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The topic of this dissertation lies at the interface between the areas of Harmonic Analysis, Partial Differential Equations, and Geometric Measure Theory, with an emphasis on the study of singular integral operators associated with second and higher order elliptic boundary value problems in non-smooth domains. The overall aim of this work is to further the development of a systematic treatment of second and higher order elliptic boundary value problems using singular integral operators. This is relevant to the theoretical and numerical treatment of boundary value problems arising in the modeling of physical phenomena such as elasticity, incompressible viscous fluid flow, electromagnetism, anisotropic plate bending, etc., in domains which may exhibit singularities at all boundary locations and all scales. Since physical domains may exhibit asperities and irregularities of a very intricate nature, we wish to develop tools and carry out such an analysis in a very general class of non-smooth domains, which is in the nature of best possible from the geometric measure theoretic point of view. The dissertation will be focused on three main, interconnected, themes: A. A systematic study of the poly-Cauchy operator in uniformly rectifiable domains in $\mathbb{C}$; B. Solvability results for the Neumann problem for the bi-Laplacian in infinite sectors in ${\mathbb{R}}^2$; C. Connections between spectral properties of layer potentials associated with second-order elliptic systems and the underlying tensor of coefficients. Theme A is based on papers [16, 17, 18] and this work is concerned with the investigation of polyanalytic functions and boundary value problems associated with (integer) powers of the Cauchy-Riemann operator in uniformly rectifiable domains in the complex plane. The goal here is to devise a higher-order analogue of the existing theory for the classical Cauchy operator in which the salient role of the Cauchy-Riemann operator $\overline{\partial}$ is now played by $\overline{\partial}^m$ for some arbitrary fixed integer $m\in{\mathbb{N}}$. This analysis includes integral representation formulas, higher-order Fatou theorems, Calderón-Zygmund theory for the poly-Cauchy operators, radiation conditions, and higher-order Hardy spaces. Theme B is based on papers [3, 19] and this regards the Neumann problem for the bi-Laplacian with $L^p$ data in infinite sectors in the plane using Mellin transform techniques, for $p\in(1,\infty)$. We reduce the problem of finding the solvability range of the integrability exponent $p$ for the $L^{p}$ biharmonic Neumann problem to solving an equation involving quadratic polynomials and trigonometric functions employing the Mellin transform technique. Additionally, we provide the range of the integrability exponent for the existence of a solution to the $L^{p}$ biharmonic Neumann problem in two-dimensional infinite sectors. The difficulty we are overcoming has to do with the fact that the Mellin symbol involves hypergeometric functions. Finally regarding theme C, based on the ongoing work in [2], the emphasis is the investigation of coefficient tensors associated with second-order elliptic operators in two dimensional infinite sectors and properties of the corresponding singular integral operators, employing Mellin transform. Concretely, we explore the relationship between distinguished coefficient tensors and $L^{p}$ spectral and Hardy kernel properties of the associated singular integral operators. / Mathematics

The Ph.D. Thesis ¿Operators on weighted spaces of holomorphic functions¿ presented here treats different areas of functional analysis such as spaces of holomorphic functions, infinite dimensional holomorphy and dynamics of operators. After a first chapter that introduces the notation, definitions and the basic results we will use throughout the thesis, the text is divided into two parts. A first one, consisting of Chapters 1 and 2, focused on a study of weighted (LB)-spaces of entire functions on Banach spaces, and a second one, corresponding to Chapters 3 and 4, where we consider differentiation and integration operators acting on different classes of weighted spaces of entire functions to study its dynamical behaviour. In what follows, we give a brief description of the different chapters: In Chapter 1, given a decreasing sequence of continuous radial weights on a Banach space X, we consider the weighted inductive limits of spaces of entire functions VH(X) and VH0(X). Weighted spaces of holomorphic functions appear naturally in the study of growth conditions of holomorphic functions and have been investigated by many authors since the work of Williams in 1967, Rubel and Shields in 1970 and Shields and Williams in 1971. We determine conditions on the family of weights to ensure