On a generalization of the extended best polynomial approximation operator in Orlicz–Lorentz spaces

In this paper, we consider the best polynomial approximation operator defined on an Orlicz–Lorentz space ${\mathrm{\Lambda }}_{w,\varphi }$, and its extension to ${\mathrm{\Lambda }}_{w,{\varphi }^{\prime }}$, where w is a non-negative continuous weight function and ${\varphi }^{\prime }$ is the derivative of ϕ, which is not required to be an Orlicz function. Our work generalizes a recent result in this field on an Orlicz–Lorentz space generated by an Orlicz function. In addition, we establish some properties and estimates for any extended best polynomial approximation.     

Unbounded operators having self-adjoint,subnormal, or hyponormal powers

We show that if a densely defined closable operator A is such that the resolvent set of A² is nonempty, then A is necessarily closed. This result is then extended to the case of a polynomial $p(A)$. We also generalize a recent result by Sebestyén–Tarcsay concerning the converse of a result by J. von Neumann. Other interesting consequences are also given. One of them is a proof that if T is a quasinormal (unbounded) operator such that $T^n$ is normal for some $n\ge 2$, then T is normal. Hence a closed subnormal operator T such that $T^n$ is normal is itself normal. We also show that if a hyponormal (nonnecessarily bounded) operator A is such that $A^p$ and $A^q$ are self-adjoint for some coprime numbers p and q, then A must be self-adjoint.     

Vanishing of higher order Alexander-type invariants of plane curves

The higher order degrees are Alexander-type invariants of complements to an affine plane curve. In this paper, we characterize the vanishing of such invariants for a curve C given as a transversal union of plane curves $C^{\prime }$ and $C^{\prime \prime }$ in terms of the finiteness and the vanishing properties of the invariants of $C^{\prime }$ and $C^{\prime \prime }$, and whether or not they are irreducible. As a consequence, we prove that the multivariable Alexander polynomial ${\mathrm{\Delta }}_C^{multi}$ is a power of $(t-1)$, and we characterize when ${\mathrm{\Delta }}_C^{multi}=1$ in terms of the defining equations of $C^{\prime }$ and $C^{\prime \prime }$. Our results impose obstructions on the class of groups that can be realized as fundamental groups of complements of a transversal union of curves.     

Theorems of Wolff–Denjoy type for semigroups of nonexpansive mappings in geodesic spaces

We show that if $S={\{f_t:Y\to Y\}}_{t\ge 0}$ is a one-parameter continuous semigroup of nonexpansive mappings acting on a complete locally compact geodesic space $(Y,d)$ that satisfies some geometric properties, then there exists $\xi \in \partial Y$ such that S converge uniformly on bounded sets of Y to ξ. In particular, our result applies to strictly convex bounded domains in ${\mathbb{R}}^n$ or ${\mathbb{C}}^n$ with respect to a large class of metrics including Hilbert's and Kobayashi's metrics.     

Self-repulsiveness of energies for closed submanifolds

We show that the regularized Riesz α-energy for smooth closed submanifolds M in ${\mathbb{R}}^n$ blows up as M degenerates to have double points if $\alpha \le -2dimM$. This gives theoretical foundation of numerical experiments to evolve surfaces to decrease the energy that have been carried out since the 90's.     

Quantifying properties (K) and (μs$\mu ^{s}$)

A Banach space X has property (K), whenever every weak* null sequence in the dual space admits a convex block subsequence ${(f_n)}_{n=1}^{\infty }$ so that $⟨f_n,x_n⟩\to 0$ as $n\to \infty$ for every weakly null sequence ${(x_n)}_{n=1}^{\infty }$ in X; X has property $({\mu }^s)$ if every weak* null sequence in $X^{\ast }$ admits a subsequence so that all of its subsequences are Cesàro convergent to 0 with respect to the Mackey topology. Both property $({\mu }^s)$ and reflexivity (or even the Grothendieck property) imply property (K). In this paper, we propose natural ways for quantifying the aforementioned properties in the spirit of recent results concerning other familiar properties of Banach spaces.     

