Showing papers in "Revista Matematica Complutense in 2014"

Pointwise properties of functions of bounded variation in metric spaces

Abstract: We study pointwise properties of functions of bounded variation on a metric space equipped with a doubling measure and a Poincare inequality. In particular, we obtain a Lebesgue type result for \(BV\) functions. We also study approximations by Lipschitz continuous functions and a version of the Leibniz rule. We give examples which show that our main result is optimal for \(BV\) functions in this generality.

Musielak–Orlicz Besov-type and Triebel–Lizorkin-type spaces

Abstract: Let \(s \in \mathbb{R}\), q ∈ (0, ∞], \(\varphi _{1},\ \varphi _{2}:\ \mathbb{R}^{n} \times [0,\infty ) \rightarrow [0,\infty )\) be two Musielak-Orlicz functions that, on the space variable, belong to the Muckenhoupt class \(\mathbb{A}_{\infty }(\mathbb{R}^{n})\) uniformly on the growth variable.

On Sobolev and Franke-Jawerth embeddings of smoothness Morrey spaces

Abstract: We study embeddings of Triebel–Lizorkin–Morrey spaces $${\mathcal {E}}^{s}_{p,u,q}({\mathbb {R}}^{d})$$ within that scale as well as to classical spaces like $$C({\mathbb {R}}^{d})$$ or $$L_r({\mathbb {R}}^{d})$$ . Here we obtain necessary and sufficient conditions for the continuity of it. Similarly we can deal with the situation when $${\mathbb {R}}^{d}$$ is replaced by a bounded smooth domain $$\Omega \subset {\mathbb {R}}^{d}$$ , now focussing, in addition, on compactness criteria. The second goal are embeddings of so-called Franke–Jawerth type, that is, $$\mathcal{N}^{s_1}_{p_1,u_1,q_1}({\mathbb {R}}^{d}) \hookrightarrow {\mathcal {E}}^{s}_{p,u,q}({\mathbb {R}}^{d}) \hookrightarrow \mathcal{N}^{s_2}_{p_2,u_2,q_2}({\mathbb {R}}^{d})$$ , where the differential dimension is fixed, $$s_1 - \frac{d}{p_1} = s_2 - \frac{d}{p_2}=s-\frac{d}{p}$$ , and $$s_1 > s>s_2$$ .

Calderón–Zygmund operators in Morrey spaces

Abstract: This paper deals with mapping properties of classical Calderon–Zygmund operators in local and global Morrey spaces.

A multiplicative convolution on the spectra of algebras of symmetric analytic functions

Abstract: We introduce a multiplicative convolution operation in the spectrum of the algebra of symmetric analytic functions of bounded type on the space \(\ell _1\) that allows us to solve an open question concerning its representation in terms of entire functions of exponential type.

Asymptotics of the spectrum of partial theta function

Abstract: The series \(\theta (q,x):=\sum _{j=0}^{\infty }q^{j(j+1)/2}x^j\) converges for \(q\in [0,1)\), \(x\in \mathbb R \), and defines a partial theta function. For any fixed \(q\in (0,1)\) it has infinitely many negative zeros. For \(q\) taking one of the spectral values \(\tilde{q}_1\), \(\tilde{q}_2\), \(\ldots \) (where \(0.3092493386\ldots =\tilde{q}_1<\tilde{q}_2<\cdots <1\), \(\lim _{j\rightarrow \infty }\tilde{q}_j=1\)) the function \(\theta (q,.)\) has a double zero \(y_j\) which is the rightmost of its real zeros (the rest of them being simple). For \(q e \tilde{q}_j\) the partial theta function has no multiple real zeros. We prove that \(\tilde{q}_j=1-(\pi /2j)+o(1/j)\) and that \(\lim _{j\rightarrow \infty }y_j=-e^{\pi }=-23.1407\ldots \).

Interpolation in variable exponent spaces

Abstract: In this paper we study both real and complex interpolation in the recently introduced scales of variable exponent Besov and Triebel–Lizorkin spaces. We also take advantage of some interpolation results to study a trace property and some pseudodifferential operators acting in the variable index Besov scale.

