For each $p > 1$ and each positive integer $m$, we give intrinsic characterizations of the restriction of the homogeneous Sobolev space $L_{p}^{m}(\mathbb{R})$ to an arbitrary closed subset $E$ of the real line. We show that the classical one-dimensional Whitney extension operator is "universal" for the scale of $L_{p}^{m}(\mathbb{R})$ spaces in the following sense: For every $p\in(1,\infty]$, it provides

The Boltzmann equation without the angular cutoff is considered when the initial data is a small perturbation of a global Maxwellian and decays algebraically in the velocity variable. We obtain a well-posedness theory in the perturbative framework for both mild and strong angular singularities. The three main ingredients in the proof are the moment propagation, the spectral gap of the linearized operator

Let $H$ be a Krull monoid with finite class group $G$ and suppose that every class contains a prime divisor. Then sets of lengths in $H$ have a well-defined structure which depends only on the class group $G$. With methods from additive combinatorics we establish a characterization of those class groups $G$ guaranteeing that all sets of lengths are (almost) arithmetical progressions.

We obtain an improved Kakeya maximal function estimate in $\mathbb R^4$ using a new geometric argument called the planebrush. A planebrush is a higher dimensional analogue of Wolff’s hairbrush, which gives effective control on the size of Besicovitch sets when the lines through a typical point concentrate into a plane. When Besicovitch sets do not have this property, the existing trilinear estimates

We pursue our work on the asymptotic regimes of the Landau–Lifshitz equation. We put the focus on the cubic Schrödinger equation, which is known to describe the dynamics of uniaxial ferromagnets in a regime of strong easy-axis anisotropy. In any dimension, we rigorously prove this claim for solutions with sufficient regularity. In this regime, we additionally classify the one-dimensional solitons of

Globally hyperbolic spacetimes-with-timelike-boundary $(\overline{M} = M \cup \partial M, g)$ are the natural class of spacetimes where regular boundary conditions (eventually asymptotic, if $\partial M$ is obtained by means of a conformal embedding) can be posed. $\partial M$ represents the naked singularities and can be identified with a part of the intrinsic causal boundary. Apart from general properties

We prove that no Brunn–Minkowski inequality from the Riemannian theories of curvature-dimension and optimal transportation can be satisfied by a strictly sub-Riemannian structure. Our proof relies on the same method as for the Heisenberg group together with new investigations by Agrachev, Barilari and Rizzi on ample normal geodesics of sub-Riemannian structures and the geodesic dimension attached to

We describe Borel and parabolic subalgebras of affine Lie superalgebras and study the Verma type modules associated to such subalgebras. We give necessary and sufficient conditions under which these modules are simple.

If $f\colon \mathbb{R}^3 \to \mathbb{R}^3$ is a uniformly quasiregular mapping with Julia set $J(f)$, a genus $g$ Cantor set, for $g\geq 1$, then for any linearizer $L$ at any repelling periodic point of $f$, the fast escaping set $A(L)$ consists of a spiders' web structure containing embedded genus $g$ tori on any sufficiently large scale. In other words, $A(L)$ contains a spiders' web of doughnuts

Let $H := \Big(\begin{smallmatrix} 1 & \mathbb{R} & \mathbb{R} \\ 0 & 1 & \mathbb{R} \\ 0 & 0 & 1 \end{smallmatrix}\Big)$ denote the Heisenberg group with the usual Carnot–Carathéodory metric $d$. It is known (since the work of Pansu and Semmes) that the metric space $(H,d)$ cannot be embedded in a bilipschitz fashion into a Hilbert space; however, from a general theorem of Assouad, for any $0 < \varepsilon

We present a coordinate-free version of Fefferman’s solution of Whitney’s extension problem in the space $C^{m−1,1}(\mathbb R^n)$. While the original argument relies on an elaborate induction on collections of partial derivatives, our proof uses the language of ideals and translation-invariant subspaces in the ring of polynomials. We emphasize the role of compactness in the proof, first in the familiar

We consider a model convex functional with orthotropic structure and super-quadratic nonstandard growth conditions. We prove that bounded local minimizers are locally Lipschitz, with no restrictions on the ratio between the highest and the lowest growth rates.

