Showing papers in "Revista Matematica Complutense in 2017"

Lebesgue inequalities for the greedy algorithm in general bases

Abstract: We present various estimates for the Lebesgue type inequalities associated with the thresholding greedy algorithm, in the case of general bases in Banach spaces. We show the optimality of the involved constants in some situations. Our results recover and slightly improve various estimates appearing earlier in the literature.

Characterization of 1-almost greedy bases

Abstract: This article closes the cycle of characterizations of greedy-like bases in the “isometric” case initiated in Albiac and Wojtaszczyk (J. Approx. Theory 138(1):65–86, 2006) with the characterization of 1-greedy bases and continued in Albiac and Ansorena (J. Approx. Theory 201:7–12, 2016) with the characterization of 1-quasi-greedy bases. Here we settle the problem of providing a characterization of 1-almost greedy bases in Banach spaces. We show that a basis in a Banach space is almost greedy with almost greedy constant equal to 1 if and only if it has Property (A). This fact permits now to state that a basis is 1-greedy if and only if it is 1-almost greedy and 1-quasi-greedy. As a by-product of our work we also provide a tight estimate of the almost greedy constant of a basis in the non-isometric case.

On the exponents of free and nearly free projective plane curves

Abstract: We show that all the possible pairs of integers occur as exponents for free or nearly free irreducible plane curves and line arrangements, by producing only two types of simple families of examples. The topology of the complements of these curves and line arrangements is also discussed, and many of them are shown not to be \(K(\pi ,1)\) spaces.

Well posedness of ODE’s and continuity equations with nonsmooth vector fields, and applications

Abstract: This paper provides an overview of the theory of flows associated to nonsmooth vector fields, describing the main developments from the seminal paper (DiPerna and Lions in Invent Math 98:511–547, 1989) till now. The problem of well posedness of ODE’s associated to vector fields arises in many fields, for instance conservation laws (via the theory of characteristics) and fluid mechanics (when looking for consistence between Eulerian and Lagrangian points of view). The theory developed so far covers many classes of vector fields and, besides uniqueness, also more quantitative aspects, as stability estimates and differentiability of the flow. Detailed lecture notes on this topic are given in Ambrosio (in: Dacorogna, Marcellini (eds) Lecture Notes in Mathematics “Calculus of variations and non-linear partial differential equations” (CIME Series, Cetraro, 2005), vol 1927, pp 2–41, 2008), Ambrosio and Crippa (Lect Notes UMI 5:3–54, 2008), Ambrosio and Crippa (Proc R Soc Edinb Sect A 144:1191–1244 2014), Ambrosio and Trevisan (Ann Fac Sci Toulouse, 2016).

Extrapolation, John-Nirenberg inequalities and characterizations of BMO in terms of Morrey type spaces

Abstract: We extend the extrapolation theory to Morrey spaces associated with Banach function spaces. Some applications by this theory such as the Fefferman–Stein vector-valued maximal inequalities are obtained. By using this extrapolation theory, we obtain the John-Nirenberg inequalities on Morrey spaces associated with Banach function spaces. In addition, by using the John-Nirenberg inequalities, we have the characterizations of the function spaces of bounded mean oscillation in terms of Morrey spaces associated with Banach function spaces.

Survey on some aspects of Lefschetz theorems in algebraic geometry

Abstract: We survey classical material around Lefschetz theorems for fundamental groups, and show the relation to parts of Deligne’s program in Weil II.

Compact embedding for p(x, t)-Sobolev spaces and existence theory to parabolic equations with p(x, t)-growth

Abstract: In this paper, we establish the compact embedding of p(x, t)-Sobolev spaces into p(x, t)-Lebesgue spaces. Moreover, we prove some existence results for nonlinear parabolic problems of the form $$\begin{aligned} \partial _tu-{\mathrm {div}\,}a(x,t,Du)=f-{\mathrm {div}\,}\left( |F|^{p(x,t)-2}F\right) \,\,\,\text {in}\,\Omega _T, \end{aligned}$$ where the vector-field \(a(x,t,\cdot )\) satisfies certain p(x, t)-growth conditions.

