Papers

We study linear and symmetric diffusion processes on $\mathbb{R}^{\mathbb{Z}^d}$ that can be seen as the Langevin dynamics of the (inhomogeneous) harmonic crystal for given conductances, with a particular focus on their stationary distributions. Despite the linearity of the interaction, we exhibit a rich variety of behaviours, depending on the disorder encoded by the conductances. Our main results provide essentially sharp criteria on the conductances ensuring that every stationary measure is reversible. We also provide examples for which these criteria fail and where either no stationary measures exist or reversible and non-reversible stationary measures coexist. Additionally, we show that the linear system can even exhibit non-trivial time-periodic behaviour and provide a spectral characterisation of the occurrence of such oscillations. The case of deterministic conductances is complemented by a study of the case of random conductances under quite general moment assumptions.

Independently orient each horizontal and vertical line of $\mathbb{Z}^2$ by a fair coin. A walker chooses one of the two lines through its current position with equal probability and takes one step in the direction of that line. Redner (1989) introduced this walk as a model of transport in an isotropic random velocity field and predicted that its root-mean-square displacement grows like $n^{2/3}$. We prove that the walk is transient almost surely. The proof is a short variational argument.

We construct a symmetric numerical semigroup $\Gamma$ of multiplicity $56$, embedding dimension $26$, and Frobenius number $181$ such that, over every field $k$, the one-dimensional Gorenstein local domain $R=k[t^\Gamma]_{\mathfrak{m}}$ has a nonprincipal ideal $I=(t^{56},t^{70})R$ for which $I\otimes_R\operatorname{Hom}_R(I,R)$ is torsion-free. This gives a counterexample to the Huneke-Wiegand conjecture in its general Gorenstein formulation. The proof reduces to an explicit finite sumset identity, interpreted through the Huneke-Iyengar-Wiegand colon criterion and Leamer's tensor-torsion formula. We also identify the tensor product with the trace ideal of $I$, compute $\operatorname{End}_R(I)=R[t^{101},t^{107}]$ and its conductor, and show that the counterexample persists under completion. A separate graded tensor-graph calculation and exact, standard-library Python verifiers accompany the paper. We explain how the example differs from the field-extension construction of Christensen, Gerko, and Iyengar.

Let $L=-\Delta+V$ be a Schrödinger operator on $\mathbb{R}^n$, where $\Delta$ is the Laplacian and $V$ satisfies the reverse Hölder inequality ${\rm RH}_q$ for some $q>n/2$. In this paper, we study the behavior of the Littlewood–Paley operators $s_L$ and $S_L$, as well as the semigroup maximal operator $T^*_L$, on the space ${\rm CMO}_L(\mathbb{R}^n)$ associated with the Schrödinger operator $L$. It is known from previous work that these operators are bounded on ${\rm BMO}_L(\mathbb{R}^n)$. Our main result shows that they are, in fact, mappings from ${\rm CMO}_L(\mathbb{R}^n)$ into itself. To prove this, we develop several equivalent characterizations of ${\rm CMO}_L(\mathbb{R}^n)$ and employ a refined decomposition that partitions the parameter interval at $r_B\rho(x_B)$, instead of the customary $r_B^2$ or $\rho(x_B)^2$. The new strategy allows us to overcome a key technical obstacle that arises when applying existing methods to the ${\rm CMO}_L$ setting.

We identify the rotational symmetry centers in the spectre tiling. We find that both 2-fold and 3-fold symmetry centers are coexisting, with two different types of 3-fold centers. Local and global symmetry operations are distinguished, and characterized in terms of substitution systems. We introduce a decomposition of the spectre tiling in terms of three metatiles named diabolo, 2-arms and 3-arms, valid at any substitution level. We propose geometrical constructions for generating patches of the spectre tiling with 2-fold and 3-fold global rotational symmetry.

