Lp Fourier multipliers on compact Lie groups

In this paper we prove Lp multiplier theorems for invariant and non-invariant operators on compact Lie groups in the spirit of the well-known Hormander-Mikhlin theorem on Rn and its variants on tori Tn. We also give applications to a-priori estimates for non-hypoelliptic operators. Already in the case of tori we get an interesting refinement of the classical multiplier theorem.Comment: 22 pages; minor correction

An open problem in complex analytic geometry arising in harmonic analysis

In this paper, an open problem in the multidimensional complex analysis is pesented that arises in the investigation of the regularity properties of Fourier integral operators and in the regularity theory for hyperbolic partial differential equations. The problem is discussed in a self-contained elementary way and some results towards its resolution are presented. A conjecture concerning the structure of appearing affine fibrations is formulated.Comment: 8 page

Hormander class of pseudo-differential operators on compact Lie groups and global hypoellipticity

In this paper we give several global characterisations of the Hormander class of pseudo-differential operators on compact Lie groups. The result is applied to give criteria for the ellipticity and the global hypoellipticity of pseudo-differential operators in terms of their matrix-valued full symbols. Several examples of the first and second order globally hypoelliptic differential operators are given. Where the global hypoelliptiticy fails, one can construct explicit examples based on the analysis of the global symbols.Comment: 20 page

On the well-posedness of weakly hyperbolic equations with time-dependent coefficients

AbstractIn this paper we analyse the Gevrey well-posedness of the Cauchy problem for weakly hyperbolic equations of general form with time-dependent coefficients. The results involve the order of lower order terms and the number of multiple roots. We also derive the corresponding well-posedness results in the space of Gevrey Beurling ultradistributions

Global elliptic estimates on symmetric spaces

The domination properties of elliptic invariant dierential operators on symmetric spaces of noncompact type are investigated. Using the relation between parametrices and fundamental solutions on symmetric space we will show that the invariant dierential operator applied to a function can be uniformly estimated by function and an elliptic operator of higher order applied to the function in Lp spaces for all 1 p 1. As a consequence, by algebraic methods we will give a simple unifying proof that derivatives of a function can be uniformly estimated by function and its Laplacian

Thermo-elasticity for anisotropic media in higher dimensions

In this note we develop tools to study the Cauchy problem for the system of thermo-elasticity in higher dimensions. The theory is developed for general homogeneous anisotropic media under non-degeneracy conditions. For degenerate cases a method of treatment is sketched and for the cases of cubic media and hexagonal media detailed studies are provided.Comment: 33 pages, 5 figure

Analytic Fourier integral operators, Monge-Ampère equation and holomorphic factorization

We will show that the factorization condition for the Fourier integral operators I X Y leads to a parametrized parabolic MongeAmpere equation In case of an analytic operator the bration by the kernels of the Hessian of phase function is shown to be analytic in a number of cases by considering more general continuation problem for the level sets of a holomorphic mapping The results are applied to obtain Lpcontinuity for translation invariant operators in Rn with n and for arbitrary Rn with dXY j

Lp-distributions on symmetric spaces

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 $n$, hold on symmetric spaces. We give estimates for the convolutions

On Lp estimates for the Laplacian on a class of symmetric spaces noncompact type

The domination properties of the Laplace operator on a class of symmetric spaces of noncompact type are investigated. Using algebraic methods we will show that derivatives of a function can be uniformly estimated by function and its Laplacian in L p spaces for all 1 p 1. We will also discuss some relative aspects of the theory of convolutions

Smoothing properties of evolution equations via canonical transforms and comparison principle

This paper describes a new approach to global smoothing problems for dispersive and non-dispersive evolution equations based on the global canonical transforms and the underlying global microlocal analysis. For this purpose, the Egorov-type theorem is established with canonical transformations in the form of a class of Fourier integral operators, and their weighted L-2-boundedness properties are derived. This allows us to globally reduce general dispersive equations to normal forms in one or two dimensions. Then, a new comparison principle for evolution equations is introduced. In particular, it allows us to relate different smoothing estimates by comparing certain expressions involving their symbols. As a result, it is shown that the majority of smoothing estimates for different equations are equivalent to each other. Moreover, new estimates as well as several refinements of known results are obtained. The proofs are considerably simplified. A comprehensive analysis is presented for smoothing estimates for dispersive equations. Applications are given to the detailed description of smoothing properties of the Schrodinger, relativistic Schrodinger, wave, Klein-Gordon and other equations