Showing papers in "Revista Matematica Complutense in 2012"

Sharp spectral stability estimates via the Lebesgue measure of domains for higher order elliptic operators

Abstract: We prove sharp stability estimates for the variation of the eigenvalues of non-negative self-adjoint elliptic operators of arbitrary even order upon variation of the open sets on which they are defined. These estimates are expressed in terms of the Lebesgue measure of the symmetric difference of the open sets. Both Dirichlet and Neumann boundary conditions are considered.

Invariants of hypersurface singularities in positive characteristic

Abstract: We study singularities f∈K[[x1,…,xn]] over an algebraically closed field K of arbitrary characteristic with respect to right respectively contact equivalence, and we establish that the finiteness of the Milnor respectively the Tjurina number is equivalent to finite determinacy. We give improved bounds for the degree of determinacy in positive characteristic. Moreover, we consider different non-degeneracy conditions of Kouchnirenko, Wall and Beelen-Pellikaan in positive characteristic, and we show that planar Newton non-degenerate singularities satisfy Milnor’s formula μ=2⋅δ−r+1. This implies the absence of wild vanishing cycles in the sense of Deligne.

On a general weighted Hardy type inequality in the variable exponent Lebesgue spaces

Abstract: We study the Hardy type, two-weight inequality for the multidimensional Hardy operator in the variable exponent Lebesgue space L p(.)(ℝ n ). We prove equivalent conditions for L p(.)→L q(.) boundness of the Hardy operator in the case of so called “mixed” exponents: q(0)≥p(0), q(∞)

Maximal functions, Riesz potentials and Sobolev embeddings on Musielak-Orlicz-Morrey spaces of variable exponent in \({\bf R}^{n}\)

Abstract: Our aim in this paper is to deal with Sobolev embeddings for Riesz potentials of variable order with functions in variable exponent Musielak-Orlicz-Morrey spaces.

Simple Neumann eigenvalues for the Laplace operator in a domain with a small hole

Abstract: Let \({\mathbb{A}}(\epsilon)\) be the annular domain obtained by removing from a bounded open domain \({\mathbb{I}}^{o}\) of ℝn a small cavity of size ϵ>0. Then we assume that for some natural index l, \(\lambda_{l}[{\mathbb{I}}^{o}]>0\) is a simple Neumann eigenvalue of −Δ in \({\mathbb{I}}^{o}\), and we show that there exists a real valued real analytic function \(\hat{\lambda }_{l}(\cdot,\cdot)\) defined in an open neighborhood of (0,0) in ℝ2 such that the lth Neumann eigenvalue \(\lambda_{l}[{\mathbb{A}}(\epsilon)]\) of −Δ in \({\mathbb{A}}(\epsilon)\) equals \(\hat{\lambda}_{l}(\epsilon,\kappa_{n}\epsilon\log\epsilon)\) and such that \(\hat{\lambda}_{l}(0,0)= \lambda_{l}[{\mathbb{I}}^{o}]\). Here κn=1 if n is even and κn=0 if n is odd. Thus in particular, we show that if n is even \(\lambda_{l}[{\mathbb {A}}(\epsilon)]\) can be expanded into a convergent double series of powers of ϵ and ϵlogϵ and that if n is odd \(\lambda_{l}[{\mathbb{A}}(\epsilon)]\) can be expanded into a convergent series of powers of ϵ. Then related statements have been proved for corresponding eigenfunctions.

A representation theorem for orthogonally additive polynomials on Riesz spaces

Abstract: The aim of this article is to prove a representation theorem for orthogonally additive polynomials in the spirit of the recent theorem on representation of orthogonally additive polynomials on Banach lattices but for the setting of Riesz spaces. To this purpose the notion of p-orthosymmetric multilinear form is introduced and it is shown to be equivalent to the orthogonally additive property of the corresponding polynomial. Then the space of positive orthogonally additive polynomials on an Archimedean Riesz space taking values on an uniformly complete Archimedean Riesz space is shown to be isomorphic to the space of positive linear forms on the n-power in the sense of Boulabiar and Buskes of the original Riesz space.

