization cannot be relaxed to $W\in\bigcup_{r\in(0,\infty)}\mathcal A_r$. The same counterexamples also show that this weaker condition $W\in\bigcup_{r\in(0,\infty)}\mathcal A_r$ is insufficient to ensure the well-definedness of $\dot B_{p,q}^{\alpha}(A,W)$. Next we characterize $\mathcal A_{p,\infty}$-matrix weights via the rescaled maximal operator, which leads naturally to a new concept of the critical rescaling index that quantitatively captures the self-improving behavior of matrix weights. In terms of this index, we obtain optimal boundedness for almost diagonal operators on the associated sequence spaces $\dot b_{p,q}^{\alpha}(A,W)$. Based on these, we further establish the molecular characterization of $\dot B_{p,q}^{\alpha}(A,W)$ and some sharp boundedness results for pseudo-differential operators on these spaces.

In this paper we provide new constructions of equiangular tight frames of size $2d$ in $\mathbb{C}^d$. We generalize the doubling construction of Fallon and Iverson to a tensor multiplication construction based on a suitable pair consisting of a complex Hadamard matrix and an equiangular tight frame. In particular, such a pair always exists whenever there is an amicable pair of real Hadamard matrices. Most notably, amicable Hadamard pairs of order $q+1$ exist for all prime powers $q\equiv 3\pmod 4$. We also find specific constructions based on a family of pairs of order 6 and on pairs whose equiangular tight frames are defined by Paley conference matrices with $q\equiv 1\pmod 4$. Finally, we provide a power construction of equiangular tight frames that generalizes the construction of Turyn for conference matrices.

Let $1<p<\infty$ and let $T$ be a positive strongly Kreiss bounded operator on an $L^p$-space. We prove that there exist constants $C,\kappa>0$ such that \[ \|T^N\| \leq C \left( \log(N+1) \right)^\kappa, \qquad N \geq 1. \] This answers a question raised by Arnold and Cuny concerning positive strongly Kreiss bounded operators on $L^p$-spaces. It also shows that the polynomial exponent appearing in the corresponding problem of Deng, Lorist and Veraar has infimum zero. The proof is elementary: positivity yields a local $\ell^p$-estimate for the orbit on intervals of length of order $\sqrt N$; a positive vector-valued lifting of $T$ then produces a self-improving estimate which can be iterated down to the logarithmic scale.

This paper develops a decision-theoretic interpretation of local expected utility on Abstract Wiener space. The framework is intended for economic acts that are naturally stochastic paths, such as income paths, consumption paths, investment payoffs, insurance losses, or laboratory stimuli unfolding over a continuum of contingencies. In this setting the object of choice is an act or payoff path, while Hermite functions serve as state-feature coordinates rather than objects of preference. Abstract Wiener space supplies a Gaussian reference measure on the Banach space of paths and a Hilbert subspace on which local utility is represented by orthogonal coefficients. The associated Wiener-integral weights are generally signed stochastic functionals; they become admissible decision weights only after a specified nonnegative normalization. This distinction clarifies the relation between the model, state-dependent utility, subjective expected utility, rank-dependent utility, and cumulative prospect theory. The main representation result gives finite primitive conditions under which projected path-valued acts admit a normalized local-utility representation. The corresponding elicitation result identifies composite utility-weight coefficients from projected acts and recovers normalized decision weights when the local utility scale is independently elicited. The paper also states testable restrictions: normalized weights must be nonnegative, add to one, preserve monotonicity, and satisfy additional rank or gain-loss restrictions when the subjective expected utility, rank-dependent utility, or cumulative prospect theory submodels are imposed. Numerical illustrations show how the representation operates in finite approximations.

