Beurling’s theorem for SL(2,$${\mathbb{R}}$$)

We prove Beurling’s theorem for the full group SL(2,). This is the master theorem in the quantitative uncertainty principle as all the other theorems of this genre follow from it.     

A Generalization Of Reifenberg’s Theorem In {\mathbb{R}}^3

In 1960 Reifenberg proved the topological disc property. He showed that a subset of which is well approximated by m-dimensional affine spaces at each point and at each (small) scale is locally a bi-H?lder image of the unit ball in . In this paper we prove that a subset of which is well approximated in the Hausdorff distance sense by one of the three standard area-minimizing cones at each point and at each (small) scale is locally a bi-H?lder deformation of a minimal cone. We also prove an analogous result for more general cones in . Received: July 2006, Revised: August 2007, Accepted: January 2008     

On Nikol’skii Inequalities for Domains in $${\mathbb {R}}^d$$

Nikol’skii inequalities for various sets of functions, domains, and weights will be discussed. Much of the work is dedicated to the class of algebraic polynomials of total degree n on a bounded convex domain D. That is, we study \(\sigma := \sigma (D,d)\) for which

$$\begin{aligned} \Vert P\Vert _{L_q(D)}\le c n^{\sigma (\frac{1}{p}-\frac{1}{q})}\Vert P\Vert _{L_p(D)},\quad 0<p\le q\le \infty , \end{aligned}$$

where P is a polynomial of total degree n. We use geometric properties of the boundary of D to determine \(\sigma (D,d)\) with the aid of comparison between domains. Computing the asymptotics of the Christoffel function of various domains is crucial in our investigation. The methods will be illustrated by the numerous examples in which the optimal \(\sigma (D,d)\) will be computed explicitly.

     

Möbius parametrizations of curves in {\mathbb{R}}^n

We use Ahlfors’ definition of Schwarzian derivative for curves in euclidean spaces to present new results about M?bius or projective parametrizations. The class of such parametrizations is invariant under compositions with M?bius transformations, and the resulting curves are simple. The analysis is based on the oscillatory behavior of the associated linear equation , where k = k(s) is the curvature as a function of arclength. Received: 24 November 2008     

Pitt’s Inequality and Logarithmic Uncertainty Principle for the Dunkl Transform on {\mathbb{R}}

We establish Pitt’s inequality and deduce Beckner’s logarithmic uncertainty principle for the Dunkl transform on \({\mathbb{R}}\) . Also, we prove Stein–Weiss inequality for the Dunkl–Bessel potentials.     

Interpolation Between $$H^{p(\cdot )}({\mathbb {R}}^n)$$ and $$L^\infty ({\mathbb {R}}^n)$$L∞(Rn): Real Method

Let \(p(\cdot ):\ {\mathbb {R}}^n\rightarrow (0,\infty )\) be a variable exponent function satisfying the globally log-Hölder continuous condition. In this article, the authors first obtain a decomposition for any distribution of the variable weak Hardy space into “good” and “bad” parts and then prove the following real interpolation theorem between the variable Hardy space \(H^{p(\cdot )}({\mathbb {R}}^n)\) and the space \(L^{\infty }({\mathbb {R}}^n)\): \((H^{p(\cdot )}(\mathbb R^n),L^{\infty }({\mathbb {R}}^n))_{\theta ,\infty }= WH^{p(\cdot )/(1-\theta )}({\mathbb {R}}^n),\quad \mathrm{where}~\theta \in (0,1), \mathrm{and}\) \(WH^{p(\cdot )/(1-\theta )}({\mathbb {R}}^n)\) denotes the variable weak Hardy space. As an application, the variable weak Hardy space \(WH^{p(\cdot )}({\mathbb {R}}^n)\) with \(p_-:=\mathop {\text {ess inf}}\limits _{x\in {{{\mathbb {R}}}^n}}p(x)\in (1,\infty )\) is proved to coincide with the variable Lebesgue space \(WL^{p(\cdot )}({\mathbb {R}}^n)\).     

在${\mathbb{R}}^m$中重对数广义律的精确速率

令\{$X$, $X_n$, $n\ge 1$\}是期望为${\mathbb{E}}X=(0,\ldots,0)_{m\times 1}$和协方差阵为${\rm Cov}(X,X)=\sigma^2I_m$的独立同分布的随机向量列, 记$S_n=\sum_{i=1}^{n}X_i$, $n\ge 1$. 对任意$d>0$和$a_n=o((\log\log n)^{-d})$, 本文研究了${{\mathbb{P}}(|S_n|\ge (\varepsilon+a_n)\sigma \sqrt{n}(\log\log n)^d)$的一类加权无穷级数的重对数广义律的精确速率.     

