Let G be a locally compact group, w be a continuous weight function on G, and L^1(G,w) be the corresponding Beurling algebra. In this thesis, we study weak amenability of L^1(G,w) and of its centre ZL^1(G,w) for non-commutative locally compact groups G. We first give examples to show that the condition that characterizes weak amenability of L^1(G,w) for commutative groups G is no longer sufficient for the non-commutative case. However, we prove that this condition remains necessary for all [IN] groups G. We also provide a necessary condition for weak amenability of L^1(G,w) of a different nature, which, among other things, allows us to obtain a number of significant results on weak amenability of l^1(F_2,w) and l^1((ax+b),w). We then study the relation between weak amenability of the algebra L^1(G,w) on a locally compact group G and the algebra L^1(G/H,^w) on the quotient group G/H of G over a closed normal subgroup H with an appropriate weight ^w induced from w. We give an example showing that L^1(G,w) may not be weakly amenable even if both L^1(G/H,^w) and L^1(H,w|_H) are weakly amenable. On the other hand, by means of constructing a generalized Bruhat function on G, we establish a sufficient condition under which weak amenability of L^1(G,w) implies that of L^1(G/H,^w). In particular, with this approach, we prove that weak amenability of the tensor product of L^1(G_1,w_1) and L^1(G_2,w_2) implies weak amenability of both Beurling algebras L^1(G_1,w_1) and L^1(G_2,w_2), provided the weights w_1, w_2 are bounded away from zero. However, given a general weight on the direct product G of G_1 and G_2, weak amenability of L^1(G,w) usually does not imply that of L^1(G_1,w|_{G_1}), even if both G_1, G_2 are commutative. We provide an example to illustrate this. While studying the centres ZL^1(G,w) of L^1(G,w), we characterize weak amenability of ZL^1(G,w) for connected [SIN] groups G, establish a necessary condition for weak amenability of ZL^1(G,w) in the case when G is an [FC] group, and give a sufficient condition for the case when G is an [FD] group. In particular, we obtain some positive results on weak amenability of ZL^1(G,w) for a compactly generated [FC] group G with a polynomial weight w. Finally, we briefly discuss the derivation problem for weighted group algebras and present a partial solution to it.

Thesis (Ph. D.)--University of Texas at Austin, 1999. / Vita. Includes bibliographical references (leaves 80-81). Available also in a digital version from Dissertation Abstracts.

Los operadores pseudodiferenciales son generalizaciones de los operadores integrales singulares y de los operadores en derivadas parciales con coeficientes variables. A cada operador le corresponde un símbolo, que es una función infinitamente diferenciable y cuyas derivadas parciales cumplen ciertas estimaciones. El próposito es introducir estos operadores en el contexto de las clases no casianalíticas de tipo Beurling, clases que recientemente han recibido mucha atención, por ser más generales y unificar teorías anteriores. La tesis consta de tres capítulos. En el primero se definen los símbolos y operadors, se estudia entre qué espacios de funciones y ultradistribuciones actúan, se prueba que la clase es cerrada por trasposición y que los operadors son pseudolocales. También se dan ejemplos naturales de operadores en este contexto: operadores diferenciales cuyos coeficientes son funciones ultradiferencciables, los operadores regularizantes y los operadores ultradiferenciales en el sentido de Komatsu, y la convolución con una solución fundamental de un operador ultradiferencial elíptico. En el segundo capítulo se introduce el cálculo simbólico, cuyo objetivo es sustituir la teoría de los operadores por una algebraica de los correspondientes símbolos. El tercer capítulo está dedicado al estudio de la hipoelipticidad, concretamente de operadores en derivadas parciales de fuerza constante cuyos coeficientes están en una clase conveniente de funciones ultradiferenciables. Se prueba que en este contexto, la hipoelipticidad coincide con la hipoelipticidad homogénea, a priori más débil. También se establece una condición suficiente para la existencia de una paramétrix / Jornet Casanova, D. (2004). Operadores pseudodiferenciales en clases no cuasianalíticas de tipo Beurling [Tesis doctoral]. Universitat Politècnica de València. https://doi.org/10.4995/Thesis/10251/54953

We review properties of the Beurling correlation function related to differentiability and functional equations. The relevance of this function is due to the fact that some properties of the Riemann zeta function can be expressed interms of it. / Se repasan algunas propiedades de la función de correlación de Beurling, que sirven para expresar ciertas propiedades de la función zeta de Riemann.

