We characterize the validity of the bilinear Hardy inequality for nonincreasing functions \[ \|f^{**} g^{**}\|_{L^q(w)} \le C \|f\|_{\Lambda^{p_1}(v_1)}\|g\|_{\Lambda^{p_2}(v_2)}, \] in terms of the weights $v_1$, $v_2$, $w$, covering the complete range of exponents $p_1,p_2,q\in (0,\infty]$. The problem is solved by reducing it into the iterated Hardy-type inequalities \begin{align*} \left( \int\limits_0^\infty \biggl( \int\limits_0^x (g^{**}(t))^\alpha \varphi(t)\,\mathrm{d}t \biggr)^\frac{\beta}{\alpha} \psi(x)\,\mathrm{d}x \right)^\frac{1}{\beta} & \le C \biggl( \int\limits_0^\infty (g^*(x))^\gamma \omega(x) \,\mathrm{d}x \biggr)^\frac{1}{\gamma}, \\ \left( \int\limits_0^\infty \biggl( \int\limits_x^\infty (g^{**}(t))^\alpha \varphi(t)\,\mathrm{d}t \biggr)^\frac{\beta}{\alpha} \psi(x)\,\mathrm{d}x \right)^\frac{1}{\beta} & \le C \biggl( \int\limits_0^\infty (g^*(x))^\gamma \omega(x) \,\mathrm{d}x \biggr)^\frac{1}{\gamma}. \end{align*} Validity of these inequalities is characterized here for $0\lt\alpha\le\beta\lt\infty$ and $0\lt\gamma\lt\infty$.

Let $S$ be an orientable surface of infinite genus with a finite number of boundary components. In this work we consider the curve complex $\mathcal{C}(S)$, the nonseparating curve complex $\mathcal{N}(S)$, and the Schmutz graph $\mathcal{G}(S)$ of $S$. When all topological ends of $S$ carry genus, we show that all elements in the automorphism groups $\operatorname{Aut}(\mathcal{C}(S))$, $\operatorname{Aut}(\mathcal{N}(S))$, and $\operatorname{Aut}(\mathcal{G}(S))$ are geometric, i.e., these groups are naturally isomorphic to the extended mapping class group $\operatorname{MCG}^{*}(S)$ of the infinite surface $S$. Finally, we study rigidity phenomena within $\operatorname{Aut}(\mathcal{C}(S))$ and $\operatorname{Aut}(\mathcal{N}(S))$.

In this paper we present a new characterization of the Sobolev space $W^{1,p}$, $1\lt p\lt \infty$ which is a higher dimensional version of a result of Waterman [32]. We also provide a new and simplified proof of a recent result of Alabern, Mateu, and Verdera [2]. Finally, we generalize the results to the case of weighted Sobolev spaces with respect to a Muckenhoupt weight.

We provide a formula for the Dubrovnik polynomial of a rational knot in terms of the entries of the tuple associated with a braid-form diagram of the knot. Our calculations can be easily carried out using a computer algebra system.

We consider critical points of the global $L^2$-norm of the second fundamental form, and of the mean curvature vector of isometric immersions of compact Riemannian manifolds into a fixed background Riemannian manifold, as functionals over the space of deformations of the immersion. We prove new gap theorems for these functionals into hyperbolic manifolds, and show that the celebrated gap theorem for minimal immersions into the standard sphere can be cast as a theorem about their critical points having constant mean curvature function, and whose second fundamental form is suitably small in relation to it. In this case, the various minimal submanifolds that occur at the pointwise upper bound on the norm of the second fundamental form are realized by manifolds of nonnegative Ricci curvature, and of these, the Einstein ones are distinguished from the others by being those that are immersed on the sphere as critical points of the first of the functionals mentioned.

We give an example, in the infinite dimensional separable Hilbert space, of a purely unrectifiable Borel set with finite nonzero one dimensional Hausdorff measure, whose projection is nonnegligible in a set of directions which is not Aronszajn null.

We investigate the following question: let $C$ be an integral curve contained in a smooth complex algebraic surface $X$; is it possible to deform $C$ in $X$ into a nodal curve while preserving its geometric genus?

We affirmatively answer it in most cases when $X$ is a Del Pezzo or Hirzebruch surface (this is due to Arbarello and Cornalba, Zariski, and Harris), and in some cases when $X$ is a $K3$ surface. Partial results are given for all surfaces with numerically trivial canonical class. We also give various examples for which the answer is negative.

In this paper we prove existence results and asymptotic behavior for strong solutions $u\in W^{2,2}_{\operatorname{loc}}(\Omega)$ of the nonlinear elliptic problem \begin{equation}\label{abstr} \begin{cases} -\Delta_{H}u+H(\nabla u)^{q}+\lambda u=f&\text{in }\Omega,\\ u\rightarrow +\infty &\text{on }\partial\Omega, \end{cases}\tag{P} \end{equation} where $H$ is a suitable norm of $\mathbb R^{n}$, $\Omega\subset\mathbb R^{n}$ is a bounded domain, $\Delta_{H}$ is the Finsler Laplacian, $1\lt q\le 2$, $\lambda>0$, and $f$ is a suitable function in $L^{\infty}_{\operatorname{loc}}$. Furthermore, we are interested in the behavior of the solutions when $\lambda\rightarrow 0^{+}$, studying the so-called ergodic problem associated to (P). A key role in order to study the ergodic problem will be played by local gradient estimates for (P).

We consider pointwise linear elliptic equations of the form $\mathrm{L}_x u_x = \eta_x$ on a smooth compact manifold where the operators $\mathrm{L}_x$ are in divergence form with real, bounded, measurable coefficients that vary in the space variable $x$. We establish $\mathrm{L}^{2}$-continuity of the solutions at $x$ whenever the coefficients of $\mathrm{L}_x$ are $\mathrm{L}^{\infty}$-continuous at $x$ and the initial datum is $\mathrm{L}^{2}$-continuous at $x$. This is obtained by reducing the continuity of solutions to a homogeneous Kato square root problem. As an application, we consider a time evolving family of metrics $\mathrm{g}_t$ that is tangential to the Ricci flow almost-everywhere along geodesics when starting with a smooth initial metric. Under the assumption that our initial metric is a rough metric on $\mathcal{M}$ with a $\mathrm{C}^{1}$ heat kernel on a ``non-singular'' nonempty open subset $\mathcal{N}$, we show that $x \mapsto \mathrm{g}_t(x)$ is continuous whenever $x \in \mathcal{N}$.

Which spaces look like an $n$-sphere through the eyes of the $n$-th Postnikov section functor and the $n$-connected cover functor? The answer is what we call the Postnikov genus of the $n$-sphere. We define in fact the notion of localization genus for any homotopical localization functor in the sense of Bousfield and Dror Farjoun. This includes exotic genus notions related for example to Neisendorfer localization, or the classical Mislin genus, which corresponds to rationalization.

Let $V$ be an arbitrary vector space over some division ring $D$, $\mathbf{L}$ a general series of subspaces of $V$ covering all of $V\backslash \{0\}$ and $S$ the full stability subgroup of $\mathbf{L}$ in $\operatorname{GL}(V)$. We prove that always the set of bounded right Engel elements of $S$ is equal to the $\omega$-th term of the upper central series of $S$ and that the set of right Engel elements of $S$ is frequently equal to the hypercentre of $S$.