Дисертації з теми "Fourier and Schur multipliers"

Cette thèse a pour but d’étudier quelques problèmes dans l'analyse harmonique sur les produits croisés tordus qui sont définis par des actions tordues d'un groupe localement compact G sur une algèbre de von Neumann M. Elle se compose de deux parties. La première porte sur les produits croisés tordus et leurs multiplicateurs de Fourier et de Schur. Nous démontrons que la propriété d’être QWEP pour l’algèbre de von Neumann tordue d’un groupe G est indépendante du 2-cocycle sous-ajacent et que les Lp-multiplicateurs de Fourier complètement bornés sur cette algèbre tordue sont aussi indépendants du 2-cocycle. Sous l’hypothèse d’une action moyennable, nous établissons plusieurs résultats de transfert entre les multiplicateurs de Fourier et de Schur sur les espaces Lp non-commutatifs du produit croisé tordu.Dans la deuxième partie, nous étudions les commutateurs de multiplicateurs de Fourier sur le produit croisé tordu d’un espace euclidien. Nous caractérisons leur appartenance à la p-classes de Schatten par celle de leurs symboles à un espace de Besov associé. Cette partie contient aussi une formule sur la trace de Dixmier qui nous donne également une caractérisation de l’appartenance de ces commutateurs à une p-classe de Schatten faible par un espace de Sobolev. En particulier, nos résultats s'appliquent au cas d’un espace euclidien quantique<br>This thesis is devoted to the study of some problems in the harmonic analysis on twisted crossed products defined by twisted actions of a locally compact group G on a von Neumann algebra M. It consists of two parts. The first concerns twisted crossed products and their Fourier and Schur multipliers. We prove that the property of being QWEP for the twisted von Neumann algebra of a group G is independent of the underlying 2-cocycle and that the completely bounded Lp-Fourier multipliers on this twisted algebra are also independent of the 2-cocycle. Under the hypothesis of an amenable action, we establish several transference results between the Fourier and Schur multipliers on the noncommutative Lp spaces of the twisted crossed product.In the second part, we study Fourier multiplier commutators on the twisted crossed product of an Euclidean space. We characterize their Schatten p-class membership by that of their symbols in the associated Besov space. In addition, this part contains a formula on the Dixmier trace, which also gives us a characterization of the weak Schatten p-class membership of these commutators by a Sobolev space. In particular, our results apply to the case of quantum Euclidean spaces

Herz–Schur multipliers of a locally compact group have a well developed theory coming from a large literature; they have proved very useful in the study of the reduced C∗-algebra of a locally compact group. There is also a rich connection to Schur multipliers,which have been studied since the early twentieth century, and have a large number of applications. We develop a theory of Herz–Schur multipliers of a C∗-dynamical system, extending the classical Herz–Schur multipliers, making Herz–Schur multiplier techniques available to study a much larger class of C∗-algebras. Furthermore, we will also introduce and study generalised Schur multipliers, and derive links between these two notions which extend the classical results describing Herz–Schur multipliers in terms of Schur multipliers. This theory will be developed in as much generality as possible, with reference to the classical motivation. After introducing all the necessary concepts we begin the investigation by defining generalised Schur multipliers. The main result is a dilation type characterisation of these multipliers; we also show how such multipliers can be represented using HilbertC∗-modules. Next we introduce and study generalised Herz–Schur multipliers, first extending a classical result involving the representation theory of SU(2), before studying how such functions are related to our generalised Schur multipliers. We give a characterisation of generalised Herz–Schur multipliers as a certain class of the generalised Schur multipliers, and obtain a description of precisely which Schur multipliers belong to this class. Finally, we consider some ways in which the generalised multipliers can arise; firstly, from the classical multipliers which provide our motivation, secondly, from the Haagerup tensor product of a C∗-algebra with itself, and finally from positivity considerations. We show that our theory behaves well with respect to positivity and give conditions under which our multipliers are automatically positive in a natural sense.

Bounded Schur multipliers were introduced and characterised several decades ago, and various applications of this algebra of functions have been discovered. More recently, research into different classes of unbounded multipliers has been carried out. In this thesis the theory of one such class, that of the local Schur multipliers, is extended in different settings. A dilation of minimal Stinespring representations of completely positive, bimodular maps on spaces of compact operators is obtained, and used to establish an unbounded version of Stinespring's Theorem. This theorem is applied to obtain a characterisation of positive local Schur multipliers. In addition, a relation is demonstrated between operator monotone functions and positive local Schur multipliers, and a description is given of positive multipliers of Toeplitz type. The theory of local multipliers is extended to the multidimensional setting, and a characterisation of such functions is obtained. Local operator multipliers are introduced as a non-commutative e analogue of local Schur multipliers and a description is provided, extending previously known results concerning completely bounded operator multipliers. Positive multipliers are defined in this setting) and characterised using elements of canonical positive cones. The two-dimensional Fourier algebra A2(G) of a compact, abelian group G is considered, and a number of results are obtained concerning the Arens product on its dual, VN(G) ®uh VN(G). It is shown that A2 (G) may be viewed as a left VN(G) ®ub VN(G)-module, and thus certain results of Eyroard are extended to the two-dimensional setting, leading to the establishment of a condition equivalent to the homeomorphic identification of the Gelfand spectrum of A2(G) with C2 .

