Although the convolution operators on Euclidean space and the Heisenberg group satisfy the same $L^p$ bounds with the same optimal constants, the former has maximizers while the latter does not. However, as work of Christ has shown, it is still possible to characterize near-maximizers. Specifically, any near-maximizing triple of the trilinear form for convolution on the Heisenberg group must be close

We show that, for each $1\le p < 2$, there exists a wild involution $\mathbb{S}^3\to \mathbb{S}^3$ in the Sobolev class $W^{1,p}(\mathbb{S}^3,\mathbb{S}^3)$.

We give a new proof of $l^2$ decoupling for the parabola inspired from efficient congruencing. Making quantitative this proof matches a bound obtained by Bourgain for the discrete restriction problem for the parabola. We illustrate similarities and differences between this new proof and efficient congruencing and the proof of decoupling by Bourgain and Demeter. We also show where tools from decoupling

Let $C$ be an irreducible smooth complex projective curve of genus $g\geq 2$ and let $C_d$ be its $d$-fold symmetric product. In this paper, we study the question of semi-orthogonal decompositions of the derived category of $C_d$. This entails investigations of the canonical system on $C_d$, in particular its base locus.

In this paper we investigate regularity aspects for solutions of the nonlinear parabolic equation \[ u_t= \Delta u^m, \quad m > 1, \] usually called the porous medium equation. More precisely, we provide sharp regularity estimates for bounded nonnegative weak solutions along the free boundary $\partial\{u > 0\}$, when the equation is universally close to the heat equation. As a consequence, local Lipschitz

For any norm on $\mathbb{R}^d$ with countably many extreme points, we prove that there is a set $E \subset \mathbb{R}^d$ of Hausdorff dimension $d$ whose distance set with respect to this norm has zero linear measure. This was previously known only for norms associated to certain finite polygons in $\mathbb{R}^2$. Similar examples exist for norms that are very well approximated by polyhedral norms

We prove several characterizations of the Hardy spaces for Fourier integral operators $\mathcal{H}^{p}_{\mathrm{FIO}}(\mathbb{R}^n)$, for $1 < p < \infty$. First we characterize $\mathcal{H}^{p}_{\mathrm{FIO}}(\mathbb{R}^n)$ in terms of $L^{p}(\mathbb{R}^n)$-norms of parabolic frequency localizations. As a corollary, any characterization of $L^{p}(\mathbb{R}^n)$ yields a corresponding version for

We study the density of the set SNA$(M,Y)$ of those Lipschitz maps from a (complete pointed) metric space $M$ to a Banach space $Y$ which strongly attain their norm (i.e., the supremum defining the Lipschitz norm is actually a maximum). We present new and somehow counterintuitive examples, and we give some applications. First, we show that SNA$(\mathbb T,Y)$ is not dense in Lip$_0(\mathbb T,Y)$ for

It is well known that the dynamics of vortices in an ideal incompressible two-dimensional fluid contained in a bounded not necessarily simply connected smooth domain is described by the Kirchhoff–Routh point vortex system. In this paper, we revisit the classical problem of how well solutions to the Euler equations approximate these vortex dynamics, and extend previous rigorous results to the case where

We study the obstacle problem for fully nonlinear elliptic operators with an anisotropic degeneracy on the gradient: \[ \left\{\begin{array}{rll} \min\left\{f-|Du|^\gamma F(D^2u),u-\phi\right\} &= 0 & \textrm{ in } \Omega,\\ u & = g & \textrm{ on } \partial \Omega, \end{array}\right. \] for some degeneracy parameter $\gamma\geq 0$, uniformly elliptic operator $F$, bounded source term $f$, and suitably

Let $\mathbb{S} \subset \mathbb{C}$ be the circle in the plane, and let $\Omega\colon \mathbb{S} \to \mathbb{S}$ be an odd bi-Lipschitz map with constant $1+\delta_\Omega$, where $\delta_\Omega\geq 0$ is small. Assume also that $\Omega$ is twice continuously differentiable. Motivated by a question raised by Mattila and Preiss, we prove the following: If a Radon measure $\mu$ has positive lower density

For any sufficiently small perturbation of a tempered exponential dichotomy, we obtain appropriate versions of the Grobman–Hartman theorem and of the Sternberg theorems both for finite and infinite regularity, thus providing, respectively, topological and smooth conjugacies. The constructions are heavily based on the existence of normal forms, with the resonances expressed in terms of the connected

We give a classification between weighted norm inequalities of strong fractional integral operators and their corresponding multi-parameter Muckenhoupt characteristics, by considering the weights to be power functions. As a result, we extend the classical Stein–Weiss theorem on product spaces.