An exterior nonlinear elliptic problem with a dynamical boundary condition

Abstract: Several results on existence, nonexistence and large-time behavior of small positive solutions $$u=u(x,t)$$ were obtained before for the equation $$-{\varDelta }u=u^p$$ , $$x\in {\mathbb R}^N_+$$ , $$t>0$$ , with a linear dynamical boundary condition. Here $${\varDelta }$$ is the N-dimensional Laplacian (in x). We study the effects of the change of the domain from the half-space to the exterior of the unit ball when $$N\ge 3$$ . We show that the critical exponent for the existence of positive solutions and the decay rate of small solutions are different. More precisely, for the half-space problem the critical exponent is $$p=(N+1)/(N-1)$$ and the decay rate is $$t^{-(N-1)}$$ , while for the exterior problem we obtain the exponent $$p=N/(N-2)$$ and the exponential rate $$e^{-(N-2)t}$$ .

Boundedness of Hardy-type operators with a kernel: integral weighted conditions for the case $0<q<1\le p<\infty$

Abstract: Boundedness of a fundamental Hardy-type operator with a kernel is characterized between weighted Lebesgue spaces $L^p(v)$ and $L^q(w)$ for $0

Optimal function spaces for the Laplace transform

Abstract: We study the action of the Laplace transform $$\mathcal L$$ on rearrangement-invariant function spaces. We focus on the optimality of the range and the domain spaces.

On the algebraicity of Puiseux series

Abstract: We deal with the algebraicity of a formal Puiseux series in terms of the properties of its coefficients. We show that the algebraicity of a Puiseux series for given bounded degrees is determined by a finite number of explicit universal polynomial formulas. Conversely, given a vanishing polynomial, there is a closed-form formula for the coefficients of the series in terms of the coefficients of the polynomial and of an initial part of the series.

“Varopoulos paradigm”: Mackey property versus metrizability in topological groups

Abstract: The class of all locally quasi-convex (lqc) abelian groups contains all locally convex vector spaces (lcs) considered as topological groups. Therefore it is natural to extend classical properties of locally convex spaces to this larger class of abelian topological groups. In the present paper we consider the following well known property of lcs: “A metrizable locally convex space carries its Mackey topology ”. This claim cannot be extended to lqc-groups in the natural way, as we have recently proved with other coauthors (Ausenhofer and de la Barrera Mayoral in J Pure Appl Algebra 216(6):1340–1347, 2012; Diaz Nieto and Martin Peinador in Descriptive Topology and Functional Analysis, Springer Proceedings in Mathematics and Statistics, Vol 80 doi:10.1007/978-3-319-05224-3_7, 2014; Dikranjan et al. in Forum Math 26:723–757, 2014). We say that an abelian group G satisfies the Varopoulos paradigm (VP) if any metrizable locally quasi-convex topology on G is the Mackey topology. In the present paper we prove that in any unbounded group there exists a lqc metrizable topology that is not Mackey. This statement (Theorem C) allows us to show that the class of groups satisfying VP coincides with the class of finite exponent groups. Thus, a property of topological nature characterizes an algebraic feature of abelian groups.

Rank two aCM bundles on the del Pezzo threefold of degree 7

Abstract: We classify indecomposable aCM bundles of rank 2 on the del Pezzo threefold of degree 7 and analyze their corresponding moduli spaces.

Dirichlet problems for the p-Laplacian with a convection term

Abstract: We consider the nonlinear Dirichlet boundary value problem Open image in new window in a bounded domain \(\Omega \subset \mathbb {R}^N\) with smooth boundary \(\partial \Omega \), where \(\Delta _p u\mathop {=}\limits ^{\mathrm{{def}}}\mathrm {div} (| abla u|^{p-2} abla u)\) with \(1 0\)). When \(h ot \equiv 0\) and \(-\infty< \lambda < \lambda _1\), we prove that there exists a weak solution \(u\in W_0^{1,p}(\Omega )\) to problem (P); this solution is unique provided \(\lambda < 0\) (without any further assumptions). When \(h\ge 0\), \(h ot \equiv 0\), and \(0\le \lambda < \lambda _1\), we show that the solution is positive and also unique.