Dans le premier chapitre, nous commençons par définir certains produits tensoriels et identifions leur dual. Nous donnons ensuite quelques propriétés des classes de Schatten. La fin du chapitre est dédiée à l’étude des espaces de Bochner à valeurs dans l'espace des opérateurs factorisables par un espace de Hilbert. Le deuxième chapitre est consacré aux multiplicateurs de Schur linéaires. Nous caractérisons les multiplicateurs bornés sur B(Lp, Lq) lorsque p est inférieur à q puis appliquons ce résultat pour obtenir de nouvelles relations d'inclusion entre espaces de multiplicateurs. Dans le troisième chapitre, nous caractérisons, au moyen de multiplicateurs de Schur linéaires, les multiplicateurs de Schur bilinéaires continus à valeurs dans l'espace des opérateurs à trace. Dans le quatrième chapitre, nous donnons divers résultats concernant les opérateurs intégraux multiples. En particulier, nous caractérisons les opérateurs intégraux triples à valeurs dans l'espace des opérateurs à trace puis nous donnons une condition nécessaire et suffisante pour qu'un opérateur intégral triple définisse une application complètement bornée sur le produit de Haagerup de l'espace des opérateurs compacts. Enfin, le cinquième chapitre est dédié à la résolution des problèmes de Peller. Nous commençons par étudier le lien entre opérateurs intégraux multiples et théorie de la perturbation pour le calcul fonctionnel des opérateurs autoadjoints pour finir par la construction de contre-exemples à ces problèmes<br>In the first chapter, we define some tensor products and we identify their dual space. Then, we give some properties of Schatten classes. The end of the chapter is dedicated to the study of Bochner spaces valued in the space of operators that can be factorized by a Hilbert space.The second chapter is dedicated to linear Schur multipliers. We characterize bounded multipliers on B(Lp, Lq) when p is less than q and then apply this result to obtain new inclusion relationships among spaces of multipliers.In the third chapter, we characterize, by means of linear Schur multipliers, continuous bilinear Schur multipliers valued in the space of trace class operators. In the fourth chapter, we give several results concerning multiple operator integrals. In particular, we characterize triple operator integrals mapping valued in trace class operators and then we give a necessary and sufficient condition for a triple operator integral to define a completely bounded map on the Haagerup tensor product of compact operators. Finally, the fifth chapter is dedicated to the resolution of Peller's problems. We first study the connection between multiple operator integrals and perturbation theory for functional calculus of selfadjoint operators and we finish with the construction of counter-examples for those problems

We study the Lp − Lq boundedness of both spectral and Fourier multi- pliers on general locally compact separable unimodular groups G. As a consequence of the established Fourier multiplier theorem we also derive a spectral multiplier theorem on general locally compact separable uni- modular groups. We then apply it to obtain embedding theorems as well as time-asymptotics for the Lp − Lq norms of the heat kernels for general positive unbounded invariant operators on G. We illustrate the obtained results for sub-Laplacians on compact Lie groups and on the Heisenberg group, as well as for higher order operators. With minor modificaitons, our proofs of Paley-type inequalities and Lp − Lq bounds of Fourier multipliers can be adapted to the setting of compact homogeneous manifolds.

In the early 1970 fs, R. Coifman and G. Weiss, generalizing the techniques introduced by A. Calderon, developed a method for transferring abstract convolution type operators, defined on general topological groups, and their respective bounds, to the so called gtransferred operators h, which are operators defined on general measure spaces. To be specific, if G is a topological group and R_x is a representation of G on some Banach space B and K is a convolution operator on G given by <br/><br/>Kf= çk(x-y) f(y) dy<br/><br/>with k an L^1 function, the transferred operator T is defined by letting <br/><br/>Tf= çk(x-y) R_xf(y) dy.<br/><br/>Transfer methods deal with the study of the preservation of properties of K that are still valid for T, mostly focusing on the preservation of boundedness on Lebesgue spaces Lp. These methods has been applied to several problems in Mathematical Analysis, and especially to the problem of restrict Fourier multipliers to closed subgroups. These techniques have been extended by other authors as N. Asmar, E. Berkson and A. Gillespie, among many others. It is worth noting however, that these prior developments have always been focused on inequalities for operators on Lebesgue spaces Lp.<br/><br/>In this thesis there are developed several transference techniques for quasi-Banach spaces more general than Lebesgue spaces Lp, as Lorentz spaces Lp, q, Orlicz-Lorentz, Lorentz-Zygmund spaces as well as for weighted Lebesgue spaces Lp(w). The most significant applications are obtained in the field of restriction of Fourier multipliers for rearrangement invariant spaces and weighted Lebesgue spaces Lp(w). Specifically, we get generalizations of the results obtained by K. De Leeuw for Fourier multipliers. There are also developed similar techniques in the context of multilinear operators of convolution type, where the basic example is the bilinear Hilbert transform, as well as for modular inequalities and inequalities arising in extrapolation