Spatial graphs with local knots

Abstract: It is shown that for any locally knotted edge of a 3-connected graph in S3, there is a ball that contains all of the local knots of that edge which is unique up to an isotopy setwise fixing the graph. This result is applied to the study of topological symmetry groups of graphs embedded in S3.

On dimension-free Sobolev imbeddings II

Abstract: We prove dimension-invariant imbedding theorems for Sobolev spaces using extrapolation means.

Characterization of a rearrangement-invariant hull of a Besov space via interpolation

Abstract: Let X be a rearrangement-invariant Banach function space over a Lipschitz domain Ω⊂ℝn. We characterize the K-functionals for the pairs (X,V1X) and (X,SX), where V1X is the reduced Sobolev space built upon X and SX is the class of measurable functions on Ω such that \(\|t\sp{-\frac{1}{n}}(f^{**}(t)-f^{*}(t))\|_{\overline{X}}<\infty\), \(\overline{X}\) being the representation space of X. Using this result, we obtain an estimate of rearrangements of a function in terms of moduli of continuity and prove its sharpness. Finally we establish sharp embeddings of general Besov spaces into Lorentz spaces and characterize the rearrangement-invariant hull of a general Besov space.

Some remarks on the planar Kouchnirenko’s theorem

Abstract: We consider different notions of non-degeneracy, as introduced by Kouchnirenko (NND), Wall (INND) and Beelen-Pellikaan (WNND) for plane curve singularities {f(x,y)=0} and introduce the new notion of weighted homogeneous Newton non-degeneracy (WHNND). It is known that the Milnor number μ resp. the delta-invariant δ can be computed by explicit formulas μN resp. δN from the Newton diagram of f if f is NND resp. WNND. It was however unknown whether the equalities μ=μN resp. δ=δN can be characterized by a certain non-degeneracy condition on f and, if so, by which one. We show that μ=μN resp. δ=δN is equivalent to INND resp. WHNND and give some applications and interesting examples related to the existence of “wild vanishing cycles”. Although the results are new in any characteristic, the main difficulties arise in positive characteristic.

Bernstein-type inequality for weakly dependent sequence and its applications

Abstract: Bernstein-type inequality and complete convergence for ρ-mixing sequence is given. By using the Bernstein-type inequality, we study the asymptotic approximation of inverse moments for ρ-mixing sequences, which generalizes the corresponding result of Wu et al. (Stat. Probab. Lett. 79(11): 1366–1371, 2009) for independent sequence.

A generalization of Puiseux’s theorem and lifting curves over invariants

Abstract: Let ρ:G→GL (V) be a rational representation of a reductive linear algebraic group G defined over ℂ on a finite dimensional complex vector space V. We show that, for any generic smooth (resp. C M ) curve c:ℝ→V//G in the categorical quotient V//G (viewed as affine variety in some ℂ n ) and for any t 0∈ℝ, there exists a positive integer N such that t↦c(t 0±(t−t 0) N ) allows a smooth (resp. C M ) lift to the representation space near t 0. (C M denotes the Denjoy–Carleman class associated with M=(M k ), which is always assumed to be logarithmically convex and derivation closed). As an application we prove that any generic smooth curve in V//G admits locally absolutely continuous (not better!) lifts. Assume that G is finite. We characterize curves admitting differentiable lifts. We show that any germ of a C ∞ curve which represents a lift of a germ of a quasianalytic C M curve in V//G is actually C M . There are applications to polar representations.

Nyman type theorem in convolution Sobolev algebras

Abstract: We characterize the dense ideals of certain convolution Sobolev algebras, on the positive half-line, as those ideals I which satisfy the Nyman conditions Z(I)=∅ and γ(I)=0. Here Z(I) is the hull of I and γ(I):=inf {inf supp (f):f∈I}.