In this paper, we establish optimal weighted norm inequalities for commutators of singular integral operators with rough kernels. While classical Calderón-Zygmund theory relies heavily on pointwise gradient smoothness, we operate under the strictly weaker log-Dini regularity condition assumed merely on the $L^{1}\left( \mathcal{S}^{n-1}\right) $ spherical restriction of the kernel. First, we prove that these rough commutators are bounded on the weighted Lebesgue spaces $L^{p}\left( w\right) $ for the full range of Muckenhoupt weights $w\in A_{p}$ $\left( 1<p<\infty \right) $. Our primary contribution establishes a sharp weighted endpoint estimate at the critical value $p=1$. For any weight $w\in A_{1}$, we demonstrate that the commutator satisfies a weak-type inequality with a precise $L\log L$ logarithmic loss, successfully recovering the classical smooth behavior in the absence of traditional kernel regularity. The proofs rely on a meticulous refinement of microlocal decompositions combined with a direct, localized sparse domination framework involving Orlicz averages. Finally, we settle the question of optimality by constructing a rigorous counterexample based on the oscillatory properties of lacunary Fourier series. This construction proves that the log-Dini condition is sharp, confirming that the logarithmic regularity cannot be relaxed without losing the operator's fundamental boundedness.

Fix $I = (0,\rho)$, where $0<\rho\leq\infty$, and let $\mathbb{P}_n(I)$ be the set of positive semidefinite $n\times n$ matrices with entries in $I$. A longstanding problem in matrix theory is to characterize the functions $f: I \to \mathbb{R}$ for which the entrywise calculus $f[A] = [f(a_{ij})]_{i,j = 1}^{n}$ preserves positive semidefiniteness for all $A \in \mathbb{P}_n(I)$. We characterize these functions exactly: if $f\in C^{2n-2}(I)$ and $\mathcal{E} = x\frac{d}{dx}$, then this holds if and only if $$ f^{(k)}(x)\geq 0 \quad (0\leq k\leq n-1) \qquad\text{and}\qquad \bigl[\mathcal{E}^{i+j}f(x)\bigr]_{i,j = 0}^{n-1}\succeq 0 $$ for every $x\in I$. Regularization then removes all a priori smoothness: for $n\geq2$, every preserver belongs to $C^{2n-4}(I)$ and the same characterization holds by interpreting the last two derivatives in the sense of distributions. As applications, we recover classical results of FitzGerald–Horn and Vasudeva, and obtain a complete classification of generalized polynomials with prescribed real exponents and arbitrary coefficients. We also determine optimal constants in entrywise domination inequalities under finite regularity, extend the sharp finite-sum thresholds of Belton–Guillot–Khare–Putinar and Khare–Tao to positive mixtures of powers, and answer a question of Khare and Tao by showing that no finite collection of matrices with entries strictly inside $I$ can detect positivity preservation on $\mathbb{P}_n(I)$.

We prove the constant-one discrete nonlinear Hausdorff-Young inequality. As a consequence, by a discrete-to-continuous limiting argument, we obtain $$ \|(\log|a_f|^{2})^{1/2}\|_{L^{p'}(\mathbb{R})} \le \|f\|_{L^{p}(\mathbb{R})},\quad 1\le p<2, $$ for all $f\in L^{p}(\mathbb{R})$, where $a_f$ denotes the transmission coefficient of the nonlinear Fourier transform. In particular, this resolves the Muscalu-Tao-Thiele uniformity problem. The proof uncovers a hidden Hilbert-space structure that reduces the nonlinear inequality to classical interpolation.

We prove that every unimodular locally compact group $G$ of noncompact Lie dimension $n$ satisfies the sharp Brunn–Minkowski inequality \[ \mu_G(XY)^{1/n}\ge\mu_G(X)^{1/n}+\mu_G(Y)^{1/n}, \] and establish a general form for arbitrary, possibly nonunimodular, locally compact groups. This fully confirms the nonabelian Brunn–Minkowski conjecture proposed by the present authors and Zhang. As an application, we obtain an isoperimetric inequality on symmetric spaces of noncompact type.