Cette thèse étudie quelques problèmes d’analyse harmonique sur les groupes quantiques compacts. Elle consiste en trois parties. La première partie présente la théorie Lp élémentaire des transformées de Fourier, les convolutions et les multiplicateurs sur les groupes quantiques compacts, y compris la théorie de Hausdorff-Young et les inégalités de Young.Dans la seconde partie, nous caractérisons les opérateurs de convolution positifs sur un groupe quantique fini qui envoient Lp dans L2, et donnons aussi quelques constructions sur les groupes quantiques compacts infinis. La méthode pour étudier les états non-dégénérés fournit une formule générale pour calculer les états idempotents associés aux images deHopf, qui généralise un travail de Banica, Franz et Skalski. La troisième partie est consacrée à l’étude des ensembles de Sidon, des ensembles _(p) et des notions associées pour les groupes quantiques compacts. Nous établissons différentes caractérisations des ensembles de Sidon, et en particulier nous démontrons que tout ensemble de Sidon est un ensemble de Sidon fort au sens de Picardello. Nous donnons quelques liens entre les ensembles de Sidon, les ensembles _(p) et les lacunarités pour les multiplicateurs de Fourier sur Lp, généralisant un travail de Blendek et Michali˘cek. Nous démontrons aussi l’existence des ensembles de type _(p) pour les systèmes orthogonaux dans les espaces Lp non commutatifs, et déduisons les propriétés correspondantes pour les groupes quantiques compacts. Nous considérons aussi les ensembles de Sidon centraux, et nous prouvons que les groupes quantiques compacts ayant les mêmes règles de fusion et les mêmes fonctions de dimension ont des ensemble de Sidon centraux identiques. Quelques exemples sont aussi étudiés dans cette thèse. Les travaux présentés dans cette thèse se basent sur deux articles de l’auteur. Le premier s’intitule “Lp-improving convolution operators on finite quantum groups” et a été accepté pour publication dans Indiana University Mathematics Journal, et le deuxième est un travail intitulé “Lacunary Fourier series for compact quantum groups” et a été publié en ligne dans Communications in Mathematical Physics<br>This thesis studies some problems in the theory of harmonic analysis on compact quantum groups. It consists of three parts. The first part presents some elementary Lp theory of Fourier transforms, convolutions and multipliers on compact quantum groups, including the Hausdorff-Young theory and Young’s inequalities. In the second part, we characterize positive convolution operators on a finite quantum group G which are Lp-improving, and also give some constructions on infinite compact quantum groups. The methods for ondegeneratestates yield a general formula for computing idempotent states associated to Hopf images, which generalizes earlier work of Banica, Franz and Skalski. The third part is devoted to the study of Sidon sets, _(p)-sets and some related notions for compact quantum groups. We establish several different characterizations of Sidon sets, and in particular prove that any Sidon set in a discrete group is a strong Sidon set in the sense of Picardello. We give several relations between Sidon sets, _(p)-sets and lacunarities for Lp-Fourier multipliers, generalizing a previous work by Blendek and Michali˘cek. We also prove the existence of _(p)-sets for orthogonal systems in noncommutative Lp-spaces, and deduce the corresponding properties for compact quantum groups. Central Sidon sets are also discussed, and it turns out that the compact quantum groups with the same fusion rules and the same dimension functions have identical central Sidon sets. Several examples are also included. The thesis is principally based on two works by the author, entitled “Lp-improvingconvolution operators on finite quantum groups” and “Lacunary Fourier series for compact quantum groups”, which have been accepted for publication in Indiana University Mathematics Journal and Communications in Mathematical Physics respectively

Nous étudions plusieurs propriétés fonctionnelles d'inconditionnalité en les exprimant à l'aide de multiplicateurs. La première partie est consacrée à l'étude de phénomènes d'inconditionnalité isométrique et presqu'isométrique dans les espaces de Banach séparables. Parmi ceux-ci, la notion la plus générale est celle de ``propriété d'approximation inconditionnelle métrique''. Nous la caractérisons parmi les espaces de Banach de cotype fini par une propriété simple d'``inconditionnalité par blocs''. En nous ramenant à des multiplicateurs de Fourier, nous étudions cette propriété dans les sous-espaces des espaces de Banach de fonctions sur le cercle qui sont engendrés par une suite de caractères $e^(int)$. Nous étudions aussi les suites basiques inconditionnelles isométriques et presqu'isométriques de caractères, en particulier les ensembles de Sidon de constante asymptotiquement 1. Nous obtenons dans chaque cas des propriétés combinatoires sur la suite. La propriété suivante des normes $L^p$ est cruciale pour notre étude: si $p$ est un entier pair, $\int |f|^p = \int (|f^(p/2)|)^2 = \sum |\widehat(f^(p/2))(n)|^2$ est une expression polynomiale en les coefficients de Fourier de $f$ et $\bar f$. Nous proposons d'ailleurs une estimation précise de la constante de Sidon des ensembles à la Hadamard. La deuxième partie étudie les multiplicateurs de Schur: nous caractérisons les suites basiques inconditionnelles isométriques d'entrées de matrice $e_(ij)$ dans la classe de Schatten $S^p$. Les propriétés combinatoires que nous obtenons portent sur les chemins dans le réseau $\N \times \N$ à sommets dans cet ensemble. La troisième partie étudie le rapport entre la croissance d'une suite d'entiers et les propriétés harmoniques et fonctionnelles de la suite de caractères associée. Nous montrons en particulier que toute suite polynomiale, ainsi que la suite des nombres premiers, contient un ensemble $\Lambda(p)$ pour tout $p$ qui n'est pas de Rosenthal.