Foundation models for ground states in spin-1/2 systems are a promising method for problems ranging from quantum chemistry to identifying new phase diagrams. Nearly all such models are currently pure-states that condition on the Hamiltonian's parameters, whose Monte Carlo samples give energy estimates according to the variational principle. In this contribution, we show that this representation is topologically obstructed. For any gapped Hamiltonian family whose ground-state bundle is non-trivial, every continuous normalized state-vector model has zero fidelity with the ground state at some parameter value in the Hamiltonian family. For that value, the energy is at least one spectral gap, $\Delta$, with an $O(\Delta)$ gap in an open-neighbourhood of that point. We show that this is a sufficient no-go also in the case of degenerate ground-state manifolds, time dynamics, and periodic systems with mixed space-time topology, demonstrating these obstructions on one- and two-qubit systems. We discuss how this causes a spike in the fidelity susceptibility, giving a numerical signature of a phase-transition where there is none. We then show that operator-valued models canonically avoid these obstructions and preserve topological information, implying a structural necessity in representation for foundation neural quantum states.

Let $G$ be a connected reductive complex algebraic group and let $q\in\mathbb{C}^{\times}$ be not a root of unity. We prove that every $(G,q)$-connection on a formal punctured disc admits a $(G,q)$-oper structure.

We prove dispersive estimates for the three-dimensional defocusing energy-critical nonlinear Schrödinger equation associated with $\mathcal L_a=-\Delta+a|x|^{-2}$. For nonnegative potentials, we extend the known finite-$p$ theory to the endpoint $L^1\to L^\infty$. For negative potentials in the global well-posedness range, set $\sigma=\frac12-\sqrt{\frac14+a}$. We obtain the free strong decay rate for $2<p<3/\sigma$ and the limiting Lorentz estimate from $L^{(3/\sigma)',1}$ to $L^{3/\sigma,\infty}$. The restriction $p<3/\sigma$ is sharp for strong Lebesgue decay. The proof combines a finite-interval bootstrap, nonlinear real interpolation on bounded energy sets, and endpoint Sobolev–Lorentz estimates adapted to $\mathcal L_a$.

In this paper we study a naturally occurring variant of Ramanujan polynomials with a Bernoulli convolution structure, $S_v$, related to the double Mordell-Tornheim series. In particular, we prove for even $v$ all zeros of $S_v$ lie on the complex unit circle, and for even $v \ge 4$ they are simple with the exception of roots at $1, -1$, which are double roots.

Linear complementary dual (LCD) codes form an important class of linear codes with applications in cryptography, classical error correction, and quantum coding theory. In this paper, we study the neighbor relation on LCD codes over finite fields and the graph induced by this relation, where two codes are adjacent whenever they intersect in codimension one. We determine the number of neighbors of an LCD code that are also LCD, and we use this result to analyze the structure of the corresponding neighbor graph. In particular, we prove its regularity over arbitrary finite fields and establish further regularity properties for its main structural subgraphs in the binary and odd-characteristic cases. These results provide a graph-theoretic framework for the study of LCD codes and reveal a strong combinatorial regularity in their neighborhood structure.

We consider $\nabla\varphi$ height functions with even and convex interaction energy $W$ on the lattice $\mathbb{Z}^d$, which are restricted to take values in the set $\{-S, \ldots, S\}$ for some integer $S \ge 1$. We study the effect of entropic repulsion, which tends to push the spin values to the middle layer. We prove that the model has a unique Gibbs measure, with exponential decay of correlations, in the cases: (a) Dimension $d=2$ at all temperatures. (b) Dimensions $d\ge3$ at all temperatures, for a wide class of $W$ with non-increasing second derivative, including the family $W(x)=|x|^p$ for $p\in[1,2]$. (c) Dimensions $d\ge 3$ at both low and high temperatures, $T\in (0,\frac{W(1)d}{4(\ln d+\ln 8)})\cup(d W(2S),\infty]$, with the normalization $W(0)=0$. At low temperatures, our proof provides an alternative to Pirogov–Sinai methods. Conversely, we exhibit a class of even and convex interaction energies $W$ which, in high dimensions and suitable temperature regimes, have multiple Gibbs measures. Though uniqueness may fail, we show that the magnetization of every Gibbs measure lies in $(-\frac{1}{2},\frac{1}{2})$. This implies the delocalization of the model restricted to take values in $\{0,1,\ldots\}$ (i.e., conditioned to lie above a floor) for all dimensions, any even and convex $W$, and all temperatures. Our methods extend to additional setups: We prove that height functions taking values in the real interval $[-1,1]$ always have a unique Gibbs measure, a result previously proved only for the quadratic interaction. For height functions taking values in $\{-S+\frac{1}{2}, \ldots, S-\frac{1}{2}\}$, $S \ge1 $ integer, we prove that the magnetization of every Gibbs measure lies in $(-1,1)$. The special case $W(x)=x^2$ of our results addresses questions left open in the work of Bricmont–El Mellouki–Fröhlich (1986).