Fix $d\geq 2$ and $s\in (0,d)$. In this paper we introduce a notion called small local action associated to a singular integral operator, which is a necessary condition for the existence of the principal value integral. Our goal is to understand the geometric properties of a measure for which an associated singular integral has small local action. We revisit Mattila's theory of symmetric measures and

We establish that for every function $u \in L^1_{\rm loc}(\Omega)$ whose distributional Laplacian $\Delta u$ is a signed Borel measure in an open set $\Omega$ in $\mathbb{R}^{N}$, the distributional gradient $\nabla u$ is differentiable almost everywhere in $\Omega$ with respect to the weak-$L^{N/(N-1)}$ Marcinkiewicz norm. We show in addition that the absolutely continuous part of $\Delta u$ with

In this paper we relate the study of actions of discrete groups over connected manifolds to that of their orbit spaces seen as differentiable stacks. We show that the orbit stack of a discrete dynamical system on a simply connected manifold encodes the dynamics up to conjugation and inversion. We also prove a generalization of this result for arbitrary discrete groups and non-simply connected manifolds

For the maximal operator $M$ on $\mathbb{R}^{d}$, and $ 1 < p,\rho < \infty$, there is a finite constant $D=D _{p, \rho }$ so that this holds. For all weights $w,\sigma$ on $\mathbb{R}^{d}$, the operator $M(\sigma \cdot)$ is bounded from $L^{p}(\sigma )\to L^{p}(w)$ if and only if the pair of weights $(w,\sigma)$ satisfy the two weight $A_{p}$ condition, and this testing inequality holds: $$\int_{Q}

The $L^q$ dimensions, for $1 < q < \infty$, quantify the degree of smoothness of a measure. We study the following problem on the real line: when does the $L^q$ dimension improve under convolution? This can be seen as a variant of the well-known $L^p$-improving property. Our main result asserts that uniformly perfect measures (which include Ahlfors-regular measures as a proper subset) have the property

We establish an integral-geometric formula for minimal two-spheres inside homogeneous three-spheres, and use it to provide a characterisation of each homogeneous metric on the three-dimensional real projective space as the unique metric with the largest possible two-systole among metrics with the same volume in its conformal class.

We prove that the Newton–Okounkov body associated to the flag $E_{\bullet}:= \{ X=X_r \supset E_r \supset \{q\} \}$, defined by the surface $X$ and the exceptional divisor $E_r$ given by any divisorial valuation of the complex projective plane $\mathbb{P}^2$, with respect to the pull-back of the line-bundle $\mathcal{O}_{\mathbb{P}^2} (1)$ is either a triangle or a quadrilateral, characterizing when

For a complete metric space $M$, we prove that the finitely supported extreme points of the unit ball of the Lipschitz-free space $\mathcal{F}{M}$ are precisely the elementary molecules $(\delta(p)-\delta(q))/d(p,q)$ defined by pairs of points $p,q$ in $M$ such that the triangle inequality $d(p,q) < d(p,r)+d(q,r)$ is strict for any $r\in M$ different from $p$ and $q$. To this end, we show that the

We prove that for pairwise co-prime numbers $k_1,\dots,k_d \geq 2$ there does not exist any infinite set of positive integers $\mathcal{A}$ such that the representation function $r_{\mathcal{A}}(n) = \# \{ (a_1, \dots, a_d) {\in} \mathcal{A}^d : k_1 a_1 + \cdots + k_d a_d = n \}$ becomes constant for $n$ large enough. This result is a particular case of our main theorem, which poses a further step

In this paper we complete the classification of the elliptic fibrations on K3 surfaces which admit a non-symplectic involution acting trivially on the Néron–Severi group. We use the geometric method introduced by Oguiso and moreover we provide a geometric construction of the fibrations classified. If the non-symplectic involution fixes at least one curve of genus 1, we relate all the elliptic fibrations

Let $\mathbb{F}_q$ be the finite field with $q$ elements and $f, g\in \mathbb{F}_q[x]$ be polynomials of degree at least one. This paper deals with the asymptotic growth of certain arithmetic functions associated to the factorization of the iterated polynomials $f(g^{(n)}(x))$ over $\mathbb{F}_q$, such as the largest degree of an irreducible factor and the number of irreducible factors. In particular

Let $\Omega$ be an open convex set in $\mathbb{R}^m$ with finite width, and with boundary $\partial \Omega$. Let $v_{\Omega}$ be the torsion function for $\Omega$, i.e., the solution of $-\Delta v=1, v|_{\partial\Omega}=0$. An upper bound is obtained for the product of $\Vert v_{\Omega}\Vert_{L^{\infty}(\Omega)}\lambda(\Omega)$, where $\lambda(\Omega)$ is the bottom of the spectrum of the Dirichlet