Cette thèse présente quelques résultats d'analyse sur les espaces Lp le plus souvent non commutatifs.La première partie exhibe de large classes de contractions sur des espaces Lp non commutatifsqui vérifient l'analogue non commutatif de la conjecture de Matsaev. De plus, cette partie fournitune comparaison entre certaines normes apparaissant naturellement dans ce domaine. La deuxièmepartie traite des fonctions carrées. Le premier résultat principal énonce que si T est un opérateurR-Ritt sur un espace Lp alors les fonctions carrées associées sont équivalentes. Le second résultatprincipal est une caractérisation de certaines estimations carrées utilisant les dilatations. La troisièmepartie de cette thèse introduit de nouvelles fonctions carrées pour les opérateurs de Ritt définis surdes espaces Lp non commutatifs. Le résultat principal est qu'en général ces fonctions carrées ne sontpas équivalentes. Cette partie contient aussi un résultat d'équivalence entre la norme usuelle et unecertaine fonction carrée. La quatrième partie introduit un analogue non commutatif de l'algèbre deFigà-Talamanca-Herz Ap(G) sur le prédual naturel de l'espace d'opérateurs Mp,cb des multiplicateursde Schur complètement bornées sur l'espace de Schatten Sp.

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.

My thesis deals with the study of elliptic PDE. It is divided into two parts, the first one concerning a nonlinear equation involving the p-Laplacian, and the second one focused on a nonlocal problem. In the first part, we study the regularity of stable solutions to a nonlinear equation involving the p-Laplacian in a bounded domain. This is the nonlinear version of the widely studied semilinear equation involving the classical Laplacian. Stable solutions to semilinear equations have been very recently proved to be bounded, and therefore smooth, up to dimension n=9 by Cabré, Figalli, Ros-Oton, and Serra. This result is known to be optimal by counterexamples in higher dimensions. In the case of the p-Laplacian, the boundedness of stable solutions is conjectured to hold up to a critical dimension depending on p. Examples of unbounded stable solutions are known if the dimension exceeds the critical one. Moreover, in the radial case or under strong assumptions on the nonlinearity, stable solutions are proved to be bounded in the optimal dimension range. We prove the boundedness of stable solutions under a new condition on n and p, which is optimal in the radial case, and more restrictive in the general one. It improves the known results in the field, and it is the first example, concerning the p-Laplacian, of a technique providing both a result in the nonradial case and the optimal result in the radial case. In the first part, we also investigate Hardy-Sobolev inequalities on hypersurfaces of Euclidean space, all containing a mean curvature term. Our motivation comes from several applications of these inequalities to the study of a priori estimates for stable solutions. Specifically, we give a simplified proof of the celebrated Michael-Simon and Allard inequality, we obtain two new forms of the Hardy inequality on hypersurfaces, and an improved Hardy inequality in the Poincaré sense. In the second part of this thesis, we deal with a Dirichlet to Neumann problem arising in a model for water waves. The system is described by a diffusion equation in a slab of fixed height, containing a weight that depends on a parameter a belonging to (-1,1). The top of the slab is endowed with a 0-Neumann condition, while on the bottom we have a Dirichlet datum and an equation involving a smooth nonlinearity. The system can also be reformulated as a nonlocal problem on the component endowed with the Dirichlet datum, by defining a suitable Dirichlet to Neumann operator. First, we prove a Liouville theorem that establishes the one dimensional symmetry of stable solutions, provided that a control on the growth of the energy associated with the problem is satisfied. As a consequence, we obtain the 1D symmetry of stable solutions to our problem in dimension 2. For n=3, we establish sharp energy estimates for both the energy minimizers and the monotone solutions, deducing the 1D symmetry of these classes of solutions, by an application of our Liouville theorem. Concerning this problem, we also investigate the nature of the associated Dirichlet to Neumann operator. First, we deduce its expression as a Fourier operator, which was known only in the case a=0. This result highlights the mixed nature of the operator, which is nonlocal, but not purely fractional. To better understand the dual behaviour of the operator, we provide a G-convergence result for an energy functional associated with the operator. Specifically, as a G-limit of our energy functional we find a mere interaction energy when a is greater than 0, and the classical perimeter when a is smaller or equal than 0. We point out that the threshold a=0 that we obtain here, as well as the G-limit behaviour for nonpositive values of a, is common to other nonlocal problems treated in the literature. On the contrary, the limit functional that we obtain in the other case appears to be new and structurally different from other nonlocal energy functionals that have been investigated in the literature.