Gross, Mansour, and Tucker [European J. Combin., 95 (2021): 103329] introduced the partial Petrial polynomial of a ribbon graph $G$, denoted by $^{\partial}{\varepsilon^{\times}_G}(z)$. For a prime bouquet $B_n$, Yan and Li [Discrete Appl. Math., 375 (2025): 281-289] determined $^{\partial}{\varepsilon^{\times}_{B_n}}(z)$ when the intersection graph $I(B_n)$ is either the complete graph or a path, and provided an equivalent condition under which the lowest degree of the nonzero coefficient in $^{\partial}{\varepsilon^{\times}_{B_n}}(z)$ is $1$. In this paper, we determine $^{\partial}{\varepsilon^{\times}_{B_n}}(z)$ when the intersection graph is a cycle. Moreover, we present a complete characterization of the prime bouquets whose lowest nonzero term in $^{\partial}{\varepsilon^{\times}_{B_n}}(z)$ is of degree $2$ and determine the partial Petrial polynomial for the prime bouquets. As corollaries, we determine $^{\partial}{\varepsilon^{\times}_{B_n}}(z)$ when $I(B_n)$ is the complete bipartite and tripartite graph.

Let $n\geq 3$, let $H\subseteq\mathbb{C}^n$ be any additive subgroup spanning $\mathbb{C}^n$, and let $0\ne t\in H$. We study Pozhidaev's central simple $n$-Lie algebra $P(H,t)=\widetilde{\mathcal A}(H,t)/\mathbb{C}e_0$ without a finite generation or discreteness assumption on $H$. Its inner derivation algebra is the simple generalized divergence-free Lie algebra $\mathcal S(0,0,n;t,H)$. We prove that its space of inner-equivariant symmetric products vanishes and that every $1/n$-derivation is a scalar multiple of the identity. Using these invariants, we show that $P(H,t)$ is not isomorphic to any simple nonabelian $n$-Lie algebra defined on the underlying spaces of the $S$, $W$, or $SW$ constructions recorded by Cantarini and Kac. We also show that $P(H,t)$ is the quotient by the constants of the derived algebra of an explicit $S$-algebra on $\mathbb{C}[H]$. Finally, we realize Pozhidaev's second construction $E(H)$ over $\mathbb{C}$ as a $W$-algebra.

We prove that the covering dimension of a metric space with curvature bounded above equals its geometric dimension, without a separability assumption. This resolves a conjecture formulated by Kleiner that goes back to Gromov's work.

We study a factorization notion for Lipschitz functions between metric spaces in which such a function is written as a composition of a Hölder function and a function that suitably “undoes” the Hölder regularity. We show simple ways to construct such “de-Höldering” factorizations. If the identity mapping on a metric space $\mathcal{M}$ admits a de-Höldering factorization through a metric space $\mathcal{Z}$ that has a conical geodesic bicombing, then the class of metric spaces from which one can extend $\mathcal{Z}$-valued Lipschitz functions is shown to be contained in the corresponding class of metric spaces for $\mathcal{M}$-valued Lipschitz functions. As a quick consequence of these abstract permanence properties, we deduce that every $L_1$-valued Lipschitz function from a subset of $\ell_2$ can be extended to a Lipschitz function that takes values in $L_1$ and is defined on all of $\ell_2$, answering a 1992 question of Ball. By work of Makarychev and Makarychev, this implies that every weighted graph has a vertex cut sparsifier of size $n$ and quality $O(\sqrt{\log n})$, improving Moitra's 2009 bound. We also show that for every metric space $\mathcal{Z}$ that has a conical geodesic bicombing, any metric transform of a metric space $\mathcal{M}$ has $\mathcal{Z}$-valued Lipschitz extension modulus at most a universal constant multiple of the $\mathcal{Z}$-valued Lipschitz extension modulus of $\mathcal{M}$ itself, improving the 2002 bound of Brudnyi and Shvartsman.