We study various partial data inverse boundary value problems for the semilinear elliptic equation $\Delta u+ a(x,u)=0$ in a domain in $\mathbb{R}^n$ by using the higher order linearization technique introduced by Lassas–Liimatainen–Lin–Salo and Feizmohammadi–Oksanen. We show that the Dirichlet-to-Neumann map of the above equation determines the Taylor series of $a(x,z)$ at $z = 0$ under general assumptions

We study the problem of recovering the singularities of a potential $q$ from backscattering data. In particular, we prove two new different estimates for the double dispersion operator $Q_2$ of backscattering, the first nonlinear term in the Born series. In the first, by measuring the regularity in the Hölder scale, we show that there is a one derivative gain in the integrablity sense for suitably

Our purpose is to explore, in the context of loop ensembles on finite graphs, the relations between combinatorial group theory, the topology of loops, loop measures, and signatures of discrete paths. We determine the distributions of the loop homotopy class, and of the first and second homologies, defined by the lower central series of the fundamental group.

We prove quantitative, matrix weighted, endpoint estimates for the matrix weighted Hardy–Littlewood maximal operator, Calderón–Zygmund operators, and commutators of CZOs with scalar BMO functions, when the matrix weight is in the class A$_1$ introduced by M. Frazier and S. Roudenko. Even in the scalar case, our estimates are sharper than the results implicit in the literature.

We prove that the natural metric universal cover of a quasihyperbolic plane domain is a Hadamard metric space with injectivity radius $\pi$. This in turn provides sharp constants for several conjectures by Väisälä.

We define in this work a notion of Young differential inclusion $$ dz_t \in F(z_t)\,dx_t, $$ for an $\alpha$-Hölder control $x$, with $\alpha > 1/2$, and give an existence result for such a differential system. As a by-product of our proof, we show that a bounded, compact-valued, $\gamma$-Hölder continuous set-valued map on the interval $[0,1]$ has a selection with finite $p$-variation, for $p > 1/\gamma$

In this paper, we study non-divergence form elliptic and parabolic equations with singular coefficients. Weighted and mixed-norm $L_p$-estimates and solvability are established under suitable partially weighted BMO conditions on the coefficients. When the coefficients are constants, the operators are reduced to extensional operators which arise in the study of fractional heat equations and fractional

The index of a Riemannian symmetric space is the minimal codimension of a proper totally geodesic submanifold. There is a conjecture for how to calculate the index. In this paper we give an affirmative answer to this conjecture for the exceptional Riemannian symmetric spaces and for the classical symmetric spaces Sp$_r(\mathbb R)/U_r$. Our methodology is new and based on the idea of using slice representations

Let $L_j={\partial}/{\partial t_j}+(a_j+ib_j)(t_j){\partial}/{\partial x}$, $j=1,\ldots,n$, be a system of complex vector fields defined on the $(n+1)$-dimensional torus, where $a_j$ and $b_j$ are real-valued functions belonging to the Gevrey class $ G^s(\mathbb{T}^1)$, $s > 1$. We present a complete characterization to the global $s$-solvability of this system in terms of diophantine properties of

We study the generalized stationary Stokes system in a bounded domain in the plane equipped with perfect slip boundary conditions. We show natural stability results in oscillatory spaces, i.e., Hölder spaces and Campanato spaces, including the border-line spaces of bounded mean oscillations (BMO) and vanishing mean oscillations (VMO). In particular, we show that, under appropriate assumptions, gradients

Motivated by the study of symplectic Lie algebroids, we focus on a type of algebroid (called an $E$-tangent bundle) which is particularly well-suited to the study of singular differential forms and their cohomology. This setting generalizes $b$-symplectic manifolds, foliated manifolds, and a wide class of Poisson manifolds. We generalize Moser’s theorem to this setting, and use it to construct symplectomorphisms

In this note we show that sufficiently collapsed, irreducible and aspherical Alexandrov 3-spaces are homeomorphic to 3-manifolds. As a corollary we obtain the validity of the Borel conjecture for sufficiently collapsed, irreducible and aspherical Alexandrov 3-spaces.