<br>Mi tesis se encaja en el estudio de las EDPs elípticas. Está dividida en dos partes: la primera trata una ecuación no-lineal con el p-Laplaciano, la segunda de un problema no-local. En la primera parte, estudiamos la regularidad de las soluciones estables de una ecuación no lineal con el p-Laplaciano en un dominio acotado. Esta ecuacion es la versión no-lineal de la ámpliamente estudiada ecuacion semilineal con el Laplaciano. Cabré, Figalli, Ros-Oton, y Serra han demostrado recientemente que las soluciones estables de las ecuaciones semilineales son acotadas, y por tanto regulares, hasta la dimensión 9. Este resultado es optimal. En el caso del p-Laplaciano, la regularidad de las soluciones estables se conjetura de ser cierta hasta una dimension critica y, de hecho, se conocen ejemplos de soluciones no acotadas cuando la dimension llega al valor critico. Además, se ha demostrado que en el caso radial o assumiendo hipótesis fuertes sobre la no-linealidad las soluciones estables son acotadas hasta la dimension critica. En el primer capítulo, demostramos que las soluciones estables son acotadas, bajo una nueva condición en n y p, que es optimal en el caso radial, y más restrictiva en el caso general. Esta investigación mejora conocidos resultados del tema y es el primer ejemplo, para el p-Laplaciano, de un método que produce un resultado para el caso general y un resultado optimal en el caso radial. En la primera parte, nos ocupamos también de las desigualdades funcionales del tipo Hardy y Sobolev sobre hipersuperfícies del espacio Euclideo, todas conteniendo un término de curvatura media. Nuestra motivación proviene de varias apliaciones que tienen estas desigualdades en el estudio de estimaciones para las soluciones estables. En detalle, damos una demostración simple de la conocida desigualdad de Michael-Simon y Allard, obtenemos dos formas nuevas de la desigualdad de Hardy sobre hipersuperfícies, y otra desigualdad de Hardy-Poincaré. En la segunda parte, nos ocupamos de un problema de Dirichlet-Neumann que emerge de un modelo para las ondas en el agua. El sistema se describe con una ecuación de difusión en una tira de altura fija, que contiene un parámetro a en (-1,1). La parte superior de la tira es dotada de una condicion 0 de Neumann, mientras en la parte inferior tenemos un dato de Dirichlet y una ecuación con una nonlinearidad regular. Este problema puede ser reformulado como una ecuación no-local sobre la componente dotada del dato de Dirichlet, definiendo un operador de Dirichlet-Neumann apropiado. Primero, demostramos un teorema del tipo Liouville, que garantiza la simetría unidimensional de las soluciones monótonas, asumiendo un control sobre el crecimiento de la energía asociada. Como consecuencia, obtenemos la simetría 1D de las soluciones estables en dimension 2. Para n=3, obtenemos estimaciónes optimales de la energía para las soluciones que minimizan la energía y para las soluciones monótonas. Estas estimaciones nos conducen a la simetría 1D de estas clases de soluciones, aplicando nuestro teorema del tipo Liouville. Relativo a este problema, estudiamos también la naturaleza del operador de Dirichlet-Neumann. Primero, deducimos su expresión como operador de Fourier, que anteriormente solo se conocía para a=0. Este resultado evidencia la naturaleza del operador, que es no-local pero no puramente fraccionaria. Estudiamos en profundidad este comportamiento mixto del operador a través del estudio de la G-convergencia de un funcional energía asociado al operador. Demostramos la G-convergencia de nuestro funcional a un límite que corresponde a una energía de interacción pura cuando a en (0,1) y al perímetro clásico cuando a en (-1,0]. El límite a=0, así como el G-límite para el régimen a en (-1,0], es común a otros problemas no-locales tratados en la literatura. Al contrario, el funcional límite en el régimen puramente no-local es nuevo y diferente a otros funciona<br>Questa tesi si occupa di equazioni differenziali alle derivate parziali di tipo ellittico. È divisa in due parti: la prima riguarda un’equazione nonlineare per il p-Laplaciano, mentre la seconda è incentrata su un problema nonlocale, che può essere formulato per mezzo di un operatore di Dirichlet-Neumann collegato con il Laplaciano frazionario. Nella prima parte, studiamo la regolarità delle soluzioni stabili dell’equazione nonlineare per il p-Laplaciano dove W è un dominio limitato, p 2 (1,+¥) e f è una nonlinearità C1. Questa equazione è la versione nonlineare dell’equazione semilineare 􀀀������������Du = f (u) in un dominio limitato W Rn, che è stata ampiamente studiata in letteratura. Molto recentemente, Cabré, Figalli, Ros-Oton, e Serra [38] hanno dimostrato che le soluzioni stabili delle equazioni semilineari sono limitate, e quindi regolari, in dimensione n 9. Questo risultato è ottimale, dato che esempi di soluzioni illimitate e stabili sono noti in dimensione n 10. Inoltre, i risultati in [38] forniscono una risposta completa ad un annoso problema aperto, proposto da Brezis e Vázquez [25], sulla regolarità delle soluzioni estremali dell’equazione 􀀀������������Du = l f (u). Queste ultime sono infatti esempi non banali di soluzioni stabili di equazioni semilineari, che possono essere limitate o illimitate in dipendenza della dimensione n, del dominio W, e della nonlinearità f . In questa tesi studiamo la limitatezza delle soluzioni stabili di (0.4), che si congettura essere vera fino alla dimensione n < p + 4p/(p 􀀀������������ 1). Sono infatti noti esempi di soluzioni stabili e illimitate quando n p + 4p/(p 􀀀������������ 1), anche quando il dominio è la palla unitaria. Inoltre, nel caso radiale o assumendo ipotesi forti sulla nonlinearità, è stato dimostrato che le soluzioni stabili di (0.4) sono limitate quando n < p + 4p/(p 􀀀������������ 1). Nel Capitolo 1 della tesi dimostriamo una nuova stima L¥ a priori per le soluzioni stabili di (0.4), assumendo una nuova condizione su n e p, che è ottimale nel caso radiale e più restrittiva nel caso generale. Il nostro risultato migliora ciò che è noto in letteratura e ed è il primo esempio di tecnica che produce sia un risultato nel caso non radiale sia il risultato ottimale nel caso radiale. Per ottenere questo risultato estendiamo al caso del p-Laplaciano una tecnica sviluppata da Cabré [30] per il caso classico del problema, con p = 2. La strategia si basa su una disuguaglianza di Hardy sugli insiemi di livello della soluzione, combinata con una disuguaglianza di tipo geometrico per le soluzioni stabili di (0.4). Nella prima parte della tesi ci occupiamo anche di disuguaglianze funzionali di tipo Hardy e Sobolev, su ipersuperfici dello spazio euclideo. Nel fare ciò siamo motivati dalle varie applicazioni di questo tipo di risultati allo studio di stime a priori per le soluzioni stabili, sia nel caso semilineare che nel caso nonlineare ...