New formulas for decreasing rearrangements and a class of Orlicz–Lorentz spaces

Abstract: Using a nonlinear version of the well known Hardy–Littlewood inequalities, we derive new formulas for decreasing rearrangements of functions and sequences in the context of convex functions. We use these formulas for deducing several properties of the modular functionals defining the function and sequence spaces \(M_{\varphi ,w}\) and \(m_{\varphi ,w}\) respectively, introduced earlier in Hudzik et al. (Proc Am Math Soc 130(6): 1645–1654, 2002) for describing the Kothe dual of ordinary Orlicz–Lorentz spaces in a large variety of cases (\(\varphi \) is an Orlicz function and \(w\) a decreasing weight). We study these \(M_{\varphi ,w}\) classes in the most general setting, where they may even not be linear, and identify their Kothe duals with ordinary (Banach) Orlicz–Lorentz spaces. We introduce a new class of rearrangement invariant Banach spaces \(\mathcal M _{\varphi ,w}\) which proves to be the Kothe biduals of the \(M_{\varphi ,w}\) classes. In the case when the class \(M_{\varphi ,w}\) is a separable quasi-Banach space, \(\mathcal M _{\varphi ,w}\) is its Banach envelope.

Widths of weighted Sobolev classes on a John domain: strong singularity at a point

Abstract: The paper is concerned with orders of Kolmogorov and linear widths of weighted Sobolev classes on a domain with John condition in a weighted Lebesgue space. It is assumed that at zero the weights have singularity, which may have effect on the orders of weights.

Stability theorems for Gagliardo–Nirenberg–Sobolev inequalities: a reduction principle to the radial case

Abstract: In this paper we investigate the quantitative stability for Gagliardo–Nirenberg–Sobolev inequalities. The main result is a reduction theorem, which states that, to solve the problem of the stability for Gagliardo-Nirenberg-Sobolev inequalities, one can consider only the class of radial decreasing functions.

Holomorphic Engel structures

Abstract: Recently there has been renewed interest among differential geometers in the theory of Engel structures. We introduce holomorphic analogues of these structures, and pose the problem of classifying projective manifolds admitting them. Besides providing their basic properties and presenting two series of examples, we classify those satisfying certain positivity conditions.

The limit as p\rightarrow \infty for the eigenvalue problem of the 1-homogeneous p-Laplacian

Abstract: In this paper we study asymptotics as \(p\rightarrow \infty \) of the Dirichlet eigenvalue problem for the \(1\)-homogeneous \(p\)-Laplacian, that is, $$\begin{aligned} \left\{ \begin{array}{ll} -\frac{1}{p} |D u|^{2-p}\mathrm{div}\,(|D u|^{p-2}Du)=\lambda u, &{}\text{ in }\;\Omega ,\\ u=0,&{}\text{ on }\;\partial \Omega . \end{array}\right. \end{aligned}$$ Here \(\Omega \) is a bounded starshaped domain in \(\mathbb{R }^n\) and \(p>n\). There exists a principal eigenvalue \(\lambda _{1,p} (\Omega )\), which is positive, and has associated a non-negative nontrivial eigenfunction. Moreover, we show that \(\lim _{p\rightarrow \infty }\lambda _{1,p}(\Omega )= \lambda _{1,\infty }(\Omega ) \), where \(\lambda _{1,\infty }(\Omega )\) is the first eigenvalue corresponding to the \(1\)-homogeneous infinity Laplacian, that is, \( -\left( D^2u\frac{Du}{|Du|}\right) \cdot \frac{Du}{|Du|} =\lambda u\).

Weighted composition followed by differentiation between Bloch-type spaces

Abstract: We investigate the behavior of \(DuC_\varphi :\mathcal {B}_\alpha \rightarrow \mathcal {B}_\beta \), that is, the product of a weighted composition operator \(uC_\varphi \) and the differentiation operator \(D\), between Bloch-type spaces with standard weights. For all \(0<\alpha ,\beta <\infty \), we characterize the boundedness and estimate the essential norm of \(DuC_\varphi \) in terms of the analytic function \(u:\mathbb {D} \rightarrow \mathbb {C}\) and the symbol function \(\varphi :\mathbb {D} \rightarrow \mathbb {D}\). As a corollary, we characterize the compactness of \(DuC_\varphi \).