We study semantic-aware remote estimation of multiple finite-state Markov sources observed by K sensors with overlapping coverage. The sensors share a time-division multiple-access uplink and differ in transmission reliability, delivery delay, and transmission budget. In each slot, the scheduler jointly selects a source and one of its monitoring sensors, or remains idle, to minimize the long-run average cost of actuation error subject to global and per-sensor transmission-frequency constraints. We formulate this problem as a finite average-cost constrained Markov decision process. We show that the transmission resource functions have rank at most K, although the global constraint may still restrict the feasible region. Consequently, the Lagrangian depends only on K effective transmission costs, and an optimal constrained solution can be represented using at most K+1 deterministic policy-recurrent-class components. We further characterize the piecewise-affine concave Lagrangian value function and derive projected dual subgradient ascent over an explicitly bounded multiplier set. Numerical results illustrate the value-function structure, the need for policy randomization in a representative instance, and the interaction between global and per-sensor transmission budgets.

We study the action of the Serre functor of smooth proper dg categories at their Hochschild-Serre algebra, and the invariant sub-aglebra of the Serre functor. As applications, we prove some theorems of categorical Torelli. Namely, let $\Ku(\X)$ be the Kuznetsov component of degree $d$ smooth hypersurface in weighted projective space $\mathbb{P}(a_0, a_1, \cdots, a_n)$, where the common maximal divisor $\gcd(d, \sum^{n}_{i=0}a_{i})=1$. We show the categorical Torelli for $\Ku(\X)$. We show that the $\mathbb{C}^{\ast}$ equivariant matrix factorization category associated with a quasi-homogeneous polynomial function $f$ that has an isolated singularity together with a twisted functor $\{1\}$ reconstructs $f$ up to an isomorphism.

We study two means on the five-dimensional Vinberg cone $\Omega$, viewed as a sparse cone of positive definite $3\times3$ matrices. The Cholesky-Vinberg mean is obtained from the simply transitive solvable group $H$ associated with $\Omega$. It is $H$-equivariant, and its interpolation curves are geodesics for two flat metric connections with torsion. One of the corresponding metrics is the canonical Hessian metric of the cone. The Log-Vinberg mean is defined by affine interpolation in the global logarithmic coordinates of the associated Vinberg algebra, or clan. The resulting Riemannian metric is complete and flat, and its weighted Fréchet means have explicit formulas. Both constructions preserve the zero pattern defining $\Omega$.

We show the following equivalence of solutions to general fractional Helmholtz equations: $u$ is a solution to $$A^su=\lambda^su$$ for some fixed $0<s<2$ and some $\lambda>0$ if and only if $u$ is a solution to the same equation for all $s\neq0$. Furthermore, we show as well that $u$ equivalently solves the logarithmic Helmholtz equation $$\log(A)u=(\log\lambda)u.$$ Here, $A$ is a nonnegative, linear operator on a Banach space $X$. In particular, our results hold whenever $A=-L$, where $L$ is the infinitesimal generator of a $C_0$-semigroup in a Banach space $X$ satisfying mild assumptions that are typical in applications. As a particular case, we recover known results for the fractional Laplacian $(-\Delta)^s$ in $\mathbb{R}^n$. More importantly, we provide a list of examples of other fractional power operators in settings where the Fourier transform is not available and for which our theorems apply. These include fractional powers and logarithms of second order elliptic operators in bounded domains, Laplace–Beltrami operators on Riemannian manifolds, space-time master equations, the discrete Laplacian, and fractional derivatives.

In this short note, we prove that, for any pair of finite groups $G$ and $H$, there exist a finite $T_0$-space $X$ and a semiflow $\varphi\colon [0,\infty)\times X\to X$ such that $\operatorname{Aut}(X)\cong G$, whereas $\operatorname{Aut}(\varphi_t(X))\cong H$ for every $t>0$. Thus, the symmetry group of a finite $T_0$-space and those of all its positive-time images can be prescribed independently.