Cette thèse a pour but d'étudier l'analyse sur les groupes quantiques compacts. Elle se compose de deux parties. La première présente la classification des semi-groupes de Markov invariants sur ces espaces homogènes quantiques. Les générateurs de ces semi-groupes sont considérés comme des opérateurs de Laplace sur ces espaces.La sphère classique , la sphère libre et la sphère semi-libérée sont considérées comme des exemples et les générateurs de semi-groupes de Markov sur ces sphères sont classés. Nous calculons aussi les dimensions spectrales des trois familles de sphères en fonction du comportement asymptotique des valeurs propres de leur opérateur de Laplace.Dans la deuxième partie, nous étudions la convergence des séries de Fourier pour les groupes non abéliens et les groupes quantiques. Il est bien connu qu'un certain nombre de propriétés d'approximation de groupes peuvent être interprétées comme des méthodes de sommation et de convergence moyenne de séries de Fourier non commutatives associées. Nous établissons un critère général d'inégalités maximales pour les identités approximatives de multiplicateurs non commutatifs de Fourier. En conséquence, nous prouvons que pour tout groupe dénombrable discret moyennable, il existe une suite de fonctions définies positives à support fini, telle que les multiplicateurs de Fourier associés sur les espaces Lp non commutatifs satisfassent à la convergence ponctuelle. Nos résultats s'appliquent également à la convergence presque partout des séries de Fourier de fonctions Lp sur des groupes compacts non-abéliens. D'autre part, nous obtenons des bornes indépendantes de la dimension pour les inégalités maximales de Hardy-Littlewood non commutatives dans l'espace à valeurs opérateurs associées à des corps convexes<br>This thesis is devoted to studying the analysis on compact quantum groups. It consists of two parts. First part presents the classification of invariant quantum Markov semigroups on these quantum homogeneous spaces. The generators of these semigroups are viewed as Laplace operators on these spaces.The classical sphere, the free sphere, and the half-liberated sphere are considered as examples and the generators of Markov semigroups on these spheres are classified. We compute spectral dimensions for the three families of spheres based on the asymptotic behavior of the eigenvalues of their Laplace operator.In the second part, we study of convergence of Fourier series for non-abelian groups and quantum groups. It is well-known that a number of approximation properties of groups can be interpreted as some summation methods and mean convergence of associated noncommutative Fourier series. We establish a general criterion of maximal inequalities for approximative identities of noncommutative Fourier multipliers. As a result, we prove that for any countable discrete amenable group, there exists a sequence of finitely supported positive definite functions, so that the associated Fourier multipliers on noncommutative Lp-spaces satisfy the pointwise convergence. Our results also apply to the almost everywhere convergence of Fourier series of Lp-functions on non-abelian compact groups. On the other hand, we obtain the dimension free bounds of noncommutative Hardy-Littlewood maximal inequalities in the operator-valued Lp space associated with convex bodies

Cette thèse de doctorat est consacrée à l'étude de quelques problèmes en analyse non commutative. Elle comprend quatre parties, allant des groupes quantiques et de l'analyse harmonique non commutative à l'information quantique. Tout d'abord, nous déterminons les états idempotents sur les groupes quantiques de Sekine, ce qui est obtenu en résolvant un système d'équations à l'aide de l'algèbre linéaire et de la théorie des nombres élémentaire. Ceci répond à une question de Franz et Skalski énoncée en 2009. Deuxièmement, nous étudions les états infiniment divisibles sur des groupes quantiques finis, c'est-à-dire, les états qui admettent une racine n-ième pour tout nge 1. Nous montrons que tout état infiniment divisible sur un groupe quantique fini est de type Poisson, c'est-à-dire qu'il peut être représenté sous la forme d'une exponentielle par rapport à un état idempotent. Troisièmement, nous donnons deux conditions suffisantes pour que les multiplicateurs de Fourier de L_p sur les algèbres de von Neumann de groupes discrets soient bornés. Très peu de résultats de ce type étaient connus auparavant. Notre idée est l'observation que, dans le cas discret, il suffit de considérer les multiplicateurs de Fourier de L_p-L_q. Enfin, dans le domaine de l'information quantique, nous confirmons une conjecture de Carlen, Frank et Lieb (puis une conjecture plus faible d'Audenaert et Datta). En conséquence, nous identifions toutes les paires (alpha, z) telles que l'alpha-z entropie relative de Rényi soit monotone sous l'action des applications complètement positives préservant la trace, ou satisfait à l'inégalité de traitement des données. La clé de la preuve est une modification d'une méthode variationnelle largement utilisée, qui permet d'obtenir des preuves simples de nombreux résultats connus<br>This PhD thesis is devoted to the study of some problems in noncommutative analysis. It consists of four parts, ranging from quantum groups and noncommutative harmonic analysis to quantum information. Firstly, we decide all the idempotent states on Sekine quantum groups, which is achieved by solving a system of equations using linear algebras and elementary number theory. This answers a question of Franz and Skalski stated in 2009. Secondly, we study the infinitely divisible states on finite quantum groups, i.e., states that admit n-th root for all nge 1. We show that every infinitely divisible state on a finite quantum group is of Poisson type, that is, it can be represented as an exponential relative to some idempotent state. Thirdly, we give two sufficient conditions for boundedness of L_p-Fourier multipliers on discrete group von Neumann algebras. Very few of such results were known before. Our idea is the observation that in the discrete case it suffices to consider L_p-L_q Fourier multipliers. Finally, in the area of quantum information, we confirm a conjecture of Carlen, Frank and Lieb (and then a weaker conjecture of Audenaert and Datta). As a consequence, we identify all the pairs (alpha,z) such that the alpha-z Rényi relative entropy is monotone under completely positive trace preserving maps, or satisfies Data Processing Inequality. The key part of the proof is a modification of a widely-used variational method. Its power yields simple proofs of many known results

