arXiv subjects
Publications and source records attributed to Michael Ruzhansky.
Basic properties of Fourier integral operators on the torus are studied by using the global representations by Fourier series instead of local representations. The results can be applied to weakly hyperbolic partial differential equations.
The notion of $L^p$-distributions is introduced on Riemannian symmetric spaces of noncompact type and their main properties are established. We use a geometric description for the topology of the space of test functions in terms of the Laplace-Beltrami operator. The techniques are based on a-priori estimates for elliptic operators. We show that structure theorems, similar to $\Rn$, hold on symmetric spaces. We give estimates for the convolutions.
In this paper a global smoothing property of Schrodinger equations is established in the critical case in dimensions two and higher. It is shown that the critical smoothing estimate is attained if the smoothing operator has some structure. This structure is related to the geometric properties of the equations. Results for critical cases for operators of higher orders as well as for hyperbolic equations are also given.
Properties of solutions of generic hyperbolic systems with multiple characteristics with diagonalizable principal part are investigated. Solutions are represented as a Picard series with terms in the form of iterated Fourier integral operators. It is shown that this series is an asymptotic expansion with respect to smoothness under quite general geometric conditions. Propagation of singularities and sharp regularity properties of solutions are obtained. Results are applied to establish regularity estimates for scalar weakly hyperbolic equations with involutive characteristics. They are also applied to derive the first and second terms of spectral asymptotics for the corresponding elliptic systems.
The local $L^2$-mapping property of Fourier integral operators has been established in Hörmander \cite{H} and in Eskin \cite{E}. In this paper, we treat the global $L^2$-boundedness for a class of operators that appears naturally in many problems. As a consequence, we will improve known global results for several classes of pseudo-differential and Fourier integral operators, as well as extend previous results of Asada and Fujiwara \cite{AF} or Kumano-go \cite{Ku}. As an application, we show a global smoothing estimate to generalized Schrödinger equations which extends the results of Ben-Artzi and Devinatz \cite{BD}, Walther \cite{Wa}, and \cite{Wa2}.