The full group of automorphisms of non-orientable unbordered Klein surfaces of topological genus 3, 4 and 5

Abstract: An important problem in the study of Riemann and Klein surfaces is determining their full automorphism groups. Up to now only very partial results are known, concerning surfaces of low genus or families of surfaces with special properties. This paper deals with non-orientable unbordered Klein surfaces. In this case the solution of the problem is known for surfaces of genus 1 and 2, and for hyperelliptic surfaces. Here we explicitly obtain the full automorphism group of all surfaces of genus 3, 4 and 5.

An additive problem in finite fields with powers of elements of large multiplicative order

Abstract: For a given finite field \(\mathbb F _q\), we study sufficient conditions to guarantee that the set \(\{\theta _1^x+\theta _2^y:\ 1\le x\le M_1,\ 1\le y\le M_2\}\) represents all the nonzero elements of \(\mathbb F _q\). We investigate the same problem for \(\theta _1^x-\theta _2^y\) and as a consequence we prove that any element in the finite field of \(q\) elements has a representation of the form \(\theta ^x-\theta ^y,\ 1\le x,y\le \sqrt{2}q^{3/4}\) whenever \(\theta \) has multiplicative order at least \(\sqrt{2} q^{3/4}\). This improves the previous known bound for a question possed by A. Odlyzko.

Manifolds with holonomy U^*(2m)

Abstract: We consider the geometry determined by a torsion-free affine connection whose holonomy lies in the subgroup \(U^*(2m)\), a real form of \(GL(2m,\mathbf {C})\), otherwise denoted by \(SL(m,\mathbf {H}) \cdot U(1)\). We show in particular how examples may be generated from quaternionic Kahler or hyperkahler manifolds with a circle action.

Topological classification of corank 1 map germs from \mathbb {R}^{3} to \mathbb {R}^{3}

Abstract: The link of a real analytic map germ \(f: (\mathbb {R}^{3}, 0) \rightarrow (\mathbb {R}^{3}, 0)\) is obtained by taking the intersection of the image with a small enough sphere \(S^2_\epsilon \) centered at the origin in \(\mathbb {R}^3\). If \(f\) is finitely determined, this link becomes a stable map from \(S^2\) to \(S^2\). In a previous work, we defined the Gauss paragraph which contains all the topological information of the link when the singular set \(S(\gamma )\) is connected. Now, starting from this point, we give a classification of some finitely determined weighted homogeneous map germs with two-jet equivalent to \((x,y,xz)\). In particular, we classify all 2-ruled map germs from \(\mathbb {R}^3\) to \(\mathbb {R}^3\).

Nonparametric estimation with mixed data types in survey sampling

Abstract: We consider the problem of finite population mean estimation with mixed data types. A model-assisted estimator based on nonparametric regression is proposed, which can handle discrete and continuous data and incorporates the sampling design in a natural manner. The proposed method shares the design-based properties of the kernel-based model-assisted estimator in the presence of continuous covariates and performs well under different scenarios in simulation experiments.

Analytic theory of curvettes and dicriticals

Abstract: The geometric theory of pencils in a germ of a smooth complex surface has been studied by several authors and their properties strongly lie on the structure of the dicritical components of their resolution, whereas the strict transform of the general element of the pencil by the resolution can be seen as a union of the so-called curvettes. This theory can be interpreted as the study of ideals (with two generators) in the ring of complex convergent power series in two variables, which is a local regular ring of dimension 2. In this work, we study properties of curvettes and dicriticals in an arbitrary local regular ring of dimension 2 (without restrictions on its characteristic or the one of its residue field). All the results are stated in purely algebraic terms though the ideas come from geometry.