Dans la première partie, nous nous intéressons à des opérateurs dont le spectre est inclus dans le cercle unité $\bbt$. Nous obtenons des résultats concernant certaines propriétés de croissance des normes $\| T^{-n} \| \, (n \geq 0)$ pour des opérateurs $T$ dont le spectre est dénombrable ou vérifie certaines conditions géométriques. Pour obtenir ces résultats, nous sommes amenés à travailler dans les espaces de fonctions<br />$$<br />A_{\omega}(\bbt) = \Big\{ f \textrm{ continue sur } \bbt : \, \big\| f \big\|_{\omega} = \sum_{n = -\infty}^{+\infty} | \widehat{f}(n) | \omega(n) < +\infty \Big\},<br />$$<br />où $\omega = \big( \omega(n) \big)_{n \in \bbz}$ est une suite de réels strictement positifs, et $\widehat{f}(n)$ désigne le $\textrm{n}^{\textrm{ième}}$ coefficient de Fourier de $f$. Lorsque la suite $\omega = \big( \omega(n) \big)_{n \in \bbz}$ est un poids, $\big( A_{\omega}(\bbt), \| \, \|_{\omega} \big)$ est une algèbre de Banach. Nous obtenons alors la caractérisation de certains idéaux fermés de $A_{\omega}(\bbt)$ pour une famille de poids. <br /><br />Dans la seconde partie, nous nous intéressons à des fermés de $\bbt$ qui sont (ou non) des ensembles d'unicité pour des espaces $\dsp A_{\omega}^{+}(\bbt) = \Big\{ f \in A_{\omega}(\bbt): \, \widehat{f}(n) = 0 \quad (n < 0) \Big\}$, où $\omega = \big( \omega(n) \big)_{n \in \bbz}$ est une suite de réels strictement positifs. Un fermé $E$ de $\bbt$ étant d'unicité pour un espace $X $ de fonctions continues sur $\bbt$, si la seule fonction dans $X$ s'annulant sur $E$ est la fonction nulle. Plus précisément, nous étudions le lien qu'il y a entre le fait qu'un fermé de $\bbt$ satisfait une condition géométrique donnée et le fait qu'il soit ou non un ensemble d'unicité pour $A_{\omega}^{+}(\bbt)$.

This thesis consists of an introduction and four papers. All four papers are devoted to problems in Number Theory. In Paper I, a special class of local ζ-functions is studied. The main theorem states that the functions have all zeros on the line Re(s)=1/2.This is a natural generalization of the result of Bump and Ng stating that the zeros of the Mellin transform of Hermite functions have Re(s)=1/2.In Paper II and Paper III we study eigenfunctions of desymmetrized quantized cat maps.If N denotes the inverse of Planck's constant, we show that the behavior of the eigenfunctions is very dependent on the arithmetic properties of N. If N is a square, then there are normalized eigenfunctions with supremum norm equal to <img src="http://www.diva-portal.org/cgi-bin/mimetex.cgi?N%5E%7B1/4%7D" />, but if N is a prime, the supremum norm of all eigenfunctions is uniformly bounded. We prove the sharp estimate <img src="http://www.diva-portal.org/cgi-bin/mimetex.cgi?%5C%7C%5Cpsi%5C%7C_%5Cinfty=O(N%5E%7B1/4%7D)" /> for all normalized eigenfunctions and all $N$ outside of a small exceptional set. For normalized eigenfunctions of the cat map (not necessarily desymmetrized), we also prove an entropy estimate and show that our functions satisfy equality in this estimate.We call a special class of eigenfunctions newforms and for most of these we are able to calculate their supremum norm explicitly.For a given <img src="http://www.diva-portal.org/cgi-bin/mimetex.cgi?N=p%5Ek" />, with k&gt;1, the newforms can be divided in two parts (leaving out a small number of them in some cases), the first half all have supremum norm about <img src="http://www.diva-portal.org/cgi-bin/mimetex.cgi?2/%5Csqrt%7B1%5Cpm%201/p%7D" /> and the supremum norm of the newforms in the second half have at most three different values, all of the order <img src="http://www.diva-portal.org/cgi-bin/mimetex.cgi?N%5E%7B1/6%7D" />. The only dependence of A is that the normalization factor is different if A has eigenvectors modulo p or not. We also calculate the joint value distribution of the absolute value of n different newforms.In Paper IV we prove a generalization of Mertens' theorem to Beurling primes, namely that \lim_{n \to \infty}\frac{1}{\ln n}\prod_{p \leq n} \left(1-p^{-1}\right)^{-1}=Ae^{\gamma}<img src="http://www.diva-portal.org/cgi-bin/mimetex.cgi?%5Clim_%7Bn%20%5Cto%20%5Cinfty%7D%5Cfrac%7B1%7D%7B%5Cln%20n%7D%5Cprod_%7Bp%20%5Cleq%20n%7D%0A%5Cleft(1-p%5E%7B-1%7D%5Cright)%5E%7B-1%7D=Ae%5E%7B%5Cgamma%7D," />where γ is Euler's constant and Ax is the asymptotic number of generalized integers less than x. Thus the limit <img src="http://www.diva-portal.org/cgi-bin/mimetex.cgi?M=%5Clim_%7Bn%5Cto%5Cinfty%7D%5Cleft(%5Csum_%7Bp%5Cle%20n%7Dp%5E%7B-1%7D-%5Cln(%5Cln%20n)%5Cright)" />exists. We also show that this limit coincides with <img src="http://www.diva-portal.org/cgi-bin/mimetex.cgi?%5Clim_%7B%5Calpha%5Cto%200%5E+%7D%0A%5Cleft(%5Csum_p%20p%5E%7B-1%7D(%5Cln%20p)%5E%7B-%5Calpha%7D-1/%5Calpha%5Cright)" /> ; for ordinary primes this claim is called Meissel's theorem. Finally we will discuss a problem posed by Beurling, namely how small |N(x)-[x] | can be made for a Beurling prime number system Q≠P, where P is the rational primes. We prove that for each c&gt;0 there exists a Q such that |N(x)-[x] | / QC 20100902