Weighted weak modular and norm inequalities for the Hardy operator in variable Lp spaces of monotone functions

Abstract: We study weak-type modular inequalities for the Hardy operator restricted to non-increasing functions on weighted Lp(⋅) spaces, where p(⋅) is a variable exponent These new estimates complete the results of Boza and Soria (J Math Anal Appl 348:383–388, 2008) where we showed some necessary and sufficient conditions on the exponent p(⋅) and on the weights to obtain weighted modular inequalities with variable exponents For this purpose, we introduced the class of weights Bp(⋅) We prove that, for exponents p(x)>1, this is also the class of weights for which the weak modular inequality holds, and a characterization is also given in the case p(x)≤1 Finally, we compare our theory with the results in Neugebauer (Stud Math 192(1):51–60, 2009), giving examples for very concrete and simple exponents which show that inequalities in norm hold true in a very general context

A singular perturbation in a linear parabolic equation with terms concentrating on the boundary

Abstract: In this paper we consider linear parabolic problems when some reaction and potential terms are concentrated in a neighborhood of a portion Γ of the boundary. This neighborhood shrinks to Γ as a parameter e goes to zero. Then we derive the limit equation which has some new terms on Γ. We also analyze the regularity and convergence of the solutions.

The Riesz-Herz equivalence for capacitary maximal functions

Abstract: We prove a Riesz-Herz estimate for the maximal function associated to a capacity C on ℝn, MCf(x)=sup Q∋xC(Q)−1∫Q|f|, which extends the equivalence (Mf)∗(t)≃f∗∗(t) for the usual Hardy-Littlewood maximal function Mf. The proof is based on an extension of the Wiener-Stein estimates for the distribution function of the maximal function, obtained using a convenient family of dyadic cubes. As a byproduct we obtain a description of the norm of the interpolation space \((L^{1},{\mathcal{L}}^{1,C})_{1/p',p}\), where \({\mathcal{L}}^{1,C}\) denotes the Morrey space based on a capacity.

On determining the calibration equations to construct model-calibration estimators of the distribution function

Abstract: The calibration approach to estimating the finite population distribution function was proposed by Rueda et al. (J. Stat. Plan. Inference 137(2):435–448, 2007). The proposed estimator is built by means of constraints that require the use of a set of fixed values. Assuming a linear regression working model, Rueda et al. (Metrika 71:33–44, 2010) considered a single fixed value for the calibration and determine the optimum value in the sense of minimum variance. Assuming more complex models, we now study the problem of determining two optimal values to achieve the best estimation under simple random sampling without replacement.

Orthogonally additive holomorphic functions on open subsets of C ( K )

Abstract: We introduce, study and characterize orthogonally additive holomorphic functions f:U -> a", where U is an open subset of C(K). We are led to consider orthogonal additivity at different points of U.

A note on strict ℂ-convexity

Abstract: We establish a relation between strict ℂ-convexity of a real hypersurface of ℂn and the behavior of its complex Gauss map. In that way we recover—with an improvement on the regularity—the known results about the topology of these hypersurfaces by using elementary differential geometric arguments. Our approach can be though of as being a complex analog of the description of strictly convex hypersurfaces in Euclidean space via Morse functions associated to pencils of hyperplanes.

Bertini-type theorems for formal functions in Grassmannians

Abstract: This paper proves that the inverse images of Schubert varieties by morphisms to Grassmannians are G3 under reasonable numerical conditions. It is well known that G3 is stronger than universal connectedness, so these results improve Debarre’s connectivity results for those inverse images (in certain cases even under milder conditions).

Heegaard surfaces for certain graphs in compressionbodies

Abstract: Let M be a compressionbody containing a properly embedded graph T (with at least one edge) such that ∂+M−T is parallel to the frontier of T∪∂−M in M. We extend methods of Hayashi and Shimokawa to show that if H is a bridge surface for T then one of the following occurs: H is stabilized, boundary stabilized, or perturbed. T contains a removable path. M is a trivial compressionbody and H−T is properly isotopic in M−T to ∂+M−T.