that the corresponding weighted space is an algebra or has polynomial Schauder decompositions. We study Hörmander algebras of entire functions defined on a Banach space and we give a description of them in terms of sequence spaces. We also focus on algebra homomorphisms between these spaces and obtain a Banach-Stone type theorem for a particular decreasing family of weights. Finally, we study the spectra of these weighted algebras, endowing them with an analytic structure, and we prove that each function f ¿ VH(X) extends naturally to an analytic function defined on the spectrum. Given an algebra homomorphism, we also investigate how the mapping induced between the spectra acts on the corresponding analytic structures and we show how in this setting composition operators have a different behavior from that for holomorphic functions of bounded type. This research is related to recent work by Carando, García, Maestre and Sevilla-Peris. The results included in this chapter are published by Beltrán in [14]. Chapter 2 is devoted to study the predual of VH(X) in order to linearize this space of entire functions. We apply Mujica¿s completeness theorem for (LB)-spaces to find a predual and to prove that VH(X) is regular and complete. We also study conditions to ensure that the equality VH0(X) = VH(X) holds. At this point, we will see some differences between the finite and the infinite dimensional cases. Finally, we give conditions which ensure that a function f defined in a subset A of X, with values in another Banach space E, and admitting certain weak extensions in a space of holomorphic functions can be holomorphically extended in the corresponding space of vector-valued functions. Most of the results obtained have been published by the author in [13]. The rest of the thesis is devoted to study the dynamical behaviour of the following three operators on weighted spaces of entire functions: the differentiation operator Df(z) = f (z), the integration operator Jf(z) = z 0 f(¿)d¿ and the Hardy operator Hf(z) = 1 z z 0 f(¿)d¿, z ¿ C. In Chapter 3 we focus on the dynamics of these operators on a wide class of weighted Banach spaces of entire functions defined by means of integrals and supremum norms: the weighted spaces of entire functions Bp,q(v), 1 ¿ p ¿ ¿, and 1 ¿ q ¿ ¿. For q = ¿ they are known as generalized weighted Bergman spaces of entire functions, denoted by Hv(C) and H0 v (C) if, in addition, p = ¿. We analyze when they are hypercyclic, chaotic, power bounded, mean ergodic or uniformly mean ergodic; thus complementing also work by Bonet and Ricker about mean ergodic multiplication operators. Moreover, for weights satisfying some conditions, we estimate the norm of the operators and study their spectrum. Special emphasis is made on exponential weights. The content of this chapter is published in [17] and [15]. For differential operators ¿(D) : Bp,q(v) ¿ Bp,q(v), whenever D : Bp,q(v) ¿ Bp,q(v) is continuous and ¿ is an entire function, we study hypercyclicity and chaos. The chapter ends with an example provided by A. Peris of a hypercyclic and uniformly mean ergodic operator. To our knowledge, this is the first example of an operator with these two properties. We thank him for giving us permission to include it in our thesis. The last chapter is devoted to the study of the dynamics of the differentiation and the integration operators on weighted inductive and projective limits of spaces of entire functions. We give sufficient conditions so that D and J are continuous on these spaces and we characterize when the differentiation operator is hypercyclic, topologically mixing or chaotic on projective limits. Finally, the dynamics of these operators is investigated in the Hörmander algebras Ap(C) and A0 p(C). The results concerning this topic are included by Bonet, Fernández and the author in [16]. / Beltrán Meneu, MJ. (2014). Operators on wighted spaces of holomorphic functions [Tesis doctoral]. Universitat Politècnica de València. https://doi.org/10.4995/Thesis/10251/36578 / Premios Extraordinarios de tesis doctorales

Cette thèse présente quelques résultats de la théorie des probabilités quantiques et de l’analyse harmonique à valeurs operateurs. La thèse est composée des trois parties.Dans la première partie, on démontre la décomposition atomique des espaces de Hardy de martingales non commutatives. On identifie aussi les interpolés complexes et réels entre les versions conditionnelles des espaces de Hardy et BMO de martingales non commutatives.La seconde partie est consacrée à l’étude des espaces de Hardy à valeurs opérateursvia la méthode d’ondellettes. Cette approche est similaire à celle du cas des martingales non commutatives. On démontre que ces espaces de Hardy sont équivalents à ceux étudiés par Tao Mei. Par conséquent, on donne une base explicite complètement