Capítulos (Análisis Matemático)

URI permanente para esta colecciónhttps://hdl.handle.net/11441/10810

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  • Acceso AbiertoCapítulo de Libro
    Doblando voy, doblando vengo
    (Publicaciones de la Universidad de Alicante, 2024-02-07) Prado Bassas, José Antonio; Universidad de Sevilla. Departamento de Análisis Matemático; Mulero, Julio; Segura Abad, Lorena; Sepulcre Martínez, Juan Matías; Universidad de Sevilla. FQM127: Análisis funcional no lineal
  • Acceso AbiertoCapítulo de Libro
    A note on the off-diagonal Muckenhoupt-Wheeden conjecture
    (World Scientific, 2016) Cruz Uribe, David; Martell Berrocal, José María; Pérez Moreno, Carlos; Universidad de Sevilla. Departamento de Análisis Matemático; Navarro Pascual, Juan Carlos; Kaidi Lhachmi, El Amin; Universidad de Sevilla. FQM-354: Análisis Real
    We obtain the off-diagonal Muckenhoupt-Wheeden conjec-ture for Calder´on-Zygmund operators. Namely, given 1 < p < q < ∞ and a pair of weights (u, v), if the Hardy-Littlewood maximal functionsatisfies the following two weight inequalities: M : Lp(v) → Lq(u) and M : Lq´(u1−q´) → Lp´(v1−p´), then any Calderón-Zygmund operator Tand its associated truncatedmaximal operator T⋆ are bounded from Lp(v) to Lq(u). Additionally, as-suming only the second estimate for Mthen Tand T* map continuouslyLp(v) into Lq,∞(u). We also consider the case of generalized Haar shiftoperators and show that their off-diagonal two weight estimates are gov-erned by the corresponding estimates for the dyadic Hardy-Littlewoodmaximal function.
  • Acceso AbiertoCapítulo de Libro
    Introduction to hyperconvex spaces
    (Springer, 2001) Espínola García, Rafael; Khamsi, Mohamed Amine; Universidad de Sevilla. Departamento de Análisis Matemático; Kirk, William Art; Sims, Brailey; Universidad de Sevilla. FQM127: Análisis Funcional no Lineal
    The notion of hyperconvexity is due to Aronszajn and Panitchpakdi (1956) who proved that a hyperconvex space is a nonexpansive absolute retract, i.e. it is a nonexpansive retract of any metric space in which it is isometrically embedded. The corresponding linear theory is well developed and associated with the names of Gleason, Goodner, Kelley and Nachbin (see for instance. The nonlinear theory is still developing. The recent interest into these spaces goes back to the results of Sine and Soardi who proved independently that fixed point property for nonexpansive mappings holds in bounded hyperconvex spaces. Since then many interesting results have been shown to hold in hyperconvex spaces.
  • Acceso AbiertoCapítulo de Libro
    El principio de Calderón-Zygmund
    (Universidad de La Rioja, 2001) Pérez Moreno, Carlos; Trujillo González, Rodrigo Francisco; Universidad de Sevilla. Departamento de Análisis Matemático; Español González, Luis; Varona Malumbres, Juan Luis; Universidad de Sevilla. FQM-354: Análisis Real
    In this note we show some estimates for multilinear commutators with vector symbol b = (b1,...,bm) defined by the expression Tbf(x) = Z Rn2 4Ym j=1 (bj (x) − bj (y)) 3 5 K(x, y)f(y) dy, where K is the kernel of any Calderón-Zygmund operator. We generalize and sharpen both classical results from Coifman, and Coifman, Rochberg and Weiss; and also more recent results from the first author. We show that these operators are intimately related to certain appropriate Orlicz type maximal function of the form ML(log L)α where the number α is related to the symbol b.
  • Acceso AbiertoCapítulo de Libro
    Los q-polinomios hipergeométricos
    (Universidad de La Rioja, 2001) Álvarez Nodarse, Renato; Universidad de Sevilla. Departamento de Análisis Matemático; Español González, Luis; Varona Malumbres, Juan Luis; Universidad de Sevilla. FQM262: Teoria de la Aproximacion
    It is well known that the q-polynomials of hypergeometric type are the polynomial solutions of a certain second order difference equation in a non-uniform lattice. In this short paper we present a modification of the proof of a Theorem by Atakishiyev, Rahman, and Suslov that characterizes the most general lattice.