Questa tesi è incentrata sullo studio di equazioni differenziali alle derivate parziali di tipo ellittico. La prima parte della tesi riguarda la regolarità delle soluzioni stabili per un'equazione nonlineare con il p-Laplaciano, in un dominio limitato dello spazio Euclideo. La tecnica è basata sull'uso di disuguaglianze di tipo Hardy-Sobolev su ipersuperfici, del quale viene approfondito lo studio. Nella seconda parte viene preso in esame un problema nonlocale di tipo Dirichlet-Neumann. Studiamo la simmetria unidimensionale di alcune sottoclassi di soluzioni stabili, ottenendo risultati in dimensione n=2, 3. Inoltre, studiamo il comportamento asintotico dell'operatore associato a questo problema nonlocale, usando tecniche di Γ-convergenza.<br>This thesis deals with the study of elliptic PDEs. The first part of the thesis is focused on the regularity of stable solutions to a nonlinear equation involving the p-Laplacian, in a bounded domain of the Euclidean space. The technique is based on Hardy-Sobolev inequalities in hypersurfaces involving the mean curvature, which are also investigated in the thesis. The second part concerns, instead, a nonlocal problem of Dirichlet-to-Neumann type. We study the one-dimensional symmetry of some subclasses of stable solutions, obtaining new results in dimensions n=2, 3. In addition, we carry out the study of the asymptotic behaviour of the operator associated with this nonlocal problem, using Γ-convergence techniques.

In the thesis are introduced and investigated spaces of Burling and of Roumieu type tempered ultradistributions, which are natural generalization of the space of Schwartz&rsquo;s tempered distributions in Denjoy-Carleman-Komatsu&rsquo;s theory of ultradistributions.&nbsp; It has been proved that the introduced spaces preserve all of the good properties Schwartz space has, among others, a remarkable one, that the Fourier transform maps continuposly the spaces into themselves.In the first chapter the necessary notation and notions are given.In the second chapter, the spaces of ultrarapidly decreasing ultradifferentiable functions and their duals, the spaces of Beurling and of Roumieu tempered ultradistributions, are introduced; their topological properties and relations with the known distribution and ultradistribution spaces and structural properties are investigated;&nbsp; characterization of&nbsp; the Hermite expansions&nbsp; and boundary value representation of the elements of the spaces are given.The spaces of multipliers of the spaces of Beurling and of Roumieu type tempered ultradistributions are determined explicitly in the third chapter.The fourth chapter is devoted to the investigation of&nbsp; Fourier, Wigner, Bargmann and Hilbert transforms on the spaces of Beurling and of Roumieu type tempered ultradistributions and their test spaces.In the fifth chapter the equivalence of classical definitions of the convolution of Beurling type ultradistributions is proved, and the equivalence of, newly introduced definitions, of ultratempered convolutions of Beurling type ultradistributions is proved.In the last chapter is given a necessary and sufficient condition for a convolutor of a space of tempered ultradistributions to be hypoelliptic in a space of integrable ultradistribution, is given, and hypoelliptic convolution equations are studied in the spaces.Bibliograpy has 70 items.<br>U ovoj tezi su proučavani prostori temperiranih ultradistribucija Beurlingovog&nbsp; i Roumieovog tipa, koji su prirodna uop&scaron;tenja prostora Schwarzovih temperiranih distribucija u Denjoy-Carleman-Komatsuovoj teoriji ultradistribucija. Dokazano je ovi prostori imaju sva dobra svojstva, koja ima i Schwarzov prostor, izmedju ostalog, značajno svojstvo da Furijeova transformacija preslikava te prostore neprekidno na same sebe.U prvom poglavlju su uvedene neophodne oznake i pojmovi.U drugom poglavlju su uvedeni prostori ultrabrzo opadajucih ultradiferencijabilnih funkcija i njihovi duali, prostori Beurlingovih i Rumieuovih temperiranih ultradistribucija; proučavana su njihova topolo&scaron;ka svojstva i veze sa poznatim prostorima distribucija i ultradistribucija, kao i strukturne osobine; date su i karakterizacije Ermitskih ekspanzija i graničnih reprezentacija elemenata tih prostora.Prostori multiplikatora Beurlingovih i Roumieuovih temperiranih ultradistribucija su okarakterisani u trećem poglavlju.Četvrto poglavlje je posvećeno proučavanju Fourierove, Wignerove, Bargmanove i Hilbertove transformacije na prostorima Beurlingovih i Rouimieovih temperiranih ultradistribucija i njihovim test prostorima.U petoj glavi je dokazana ekvivalentnost klasičnih definicija konvolucije na Beurlingovim prostorima ultradistribucija, kao i ekvivalentnost novouvedenih definicija ultratemperirane konvolucije ultradistribucija Beurlingovog tipa.U poslednjoj glavi je dat potreban i dovoljan uslov da konvolutor prostora temperiranih ultradistribucija bude hipoeliptičan u prostoru integrabilnih ultradistribucija i razmatrane su neke konvolucione jednačine u tom prostoru.Bibliografija ima 70 bibliografskih jedinica.