inconditionnelle pour l’espace de Hardy H1(R), muni d’une structure d’espace d’opérateurs naturelle. La troisième partie porte sur l’analyse harmonique sur le tore quantique. On établit les inégalités maximales pour diverses moyennes de sommation des séries de Fourier définies sur le tore quantique et obtient les théorèmes de convergence ponctuelle correspondant. En particulier, on obtient un analogue non commutative du théorème classique de Stein sur les moyennes de Bochner-Riesz. Ensuite, on démontre que les multiplicateurs de Fourier complètement bornés sur le tore quantique coïncident à ceux définis sur le tore classique. Finalement, on présente la théorie des espaces de Hardy et montre que ces espaces possèdent les propriétés des espaces de Hardy usuels. En particulier, on établit la dualité entre H1 et BMO. / This thesis presents some results in quantum probability and operator-valued harmonicanalysis. The main results obtained in the thesis are contained in the following three parts:In first part, we prove the atomic decomposition for the Hardy spaces h1 and H1 of noncommutative martingales. We also establish that interpolation results on the conditionedHardy spaces of noncommutative martingales. The second part is devoted to studying operator-valued Hardy spaces via Meyer’s wavelet method. It turns out that this way of approaching these spaces is parallel to that in the noncommutative martingale case. We also show that these Hardy spaces coincide with those introduced and studied by Tao Mei in [52]. As a consequence, we give an explicit completely unconditional base for Hardy spaces H1(R) equipped with a natural operator space structure. The third part deals with with harmonic analysis on quantum tori. We first establish the maximal inequalities for several means of Fourier series defined on quantum tori and obtain the corresponding pointwise convergence theorems. In particular, we prove the noncommutative analogue of the classical Stein theorem on Bochner-Riesz means. Then we prove that Lp completely bounded Fourier multipliers on quantum tori coincide with those on classical tori with equal cb-norms. Finally, we present the H1-BMO and Littlewood- Paley theories associated with the circular Poisson semigroup over quantum tori.

Ce mémoire est divisé en deux parties : Partie 1 : Nous obtenons des résultats d'existence pour des problèmes au limite mettant en jeu des opérateurs polyharmoniques conformément invariants. Nous nous plaçons indifféremment dans le cas d'une variété riemannienne avec ou sans bord. En particulier, nous montrons que la meilleure constante de Sobolev sur les variétés est exactement la constante euclidienne. En conséquence, nous montrons l'existence d'une solution d'énergie minimale lorsque la fonctionnelle descend en-dessous d'un seuil quantifié. Puis nous montrons l'existence de solutions de haute énergie en utilisant la méthode topologique de Coron. Nous généralisons la décomposition des suites de Palais-Smale comme somme de bulles sur une variété avec ou sans bord : il s'agit d'un résultat dans l'esprit du célèbre théorème de Struwe en 1984. Nous obtenons aussi une version du lemme de compacité-concentration de Pierre-Louis Lions sur les variétés. Partie 2 : Dans cette partie, nous effectuons une analyse de blow-up pour une équation de Hardy-Sobolev à croissance critique et à singularité évanescente au bord. En supposant que l'équation limite n'admet pas de solution minimisante, nous étudions le comportement asymptotique d’une suite de solutions de l'équation perturbée. Ici, la perturbation est la singularité à l'origine. Dans un premier temps, nous obtenons un contrôle ponctuel optimal de la suite de solutions. Dans un second temps, nous obtenons des informations précises sur le point d'explosion en utilisant une identité de Pohozaev / This memoir can be divided into two parts: Part 1: In this part we obtain some existence results for conformally invariant polyharmonic boundary value problems on a compact Riemannian manifold with or without boundary. In particular we show that the best constant of the Sobolev embedding on manifolds is same as the euclidean one, and as a consequence prove the existence of minimum energy solutions when the energy functionnal goes below a quantified threshold. Next we show the existence of high energy solution using the topological method of Coron. We generalize the decomposition of Palais Smale sequences as a sum of bubble on manifolds with or without boundary, a result in the spirit of Struwe's celebrated 1984 result and also an extension of PL Lions concentration compactness result on manifolds. Part2: In this part we do a blow-up analysis of the nonlinear elliptic Hardy-Sobolev equation with critical growth and vanishing boundary singularity. We assume that our equation does not admit minimising solutions, and study the asymptotic behaviour of a sequence of solution to the perturbed equation. Here the perturbation is the singularity at the origin. First we obtain optimal pointwise controlon the sequence and then obtain more precise informations on the localization of the blow-up point using the Pohozaev identity