A complex orthogonal (geometric) structure on a complex manifold is a geometric structure locally modelled on a non-degenerate quadric. One of the first examples of such a structure on a compact manifold of dimension three was constructed by Guillot. In this paper, we show that the same manifold carries a family of uniformizable complex orthogonal (geometric) structures which includes Guillot’s structure;

We introduce a spectrum of monotone coarse invariants for metric measure spaces called Poincaré profiles. The two extremes of this spectrum determine the growth of the space, and the separation profile as defined by Benjamini–Schramm–Timár. In this paper we focus on properties of the Poincaré profiles of groups with polynomial growth, and of hyperbolic spaces, where we deduce a connection between these

In this paper, we establish the following Stein–Weiss inequality with the fractional Poisson kernel: \begin{align*} (\star)\qquad \int_{\mathbb{R}^n_{+}} \int_{\partial\mathbb{R}^n_{+}} |\xi|^{-\alpha} f(\xi) &\,P(x,\xi,\gamma)\, g(x)\, |x|^{-\beta}\, d\xi\, dx \\ &\leq C_{n,\alpha,\beta,p,q'}\, \|g\|_{L^{q'}(\mathbb{R}^n_{+})}\, \|f\|_{L^p(\partial \mathbb{R}^{n}_{+})}, \end{align*} where $$ P(x,\xi

The 7-dimensional link $K$ of a weighted homogeneous hypersurface on the round 9-sphere in $\mathbb{C}^5$ has a nontrivial null Sasakian structure which is contact Calabi–Yau, in many cases. It admits a canonical co-calibrated $\mathrm G_2$-structure $\varphi$ induced by the Calabi–Yau 3-orbifold basic geometry. We distinguish these pairs $(K,\varphi)$ by the Crowley–Nordström $\mathbb{Z}_{48}$-valued

In this paper we prove the uniqueness of the saddle-shaped solution $u\colon \mathbb{R}^{2m} \to \mathbb{R}$ to the semilinear nonlocal elliptic equation $(-\Delta)^\gamma u = f(u)$ in $\mathbb{R}^{2m}$, where $\gamma \in (0,1)$ and $f$ is of Allen–Cahn type. Moreover, we prove that this solution is stable if $2m\geq 14$. As a consequence of this result and the connection of the problem with nonlocal

A free boundary optimization problem involving the $\Phi$-Laplacian in Orlicz–Sobolev spaces is considered for the case where $\Phi$ does not satisfy the natural conditions introduced by Lieberman. A minimizer $u\Phi$ having non-degeneracy at the free boundary is proved to exist and some important consequences are established, namely, the Lipschitz regularity of $u\Phi$ along the free boundary, that

In the planar setting, the Radó–Kneser–Choquet theorem states that a harmonic map from the unit disk onto a Jordan domain bounded by a convex curve is a diffeomorphism provided that the boundary mapping is a homeomorphism. We prove the injectivity criterion of Radó–Kneser–Choquet for $p$-harmonic mappings between Riemannian surfaces. In our proof of the injectivity criterion we approximate the $p$-harmonic

We prove a dichotomy theorem about compact Kähler manifolds admitting nontrivial real-holomorphic geodesic gradient vector fields, which has the following consequence: either such a manifold satisfies an additional integrability condition, or through every zero of the real-holomorphic geodesic gradient there passes an uncountable family of totally geodesic, holomorphically immersed complex projective

We consider the discrepancy of the integer lattice with respect to the collection of all translated copies of a dilated convex body having a finite number of flat, possibly non-smooth, points in its boundary. We estimate the $L^p$ norm of the discrepancy with respect to the translation variable, as the dilation parameter goes to infinity. If there is a single flat point with normal in a rational direction

The purpose of this paper is to establish some Adams–Moser–Trudinger inequalities, which are the borderline cases of the Sobolev embedding, in the hyperbolic space $\mathbb H^n$. First, we prove a sharp Adams’ inequality of order two with the exact growth condition in $\mathbb H^n$. Then we use it to derive a sharp Adams-type inequality and an Adachi–Tanakat-ype inequality. We also prove a sharp Adams-type

We study regularity for solutions of quasilinear elliptic equations of the form $\mathrm {div}\mathbf{A}(x,u,\nabla u)=\mathrm {div}\mathbf{F}$ in bounded domains in $\mathbb{R}^n$. The vector field $\mathbf{A}$ is assumed to be continuous in $u$, and its growth in $\nabla u$ is like that of the $p$-Laplace operator. We establish interior gradient estimates in weighted Morrey spaces for weak solutions

Let $\{\phi_t\}$ be a semigroup of holomorphic self-maps of the unit disc $\mathbb{D}$ with Denjoy–Wolff point $\tau\in \partial\mathbb{D}$. We study the rate of convergence of the semigroup to $\tau$, that is, given $z\in \overline{\mathbb{D}}$, we discuss the behavior of $|\phi_{t}(z)-\tau|$ as $t$ goes to $+\infty$.