In this thesis we deal with two problems in harmonic analysis. In the first problem we discuss weighted norm inequalities for Weyl multipliers satisfying Mauceri’s condition. As an application, we prove certain multiplier theorems on the Heisenberg group and also show in the context of a theorem of Weis on operator valued Fourier multipliers that the R-boundedness of the derivative of the multiplier is not necessary for the boundedness of the multiplier transform. In the second problem we deal with a variation of a theorem of Mauceri concerning the Lp bound-edness of operators Mwhich are known to be bounded on L2 .We obtain sufficient conditions on the kernel of the operaor Mso that it satisfies weighted Lp estimates. As an application we prove Lp boundedness of Hermite pseudo-multipliers.

In this thesis we deal with two problems in harmonic analysis. In the first problem we discuss weighted norm inequalities for Weyl multipliers satisfying Mauceri’s condition. As an application, we prove certain multiplier theorems on the Heisenberg group and also show in the context of a theorem of Weis on operator valued Fourier multipliers that the R-boundedness of the derivative of the multiplier is not necessary for the boundedness of the multiplier transform. In the second problem we deal with a variation of a theorem of Mauceri concerning the Lp bound-edness of operators Mwhich are known to be bounded on L2 .We obtain sufficient conditions on the kernel of the operaor Mso that it satisfies weighted Lp estimates. As an application we prove Lp boundedness of Hermite pseudo-multipliers.

Thesis (Ph.D.)--University of Illinois at Urbana-Champaign, 2008.<br>Source: Dissertation Abstracts International, Volume: 69-11, Section: B, page: 6848. Adviser: Zhong-Jin Ruan. Includes bibliographical references (leaves 77-80) Available on microfilm from Pro Quest Information and Learning.

碩士<br>國立高雄應用科技大學<br>電子工程系<br>98<br>Due to the popularity of the communication systems, the Fourier transform is still one of research and development topics of wired and wireless communication. The high-speed computing of the discrete Fourier transform is very important in the real-time signal processing system. So many fast Fourier transform (FFT) algorithm are developed. Because of the regularity of FFT algorithm, it is very suitable for the implementation by using hardware circuits. Most of the developed algorithms reduce the computational complexity. In this thesis, we use radix-22 algorithm which can reduce the computational complexity from to . In this study, we compare various circuit architectures of fast Fourier transform in the view of hardware regularity, needed memory space and the number of computing operation. Finally, we adopt the radix-22 algorithm and the single-path delay feedback (SDF) architecture to implement the high-performance FFT processor. The conventional complex multiplier, multiplier-less canonical signed digit (CSD) complex multiplier and coordinate rotation digital computer (Cordic) architecture are used to realize pipelined fast Fourier transform processors. To reduce the error, we also realize our complex multiplier computing circuits with the double rounding technique. The Xilinx ISE software is used to synthesis the hardware, it shows that the 16-point FFT processor based on the multiplier-less canonical signed digit (CSD) complex multiplier can achieve the advantages of less needed hardware and moderate accuracy.

This doctoral thesis consist of four parts, in which the properties of operators on nonreflexive spaces of smooth functions are investigated.<br>Moja rozprawa doktorska składa się z rezultatów, które uzyskałem badając własności operatorów na niere eksywnych przestrzeniach funkcji gładkich. W dziedzinie analizy funkcjonalnej jednymi z najbardziej interesujących przykładów przestrzeni są przestrzenie funkcji analitycznych (n.p. przestrzenie Hardy’ego) i przestrzenie funkcji gładkich (n.p. przestrzenie Sobolewa, przestrzenie Biesowa). W odróżnieniu od przestrzeni Hardy’ego, gdzie własności operatorów są dobrze zbadane, nasza wiedza na temat własności operatorów w nierefleksywnych przestrzeniach Sobolewa jest wciąż niezadowalająca.

We prove a version of the Riesz-Thorin interpolation theorem for some types of weighted variable Lebesgue spaces. In order to do this we use the theory developed by Calderón in his 1964 article, together with some Banach function space theory. Using our version of the Riesz-Thorin theorem, we prove a version of the Stechkin inequality forweighted variable Lebesgue spaces, allowing us to define algebras of Fourier multipliers arising from functions of bounded variation. After analyzing the invertibility of Fourier convolution operators with piecewise continuous symbols, we shift our attention to slowly oscillating Fourier multipliers, finishing with a proof that the image in the Calkin algebra of the algebra of convolution type operators with slowly oscillating data is commutative.<br>Provamos uma versão do teorema de interpolação de Riesz-Thorin para alguns tipos de espaços de Lebesgue com expoente variável e peso. De forma a atingir este objectivo, usamos a teoria desenvolvida por Calderón no seu artigo de 1964. Usando a versão do teorema de Riesz-Thorin obtida, provamos uma versão da desigualdade de Stechkin para espaços de Lebesgue com expoente variável e peso. Isto permite-nos definir álgebras de multiplicadores de Fourier associados a funções de variação limitada. Após analisada a invertibilidade dos operadores de convolução com símbolos contínuos por troços, deslocamos a nossa atenção para multiplicadores de Fourier fracamente oscilantes. Terminamos com a prova de que a imagem na álgebra de Calkin da álgebra de operadores tipo convolução com dados fracamente oscilantes é comutativa.