Integral extension of multiplier operators in A(G)

Abstract: We investigate operator ideal properties of multiplier operators Tϕ (via ϕ∈l∞(Γ) with Γ the dual group) acting in the space A(G) of all integrable functions on an infinite, compact abelian group G whose Fourier coefficients are absolutely summable. Of interest is when Tϕ is 1-summing, as this corresponds to \(\varphi= \widehat {f}\) with f square integrable relative to Haar measure. Precisely then there is an optimal Banach function space L1(mϕ) available which contains A(G) densely and continuously and such that Tϕ has a continuous A(G)-valued linear extension \(I_{m_{\varphi}}\) to L1(mϕ). Relevant for studying L1(mϕ) and \(I_{m_{\varphi}}\) is the space Sp, 1≤p≤∞, of all functions in Lp(G) whose Fourier series is unconditionally convergent in Lp(G). Amongst other things, it is shown that \(I_{m_{\varphi}}\) is 1-summing iff Tϕ is nuclear iff the vector measure \(m_{\varphi}(E) := T_{\varphi}( \chi _{{}_{E}}) \) has finite variation. Moreover, L1(mϕ) is a homogeneous Banach space. Hence, it is also a Banach algebra and an L1(G)-module under convolution.

Real logarithmic models for real analytic foliations in the plane

Abstract: Let S be a germ of a holomorphic curve at (ℂ2,0) with finitely many branches S 1,…,S r and let ${\mathcal{I}}=(I_{1},\ldots,I_{r}) \in {\mathbb{C}}^{r}$ . We show that there exists a non-dicritical holomorphic foliation of logarithmic type at 0∈ℂ2 whose set of separatrices is S and having index I i along S i in the sense of Lins Neto (Lecture Notes in Math. 1345, 192–232, 1988) if the following (necessary) condition holds: after a reduction of singularities π:M→(ℂ2,0) of S, the vector ${\mathcal{I}}$ gives rise, by the usual rules of transformation of indices by blowing-ups, to systems of indices along components of the total transform $\bar{S}$ of S at points of the divisor E=π −1(0) satisfying: (a) at any singular point of $\bar{S}$ the two indices along the branches of $\bar{S}$ do not belong to ℚ≥0 and they are mutually inverse; (b) the sum of the indices along a component D of E for all points in D is equal to the self-intersection of D in M. This construction is used to show the existence of logarithmic models of real analytic foliations which are real generalized curves. Applications to real center-focus foliations are considered.

Determination of the real poles of the Igusa zeta function for curves

Abstract: The numerical data of an embedded resolution determine the candidate poles of Igusa’s p-adic zeta function. We determine in complete generality which real candidate poles are actual poles in the curve case.

Schur-Szegö composition of entire functions

Abstract: For any pair of algebraic polynomials $A(x) =\sum _{k=0}^ n{n\choose k} a_kx^k$ and $B(x) =\sum _{k=0}^ n{n\choose k} b_kx^k$, their Schur-Szeg\H{o} composition is defined by $(A^*_nB)(x) =\sum _{k=0}^ n{n\choose k} a_kb_kx^k$. Motivated by some recent results which show that every polynomial $P(x)$ of degree $n$ with $P(−1) = 0$ can be represented as $K_{a_1}^∗_n\cdots ^∗_n K_{a_{n−1}}$ with $K_a := (x + 1)^{n−1}(x + a)$, we introduce the notion of Schur-Szeg\H{o} composition of formal power series and study its properties in the case when the series represents an entire function. We also concentrate on the special case of composition of functions of the form $e^xP(x)$, where $P(x)$ is an algebraic polynomial and investigate its properties in detail.

Geometrical study of prime order automorphisms on Klein surfaces

Abstract: We consider (n,q)-gonal Klein surfaces of algebraic genus p≥2 for which n≥p and n prime. We exclude the case n=2 corresponding to q-hyperelliptic surfaces, so n≥3. We study geometrical conditions on fundamental polygons of the NEC groups which uniformize these surfaces. From these conditions we give a way to construct such surfaces and obtain parameters which characterize them.

Semigroup estimates and stability/instability results for the linearized three waves interaction equations

Abstract: We consider the three waves interaction system and its linearization (the “pump-wave approximation”). We give some estimates on the semigroup as well as stability or instability results for the linearized problem in suitable norms. We work in the whole space and with periodic boundary condition, and our analysis relies on energy estimates and not on the complete integrability of the system.