Elliptic Weingarten hypersurfaces of Riemannian products

Let $M^n$ be either a simply connected space form or a rank-one symmetric space of the noncompact type. We consider Weingarten hypersurfaces of $M\times \mathbb{R}$, which are those whose principal curvatures $k_1,\cdots ,k_n$ and angle function $𝛩$ satisfy a relation $W(k_1,\cdots ,k_n,{𝛩}^2)=0$, being W a differentiable function which is symmetric with respect to $k_1,\cdots ,k_n$. When $\partial W/\partial k_i>0$ on the positive cone of ${\mathbb{R}}^n$, a strictly convex Weingarten hypersurface determined by W is said to be elliptic. We show that, for a certain class of Weingarten functions W, there exist rotational strictly convex Weingarten hypersurfaces of $M\times \mathbb{R}$ which are either topological spheres or entire graphs over M. We establish a Jellett–Liebmann-type theorem by showing that a compact, connected and elliptic Weingarten hypersurface of either ${\mathbb{S}}^n\times \mathbb{R}$ or ${\mathbb{H}}^n\times \mathbb{R}$ is a rotational embedded sphere. Other uniqueness results for complete elliptic Weingarten hypersurfaces of these ambient spaces are obtained. We also obtain existence results for constant scalar curvature hypersurfaces of ${\mathbb{S}}^n\times \mathbb{R}$ and ${\mathbb{H}}^n\times \mathbb{R}$ which are either rotational or invariant by translations (parabolic or hyperbolic). We apply our methods to give new proofs of the main results by Manfio and Tojeiro on the classification of constant sectional curvature hypersurfaces of ${\mathbb{S}}^n\times \mathbb{R}$ and ${\mathbb{H}}^n\times \mathbb{R}$.     

Invariant hypercomplex structures and algebraic curves

We show that $U(k)$-invariant hypercomplex structures on (open subsets) of regular semisimple adjoint orbits in $\mathfrak{g}\mathfrak{l}(k,\mathbb{C})$ correspond to algebraic curves C of genus ${(k-1)}^2$, equipped with a flat projection $\pi :C\to {\mathbb{P}}^1$ of degree k, and an antiholomorphic involution $\sigma :C\to C$ covering the antipodal map on ${\mathbb{P}}^1$.     

On topological and combinatorial structures of pointed stable curves over algebraically closed fields of positive characteristic

In this paper, we study anabelian geometry of curves over algebraically closed fields of positive characteristic. Let $X^{•}=(X,D_X)$ be a pointed stable curve over an algebraically closed field of characteristic $p>0$ and ${\mathrm{\Pi }}_{X^{•}}$ the admissible fundamental group of $X^{•}$. We prove that there exists a group-theoretical algorithm, whose input datum is the admissible fundamental group ${\mathrm{\Pi }}_{X^{•}}$, and whose output data are the topological and the combinatorial structures associated with $X^{•}$. This result can be regarded as a mono-anabelian version of the combinatorial Grothendieck conjecture in the positive characteristic. Moreover, by applying this result, we construct clutching maps for moduli spaces of admissible fundamental groups.     

On the Galois covers of degenerations of surfaces of minimal degree

We investigate the topological structures of Galois covers of surfaces of minimal degree (i.e., degree n) in ${\mathbb{CP}}^{n+1}$. We prove that for $n\ge 5$, the Galois covers of any surfaces of minimal degree are simply-connected surfaces of general type.     

A Runge theorem for generalized analytic functions

We show an approximation theorem of Runge type for solutions of the generalized Vekua equation $Lu=Au+B\overline{u}$, where L belongs to a class of degenerate elliptic planar vector fields and $A,B\in L^p$. To prove the theorem, we make use of an integral representation for the solutions of the generalized Vekua equation valid on relatively compact sets. As an application, we study the global solvability of the equation $Lu=Au+B\overline{u}+f$ with $f\in L^p$ and some of its consequences.     