Cette thèse a pour objectif d’étudier la tragédie du XVIIe siècle en France avec la notion de faute tragique, hamartia, et de montrer l’originalité et la diversité des tragédies classiques. Commentée dans la Poétique d’Aristote, l’hamartia se définit comme une notion médiane entre un délit volontaire et une malchance. Coupable d’une hamartia, le héros n’en est pas pour autant entièrement responsable malgré sa prise d’initiative, dans la mesure où la faute n’est pas due à l’intention perverse. Parce que le malheur déclenché par une hamartia, apparaît disproportionné à l’intention du coupable, les spectateurs éprouvent de la compassion envers le héros infortuné. Dans la mesure où la compassion est une des émotions essentielles de la tragédie au même titre que la terreur – en effet, la catharsis consiste en l’épuration de ces émotions –, la faute tragique constitue à cet égard un des composants sine qua non de la tragédie. Mais arrivée au XVIIe siècle en France, la faute garde-t-elle le même statut ? Pour répondre à cette question, nous examinons les tragédies de sept dramaturges, Alexandre Hardy, Pierre Du Ryer, Jean Rotrou, Tristan L’Hermite, Pierre Corneille, Jean Racine et Jean-Galbert de Campistron, ce qui permet d’étudier le XVIIe siècle en entier. Le premier chapitre sera consacré à l’examen de la notion de faute dans les écrits théoriques. Dans le deuxième chapitre, nous nous intéresserons à la fabrique du héros coupable. Le troisième chapitre portera sur la nature de la faute. Et nous étudierons, dans le quatrième chapitre, le statut de la faute sur le plan dramaturgique, avant d’examiner, dans le dernier chapitre, les problèmes moraux. Ce travail révèlera l’importance de la Poétique d’Aristote dans la tragédie du XVIIe siècle en France. / The purpose of this doctoral thesis is to study the French tragedy in the seventeenth century researching into the notion of tragic flaw, hamartia, and to show the originality and the diversity of the French classical tragedy. This notion, commented by Aristotle in "Poetics" originally differs from either an intentional crime or an accidental one. The tragic flaw presupposes the participation of an agent without denying the presence of fortuity. Although a tragic hero is responsible for his misfortune, as it proceeded from his fault, the result exceeds his intention. And this disparity between intention and misfortune makes the audience feel compassion for the hero suffering from his misfortune. If this compassion is one of the emotions caused only by tragedies, the tragic flaw constitutes in this respect an essential element of tragedy. But is the concept of fault identical in the Christendom society of the seventeenth century? To answer this question, I deal with the tragedies of seven dramaturges, Alexandre Hardy, Pierre Du Ryer, Jean Rotrou, Tristan L’Hermite, Pierre Corneille, Jean Racine and Jean-Galbert de Campistron, to cover the seventeenth century. In the first section of this work, I examine the notion of fault in the theoretical texts. The second section consists in studying the tragic figure. The third section is about the nature of the fault, the private fault and the politic fault. The fourth section concerns the status of fault on the dramaturgical level, before examining moral questions in the final section. This work reveals the importance of Aristotle’s Poetics in the French tragedy of the seventeenth century.

Ce mémoire est consacré à des résultats d'analyse harmonique réelle dans des cadres géométriques discrets (graphes) ou continus (groupes de Lie).Soit $\Gamma$ un graphe (ensemble de sommets et d'arêtes) muni d'un laplacien discret $\Delta=I-P$, où $P$ est un opérateur de Markov.Sous des hypothèses géométriques convenables sur $\Gamma$, nous montrons la continuité $L^p$ de fonctionnelles de Littlewood-Paley fractionnaires. Nous introduisons des espaces de Hardy $H^1$ de fonctions et de $1$-formes différentielles sur $\Gamma$, dont nous donnons plusieurs caractérisations, en supposant seulement la propriété de doublement pour le volume des boules de $\Gamma$. Nous en déduisons la continuité de la transformée de Riesz sur $H^1$. En supposant de plus des estimations supérieures ponctuelles (gaussiennes ou sous-gaussiennes) sur les itérées du noyau de l'opérateur $P$, nous obtenons aussi la continuité de la transformée de Riesz sur $L^p$ pour $1<p<2$.Nous considérons également l'espace de Besov $B^{p,q}_\alpha(G)$ sur un groupe de Lie unimodulaire $G$ muni d'un sous-laplacien $\Delta$. En utilisant