We study, with respect to the parameter $q\neq0$, the following Schrödinger-Bopp-Podolsky system in $\mathbb R^{3}$ \begin{equation*} \begin{cases} -\Delta u+\omega u+q^2\phi u=|u|^{p-2}u, \\ -\Delta \phi+a^2\Delta^2 \phi = 4\pi u^2, \end{cases} \end{equation*} where $p\in(2,3]$, $\omega>0$, $a\geq0$ are fixed. We prove, by means of the fibering approach, that the system has no solutions at all for large values of $q$ and has two radial solutions for small $q$'s. We give also qualitative properties about the energy level of the solutions and a variational characterization of these extremal values of $q$. Our results recover and improve some results in [2,5].

In the setting of two-step Carnot groups we show a “cone property” for horizontally convex sets. Namely, we prove that, given a horizontally convex set $C$, a pair of points $P\in \partial C$ and $Q\in \operatorname{int} (C)$, both belonging to a horizontal line $\ell$, then an open truncated subRiemannian cone around $\ell$ and with vertex at $P$ is contained in $C$.

We apply our result to the problem of classification of horizontally monotone sets in Carnot groups. We are able to show that monotone sets in the direct product $\mathbb{H}\times\mathbb{R}$ of the Heisenberg group with the real line have hyperplanes as boundaries.

In this work we construct some irreducible components of the space of two-dimensional holomorphic foliations on $\mathbb{P}^n$ associated to some algebraic representations of the affine Lie algebra $\mathfrak{aff}(\mathbb{C})$. We give a description of the generalized Kupka components, obtaining a classification of them in terms of the degree of the foliations, in both cases $n=3$ and $n=4$.

We extend the theory of weighted norm inequalities on variable Lebesgue spaces to the case of bilinear operators. We introduce a bilinear version of the variable $\mathcal{A}_{p(\cdot)}$ condition and show that it is necessary and sufficient for the bilinear maximal operator to satisfy a weighted norm inequality. Our work generalizes the linear results of the first author, Fiorenza, and Neugebauer [7] in the variable Lebesgue spaces and the bilinear results of Lerner et al. [22] in the classical Lebesgue spaces. As an application we prove weighted norm inequalities for bilinear singular integral operators in the variable Lebesgue spaces.

We show that the universal groupoid of an inverse semigroup $S$ is topologically (measurewise) amenable if and only if $S$ is hyperfinite and all members of a family of subsemigroups of $S$ indexed by the spectrum of the commutative $C^*$-algebra $C^*(E_S)$ on the idempotents $E_S$ of $S$ are amenable. Thereby we solve some problems raised by A. L. T. Paterson.

For a reduced curve $C:f=0$ in the complex projective plane $\mathbb{P}^2$, we study the set of jumping lines for the rank two vector bundle $T\langle C \rangle $ on $\mathbb{P}^2$ whose sections are the logarithmic vector fields along $C$. We point out the relations of these jumping lines with the Lefschetz type properties of the Jacobian module of $f$ and with the Bourbaki ideal of the module of Jacobian syzygies of $f$. In particular, when the vector bundle $T\langle C \rangle $ is unstable, a line is a jumping line if and only if it meets the $0$-dimensional subscheme defined by this Bourbaki ideal, a result going back to Schwarzenberger. Other classical general results by Barth, Hartshorne, and Hulek resurface in the study of this special class of rank two vector bundles.

We will prove that over commutative rings the silting property of $2$-term complexes induced by morphisms between projective modules is preserved and reflected by faithfully flat extensions.

Let $f\colon(\mathbb R^n,0)\to(\mathbb R^p,0)$ be a real analytic map germ with isolated instability. We prove that if $n=2$ and $p=2,3$, then $f$ is finitely $C^0$-determined. This result can be seen as a weaker real counterpart of Mather-Gaffney finite determinacy criterion.

We study the special values of the triple product $p$-adic $L$-function constructed by Darmon and Rotger at all classical points outside the region of interpolation. We propose conjectural formulas for these values that can be seen as extending the Elliptic Stark Conjecture, and we provide theoretical evidence for them by proving some particular cases.

We obtain symmetry results for solutions of an elliptic system of equation possessing a cooperative structure. The domain in which the problem is set may possess “holes” or “small vacancies” (measured in terms of capacity) along which the solution may diverge.

The method of proof relies on the moving plane technique, which needs to be suitably adapted here to take care of the complications arising from the vacancies in the domain and the analytic structure of the elliptic system.

In this note we determine all possible dominations between different products of manifolds when none of the factors of the codomain is dominated by products. As a consequence, we determine the finiteness of every product-associated functorial semi-norm on the fundamental classes of the aforementioned products. These results give partial answers to questions of M. Gromov.

Consider $m \in \mathbb{N}$ and $\beta \in (1, m + 1]$. Assume that $a\in \mathbb{R}$ can be represented in base $\beta$ using a development in series $a = \sum^{\infty}_{n = 1}x(n)\beta^{-n}$, where the sequence $x = (x(n))_{n \in \mathbb{N}}$ takes values in the alphabet $\mathcal{A}_m := \{0, \dotsc, m\}$. The above expression is called the $\beta$-expansion of $a$ and it is not necessarily unique. We are interested in sequences $x = (x(n))_{n \in \mathbb{N}} \in \mathcal{A}_m^\mathbb{N}$ which are associated to all possible values $a$ which have a unique expansion. We denote the set of such $x$ (with some more technical restrictions) by $X_{m,\beta} \subset\mathcal{A}_m^\mathbb{N}$. The space $X_{m, \beta}$ is called the symmetric $\beta$-shift associated to the pair $(m, \beta)$. It is invariant by the shift map but in general it is not a subshift of finite type. Given a Hölder continuous potential $A\colon X_{m, \beta} \to\mathbb{R}$, we consider the Ruelle operator $\mathcal{L}_A$ and we show the existence of a positive eigenfunction $\psi_A$ and an eigenmeasure $\rho_A$ for some values of $m$ and $\beta$. We also consider a variational principle of pressure. Moreover, we prove that the family of entropies $(h(\mu_{tA}))_{t>0}$ converges, when $t \to\infty$, to the maximal value among the set of all possible values of entropy of all $A$-maximizing probabilities.

We provide an explicit technical framework for proving very general two-weight commutator estimates in arbitrary parameters. The aim is to both clarify existing literature, which often explicitly focuses on two parameters only, and to extend very recent results to the full generality of arbitrary parameters. More specifically, we study two-weight commutator estimates – Bloom type estimates – in the multi-parameter setting involving weighted product $\operatorname{BMO}$ and little $\operatorname{BMO}$ spaces, and their combinations.