Algebraic models of symmetric Nash sets

Abstract: The aim of this paper is to prove the existence of algebraic models for Nash sets having suitable symmetries. Given a Nash set $$M \subset {\mathbb {R}}^n$$ , we say that $$M$$ is specular if it is symmetric with respect to an affine subspace $$L$$ of $${\mathbb {R}}^n$$ and $$M \cap L=\emptyset $$ . If $$M$$ is symmetric with respect to a point of $${\mathbb {R}}^n$$ , we call $$M$$ centrally symmetric. We prove that every specular compact Nash set is Nash isomorphic to a specular real algebraic set and every specular noncompact Nash set is semialgebraically homeomorphic to a specular real algebraic set. The same is true replacing “specular” with “centrally symmetric”, provided the Nash set we consider is equal to the union of connected components of a real algebraic set. Less accurate results hold when such a union is symmetric with respect to a plane of positive dimension and it intersects that plane. The algebraic models for symmetric Nash sets $$M$$ we construct are symmetric. If the local semialgebraic dimension of $$M$$ is constant and positive, then we are able to prove that the set of birationally nonisomorphic symmetric algebraic models for $$M$$ has the power of continuum.

Simply connected open 3-manifolds with rigid genus one ends

Abstract: We construct uncountably many simply connected open 3-manifolds with genus one ends homeomorphic to the Cantor set. Each constructed manifold has the property that any self homeomorphism of the manifold (which necessarily extends to a homeomorphism of the ends) fixes the ends pointwise. These manifolds are complements of rigid generalized Bing–Whitehead (BW) Cantor sets. Previous examples of rigid Cantor sets with simply connected complement in $$R^{3}$$ had infinite genus and it was an open question as to whether finite genus examples existed. The examples here exhibit the minimum possible genus, genus one. These rigid generalized BW Cantor sets are constructed using variable numbers of Bing and Whitehead links. Our previous result with Željko determining when BW Cantor sets are equivalently embedded in $$R^{3}$$ extends to the generalized construction. This characterization is used to prove rigidity and to distinguish the uncountably many examples.

Functions of bounded second p-variation

Abstract: The generalized functionals of Merentes type generate a scale of spaces connecting the class of functions of bounded second \(p\)-variation with the Sobolev space of functions with p-integrable second derivative. We prove some limiting relations for these functionals as well as sharp estimates in terms of the fractional modulus of order \(2-1/p\). These results extend the results in Lind (Math Inequal Appl 16:2139, 2013) for functions of bounded variation but are not consequence of the last.

On the multilinear Hausdorff problem of moments

Abstract: Given a multi-index sequence \(\mu _\mathbf{k }\), \(\mathbf k = (k_1, \ldots , k_n) \in \mathbb N _0^n\), necessary and sufficient conditions are given for the existence of a regular Borel polymeasure \(\gamma \) on the unit interval \(I= [0,1]\) such that \(\mu _\mathbf{k } = \int _{I^n} t_1^{k_1}\otimes \cdots \otimes t_n^{k_n} \, \gamma \). This problem will be called the weak multilinear Hausdorff problemof moments for \(\mu _\mathbf{k }\). Comparison with classical results will allow us to relate the weak multilinear Hausdorff problem with the multivariate Hausdorff problem. A solution to the strong multilinear Hausdorff problem of moments will be provided by exhibiting necessary and sufficient conditions for the existence of a Radon measure \(\mu \) on \([0,1]\) such that \(L_\mu (f_1,\ldots , f_n) = \int _{I} f_1(t) \cdots f_n(t) \, \mu (dt)\) where \(L_\mu \) is the \(n\)-linear moment functional on the space of continuous functions on the unit interval defined by the sequence \(\mu _\mathbf{k }\). Finally the previous results will be used to provide a characterization of a class of weakly harmonizable stochastic processes with bimeasures supported on compact sets.