We prove several inequalities involving the Fourier transform of functions which are compactly supported. The constraint that the functions have compact support is a simplifying feature which is desirable in applications, but there is a trade-off in control of other relevant quantities-- such as the mass of the function. With applications in mind, we prove inequalities which quantify these types of trade-offs. / text

In this thesis, we have dealt primarily with two function algebras. The first one is the space of all holomorphic functions on the unit disc D in the complex plane which are continuous up to the boundary, denoted by A(D). The second one is H1(D), the space of all bounded analytic functions on D. We study results that characterize their maximal ideals. We start with necessary definitions and recall some useful results. In particular, the factorization of Hp functions in terms of Blaschke products, inner and outer functions is stated. Using this factorization, we provide an exposition of a beautiful result, originally by Beurling and rediscovered by Rudin, on the closed ideals of A(D). A maximality theorem by Wermer, which proves that A(D) is itself a maximal closed ideal of H1(D) is proved next. In chapter three, we expand our horizon and look at H1(D) as a dual space to characterize its weak-* closed maximal ideals. In the process we come across the shift operator and a theorem by Beurling, on the shift invariant subspaces of H2(D). We return in our quest to find out more about the maximal ideals of H1(D). The corona theorem states that the maximal ideals of the form Mτ = {ƒ ε H1(D) : ƒ (τ)=0} where τ is in D, are dense in the space of maximal ideals equipped with the Gelfand topology. We describe two approaches to the theorem, one that uses a lemma by Carleson on the existence and special properties of a contour in D. This is followed by a shorter and much more elegant proof by Wolff that uses elementary properties of Hp functions to achieve the same end. We conclude by presenting a proof of the Toeplitz corona theorem.

Dans cette thèse, nous nous intéressons à la description des idéaux fermés de certaines algèbres de fonctions analytiques sur le disque et le polydisque unité.

The main theme of this thesis is to study the isometric algebra isomorphisms and the bipositive algebra isomorphisms between various Banach algebras associated with locally compact groups. Let $LUC(G)$ denote the $C^*$-algebra of left uniformly continuous functions with the uniform norm and let $C_0(G)^{\perp}$ denote the annihilator of $C_0(G)$ in $LUC(G)^*$. In Chapter 2 of this thesis, among other results, we show that if $G$ is a locally compact group and $H$ is a discrete group then whenever there exists a weak-star continuous isometric isomorphism between $C_0(G)^{\perp}$ and $C_0(H)^{\perp}$, $G$ is isomorphic to $H$ as a topological group. In particular, when $H$ is discrete $C_0(H)^{\perp}$ determines $H$ within the class of locally compact topological groups. In Chapter 3 of this thesis, we show that if $M(G,\omega_1)$ (the weighted measure algebra on $G$) is isometrically algebra isomorphic to $M(H,\omega_2)$, then the underlying weighted groups are isomorphic, i.e. there exists an isomorphism of topological groups $\phi:G\to H$ such that $\small{\displaystyle{\frac{\omega_1}{\omega_2\circ\phi}}}$ is multiplicative. Similarly, we show that any weighted locally compact group $(G,\omega)$ is completely determined by its Beurling group algebra $L^1(G,\omega)$, $LUC(G,\omega^{-1})^*$ and $L^1(G,\omega)^{**}$, when the two last algebras are equipped with an Arens product. Here, $LUC(G,\omega^{-1})$ is the weighted analogue of $LUC(G)$, for weighted locally compact groups. In Chapter 4 of this thesis, we show that the order structure combined with the algebra structure of each of the Banach algebras $L^1(G,\omega)$, $M(G,\omega)$, $LUC(G,\omega^{-1})^*$ and $L^1(G,\omega)^{**}$ completely determines the underlying topological group structure together with a constraint on the weight. In particular, we obtain new proofs for a previously known result of Kawada and results of Farhadi as special cases of our results. Finally, we provide an example of a bipositive algebra isomorphism between Beurling measure algebras that is not an isometry. We conclude this thesis with a selective list of open problems. / February 2016