A family $\mathcal{A}$ of $k$-subsets of $\{1,2,\dots, N\}$ is a Sidon system if the sumsets $A+B$, $A,B\in \mathcal{A}$ are pairwise distinct. We show that the largest cardinality $F_k(N)$ of a Sidon system of $k$-subsets of $[N]$ satisfies $F_k(N)\le {N-1\choose k-1}+N-k$ and the asymptotic lower bound $F_k(N)=\Omega_k(N^{k-1})$. More precise bounds on $F_k(N)$ are obtained for $k\le 3$. We also

Based on the quasiconformal theory of the universal Teichmüller space, we introduce the Teichmüller space of diffeomorphisms of the unit circle with $\alpha$-Hölder continuous derivatives as a subspace of the universal Teichmüller space. We characterize such a diffeomorphism quantitatively in terms of the complex dilatation of its quasiconformal extension and the Schwarzian derivative given by the

In non-variational two-phase free boundary problems for harmonic measure, we examine how the relationship between the interior and exterior harmonic measures of a domain $\Omega \subset \mathbb R^n$ influences the geometry of its boundary. This type of free boundary problem was initially studied by Kenig and Toro in 2006, and was further examined in a series of separate and joint investigations by

This paper is concerned with the following quasilinear Schrödinger equation: $$-\Delta u+V(x)u-\frac{1}{2}\Delta (u^2)u= g(u), \quad x\in \mathbb{R}^N,$$ where $N\ge 3$, $V\in \mathcal{C}(\mathbb R^N,[0,\infty))$ and $g\in \mathcal{C}(\mathbb{R}, \mathbb{R})$ is superlinear at infinity. By using variational and some new analytic techniques, we prove the above problem admits ground state solutions under

In this article, we investigate the quantitative unique continuation properties of real-valued solutions to elliptic equations in the plane. Under a general set of assumptions on the operator, we establish quantitative forms of Landis’ conjecture. Of note, we prove a version of Landis’ conjecture for solutions to $−\Delta u + Vu = 0$, where $V$ is a bounded function whose negative part exhibits polynomial

Braces were introduced by Rump to study involutive non-degenerate set-theoretic solutions of the Yang–Baxter equation. A constructive method for producing all such finite solutions from a description of all finite left braces has been recently discovered. It is thus a fundamental problem to construct and classify all simple left braces, as they can be considered as building blocks for the general theory

In this paper we prove a Pohozaev identity for the weighted anisotropic $p$-Laplace operator. As an application of the identity, we deduce the nonexistence of nontrivial solutions of the Dirichlet problem for the weighted anisotropic $p$-Laplacian in star-shaped domains of $\mathbb R^n$. We also provide an upper bound estimate for the first Dirichet eigenvalue of the anisotropic $p$-Laplacian on bounded

Let $\Omega \subset {\mathbb R}^N$ ($N \geq 3$) be a $C^2$ bounded domain, and let $\delta$ be the distance to $\partial \Omega$. In this paper, we study positive solutions of the equation $(\star)\ -L_\mu u+ g(|\nabla u|) = 0$ in $\Omega$, where $L_\mu=\Delta + {\mu}/{\delta^2} $, $\mu \in (0,{1}/{4}]$ and $g$ is a continuous, nondecreasing function on ${\mathbb R}_+$. We prove that if $g$ satisfies

This paper studies boundary properties of bounded functions with bounded $\overline\partial_J$ differential on strictly pseudoconvex domains in an almost complex manifold.

In this article, we will show the scattering of the cubic defocusing nonlinear Schrödinger equation on the waveguide $\mathbb{R}^2\times \mathbb{T}$ in $H^1$. We first establish the linear profile decomposition in $H^{1}(\mathbb{R}^2 \times \mathbb{T})$ motivated by the linear profile decomposition of the mass-critical Schrödinger equation in $L^2(\mathbb{R}^2)$. Then by using the solution of the cubic

The defining equation $$(\ast)\qquad \dot \omega_t=-F'(\omega_t)$$ of a gradient flow is kinetic in essence. This article explores some dynamical (rather than kinetic) features of gradient flows (i) by embedding equation $(\ast)$ into the family of slowed down gradient flow equations: $\dot \omega^{\varepsilon}_t=- \varepsilon F'( \omega ^{ \varepsilon}_t)$, where $\varepsilon > 0$, and (ii) by considering

Let $R$ be a nil algebra over a field of characteristic 0, and let $\delta$ be a derivation of $R$. Then the differential polynomial ring $R[X, \delta]$ cannot be mapped onto a unital simple ring homomorphically.