  • Acceso AbiertoCapítulo de Libro
    On the sign of the real part of the Riemann zeta-function
    (Springer, 2013) Arias de Reyna Martínez, Juan; Brent, Richard P.; Lune, Jan van de; Universidad de Sevilla. Departamento de Análisis Matemático; Borwein, Jonathan M.; Shparlinski, Igor; Zudilin, Wadim; Universidad de Sevilla. FQM104: Analisis Matematico
    We consider the distribution of argζ(σ +it) on fixed lines σ > 1/2, and in particular the density d(σ) = lim T→+∞ 1/2T |{t ∈ [−T,+T] : |argζ(σ +it)| > π/2}|, and the closely related density d−(σ) = lim T→+∞ 1/2T |{t ∈ [−T,+T] : ℜζ(σ +it) < 0}|. Using classical results of Bohr and Jessen, we obtain an explicit expression for the characteristic function ψσ(x) associated with argζ(σ + it). We give explicit expressions for d(σ) and d−(σ) in terms of ψσ(x). Finally, we give a practical algorithm for evaluating these expressions to obtain accurate numerical values of d(σ) and d−(σ).
  • Acceso AbiertoCapítulo de Libro
    Refined size estimates for Furstenberg sets via Hausdorff measures: a survey of some recent results
    (Springer, 2014) Rela, Ezequiel; Universidad de Sevilla. Departamento de Análisis Matemático; Cepedello Boiso, Manuel; Hedenmalm, Håkan; Kaashoek, Marinus A.; Montes Rodríguez, Alfonso; Treil, Sergei
    In this survey we collect and discuss some recent results on the so called “Furstenberg set problem”, which in its classical form concerns the estimates of the Hausdorff dimension (dimH) of the sets in the Fα-class: for a given α ∈ (0, 1], a set E ⊆ R2 is in the Fα-class if for each e ∈ S there exists a unit line segment `e in the direction of e such that dimH(` ∩ E) ≥ α. For α = 1, this problem is essentially equivalent to the “Kakeya needle problem”. Define γ(α) = inf {dimH(E) : E ∈ Fα}. The best known results on γ(α) are the following inequalities: max {1/2 + α; 2α} ≤ γ(α) ≤ (1 + 3α)/2. In this work we approach this problem from a more general point of view, in terms of a generalized Hausdorff measure Hh associated with the dimension function h. We define the class Fh of Furstenberg sets associated to a given dimension function h. The natural requirement for a set E to belong to Fh, is that Hh(`e ∩ E) > 0 for each direction. We generalize the known results in terms of “logarithmic gaps” and obtain analogues to the estimates given above. Moreover, these analogues allow us to extend our results to the endpoint α = 0. For the upper bounds we exhibit an explicit construction of Fh-sets which are small enough. To that end we adapt and prove some results on Diophantine Approximation about the the dimension of a set of “well approximable numbers”. We also obtain results about the dimension of Furstenberg sets in the class Fαβ, defined analogously to the class Fα but only for a fractal set L ⊂ S of directions such that dimH(L) ≥ β. We prove analogous inequalities reflecting the interplay between α and β. This problem is also studied in the general scenario of Hausdorff measures.
  • Acceso AbiertoCapítulo de Libro
    Improving bounds for singular operators via sharp reverse Hölder inequality for A∞
    (Springer, 2013) Ortiz Caraballo, Carmen María; Pérez Moreno, Carlos; Rela, Ezequiel; Universidad de Sevilla. Departamento de Análisis Matemático; Universidad de Sevilla. FQM-354 Análisis Real
    In this expository article we collect and discuss some recent results on different consequences of a Sharp Reverse Hölder Inequality for A∞ weights. For two given operators T and S, we study Lp(w) bounds of CoifmanFefferman type: kT fkLp(w) ≤ cn,w,pkSfkLp(w), that can be understood as a way to control T by S. We will focus on a quantitative analysis of the constants involved and show that we can improve classical results regarding the dependence on the weight w in terms of Wilson’s A∞ constant [w]A∞ := sup Q 1 w(Q) Z Q M(wχQ). We will also exhibit recent improvements on the problem of finding sharp constants for weighted norm inequalities involving several singular operators In the same spirit as in T. Hytönen and C. Perez, Sharp weighted bounds involving A∞, we obtain mixed A1-A∞ estimates for the commutator [b, T] and for its higher order analogue Tk b. A common ingredient in the proofs presented here is a recent improvement of the Reverse Hölder Inequality for A∞ weights involving Wilson’s constant from T. Hytönen and C. Perez, Sharp weighted bounds involving A∞.
  • Acceso AbiertoCapítulo de Libro
    Existence and approximation of fixed points of right Bregman nonexpansive operators
    (Springer, 2013-07-29) Martín Márquez, Victoria; Reich, Simeon; Sabach, Shoham; Universidad de Sevilla. Departamento de Análisis Matemático; Universidad de Sevilla. FQM127: Análisis Funcional no Lineal
    We study the existence and approximation of fixed points of right Bregman nonexpansive operators in reflexive Banach space. We present, in particular, necessary and sufficient conditions for the existence of fixed points and an implicit scheme for approximating them.