The Roles Played by Order of Convexity or Starlikeness and the Bloch Condition in the Extension of Mappings from the Disk to the Ball

Let ${G: \mathbb {C}^{n-1} \rightarrow \mathbb {C}}$ be holomorphic such that G(0)?=?0 and DG(0)?=?0. When f is a convex (resp. starlike) normalized (f(0)?=?0, f??(0)?=?1) univalent mapping of the unit disk ${\mathbb {D}}$ in ${\mathbb {C}}$ , then the extension of f to the Euclidean unit ball ${\mathbb {B}}$ in ${\mathbb {C}^n}$ given by ${\Phi_G(f)(z)=(f(z_1)+G(\sqrt{f^{\prime}(z_1)} \, \hat{z}),\sqrt{f^{\prime}(z_1)}\, \hat{z})}$ , ${\hat{z}=(z_2,\dots,z_n) \in \mathbb {C}^{n-1}}$ , is known to be convex (resp. starlike) if G is a homogeneous polynomial of degree 2 with sufficiently small norm. Conversely, it is known that G cannot have terms of degree greater than 2 in its expansion about 0 in order for ${\Phi_G(f)}$ to be convex (resp. starlike), in general. We examine whether the restriction that f be either convex or starlike of a certain order ${\alpha \in (0,1]}$ allows, in general, for G to contain terms of degree greater than 2 and still have ${\Phi_G(f)}$ maintain the respective geometric property. Related extension results for convex and starlike Bloch mappings are also given.     

On generalized Dhombres equations with nonconstant polynomial solutions in the complex plane

We characterize functional equations of the form ${f(zf(z))=f(z)^{k+1},z\in\mathbb {C}}$ , with ${k\in\mathbb N}$ , like those generalized Dhombres equations ${f(zf(z))=\varphi (f(z))}$ , ${z\in\mathbb C}$ , with given entire function ${\varphi}$ , which have a nonconstant polynomial solution f.     

Saddle solutions for bistable symmetric semilinear elliptic equations

This paper concerns the existence and asymptotic characterization of saddle solutions in ${\mathbb {R}^{3}}$ for semilinear elliptic equations of the form $$-\Delta u + W'(u) = 0,\quad (x, y, z) \in {\mathbb {R}^{3}} \qquad\qquad\qquad (0.1)$$ where ${W \in \mathcal{C}^{3}(\mathbb {R})}$ is a double well symmetric potential, i.e. it satisfies W(?s) =  W(s) for ${s \in \mathbb {R},W(s) > 0}$ for ${s \in (-1,1)}$ , ${W(\pm 1) = 0}$ and ${W''(\pm 1) > 0}$ . Denoted with ${\theta_{2}}$ the saddle planar solution of (0.1), we show the existence of a unique solution ${\theta_{3} \in {\mathcal{C}^{2}}(\mathbb {R}^{3})}$ which is odd with respect to each variable, symmetric with respect to the diagonal planes, verifies ${0 < \theta_{3}(x,y,z) < 1}$ for x, y, z >  0 and ${\theta_{3}(x, y, z) \to_{z \to + \infty} \theta_{2}(x, y)}$ uniformly with respect to ${(x, y) \in \mathbb {R}^{2}}$ .     

Some Remarks to the Formal and Local Theory of the Generalized Dhombres Functional Equation

We are looking for local analytic respectively formal solutions of the generalized Dhombres functional equation ${f(zf(z))=\varphi(f(z))}$ in the complex domain. First we give two proofs of the existence theorem about solutions f with f(0) = w ₀ and ${w_0 \in \mathbb{C}^\star {\setminus}\mathbb{E}}$ where ${\mathbb{E}}$ denotes the group of complex roots of 1. Afterwards we represent solutions f by means of infinite products where we use on the one hand the canonical convergence of complex analysis, on the other hand we show how solutions converge with respect to the weak topology. In this section we also study solutions where the initial value z ₀ is different from zero.     

Generalized Complex Spherical Harmonics, Frame Functions, and Gleason Theorem

Consider a finite dimensional complex Hilbert space ${\mathcal{H}}$ , with ${dim(\mathcal{H}) \geq 3}$ , define ${\mathbb{S}(\mathcal{H}):= \{x\in \mathcal{H} \:|\: \|x\|=1\}}$ , and let ${\nu_\mathcal{H}}$ be the unique regular Borel positive measure invariant under the action of the unitary operators in ${\mathcal{H}}$ , with ${\nu_\mathcal{H}(\mathbb{S}(\mathcal{H}))=1}$ . We prove that if a complex frame function ${f : \mathbb{S}(\mathcal{H})\to \mathbb{C}}$ satisfies ${f \in \mathbb{L}^2(\mathbb{S}(\mathcal{H}), \nu_\mathcal{H})}$ , then it verifies Gleason’s statement: there is a unique linear operator ${A: \mathcal{H} \to \mathcal{H}}$ such that ${f(u) = \langle u| A u\rangle}$ for every ${u \in \mathbb{S}(\mathcal{H}).\,A}$ is Hermitean when f is real. No boundedness requirement is thus assumed on f a priori.     