Classification of solutions to the higher order Liouville’s equation on {\mathbb{R}^{2m}}

We classify the solutions to the equation (−Δ) ^m u = (2m − 1)!e ^2mu on giving rise to a metric with finite total Q-curvature in terms of analytic and geometric properties. The analytic conditions involve the growth rate of u and the asymptotic behaviour of Δu at infinity. As a consequence we give a geometric characterization in terms of the scalar curvature of the metric at infinity, and we observe that the pull-back of this metric to S ^2m via the stereographic projection can be extended to a smooth Riemannian metric if and only if it is round.     

Time-periodic solutions of the compressible Navier–Stokes equations in {\mathbb{R}^{4}}

This paper deals with the existence of time-periodic solutions to the compressible Navier–Stokes equations effected by general form external force in \({\mathbb{R}^{N}}\) with \({N = 4}\). Using a fixed point method, we establish the existence and uniqueness of time-periodic solutions. This paper extends Ma, UKai, Yang’s result [5], in which, the existence is obtained when the space dimension \({N \ge 5}\).     

Uniformisation de l’espace des feuilles de certains feuilletages de codimension un

This paper deals with codimension one (may be singular) foliations on compact Kälher manifoldswhose conormal bundle is assumed to be pseudo-effective. Using currents with minimal singularities, we show that one can endow the space of leaves with a metric of constant non positive curvature wich may degenerate on a “rigidly” embedded invariant hypersurface.     

Deitmar’s versus To?n-Vaquié’s schemes over {\mathbb{F}_{1}}

Deitmar introduced schemes over ${\mathbb {F}_{1}}$ , the so-called “field with one element”, as certain spaces with an attached sheaf of monoids, generalizing the definition of schemes as ringed spaces. On the other hand, To?n and Vaquié defined them as particular Zariski sheaves over the opposite category of monoids, generalizing the definition of schemes as functors of points. We show the equivalence between Deitmar’s and To?n-Vaquiés notions and establish an analog of the classical case of schemes over ${\mathbb {Z}}$ . This result has been assumed by the leading experts on ${\mathbb {F}_{1}}$ , but no proof was given. During the proof, we also conclude some new basic results on commutative algebra of monoids, such as a characterization of local flat epimorphisms and of flat epimorphisms of finite presentation. We also inspect the base-change functors from the category of schemes over ${\mathbb {F}_{1}}$ to the category of schemes over ${\mathbb {Z}}$ .     

Fredholm and regularity theory of Douglis–Nirenberg elliptic systems on {\mathbb{R}^{N}}

We give a fairly complete exposition of the Fredholm properties of the Douglis–Nirenberg elliptic systems on ${\mathbb{R}^{N}}$ in the classical (unweighted) L ^p Sobolev spaces and under “minimal” assumptions about the coefficients. These assumptions rule out the use of classical pseudodifferential operator theory, although it is indirectly of assistance in places. After generalizing a necessary and sufficient condition for Fredholmness, already known in special cases, various invariance properties are established (index, null space, etc.), with respect to p and the Douglis–Nirenberg numbers. Among other things, this requires getting around the problem that the L ^p spaces are not ordered by inclusion. In turn, with some work, invariance leads to a regularity theory more general than what can be obtained by the method of differential quotients.     

Sur la densité des représentations cristallines de \text{ Gal}(\overline{\mathbb Q }_p/\mathbb Q _p)

Let ${\mathfrak X }_d$ be the $p$ -adic analytic space classifying the semisimple continuous representations $\text{ Gal}(\overline{\mathbb Q }_p/\mathbb Q _p) \rightarrow \mathrm GL _d(\overline{\mathbb Q }_p)$ . We show that the crystalline representations are Zarski-dense in many irreducible components of ${\mathfrak X }_d$ , including the components made of residually irreducible representations. This extends to any dimension $d$ previous results of Colmez and Kisin for $d = 2$ . For this we construct an analogue of the infinite fern of Gouvêa–Mazur in this context, based on a study of analytic families of trianguline $(\varphi ,\Gamma )$ -modules over the Robba ring. We show in particular the existence of a universal family of (framed, regular) trianguline $(\varphi ,\Gamma )$ -modules, as well as the density of the crystalline $(\varphi ,\Gamma )$ -modules in this family. These results may be viewed as a local analogue of the theory of $p$ -adic families of finite slope automorphic forms and they are new already in dimension $2$ . The technical heart of the paper is a collection of results about the Fontaine–Herr cohomology of families of trianguline $(\varphi ,\Gamma )$ -modules.     

