Doctor of Philosophy / Department of Mathematics / Virginia Naibo / Investigations of pseudodifferential operators are useful in a variety of applications. These include finding solutions or estimates of solutions to certain partial differential equations, studying boundedness properties of commutators and paraproducts, and obtaining fractional Leibniz rules. A pseudodifferential operator is given through integration involving the Fourier transform of the arguments and a function called a symbol. Pseudodifferential operators were first studied in the linear case and results were obtained to advance both the theory and applicability of these operators. More recently, significant progress has been made in the study of bilinear, and more generally multilinear, pseudodifferential operators. Of special interest are boundedness properties of bilinear pseudodifferential operators which have been examined in a variety of function spaces. Since determining factors in the boundedness of these operators are connected to properties of the corresponding symbols, significant effort has been directed at categorizing the symbols according to size and decay conditions as well as at establishing the associated symbolic calculus. One such category, the bilinear Hörmander classes, plays a vital role in results concerning the boundedness of bilinear pseudodifferential operators in the setting of Lebesgue spaces in particular. The new results in this work focus on the study of bilinear pseudodifferential operators with symbols in weighted Besov spaces of product type. Unlike the Hörmander classes, symbols in these Besov spaces are not required to possess in finitely many derivatives satisfying size or decay conditions. Even without this much smoothness, boundedness properties on Lebesgue spaces are obtained for bilinear operators with symbols in certain Besov spaces. Important tools in the proofs of these new results include the demonstration of appropriate estimates and the development of a symbolic calculus for some of the Besov spaces along with duality arguments. In addition to the new boundedness results and as a byproduct of studying operators with symbols in Besov spaces, it is possible to quantify the smoothness of the symbols, in terms of the conditions that define the Hörmander classes, that is sufficient for boundedness of the operators in the context of Lebesgue spaces.

The common topic of this thesis is boundedness of integral and supremal operators between weighted function spaces. The first type of results are characterizations of boundedness of a convolution-type operator between general weighted Lorentz spaces. Weighted Young-type convolution inequalities are obtained and an optimality property of involved domain spaces is proved. Additional provided information includes an overview of basic properties of some new function spaces appearing in the proven inequalities. In the next part, product-based bilinear and multilinear Hardy-type operators are investigated. It is characterized when a bilinear Hardy operator inequality holds either for all nonnegative or all nonnegative and nonincreasing functions on the real semiaxis. The proof technique is based on a reduction of the bilinear problems to linear ones to which known weighted inequalities are applicable. Further objects of study are iterated supremal and integral Hardy operators, a basic Hardy operator with a kernel and applications of these to more complicated weighted problems and embeddings of generalized Lorentz spaces. Several open problems related to missing cases of parameters are solved, thus completing the theory of the involved fundamental Hardy-type operators. / Operators acting on function spaces are classical subjects of study in functional analysis. This thesis contributes to the research on this topic, focusing particularly on integral and supremal operators and weighted function spaces. Proving boundedness conditions of a convolution-type operator between weighted Lorentz spaces is the first type of a problem investigated here. The results have a form of weighted Young-type convolution inequalities, addressing also optimality properties of involved domain spaces. In addition to that, the outcome includes an overview of basic properties of some new function spaces appearing in the proven inequalities.  Product-based bilinear and multilinear Hardy-type operators are another matter of focus. It is characterized when a bilinear Hardy operator inequality holds either for all nonnegative or all nonnegative and nonincreasing functions on the real semiaxis. The proof technique is based on a reduction of the bilinear problems to linear ones to which known weighted inequalities are applicable.  The last part of the presented work concerns iterated supremal and integral Hardy operators, a basic Hardy operator with a kernel and applications of these to more complicated weighted problems and embeddings of generalized Lorentz spaces. Several open problems related to missing cases of parameters are solved, completing the theory of the involved fundamental Hardy-type operators. / <p>Artikel 9 publicerad i avhandlingen som manuskript med samma titel.</p>

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In this manuscript we study the action of one-dimensional integral operators on rearrangement-invariant Banach function spaces. Our principal goal is to characterize optimal target and optimal domain spaces corresponding to given spaces within the category of rearrangement-invariant Banach function spaces as well as to establish pointwise estimates of the non-increasing rearrangement of a given operator applied on a given function. We apply these general results to proving optimality relations between special rearrangement-invariant spaces. We pay special attention to the Laplace transform, which is a pivotal example of the operators in question. Powered by TCPDF (www.tcpdf.org)

Cette thèse, comporte deux parties distinctes.Dans la première partie, on étudie l'existence et l'inexistence d'une inégalité qu'on a appelée l'inégalité de Hopf Uniforme (IHU), pour une équation linéaire de la forme Lv = f à coefficients bornés mesurables et sous les conditions de Dirichlet homogènes. L'IHU est une variante du principe de maximum, on l'a appliquée dans la preuve de la régularité W1;p 0 pour un problème semi-linéaire singulier : Lu = F(u) où les coefficients de L sont dans l'espace vmor (fonctions à oscillation moyenne évanescente) et F(u) est singulier en u = 0 F(0) = +∞. De plus, si les coefficients sont lipschitziens, on prouve que la régularité optimale du gradient de la solution u est bmor (fonctions à oscillation moyenne bornée i.e Grad u dans bmor).Dans la seconde partie, on s'intéresse à la régularité du système d'élasticité (équations stationnaires des ondes élastiques) avec une fonction source singulière au sens qu'elle n’est qu'intégrable par rapport à la fonction distance au bord du domaine. Via la dualité, nous montrons, selon ~f , que le problème admet une solution dite très faible dont le gradient n'est pas nécessairement intégrable sur tout le domaine mais uniquement localement. Nous déterminons aussi les fonctions vectorielles ~f pour lesquelles, ~u a son gradient intégrable sur tout l'espace de travail. / We discuss the existence and non existence of the so called Hopf uniform Inequality (variant of a maximum principle) for the linear equation Lv = f with measurable coefficients and under the homogeneous Dirichlet Boundary condition. Then we apply such inequality to prove the W1;p 0 -regularity of a semi linear problem Lu = F(u), singular at u = 0, with the coefficients of the main operator of L in the space of vanishing mean oscillation. Moreover, when those coefficients are Lipschitz, we show that the gradient of the solution is at most in the space of bounded mean oscillation : bmor. In the last part of this thesis, we are concerned with the linear easticity system (Stationnary equation of the waves elasticity). But, here the second terms varies with respect to the distance function until the boundary.Using the duality method, we study the regularity of the solution of the elasticity system for the data belonging to various weighted spaces.