Functional inequalities related to Mahler’s conjecture

Extending a result of Meyer and Reisner (Monatsh Math 125:219–227, 1998), we prove that if ${g: \mathbb{R}\to \mathbb{R}_+}$ is a function which is concave on its support, then for every m > 0 and every ${z\in\mathbb{R}}$ such that g(z) > 0, one has $$ \int\limits_{\mathbb{R}} g(x)^mdx\int\limits_{\mathbb{R}} (g^{*z}(y))^m dy\ge \frac{(m+2)^{m+2}}{(m+1)^{m+3}},$$ where for ${y\in \mathbb{R}}$ , ${g^{*z}(y)=\inf_x \frac{(1-(x-z)y)_+}{g(x)}}$ . It is shown how this inequality is related to a special case of Mahler’s conjecture (or inverse Santaló inequality) for convex bodies. The same ideas are applied to give a new (and simple) proof of the exact estimate of the functional inverse Santaló inequality in dimension 1 given in Fradelizi and Meyer (Adv Math 218:1430–1452, 2008). Namely, if ${\phi:\mathbb{R}\to\mathbb{R}\cup\{+\infty\}}$ is a convex function such that ${0 < \int e^{-\phi} < +\infty}$ then, for every ${z\in\mathbb{R}}$ such that ${\phi(z) < +\infty}$ , one has $$ \int\limits_{\mathbb{R}}e^{-\phi}\int\limits_{\mathbb{R}} e^{-\mathcal{L}^z\phi}\ge e,$$ where ${\mathcal {L}^z\phi}$ is the Legendre transform of ${\phi}$ with respect to z.     

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 ${\mathbb {R}^{3}}$ .     

On the topology of semi-algebraic functions on closed semi-algebraic sets

We consider a closed semi-algebraic set ${X \subset \mathbb{R}^n}$ and a C ² semi-algebraic function ${f : \mathbb{R}^n \rightarrow\mathbb{R}}$ such that ${f_{\vert X}}$ has a finite number of critical points. We relate the topology of X to the topology of the sets ${X \cap \{ f * \alpha \}}$ , where ${* \in \{\le,=,\ge \}}$ and ${\alpha \in \mathbb{R}}$ , and the indices of the critical points of ${f_{\vert X}}$ and ${-f_{\vert X}}$ . We also relate the topology of X to the topology of the links at infinity of the sets ${X \cap \{ f * \alpha\}}$ and the indices of these critical points. We give applications when ${X=\mathbb{R}^n}$ and when f is a generic linear function.     

Compact Hankel Operators on Generalized Bergman Spaces of the Polydisc

Let ${\vartheta}$ be a measure on the polydisc ${\mathbb{D}^n}$ which is the product of n regular Borel probability measures so that ${\vartheta([r,1)^n\times\mathbb{T}^n) >0 }$ for all 0 < r < 1. The Bergman space ${A^2_{\vartheta}}$ consists of all holomorphic functions that are square integrable with respect to ${\vartheta}$ . In one dimension, it is well known that if f is continuous on the closed disc ${\overline{\mathbb{D}}}$ , then the Hankel operator H _f is compact on ${A^2_\vartheta}$ . In this paper we show that for n ≥ 2 and f a continuous function on ${{\overline{\mathbb{D}}}^n}$ , H _f is compact on ${A^2_\vartheta}$ if and only if there is a decomposition f = h + g, where h belongs to ${A^2_\vartheta}$ and ${\lim_{z\to\partial\mathbb{D}^n}g(z)=0}$ .     

Cohomology of Some Complex Laminations

In this paper we solve the ${\overline{\partial }}$ -problem along the leaves for two types of laminations: (i) Some open sets Ω of ${{\mathbb C}\times B}$ (where B is any differentiable manifold) endowed with the canonical foliation that is, the foliation whose leaves are the sections ${\Omega ^t=\{ z\in {\mathbb C}:(z,t)\in \Omega \}}$ . We construct a solution to the equation ${\overline{\partial }h=fd\overline z}$ for any function ${f:\Omega\longrightarrow {\mathbb C}}$ of class ${C^{s}\,(s\in \mathbb{N}\cup\{ \infty \}),\,C^\infty}$ along the leaves and satisfies some growth conditions near the singularities. (ii) A complex lamination by Riemann surfaces obtained by suspending a homeomorphism of a closed set of the Euclidean space ${\mathbb{C}\times \mathbb{R}}$ .     

