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Certain subquotients of Vec(R)-modules of pseudodifferential operators from one tensor density module to another are categorized, giving necessary and sufficient conditions under which two such subquotients are equivalent as Vec(R)-representations. These subquotients split under the projective subalgebra, a copy of ????2, when the members of their composition series have distinct Casimir eigenvalues. Results were obtained using the explicit description of the action of Vec(R) with respect to this splitting. In the length five case, the equivalence classes of the subquotients are determined by two invariants. In an appropriate coordinate system, the level curves of one of these invariants are a pencil of conics, and those of the other are a pencil of cubics.

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.

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O cálculo funcional de operadores em espaços de Banach tem uma longa história, sendo inicialmente desenvolvido por F. Riesz, N. Dunford entre outros. Em 1986, uma importante contribuição foi feita por Alan McIntosh, que definiu um cálculo funcional holomorfo de operadores setoriais e destacou uma importante classe de operadores setoriais desses operadores: a dos operadores com cálculo funcional holomorfo limitado (CFHL). Do ponto de vista de operadores diferenciais e pseudodiferenciais, alguns elementos envolvidos neste cálculo já estavam presentes nos trabalhos de R. T. Seeley sobre potências complexas de operadores diferenciais elípticos. Mais tarde mostrou-se que diversos operadores possuem CFHL. Um artigo recente nesta direção e base para esta dissertação foi publicado por Bilyj, Schrohe e Seiler. Neste trabalho mostraremos que certos operadores pseudodiferenciais, agindo em espaços de Banach apropriados, são setoriais e possuem CFHL. Para isso faremos o estudo da álgebra dos símbolos de ordem zero e utilizaremos uma construção para a parametriz do resolvente. A apresentação procura ser uma versão mais didática do artigo de Bilyj, Schrohe e Seiler. Além disso, fazemos certas adaptações nas demonstrações com o propósito de facilitar a compreensão dos argumentos. Também vamos apresentar aplicações do resultado obtido. / Functional calculus for operators acting on Banach Spaces has a long history. It was initially developed by F. Riesz, N. Dunford among others. In 1986, an important contribution was made by Alan McIntosh who defined a holomorphic functional calculus for sectorial operators and put on the scene an important class of sectorial operators, namely, operators with a bounded holomorphic functional calculus (BHFC). From the point of view of differential and pseudodifferential operators, some elements treated in this calculus were already in the works of R. T. Seeley about complex powers of elliptic differential operators. Later it was shown that several operators have BHFC. A recent paper in this direction, and the one on which this dissertation is based, was published by Bilyj, Schrohe and Seiler. In this work we show that certain pseudodifferential operators, acting on appropriate Banach spaces, are sectorial and have BHFC. For this we will study the algebra of symbols of order zero and use a construction for the parametrix. This presentation aims to explore and detail the paper of Bilyj, Schrohe and Seiler. Furthermore, we make adaptations in the proofs in order to clarify the argument. We also show applications of the obtained results.

The paper is devoted to the construction of the exponential function of a matrix pseudo-differential operator which do not satisfy any of the known theorems (see, Sec.8 Ch.VIII and Sec.2 Ch.XI of [17]). The applications to the construction of the fundamental solution for the Cauchy problem for the hyperbolic operators with the characteristics of variable multiplicity are given, too.

We construct elliptic theory in the subspaces, determined by pseudodifferential projections. The finiteness theorem as well as index formula are obtained for elliptic operators acting in the subspaces. Topological (K-theoretic) aspects of the theory are studied in detail.

We study mixed boundary value problems for an elliptic operator A on a manifold X with boundary Y / i.e., Au = f in int X, T±u = g± on int Y±, where Y is subdivided into subsets Y± with an interface Z and boundary conditions T± on Y± that are Shapiro-Lopatinskij elliptic up to Z from the respective sides. We assume that Z ⊂ Y is a manifold with conical singularity v. As an example we consider the Zaremba problem, where A is the Laplacian and T− Dirichlet, T+ Neumann conditions. The problem is treated as a corner boundary value problem near v which is the new point and the main difficulty in this paper. Outside v the problem belongs to the edge calculus as is shown in [3]. With a mixed problem we associate Fredholm operators in weighted corner Sobolev spaces with double weights, under suitable edge conditions along Z {v} of trace and potential type. We construct parametrices within the calculus and establish the regularity of solutions.

We establish a parameter-dependent pseudo-differential calculus on an infinite cylinder, regarded as a manifold with conical exits to infinity. The parameters are involved in edge-degenerate form, and we formulate the operators in terms of operator-valued amplitude functions.

I propose a computationally efficient method to approximate the inverse of the normal operator arising in the multi-parameter linearized inverse problem for reflection seismology in two and three spatial dimensions. Solving the inverse problem using direct matrix methods like Gaussian elimination is computationally infeasible. In fact, the application of the normal operator requires solving large scale PDE problems. However, under certain conditions, the normal operator is a matrix of pseudodifferential operators. This manuscript shows how to generalize Cramer's rule for matrices to approximate the inverse of a matrix of pseudodifferential operators. Approximating the solution to the normal equations proceeds in two steps: (1) First, a series of applications of the normal operator to specific permutations of the right hand side. This step yields a phase-space scaling of the solution. Phase space scalings are scalings in both physical space and Fourier space. Second, a correction for the phase space scaling. This step requires applying the normal operator once more. The cost of approximating the inverse is a few applications of the normal operator (one for one parameter, two for two parameters, six for three parameters). The approximate inverse is an adequately accurate solution to the linearized inverse problem when it is capable of fitting the data to a prescribed precision. Otherwise, the approximate inverse of the normal operator might be used to precondition Krylov subspace methods in order to refine the data fit. I validate the method on a linearized version of the Marmousi model for constant density acoustics for the one-parameter problem. For the two parameter problem, the inversion of a variable density acoustics layered model corroborates the success of the proposed method. Furthermore, this example details the various steps of the method. I also apply the method to a 1D section of the Marmousi model to test the behavior of the method on complex two-parameter layered models.