Basic properties of multiplication and composition operators between distinct Orlicz spaces

Abstract: First, we present some simple (and easily verifiable) necessary conditions and sufficient conditions for boundedness of the multiplication operator \(M_u\) and composition operator \(C_T\) acting from Orlicz space \(L^{\Phi _1}(\Omega )\) into Orlicz space \(L^{\Phi _2}(\Omega )\) over arbitrary complete, \(\sigma \)-finite measure space \((\Omega ,\Sigma ,\mu )\). Next, we investigate the problem of conditions on the generating Young functions, the function u, and/or the function \(h=d(\mu \circ T^{-1})/d\mu \), under which the operators \(M_u\) and \(C_T\) are of closed range or finite rank. Finally, we give necessary and sufficient conditions for boundedness of the operators \(M_u\) and \(C_T\) in terms of techniques developed within the theory of Musielak–Orlicz spaces.

Functions that are Lipschitz in the small

Abstract: Let X and Y be metric spaces. A function \(f:X\rightarrow Y\) is said to be Lipschitz in the small if there are \(r> 0\) and \(K<\infty \) so that \(d(f(u),f(v)) \le Kd(u,v)\) for any \(u,v \in X\) with \(d(u,v) \le r\). We find necessary and sufficient conditions on a subset A of X such that \(f_{|A}\) is Lipschitz for every function f that is Lipschitz in the small on X. We also find necessary and sufficient conditions on X for \({\text {*}}{LS}\left( X\right) \) to be linearly order isomorphic to \({\text {Lip}}(Y)\) for some metric space Y.

On the viscosity solutions to a class of nonlinear degenerate parabolic differential equations

Abstract: In this work, we show existence and uniqueness of positive solutions of \(H(Du, D^2u)+\chi (t)|Du|^\Gamma -f(u)u_t=0\) in \(\Omega \times (0, T)\) and \(u=h\) on its parabolic boundary. The operator H satisfies certain homogeneity conditions, \(\Gamma >0\) and depends on the degree of homogeneity of \(H, f>0\), increasing and meets a concavity condition. We also consider the case \(f\equiv 1\) and prove existence of solutions without sign restrictions.

Birationality of moduli spaces of twisted $\mathrm{U}(p,q)$-Higgs bundles

Abstract: A $$\mathrm {U}(p,q)$$ -Higgs bundle on a Riemann surface (twisted by a line bundle) consists of a pair of holomorphic vector bundles, together with a pair of (twisted) maps between them. Their moduli spaces depend on a real parameter $$\alpha $$ . In this paper we study wall crossing for the moduli spaces of $$\alpha $$ -polystable twisted $$\mathrm {U}(p,q)$$ -Higgs bundles. Our main result is that the moduli spaces are birational for a certain range of the parameter and we deduce irreducibility results using known results on Higgs bundles. Quiver bundles and the Hitchin–Kobayashi correspondence play an essential role.

A note on the Fefferman–Stein inequality on Morrey spaces

Abstract: In this paper, we discuss the Fefferman–Stein inequality on Morrey spaces and give a sufficient and necessary condition for which the inequality holds. Further, we give an example of weights such that the Fefferman–Stein inequality on Morrey spaces fails. We also consider the multilinear setting.

Conformal mapping asymptotics at a cusp

Abstract: We describe the asymptotic behavior of the mapping function at an analytic cusp compared with Kaiser’s results for cusps with small perturbation of angles and the known explicit formulae for cusps with circular boundary curves. Saying “analytic cusp” here we mean that the boundary curves are real analytic away probably from the cusp. We propose a boundary curve parametrization by generalized power series which allows us to give explicit representations for locally univalent mapping functions with given asymptotic properties and for cusp boundary curves having an arbitrary order of tangency.

Intrinsicness of the Newton polygon for smooth curves on $\mathbb{P}^1\times \mathbb{P}^1$

Abstract: Let C be a smooth projective curve in $${\mathbb {P}}^1 \times {\mathbb {P}}^1$$ of genus $$g e 4$$ , and assume that it is birationally equivalent to a curve defined by a Laurent polynomial that is non-degenerate with respect to its Newton polygon $$\Delta $$ . Then we show that the convex hull $$\Delta ^{(1)}$$ of the interior lattice points of $$\Delta $$ is a standard rectangle, up to a unimodular transformation. Our main auxiliary result, which we believe to be interesting in its own right, is that the first scrollar Betti numbers of $$\Delta $$ -non-degenerate curves are encoded in the combinatorics of $$\Delta ^{(1)}$$ , if $$\Delta $$ satisfies some mild conditions.