Donsker-type theorems for nonparametric maximum likelihood estimators

Let ${\mathcal{P}}$ be a nonparametric probability model consisting of smooth probability densities and let ${\hat{p}_{n}}$ be the corresponding maximum likelihood estimator based on n independent observations each distributed according to the law ${\mathbb{P}}$ . With $\hat{\mathbb{P}}_{n}$ denoting the measure induced by the density ${\hat{p}_{n}}$ , define the stochastic process ${\hat{\nu}}_{n}: f\longmapsto \sqrt{n} \int fd({\hat{\mathbb{P}}}_{n} -\mathbb{P})$ where f ranges over some function class ${\mathcal{F}}$ . We give a general condition for Donsker classes ${\mathcal{F}}$ implying that the stochastic process $\hat{\nu}_{n}$ is asymptotically equivalent to the empirical process in the space ${\ell ^{\infty }(\mathcal{F})}$ of bounded functions on ${ \mathcal{F}}$ . This implies in particular that $\hat{\nu}_{n}$ converges in law in ${\ell ^{\infty }(\mathcal{F})}$ to a mean zero Gaussian process. We verify the general condition for a large family of Donsker classes ${\mathcal{ F}}$ . We give a number of applications: convergence of the probability measure ${\hat{\mathbb{P}}_{n}}$ to ${\mathbb{P}}$ at rate ${\sqrt{n}}$ in certain metrics metrizing the topology of weak(-star) convergence; a unified treatment of convergence rates of the MLE in a continuous scale of Sobolev-norms; ${\sqrt{n}}$ -efficient estimation of nonlinear functionals defined on ${\mathcal{P}}$ ; limit theorems at rate ${\sqrt{n}}$ for the maximum likelihood estimator of the convolution product ${\mathbb{P\ast P}}$ .     

Uniform existential interpretation of arithmetic in rings of functions of positive characteristic

We show that first order integer arithmetic is uniformly positive-existentially interpretable in large classes of (subrings of) function fields of positive characteristic over some languages that contain the language of rings. One of the main intermediate results is a positive existential definition (in these classes), uniform among all characteristics p, of the binary relation “ $y=x^{p^{s}}$ or $x=y^{p^{s}}$ for some integer s≥0”. A natural consequence of our work is that there is no algorithm to decide whether or not a system of polynomial equations over $\mathbb {Z}[z]$ has solutions in all but finitely many polynomial rings $\mathbb {F}_{p}[z]$ . Analogous consequences are deduced for the rational function fields $\mathbb {F}_{p}(z)$ , over languages with a predicate for the valuation ring at zero.     

On the Linear Dependence of a Finite Set of Additive Functions

In this note we prove the following: Let n?≥ 2 be a fixed integer. A system of additive functions ${A_{1},A_{2},\ldots,A_{n}:\mathbb{R} \to\mathbb{R}}$ is linearly dependent (as elements of the ${\mathbb{R}}$ vector space ${\mathbb{R}^{\mathbb{R}}}$ ), if and only if, there exists an indefinite quadratic form ${Q:\mathbb{R}^{n}\to\mathbb{R} }$ such that ${Q(A_{1}(x),A_{2}(x),\ldots,A_{n}(x))\geq 0}$ or ${Q(A_{1}(x),A_{2}(x),\ldots,A_{n}(x))\leq 0}$ holds for all ${x\in\mathbb{R}}$ .     

Bent functions embedded into the recursive framework of {\mathbb{Z}} -bent functions

Suppose that n is even. Let ${\mathbb{F}_2}$ denote the two-element field and ${\mathbb{Z}}$ the set of integers. Bent functions can be defined as ± 1-valued functions on ${\mathbb{F}_2^n}$ with ± 1-valued Fourier transform. More generally we call a mapping f on ${\mathbb{F}_2^n}$ a ${\mathbb{Z}}$ -bent function if both f and its Fourier transform ${\widehat{f}}$ are integer-valued. ${\mathbb{Z}}$ -bent functions f are separated into different levels, depending on the size of the maximal absolute value attained by f and ${\widehat{f}}$ . It is shown how ${\mathbb{Z}}$ -bent functions of lower level can be built up recursively by gluing together ${\mathbb{Z}}$ -bent functions of higher level. This recursion comes down at level zero, containing the usual bent functions. In the present paper we start to study bent functions in the framework of ${\mathbb{Z}}$ -bent functions and give some guidelines for further research.     