Nakamura [J. Comput. Appl. Math. 131 (2001)] proved that the arithmetic–harmonic matrix iteration converges quadratically to the geometric matrix mean, the Riemannian midpoint of the affine-invariant metric on positive-definite matrices. We consider the more general problem of when a Riemannian midpoint is a Gauss compound mean. A Riemannian metric and an affine connection determine three midpoint maps associated with the connection, its metric dual, and the Levi–Civita connection. We show that the Gauss composition of the first two equals the Levi–Civita midpoint if and only if the latter is invariant under one step of the iteration. We give a sufficient condition for this invariance in terms of an isometry which acts as the point reflection about the Levi–Civita midpoint and exchanges the connection with its dual. Under this invariance, the iterations converge quadratically for sufficiently close initial pairs. Nakamura's iteration is an example of this criterion. The criterion also applies to dual pairs of matrix power means whose Gauss composition is the geometric matrix mean. We give a dually flat Hessian example which shows that duality alone does not suffice. We also give non-flat examples with a Euclidean metric and a parallel cubic form in every dimension larger than one. In dimension one, we characterize the invariance completely by using the Matkowski–Sutô equation. For Euclidean dual pairs, we conjecture that the invariance implies that the cubic form is constant. We prove this conjecture in dimension one and for scalar multiples of a constant cubic form.

We consider generalised subshifts generated by continuous substitution on compact Hausdorff alphabets. Although primitivity still implies minimality of the subshift generated by the substitution, Durand, Ormes and Petite showed, in contrast to the finite case, that primitivity no longer implies unique ergodicity, by constructing counterexamples with Cantor alphabet. Here we show that, even for the arguably simplest case of the one-point compactification of the natural numbers, primitivity is still insufficient for unique ergodicity, or even the existence of a natural length function (NLF). In previous work with Mañibo and Rust we showed that, for irreducible substitutions, unique ergodicity and existence of a NLF follow from strong power convergence of the renormalised substitution operator \(T\); criteria were also developed that can sometimes confirm strong power convergence. Here we show the converse, so that strong power convergence of \(T\) is equivalent to having both unique ergodicity and a NLF. We then consider the case of alphabets with only finitely many accumulation points, showing how upper bounds (and usually an exact formula) for the essential spectral radius of \(T\) can be derived from associated finite substitutions, determined by the behaviour of substitution of the accumulation points. For many primitive substitutions this can be used to prove quasi-compactness of \(T\), and thus unique ergodicity. For primitive substitutions on alphabets with at least one isolated point, we show that strong power convergence and quasi-compactness of \(T\) are equivalent and, in fact, that these properties are equivalent to iteration of substitution growing words in length uniformly across all seeds in the alphabet.

We study the one-dimensional time-independent discrete Dirac equation on graphs with discrete edges. Both continuous and discrete formulations of the graphs are considered, with particular focus on star graphs with three edges as a simple example. Exact solutions for eigenstates and spectra are derived and compared with the continuum case, showing good agreement in tabulated results. This approach offers a systematic framework for analyzing relativistic quantum transport on branched structures relevant to nanostructures and quantum device design.

We review some topics in "exceptional mathematics” from the perspective of 3-dimensional geometry: the octonions $\mathbb{O}$, the split octonions $\mathbb{O}'$, the bioctonions $\mathbb{O}_\mathbb{C} \cong \mathbb{C} \otimes_\mathbb{R} \mathbb{O}$, the complex Albert algebra $\mathfrak{h}_3(\mathbb{O}_\mathbb{C})$, and the complex form of the exceptional Lie algebra $\mathfrak{e}_6$. We show how to functorially build an algebra isomorphic to $\mathbb{O}$ from any 3d complex vector space equipped with an inner product and complex volume form. Similarly, we build one isomorphic to $\mathbb{O}'$ starting from a 3d real vector space equipped with a volume form, and one isomorphic to $\mathbb{O}_\mathbb{C}$ starting from a 3d complex vector space equipped with a complex volume form. We give applications to 3-dimensional real and complex manifolds. Finally, we describe how to build an Jordan algebra isomorphic to $\mathfrak{h}_3(\mathbb{O}_\mathbb{C})$ starting from three 3d complex vector spaces equipped with complex volume forms. This last construction gives a nice explicit description of the complex Lie algebra $\mathfrak{e}_6$ and its subalgebra $\mathfrak{sl}(3,\mathbb{C}) \oplus \mathfrak{sl}(3,\mathbb{C}) \oplus \mathfrak{sl}(3,\mathbb{C})$.

We explore the regularity of energy maximizers for the Lagrangian of a periodic dispersion managed fiber optic at fixed intensity, on a torus of length $L$ and with vanishing average dispersion. We show that the Fourier coefficients decay at a polynomial rate, and then upgrade this to exponential decay, so that the maximizers are analytic in space for large enough $L$. In addition, by an asymptotic comparison to the optimizers on the real line, we prove that the solutions are non-trivial. We also consider a conjecture that the maximizer necessarily has an underlying symmetry inherent to both the energy functional and the resulting Euler-Lagrange equation. All of our results are supported with illustrative numerical experiments.