The exceptional zero phenomenon has been widely studied in the realm of $p$-adic $L$-functions, where the starting point lies in the foundational works of Ferrero–Greenberg, Katz, Gross, or Mazur–Tate–Teitelbaum, among others. This phenomenon also appears in the study of Euler systems, and in this case one is led to study higher order derivatives of cohomology classes in order to extract the arithmetic

In this paper, we establish a large sieve inequality of Elliott–Montgomery–Vaughan type for Maass forms on GL($n,\mathbb R$) and explore three applications.

A conjecture due to Cilleruelo states that for an irreducible polynomial $f$ with integer coefficients of degree $d\geq 2$, the least common multiple $L_f(N)$ of the sequence $f(1), f(2), \dots, f(N)$ has asymptotic growth $\log L_f(N)\sim (d-1)N\log N$ as $N\to \infty$. We establish a version of this conjecture for almost all shifts of a fixed polynomial, the range of $N$ depending on the range of

In this short note we describe an alternative global version of the twisting procedure used by Dolgushev to prove formality theorems. This allows us to describe the maps of Fedosov resolutions, which are key factors of the formality morphisms, in terms of a twist of the fiberwise quasi-isomorphisms induced by the local formality theorems proved by Kontsevich and Shoikhet. The key point consists in

For a given planar convex body $K$, a bisection of $K$ is a decomposition of $K$ into two closed sets $A, B$ so that $A \cap B$ is an injective continuous curve connecting exactly two boundary points of $K$. Consider a bisection of $K$ minimizing, over all bisections, the maximum diameter (resp., maximum width) of the sets in the decomposition. In this note, we study some properties of these minimizing

For any $x\in\mathbb{R}^d$, $d\geq 2$, denote by $\pi^x\colon \mathbb{R}^d\backslash\{x\}\rightarrow S^{d-1}$ the radial projection $$ \pi^x(y)=\frac{y-x}{|y-x|}\cdot $$ Given a Borel set $E\subset\mathbb R^d$ with $\mathrm{dim}_{\mathcal{H}} E\leq d-1$, in this paper we investigate for how many $x\in \mathbb{R}^d$ the radial projection $\pi^x$ preserves the Hausdorff dimension of $E$, namely whether

We classify totally geodesic submanifolds of Damek–Ricci spaces and show that they are either subgroups (“smaller” Damek–Ricci spaces) or isometric to rank-one symmetric spaces of negative curvature.

Let $\mathbb{F}_q$ be an arbitrary finite field of order $q$ and let $M_2(\mathbb F_q)$ be the ring of all $2\times 2$ matrices with entries in $\mathbb F_q$. In this article, we study $\mathrm{det} S$ for certain types of subsets $S \subset M_2(\mathbb F_q)$. For $i\in \mathbb{F}_q$, let $D_i$ be the subset of $M_2(\mathbb F_q)$ defined by $ D_i := \{x\in M_2(\mathbb F_q): \det(x)=i\}.$ We first show

In this paper, we show that if $R$ is a compact Riemann surface and $M=R\setminus\bigcup_i D_i$ is a domain in $R$ whose complement is a union of countably many pairwise disjoint smoothly bounded closed discs, $D_i$, then there is a complete conformal minimal immersion $X\colon M\to\mathbb{R}^3$, extending to a continuous map $X\colon \overline M\to\mathbb{R}^3$, such that $X(bM)=\bigcup_i X(bD_i)$

Motivated by applications to renewal theory, Erdős, de Bruijn and Kingman posed a problem on boundedness of reciprocals $(1 − z)/(1 − F(z))$ in the unit disc for probability generating functions $F(z)$. This problem was solved by Ibragimov in 1975 by constructing a counterexample. In this paper, we provide much stronger counterexamples showing that the problem does not allow for a positive answer even

We prove the existence of horizontal Jenkins–Serrin graphs that are translating solitons of the mean curvature flow in Riemannian product manifolds $M \times \mathbb R$. Moreover, we give examples of these graphs in the cases of $\mathbb R^3$ and $\mathbb H^2 \times \mathbb R$.

We introduce new spaces of martingales that behave like $L_1$-based Sobolev spaces on $\mathbb R^d$. We prove a martingale analog of Van Schaftingen’s theorem and give sharp estimates on the lower Hausdorff dimension of the terminal distributions of these martingales. We also provide martingale analogs of trace theorems for Sobolev functions.