Singular Solutions of the Generalized Dhombres Functional Equation

We consider singular solutions of the functional equation ${f(xf(x)) = \varphi (f(x))}$ where ${\varphi}$ is a given and f an unknown continuous map ${\mathbb R_{+} \rightarrow \mathbb R_{+}}$ . A solution f is regular if the sets ${R_f \cap (0, 1]}$ and ${R_f \cap [1, \infty)}$ , where R _f is the range of f, are ${\varphi}$ -invariant; otherwise f is singular. We show that for singular solutions the associated dynamical system ${({R_f}, \varphi|_{R_f})}$ can have strange properties unknown for the regular solutions. In particular, we show that ${\varphi |_{R_f}}$ can have a periodic point of period 3 and hence can be chaotic in a strong sense. We also provide an effective method of construction of singular solutions.     

Weighted Composition Operators from the Minimal Möbius Invariant Space into the Bloch Space

Let ${{\varphi}}$ be an analytic self-map of the open unit disk ${{\mathbb{D}}}$ in the complex plane ${{\mathbb{C}, H(\mathbb{D})}}$ the space of complex-valued analytic functions on ${{\mathbb{D}}}$ , and let u be a fixed function in ${{H(\mathbb{D})}}$ . The weighted composition operator is defined by $$(uC_{\varphi}f)(z) = u(z)f({\varphi}(z)), \quad z \in \mathbb{D}, f \in H(\mathbb{D}).$$ In this paper, we study the boundedness and the compactness of the weighted composition operators from the minimal Möbius invariant space into the Bloch space and the little Bloch space.     

Congruence Properties of Binary Partition Functions

Let ${\mathcal{A}}$ be a finite subset of ${\mathbb{N}}$ containing 0, and let f (n) denote the number of ways to write n in the form ${\sum \varepsilon _{j}2^{j}}$ , where ${\varepsilon _{j} \epsilon \mathcal{A}}$ . We show that there exists a computable ${T = T (\mathcal{A})}$ so that the sequence (f (n) mod 2) is periodic with period T. Variations and generalizations of this problem are also discussed.     

Boundedness and Compactness of Pseudodifferential Operators with Non-Regular Symbols on Weighted Lebesgue Spaces

Applying the boundedness on weighted Lebesgue spaces of the maximal singular integral operator S _* related to the Carleson?CHunt theorem on almost everywhere convergence, we study the boundedness and compactness of pseudodifferential operators a(x, D) with non-regular symbols in ${L^\infty(\mathbb{R}, V(\mathbb{R})), PC(\overline{\mathbb{R}}, V(\mathbb{R}))}$ and ${\Lambda_\gamma(\mathbb{R}, V_d(\mathbb{R}))}$ on the weighted Lebesgue spaces ${L^p(\mathbb{R},w)}$ , with 1?< p <? ?? and ${w\in A_p(\mathbb{R})}$ . The Banach algebras ${L^\infty(\mathbb{R}, V(\mathbb{R}))}$ and ${PC(\overline{\mathbb{R}}, V(\mathbb{R}))}$ consist, respectively, of all bounded measurable or piecewise continuous ${V(\mathbb{R})}$ -valued functions on ${\mathbb{R}}$ where ${V(\mathbb{R})}$ is the Banach algebra of all functions on ${\mathbb{R}}$ of bounded total variation, and the Banach algebra ${\Lambda_\gamma(\mathbb{R}, V_d(\mathbb{R}))}$ consists of all Lipschitz ${V_d(\mathbb{R})}$ -valued functions of exponent ${\gamma \in (0,1]}$ on ${\mathbb{R}}$ where ${V_d(\mathbb{R})}$ is the Banach algebra of all functions on ${\mathbb{R}}$ of bounded variation on dyadic shells. Finally, for the Banach algebra ${\mathfrak{A}_{p,w}}$ generated by all pseudodifferential operators a(x, D) with symbols ${a(x, \lambda) \in PC(\overline{\mathbb{R}}, V(\mathbb{R}))}$ on the space ${L^p(\mathbb{R}, w)}$ , we construct a non-commutative Fredholm symbol calculus and give a Fredholm criterion for the operators ${A \in \mathfrak{A}_{p,w}}$ .     

Classification of rotational special Weingarten surfaces of minimal type in {\mathbb{S}^2 \times \mathbb{R}} and {\mathbb{H}^2 \times \mathbb{R}}

In this paper we classify the complete rotational special Weingarten surfaces in ${\mathbb{S}^2 \times \mathbb{R}}$ and ${\mathbb{H}^2 \times \mathbb{R}}$ ; i.e. rotational surfaces in ${\mathbb{S}^2 \times \mathbb{R}}$ and ${\mathbb{H}^2 \times \mathbb{R}}$ whose mean curvature H and extrinsic curvature K _e satisfy H = f(H ² ? K _e ), for some function ${f \in \mathcal{C}^1([0,+\infty))}$ such that f(0) = 0 and 4x(f′(x))² < 1 for any x ≥ 0. Furthermore we show the existence of non-complete examples of such surfaces.