Xiuxiong Chen proved the existence of $C^{1, \bar 1}$ geodesics in the space of Kähler potentials and the convexity of the $J$-functional along $C^{1, \bar 1}$ geodesics. Analogous results were proved by Collins and Yau for the hypercritical Leung–Yau–Zaslow equation. In this paper, we study a similar picture for generalised Monge–Ampère equations associated with two right-Noetherian polynomials in proper position.

Recently, increasing attention has been paid to finite element discretizations of vector-valued flow problems posed on curved surfaces. In this work, we study a surface Stokes system defined on a two-dimensional manifold embedded in three-dimensional space. The surface velocity field is approximated using a nonconforming Crouzeix-Raviart finite element on a polyhedral approximation of the surface, while the pressure is discretized by piecewise constant functions. The governing equations involve the symmetric strain-rate tensor, whose approximation with Crouzeix-Raviart finite elements leads to spurious oscillations in the velocity field because the discrete Korn inequality fails on the nonconforming space. We therefore stabilize the momentum equation by adding an edge-jump penalty term, which restores the coervity of the bilinear term as the discrete Korn inequality holds for this stabilized version. Additionally, we establish that this finite element pair of velocity and pressure spaces satisfies the discrete inf-sup compatibility condition. Furthermore, we derive optimal a-priori error estimates in both the energy norm and the $L^2$-norm. These theoretical results are corroborated by numerical experiments.

This paper studies a linear stochastic delay differential equation driven by fractional Brownian motion and involving a finite number of discrete delays. The model combines two sources of memory: delayed feedback in the drift and temporal dependence induced by the Hurst parameter of the fractional Brownian motion. We first introduce the fundamental solution associated with the corresponding deterministic delay equation and use it to obtain an explicit representation of the solution. The stochastic convolution with respect to fractional Brownian motion is defined pathwise as a Riemann–Stieltjes integral, using the finite-variation properties of the fundamental solution. This representation allows us to compute the mean and auto-covariance function of the process. Under a delay-independent stability condition on the drift coefficients, we prove that the fundamental solution decays exponentially and derive the long-time behaviour of the stochastic delay equation. In particular, the solution converges in finite-dimensional distributions to a stationary Gaussian process, whose limiting mean and covariance are given explicitly. When the Hurst parameter satisfies \(H>1/2\), we show that the limiting covariance preserves the long-memory behaviour of the driving fractional Brownian motion and decays asymptotically as \(h^{2H-2}\). Finally, by extending the noise to a double-sided fractional Brownian motion, we construct a stationary solution of the equation and prove its uniqueness. The results provide a tractable framework for modelling stationary systems with both delay effects and fractional memory, with potential applications to stochastic volatility, energy modelling, and other time series exhibiting persistence.

We explore some methods to obtain cotorsion pairs and model category structures from a Frobenius pair. Specifically, we consider a type of Frobenius pair $(\mathcal{X},\omega)$ in an abelian category $\mathfrak{C}$ for which there exists a nonnegative integer $m$ such that every object in the right orthogonal complement $\mathcal{X}^{\perp_1}$ of $\mathcal{X}$ under $\text{Ext}^1_{\mathfrak{C}}(-,\sim)$ has a special $\omega^\wedge_m$-precover and a special $[\omega^\wedge_m]^{\perp_1}$-preenvelope, where $\omega^\wedge_m$ denotes the class of objects in $\mathfrak{C}$ with resolution dimension relative to $\omega$ at most $m$. The cotorsion pairs and model structures obtained from these Frobenius pairs will involve the classes $\mathcal{X}$, $\omega$, $\mathcal{X}^\wedge_m$, $\omega^\wedge_m$, and their orthogonal complements, in the description of its cofibrant, fibrant and trivial objects. As applications of our results, we obtain several cotorsion pairs and model structures related to relative and absolute Gorenstein homological dimensions.

We prove the boundedness of multilinear Hilbert transforms along moment curves by establishing uniform estimates for both translated and untranslated multilinear paraproducts and by applying a Sobolev smoothing inequality. As a byproduct of our method, we obtain the same bounds for multilinear maximal functions along moment curves.