Normalized ground state for the Sobolev critical Schrödinger equation involving Hardy term with combined nonlinearities

In this paper, we study the existence and properties of normalized solutions for the following Sobolev critical Schrödinger equation involving Hardy term: $$-\mathrm{\Delta }u-\frac{\mu }{{|x|}^2}u=\lambda u+{|u|}^{2^{\ast }-2}u+\nu {|u|}^{p-2}u\text{in}{\mathbb{R}}^N,N\ge 3,$$ with prescribed mass $${\int }_{{\mathbb{R}}^N}{u}^2=a^2,$$ where 2* is the Sobolev critical exponent. For a L²-subcritical, L²-critical, or L²-supercritical perturbation ${\nu |u|}^{p-2}u$, we prove several existence results of normalized ground state when $\nu \ge 0$ and non-existence results when $\nu \le 0$. Furthermore, we also consider the asymptotic behavior of the normalized solutions u as $\mu \to 0$ or $\nu \to 0$.     

On the volume functional of compact manifolds with harmonic Weyl tensor

The main aim of this article is to give the complete classification of critical metrics of the volume functional on a compact manifold M with boundary $\partial M$ and under the harmonic Weyl tensor condition. In particular, we prove that a critical metric with a harmonic Weyl tensor on a simply connected compact manifold with the boundary isometric to a standard sphere ${\mathbb{S}}^{n-1}$ must be isometric to a geodesic ball in a simply connected space form ${\mathbb{R}}^n$, ${\mathbb{H}}^n$, and ${\mathbb{S}}^n$. To this end, we first conclude the classification of such critical metrics under the Bach-flat assumption and then we prove that both geometric conditions are equivalent in this situation.     

Dimension-free square function estimates for Dunkl operators

Dunkl operators may be regarded as differential-difference operators parameterized by finite reflection groups and multiplicity functions. In this paper, the Littlewood–Paley square function for Dunkl heat flows in ${\mathbb{R}}^d$ is introduced by employing the full “gradient” induced by the corresponding carré du champ operator and then the $L^p$ boundedness is studied for all $p\in (1,\infty )$. For $p\in (1,2]$, we successfully adapt Stein's heat flows approach to overcome the difficulty caused by the difference part of the Dunkl operator and establish the $L^p$ boundedness, while for $p\in [2,\infty )$, we restrict to a particular case when the corresponding Weyl group is isomorphic to ${\mathbb{Z}}_2^d$ and apply a probabilistic method to prove the $L^p$ boundedness. In the latter case, the curvature-dimension inequality for Dunkl operators in the sense of Bakry–Emery, which may be of independent interest, plays a crucial role. The results are dimension-free.     

Thom property and Milnor–Lê fibration for analytic maps

Let (X, 0) be the germ of either a subanalytic set $X\subset {\mathbb{R}}^n$ or a complex analytic space $X\subset {\mathbb{C}}^n$, and let $f:(X,0)\to ({\mathbb{K}}^k,0)$ be a $\mathbb{K}$-analytic map-germ, with $\mathbb{K}=\mathbb{R}$ or $\mathbb{C}$, respectively. When $k=1$, there is a well-known topological locally trivial fibration associated with f, called the Milnor–Lê fibration of f, which is one of the main pillars in the study of singularities of maps and spaces. However, when $k>1$ that is not always the case. In this paper, we give conditions which guarantee that the image of f is well-defined as a set-germ, and that f admits a Milnor–Lê fibration. We also give conditions for f to have the Thom property. Finally, we apply our results to mixed function-germs of type $f\overline{g}:(X,0)\to (\mathbb{C},0)$ on a complex analytic surface $X\subset {\mathbb{C}}^n$ with arbitrary singularity.     

An interpolation problem in the Denjoy–Carleman classes

Inspired by some iterative algorithms useful for proving the real analyticity (or the Gevrey regularity) of a solution of a linear partial differential equation with real-analytic coefficients, we consider the following question. Given a smooth function defined on $[a,b]\subset \mathbb{R}$ and given an increasing divergent sequence $d_n$ of positive integers such that the derivative of order $d_n$ of f has a growth of the type $M_{d_n}$, when can we deduce that f is a function in the Denjoy–Carleman class $C^M([a,b])$? We provide a positive result and show that a suitable condition on the gaps between the terms of the sequence $d_n$ is needed.