In this paper, we consider a blow-up solution $u(t)$ (close to the soliton manifold) to the $L^2$-critical gKdV equation $\partial_tu+(u_{xx}+u^5)_x=0$, with finite blow-up time $T < +\infty$. We expect to construct a natural extension of $u(t)$ after the blow-up time. To do this, we consider the solution $u_{\gamma}(t)$ to the saturated $L^2$-critical gKdV equation $\partial_tu+(u_{xx}+u^5-\gamma

In the spirit of the classical work of P. H. Rabinowitz on nonlinear Schrödinger equations, we prove existence of mountain-pass solutions and least energy solutions to the nonlinear Schrödinger–Poisson system $$\left\{\begin{array}{lll} - \Delta u+ u + \rho (x) \phi u = |u|^{p-1} u, &x\in \mathbb{R}^3, \\ -\Delta \phi=\rho(x) u^2, & x\in \mathbb{R}^3, \end{array}\right.$$ under different assumptions

We show that there exists an integrable function on the $n$-sphere $(n \geq 2)$, whose Cesàro $(C, (n − 1)/2)$ means with respect to the spherical harmonic expansion diverge unboundedly almost everywhere. This extends results of Stein (1961) for flat tori and complements the work of Taibleson (1985) for spheres. We also study relations among Cesàro, Riesz, and Bochner–Riesz means.

We study the cohomological equation associated to linear cocycles on semi simple Lie groups $\mathcal G$ over hyperbolic dynamics. We give sufficient conditions for the solution of the cohomological equation of fiber-bunched cocycles to be unique and for the Hölder conjugacy class of the cocycle to coincide with $C^\nu(M,\mathcal G)$. In particular, we prove that there exists an open and dense subset

We describe a Lefschetz fibration of genus one on the disk cotangent bundle of any closed orientable surface $\Sigma$. As a corollary, we obtain an explicit genus one open book decomposition adapted to the canonical contact structure on the unit cotangent bundle of $\Sigma$.

In this paper we study a Bloch–Torrey regularization of the Rosensweig system for ferrofluids. The scope of this paper is twofold. First of all, we investigate the existence and uniqueness of Leray–Hopf solutions of this model in the whole space $\mathbb R^2$. In the second part of this paper we investigate both the long-time behavior of weak solutions and the propagation of Sobolev regularities in

It is known that the compactness of the commutators of point-wise multiplication with bilinear homogeneous Calderón–Zygmund operators acting on product of Lebesgue spaces is characterized by the multiplying function being in the space CMO. This space is the closure in BMO of its subspace of smooth functions with compact support. It is shown in this work that for bilinear Calderón–Zygmund operators

We discuss the Hörmander multiplier theorem for $L^p$ boundedness of Fourier multipliers in which the multiplier belongs to a fractional Sobolev space with smoothness $s$. We show that this theorem does not hold in the limiting case $|1/p-1/2|=s/n$.

We consider the anisotropic version of an extension problem studied by Caffarelli and Silvestre. We present the anisotropic fractional Laplacian and prove the Almgren’s frequency formula obtained by Caffarelli and Silvestre for the anisotropic case.

We consider a partially overdetermined problem in a sector-like domain $\Omega$ in a cone $\Sigma$ in $\mathbb{R}^N$, $N\geq 2$, and prove a rigidity result of Serrin type by showing that the existence of a solution implies that $\Omega$ is a spherical sector, under a convexity assumption on the cone. We also consider the related question of characterizing constant mean curvature compact surfaces $\Gamma$

We construct Zariski K3 surfaces of Artin invariant 1, 2 and 3 in many characteristics. In particular, we prove that any supersingular Kummer surface is Zariski if $p\not\equiv 1$ mod 12. Our methods combine different approaches such as quotients by the group scheme $\alpha_p$, Kummer surfaces, and automorphisms of hyperelliptic curves.

We study the Morse index of self-shrinkers for the mean curvature flow and, more generally, of $f$-minimal hypersurfaces in a weighted Euclidean space endowed with a convex weight. When the hypersurface is compact, we show that the index is bounded from below by an affine function of its first Betti number. When the first Betti number is large, this improves index estimates known in literature. In