arXiv subjects
Publications and source records attributed to Michael Ruzhansky.
In this note we provide dispersive estimates for Fourier integrals with parameter-dependent phase functions in terms of geometric quantities of associated families of Fresnel surfaces. The results are based on a multi-dimensional van der Corput lemma due to the first author. Applications to dispersive estimates for hyperbolic systems and scalar higher order hyperbolic equations are also discussed.
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.
The aim of this paper is to establish time decay properties and dispersive estimates for strictly hyperbolic equations with homogeneous symbols and with time-dependent coefficients whose derivatives are integrable. For this purpose, the method of asymptotic integration is developed for such equations and representation formulae for solutions are obtained. These formulae are analysed further to obtain time decay of Lp-Lq norms of propagators for the corresponding Cauchy problems. It turns out that the decay rates can be expressed in terms of certain geometric indices of the limiting equation and we carry out the thorough analysis of this relation. This provides a comprehensive view on asymptotic properties of solutions to time-perturbations of hyperbolic equations with constant coefficients. Moreover, we also obtain the time decay rate of the Lp-Lq estimates for equations of these kinds, so the time well-posedness of the corresponding nonlinear equations with additional semilinearity can be treated by standard Strichartz estimates.
Dispersive and Strichartz estimates for solutions to general strictly hyperbolic partial differential equations with constant coefficients are considered. The global time decay estimates of $L^p-L^q$ norms of propagators are obtained, and it is shown how the time decay rates depend on the geometry of the problem. The frequency space is separated in several zones each giving a certain decay rate. Geometric conditions on characteristics responsible for the particular decay are identified and investigated. Thus, a comprehensive analysis is carried out for strictly hyperbolic equations of high orders with lower order terms of a general form. Results are applied to establish time decay estimates for the Fokker-Planck equation and for semilinear hyperbolic equations.
In this paper we establish global Lp regularity properties of Fourier integral operators. The orders of decay of the amplitude are determined for operators to be bounded on $L^p(\Rn)$, $1<p<\infty$, as well as to be bounded from Hardy space $H^1(\Rn)$ to $L^1(\Rn)$. The obtained results extend local $L^p$ regularity properties of Fourier integral operators established by Seeger, Sogge and Stein (1991) as well as global $L^2(\Rn)$ results of Asada and Fujiwara (1978) and Ruzhansky and Sugimoto (2006), to the global setting of $L^p(\Rn)$. Global boundedness in weighted Sobolev spaces $W^{σ,p}_s(\Rn)$ is also established. The techniques used in the proofs are the space dependent dyadic decomposition and the global calculi developed by Ruzhansky and Sugimoto (2006) and Coriasco (1999).
Global quantization of pseudo-differential operators on compact Lie groups is introduced relying on the representation theory of the group rather than on expressions in local coordinates. Operators on the 3-dimensional sphere and on group SU(2) are analysed in detail. A new class of globally defined symbols is introduced giving rise to the usual Hormander's classes of operators $Ψ^m(G)$, $Ψ^m(S^3)$ and $Ψ^m(SU(2))$. Properties of the new class and symbolic calculus are analysed. Properties of symbols as well as $L^2$-boundedness and Sobolev $L^2$--boundedness of operators in this global quantization are established on general compact Lie groups.
Pseudo-differential and Fourier series operators on the n-torus are analyzed by using global representations by Fourier series instead of local representations in coordinate charts. Toroidal symbols are investigated and the correspondence between toroidal and Euclidean symbols of pseudo-differential operators is established. Periodization of operators and hyperbolic partial differential equations is discussed. Fourier series operators, which are analogues of Fourier integral operators on the torus, are introduced, and formulae for their compositions with pseudo-differential operators are derived. It is shown that pseudo-differential and Fourier series operators are bounded on $L^2$ under certain conditions on their phases and amplitudes.
In this paper various properties of global and local changes of variables as well as properties of canonical transforms are investigated on modulation and Wiener amalgam spaces. We establish several relations among localisations of modulation and Wiener amalgam spaces and, as a consequence, we obtain several versions of local and global Beurling-Helson type theorems. We also establish a number of positive results such as local boundedness of canonical transforms on modulation spaces, properties of homogeneous changes of variables, and local continuity of Fourier integral operators on Fourier Lebesgue spaces. Finally, counterparts of these results are discussed for spaces on the torus as well as for weighted spaces.
In this paper we will outline elements of the global calculus of seudo-differential operators on the group SU(2). This is a part of a more general approach to pseudo-differential operators on compact Lie groups that will appear in the forthcoming book by the authors.
This paper describes a new comparison principle that can be used for the comparison of space-time estimates for dispersive equations. In particular, results are applied to the global smoothing estimates for several classes of dispersive partial differential equations.
The aim of this paper is to give a review of local and global properties of Fourier integral operators with real and complex phases, in local $L^p$, global $L^2$, and in Colombeau's spaces.
A necessary and sufficient condition for the resolution of the weak extension problem is given. This criterion is applied to also give a criterion for the solvability of the classical Bochner's extension problem in the $L^p$-category. The solution of the $L^p$-extension problem by Bochner giving the relation between the order of the operator, the dimension, and index $p$, for which the $L^p$-extension property holds, can be viewed as a subcritical case of the general $L^p$-extension problem. In general, this property fails in some critical and in all supercritical cases. In this paper, the $L^p$-extension problem is investigated for operators of all orders and for all $1\leq p\leq\infty$. Necessary and sufficient conditions on the subset of $L^p$ are given for which the $L^p$-extension property still holds, in the critical and supercritical cases.
In this paper we develop elements of the global calculus of Fourier integral operators in $R^n$ under minimal decay assumptions on phases and amplitudes. We also establish global weighted Sobolev $L^2$ estimates for a class of Fourier integral operators that appears in the analysis of global smoothing problems for dispersive partial differential equations. As an application, we exhibit a new type of smoothing estimates for hyperbolic equations, where the decay of data in space is quantitatively translated into the time decay of solutions.
In this paper we present a multidimensional version of the van der Corput lemma where the decay of the oscillatory integral is gained with respect to all space variables, connecting the standard one-dimensional van der Corput lemma with the stationary phase method.
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.
The 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 L2-boundedness properties are derived. This allows us to globally reduce general dispersive equations to normal forms in one or two dimensions. Then, several comparison principles are introduced 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 of smoothing estimates for dispersive and also non-dispersive equations with constant coefficients is presented. Applications are given to the detailed description of smoothing properties of the Schrodinger, relativistic Schrodinger, wave, Klein-Gordon, and other equations. Critical cases of some estimates and their relation to the trace estimates are discussed.
The aim of this article is to describe asymptotic profiles for the Kirchhoff equation, and to establish time decay properties and dispersive estimates for Kirchhoff equations. For this purpose, the method of asymptotic integration is developed for the corresponding linear equations and representation formulae for their solutions are obtained. These formulae are analysed further to obtain the time decay rate of $L^p$--$L^q$ norms of propagators for the corresponding Cauchy problems.
Global time estimates of Lp-Lq norms of solutions to general strictly hyperbolic partial differential equations are considered. The case of special interest in this paper are equations exhibiting the dissipative behaviour. Results are applied to discuss time decay estimates for Fokker-Planck equations and for wave type equations with negative mass.