A geometric definition of Gabrielov numbers

Abstract: Gabrielov numbers describe certain Coxeter–Dynkin diagrams of the 14 exceptional unimodal singularities and play a role in Arnold’s strange duality. In a previous paper, the authors defined Gabrielov numbers of a cusp singularity with an action of a finite abelian subgroup $$G$$ of $$\mathrm{SL}(3,{\mathbb {C}})$$ using the Gabrielov numbers of the cusp singularity and data of the group $$G$$ . Here we consider a crepant resolution $$Y \rightarrow {\mathbb {C}}^3/G$$ and the preimage $$Z$$ of the image of the Milnor fibre of the cusp singularity under the natural projection $${\mathbb {C}}^3 \rightarrow {\mathbb {C}}^3/G$$ . Using the McKay correspondence, we compute the homology of the pair $$(Y,Z)$$ . We construct a basis of the relative homology group $$H_3(Y,Z;{\mathbb {Q}})$$ with a Coxeter–Dynkin diagram where one can read off the Gabrielov numbers.

The periodic complex method in interpolation spaces

Abstract: In this paper we consider the “periodic” variant of the complex interpolation method, apparently first studied by Peetre (Rend Sem Mat Univ Padova 46:173–190, 1971). Cwikel showed (Indiana Univ Math J 27:1005–1009, 1978) that using functions with a given period \(i\lambda \) in the complex method construction introduced and studied by Calderon (Studia Math 24:113–190, 1964), one may construct the same interpolation spaces as in the “regular” complex method, up to equivalence of norms. Cwikel also showed that one of the constants of this equivalence will, in some cases, “blow up” as \(\lambda \rightarrow 0\). (The other constant is obviously bounded by 1.) We show that this same equivalence constant approaches \(1\) as \(\lambda \rightarrow \infty \). Intuitively, this means that when applying the complex method of Calderon, it makes a very small difference if one restricts oneself to periodic functions, provided that the period is very large.

On the topology of the image by a morphism of plane curve singularities

Abstract: For a finite morphism $$\varphi =(f,g)$$ from the plane to the plane we describe the topology of the image of a branch in the source by the use of iterated pencils of analytic functions, constructed inductively in a natural way starting from the components of the map. In the case of the study of the topology of the discriminant curve, image by $$\varphi $$ of the critical locus of the map, we show that the special fibres of the pencil $$ \langle f,g\rangle $$ suffice to determine the topological type of each branch of the discriminant curve. This is due to the known relations that exist between the branches of the critical locus of $$\varphi $$ and the special fibres.

Stabilization of the wave equation in a polygonal domain with cracks

Abstract: The stabilization of the wave equation in a polygonal domain with cracks is analyzed. Using the multiplier method, we show that a boundary stabilization augmented by an internal one concentrated in a small neighbourhood of the cracks lead to the exponential stability of the problem.

Entropy numbers of operators factoring through general diagonal operators

Abstract: In this paper we study the entropy numbers of composition operators \(S=TD\) where \(D\) is a diagonal operator generated by a sequence belonging to some generalized Lorentz sequence space and \(T\) is a linear bounded operator with image in a Banach space \(Y\). We highlight the special case of this setting where \(Y\) is a Banach space of type \(p\). Results can be used to obtain entropy estimates of absolutely convex hulls in Banach spaces of type \(p\).

The variable exponent BV-Sobolev capacity

Abstract: In this article we study basic properties of the mixed BV-Sobolev capacity with variable exponent \(p\). We give an alternative way to define the mixed type BV-Sobolev-space which was originally introduced by Harjulehto, Hasto, and Latvala. Our definition is based on relaxing the \(p\)-energy functional with respect to the Lebesgue space topology. We prove that this procedure produces a Banach space that coincides with the space defined by Harjulehto et al. for a bounded domain \(\Omega \) and a log-Holder continuous exponent \(p\). Then we show that this induces a type of variable exponent BV-capacity and that this is a Choquet capacity with many usual properties. Finally we prove that if \(p\) is log-Holder continuous, then this capacity has the same null sets as the variable exponent Sobolev capacity.

A note on a topological approach to the \mu -constant problem in dimension 2

Abstract: We provide an example, which shows that studying homological and homotopical properties of cobordisms between arbitrary, that is not necessarily negative, graph manifolds is not enough to prove the \(\mu \)-constant conjecture of Le Dũng Trang in complex dimension 2.