The Helmholtz Operator on Higher Dimensional M?bius Strips Embedded in {\mathbb{R}^{4}}

In this paper we present explicit formulas for the fundamental solution to the Helmholtz operator on a higher-dimensional analogue of the M?bius strip in three real variables (embedded in ${\mathbb{R}^{4}}$ ) with values in distinct pinor bundles. Herefore we use an approach that uses classical harmonic analysis methods combined with some Clifford analysis tools and adapt it to this special geometry. The fundamental solution is described in terms of generalizations of the Weierstrass ${\wp}$ -function that are adapted to the context of these geometries. As our main result we present an analytic integral representation formula to express the solutions of the inhomogeneous time-independent Klein-Gordon problem on M?bius strips.     

Lp estimates for non-smooth bilinear Littlewood–Paley square functions on $${\mathbb{R}}$$

In this work, we study some non-smooth bilinear analogues of linear Littlewood–Paley square functions on the real line. We prove boundedness-properties in Lebesgue spaces for them. Let us consider the functions \({\phi_{n}}\) satisfying \({\widehat{\phi_n}(\xi)={\bf 1}_{[n,n+1]}(\xi)}\) and define the bilinear operator \({S_n(f,g)(x):=\int f(x+y)g(x-y) \phi_n(y) dy}\) . These bilinear operators are closely related to the bilinear Hilbert transforms. Then for exponents \({p,q,r'\in[2,\infty)}\) satisfying \({\frac{1}{p}+\frac{1}{q}=\frac{1}{r}}\) , we prove that

$\left\| \left( \sum_{n\in \mathbb{Z}}\left|S_n(f,g) \right|^2 \right)^{1/2}\right\|_{L^{r}(\mathbb{R})}\lesssim \|f\|_{L^p(\mathbb{R})}\|g\|_{L^q(\mathbb{R})}.$

     

Borsuk’s Partition Problem in $$(mathbb{R}^{n},ell_{p})$$

Mathematical Notes - For a bounded set $$X$$ with diameter $$d_{C}(X)$$ in a finite-dimensional normed space with an origin-symmetric convex body $$C$$ as the unit ball, the Borsuk number of $$X$$...     

On the Fučik point spectrum for Schrödinger operators on {\mathbb{R}}^{N}

We investigate the Fučik point spectrum of the Schr?dinger operator when the potential V_λ has a steep potential well for sufficiently large parameter λ > 0. It is allowed that S_λ has essential spectrum with finitely many eigenvalues below the infimum of . We construct the first nontrivial curve in the Fučik point spectrum by minimax methods and show some qualitative properties of the curve and the corresponding eigenfunctions. As applications we establish some results on existence of multiple solutions for nonlinear Schr?dinger equations with jumping nonlinearity.      

Existence of Isoperimetric Sets with Densities “Converging from Below” on $${\mathbb {R}}^N$$

In this paper, we consider the isoperimetric problem in the space \({\mathbb {R}}^N\) with a density. Our result states that, if the density f is lower semi-continuous and converges to a limit \(a>0\) at infinity, with \(f\le a\) far from the origin, then isoperimetric sets exist for all volumes. Several known results or counterexamples show that the present result is essentially sharp. The special case of our result for radial and increasing densities positively answers a conjecture of Morgan and Pratelli (Ann Glob Anal Geom 43(4):331–365, 2013.     

Layered solutions with multiple asymptotes for non autonomous Allen–Cahn equations in {\mathbb {R}^{3}}

We consider a class of semilinear elliptic equations of the form $$ \label{eq:abs}-\Delta u(x,y,z)+a(x)W'(u(x,y,z))=0,\quad (x,y,z)\in\mathbb {R}^{3},$$ where ${a:\mathbb {R} \to \mathbb {R}}$ is a periodic, positive, even function and, in the simplest case, ${W : \mathbb {R} \to \mathbb {R}}$ is a double well even potential. Under non degeneracy conditions on the set of minimal solutions to the one dimensional heteroclinic problem $$-\ddot q(x)+a(x)W^{\prime}(q(x))=0,\ x\in\mathbb {R},\quad q(x)\to\pm1\,{\rm as}\, x\to \pm\infty,$$ we show, via variational methods the existence of infinitely many geometrically distinct solutions u of (0.1) verifying u(x, y, z) → ± 1 as x → ± ∞ uniformly with respect to ${(y, z) \in \mathbb {R}^{2}}$ and such that ${\partial_{y}u \not \equiv0, \partial_{z}u \not\equiv 0}$ in ${\mathbb {R}^{3}}$ .