Long time behavior of stochastic parabolic problems with white noise in materials with thermal memory

Abstract: The existence and limiting behavior of the solutions of stochastic parabolic problems with thermal memory are investigated in the cases that the nonlinear term satisfies subcritical and critical growth conditions. The existence, uniqueness and continuity of solutions is proved by a semigroup method and the Lax–Milgram theorem, then the dynamics of solutions is analyzed by a priori estimates. In particular, the existence of pullback random attractors for the random dynamical system associated to the problem is established and the upper semi-continuity of the pullback random attractors is verified.

Non-linear plank problems and polynomial inequalities

Abstract: We study lower bounds for the norm of the product of polynomials and their applications to the so called plank problem. We are particularly interested in polynomials on finite dimensional Banach spaces, in which case our results improve previous works for large numbers of polynomials.

Intrinsic atomic characterization of 2-microlocal spaces with variable exponents on domains

Abstract: We provide an intrinsic atomic characterization for 2-microlocal Besov and Triebel–Lizorkin spaces with variable integrability on domains, \(B_{p(\cdot ),q(\cdot )}^{\varvec{w}}(\varOmega )\) and \(F_{p(\cdot ),q(\cdot )}^{\varvec{w}}(\varOmega )\), where \(\varOmega \) is a regular domain. We make use of the non-smooth atomic decomposition result obtained in Goncalves and Kempka (J Math Anal Appl 434:1875–1890, 2016) for these spaces to get the main result.

Second order linear parabolic equations in uniform spaces in RN

Abstract: We solve second order parabolic equations with nonsmooth coefficients and initial data in suitable uniform spaces. We also show the smoothing effect of the corresponding analytic semigroup depending on the integrability properties of the coefficients. Robustness with respect to perturbations in the coefficients is also obtained.

Quasi-stationary ferromagnetic problem for thin multi-structures

Abstract: In this paper we study the asymptotic behavior of the solutions of time dependent micromagnetism problem in a multi-domain consisting of a thin-wire in junction with a thin film. We assume that the volumes of the two parts composing each multi-structure vanish with same rate. We obtain a 1D limit problem on the thin-wire and a 2D limit problem on the thin film, and the two limit problems are uncoupled. The limit problem remains non-convex, but now it becomes completely local.

Non-Lipschitz differentiable functions on slit domains

Abstract: It is proved the existence of large algebraic structures—including large vector subspaces or infinitely generated free algebras—inside the family of non-Lipschitz differentiable real functions with bounded gradient defined on special non-convex plane domains. In particular, this yields that there are many differentiable functions on plane domains that do not satisfy the mean value theorem.

Nice operators into $$L_1$$ L 1 -preduals

Abstract: A Banach space X is said to be “nice” if every extreme operator from any Banach space into X is a nice operator (that is, its adjoint preserves extreme points). We prove that a nice \(L_1\)-predual space is isometrically isomorphic to \(c_0(I)\) for some set I. In fact, by using the structure topology, we get a more general result which allows us to conclude that there exist extreme non-nice operators into certain spaces of continuous affine functions which are not \(L_1\)-preduals.

Basis properties of Lindqvist–Peetre functions in L^r(0,1)^n

Abstract: We prove that the generalized p-trigonometric functions of Lindqvist and Peetre form a basis in the Lebesgue space \(L^r(0,1)^n\) for any \(r\in (1,\infty )\), provided \(n\le 3\) and \(p>p_n\ge 1\).

On the hybrids of k-spacing empirical and partial sum processes

Abstract: The present paper is devoted to the study of the hybrids of k-spacing empirical and partial sum processes. In the first part, we present the gaussian approximation of these processes. In the second part, we obtain new results on the precise asymptotics in the law of the logarithm related to complete convergence and a.s. convergence, under some mild conditions, for the hybrids of k-spacing empirical and partial sum processes.