des estimations du noyau de la chaleur associé à $\Delta$, nous donnons plusieurs caractérisations des espaces de Besov, et montrons une propriété d'algèbre pour $B^{p,q}_\alpha(G) \cap L^\infty(G)$, pour $\alpha>0$, $1\leq p\leq+\infty$ et $1\leq q\leq +\infty$. Les résultats sont valables en croissance polynomiale ou exponentielle du volume des boules. / This thesis is devoted to results in real harmonic analysis in discrete (graphs) or continuous (Lie groups) geometric contexts.Let $\Gamma$ be a graph (a set of vertices and edges) equipped with a discrete laplacian $\Delta=I-P$, where $P$ is a Markov operator.Under suitable geometric assumptions on $\Gamma$, we show the $L^p$ boundedness of fractional Littlewood-Paley functionals. We introduce $H^1$ Hardy spaces of functions and of $1$-differential forms on $\Gamma$, giving several characterizations of these spaces, only assuming the doubling property for the volumes of balls in $\Gamma$. As a consequence, we derive the $H^1$ boundedness of the Riesz transform. Assuming furthermore pointwise upper bounds for the kernel (Gaussian of subgaussian upper bounds) on the iterates of the kernel of $P$, we also establish the $L^p$ boundedness of the Riesz transform for $1<p<2$.We also consider the Besov space $B^{p,q}_\alpha(G)$ on a unimodular Lie group $G$ equipped with a sublaplacian $\Delta$.Using estimates of the heat kernel associated with $\Delta$, we give several characterizations of Besov spaces, and show an algebra property for $B^{p,q}_\alpha(G) \cap L^\infty(G)$ for $\alpha>0$, $1\leq p\leq+\infty$ and $1\leq q\leq +\infty$.These results hold for polynomial as well as for exponential volume growth of balls.

Text in Afrikaans / Hierdie navorsing handel oor ouderdom en geslag as veranderlikes in die salutogenese paradigma. Die salutogenese paradigma het sy ontstaan en ontwikkeling aan verskeie navorsingsperspektiewe te danke. Daar is reeds op internasionale gebied breedvoerig navorsing oor hierdie paradigma gedoen. In die Suid-Afrika is die navorsing van Strumpfer en Wissing goed bekend. In hierdie navorsing is daar deur middel van faktorontleding gepoog om die onderliggende dimensies van die konstrukte gevoel van koherensie en geharde persoonlikheid bloot te le. Daar is onderskeidelik twee duidelike faktore vir beide konstrukte bepaal. Die faktore op die Lewensorientasievraelys (OLQ) het noue ooreenstemming getoon met die komponente betekenisvolheid en hanteerbaarheid (OLQ1 ), en verstaanbaarheid (OLQ2) soos deur Antonovsky (1987) bespreek. Die faktore op die "Personal Views Survey" (PVS) het ooreenstemming getoon met die komponente verbintenis en beheer (PVS1) en uitdaging (PVS2), soos deur Kobasa (1979) daargestel. Hierdie navorsingsresultate toon verder dat ouderdom wel die mate van gevoel van koherensie wat 'n individu mag ervaar, kan be"invloed. Alhoewel geslag in 'n mindere mate 'n invloed op die mate van gevoel van koherensie getoon het, behoort geslag (in perspektief van die totale steekproef beskou) nie 'n bepalende invloed uit te oefen nie. In terme van die mate van geharde persoonlikheid wat 'n individu mag ervaar, het ouderdom en geslag geen invloed getoon nie. Daar is ook geen interaksie-effek tussen ouderdom en geslag en die onderskeie konstrukte vasgestel nie. / This research project deals with age and gender as variables in the salutogenesis paradigm. The salutogenesis paradigm owes its origin and development to various research projects. Research in the international field has already been done on this paradigm on a wide sphere. The research of Strumpfer and Wissing is well known in South Africa. In this area of research an attempt is made through the analysis of factors to expose the underlying dimensions of the construct sense of coherence and the construct hardy personality. Two certain factors for both constructs have been indicated. The factors influencing the Orientation to Life Questionnaire indicated a close resemblance with the components of meaningfulness and manageability (OLQ1), and comprehensibility (OLQ2), discussed by Antonovsky (1987). The factors of the Personal Views Survey (PVS) demonstrated a similarity with the components commitment and control (PVS 1) and challenge (PVS2), as stated by Kobasa (1979). The results of this research demonstrates that age may well influence the measure of the sense of coherence which an individual may experience. Although gender indicated a minor measure of influence on the degree of sense of coherence, gender should not (in perspective of this research findings) have a deciding influence. Age and gender indicated no deciding influence in the measure of hardy personality experienced by an individual. Age and gender did not manifest any interaction in the various constructs. / Industrial and Organisational Psychology / M. Com. (Bedryfsielkunde)