The present paper pioneers the study of the Dirichlet problem with $L^q$ boundary data for second order operators with complex coefficients in domains with lower dimensional boundaries, e.g., in $\Omega := \mathbb{R}^n \setminus \mathbb{R}^d$, with $d < n-1$. Following results of David, Feneuil and Mayboroda, we introduce an appropriate degenerate elliptic operator and show that the Dirichlet problem

In his 1990 Inventiones paper, P. Jones characterized subsets of rectifiable curves in the plane, using a multiscale sum of what is now known as Jones $\beta$ -numbers, numbers measuring flatness in a given scale and location. This work was generalized to $\mathbb R^n$ by Okikiolu, to Hilbert space by the second author, and has many variants in a variety of metric settings. Notably, in 2005, Hahlomaa

We obtain, up to a flat boundary, regularity results in parabolic Hölder spaces for viscosity solutions of fully nonlinear parabolic equations with oblique boundary conditions.

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

We obtain a two weight local $Tb$ theorem for any elliptic and gradient elliptic fractional singular integral operator $T^{\alpha}$ on the real line $\mathbb{R}$, and any pair of locally finite positive Borel measures $(\sigma,\omega)$ on $\mathbb{R}$. The Hilbert transform is included in the case $\alpha = 0$, and is bounded from $L^{2}(\sigma)$ to $L^{2}(\omega)$ if and only if the Muckenhoupt and

Let $F$ be a binary form with integer coefficients, non-zero discriminant and degree $d$ with $d$ at least $3$ and let $r$ denote the largest degree of an irreducible factor of $F$ over the rationals. Let $k$ be an integer with $k \geq 2$ and suppose that there is no prime $p$ such that $p^k$ divides $F(a,b)$ for all pairs of integers $(a,b)$. Let $R_{F,k}(Z)$ denote the number of $k$-free integers

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

The aim of this paper is to prove the existence of radially symmetric $k$-admissible solutions for the following Dirichlet problem associated with the $k$-th Hessian operator: $$\left\{\begin{aligned} &\left.\begin{aligned} &S_k[u]=f(x,-u)\\ &u<0 \\ \end{aligned}\right\}&\mbox{in }& B,\\ &\;u=0&\mbox{on }& \partial B, \end{aligned}\right.$$ where $B$ is the unit ball of $\mathbb{R}^{N},$ $N=2k$ $(k\in

We study the Obata equation with Robin boundary condition $\partial f/\partial \nu + af = 0$ on manifolds with boundary, where $a$ is a non-zero constant. Dirichlet and Neumann boundary conditions were previously studied by Reilly, Escobar and Xia. Compared with their results, the sign of $a$ plays an important role here. The new discovery shows besides spherical domains, there are other manifolds

We study locally compact metric spaces that enjoy various forms of homogeneity with respect to Möbius self-homeomorphisms. We investigate connections between such homogeneity and the combination of isometric homogeneity with invertibility. In particular, we provide a new characterization of snowflakes of boundaries of rank-one symmetric spaces of non-compact type among locally compact and connected

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

Fix $m\geq 0$, and let $A=(A_{ij}(x))_{1 \leq i \leq N, 1\leq j \leq M}$ be a matrix of semialgebraic functions on $\mathbb{R}^n$ or on a compact subset $E \subset \mathbb{R}^n$. Given $f=(f_1,\ldots,f_N) \in C^\infty(\mathbb{R}^n, \mathbb{R}^N)$, we consider the following system of equations: \[ \sum_{j=1}^M A_{ij} (x) F_j (x) = f_i (x) \quad\text{for } i =1,\ldots, N. \] In this paper, we give algorithms

Let $\mathscr{R}$ denote the ring of real polynomials on $\mathbb{R}^{n}$. Fix $m\geq 0$, and let $A_{1},\ldots,A_{M}\in\mathscr{R}$. The \emph{$C^{m}$-closure} of $(A_{1},\ldots,A_{M})$, denoted here by $[A_{1},\ldots,A_{M};C^{m}]$, is the ideal of all $f\in \mathscr{R}$ expressible in the form $f=F_{1}A_{1}+\cdots +F_{M}A_{M}$ with each $F_{i}\in C^{m}(\mathbb{R}^{n})$. In this paper we exhibit an

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