Papers by Michael Ruzhansky
Comptes Rendus. Mathématique
In this note we present a notion of harmonic oscillator on the Heisenberg group H n which forms t... more In this note we present a notion of harmonic oscillator on the Heisenberg group H n which forms the natural analogue of the harmonic oscillator on R n under a few reasonable assumptions: the harmonic oscillator on H n should be a negative sum of squares of operators related to the sub-Laplacian on H n , essentially self-adjoint with purely discrete spectrum, and its eigenvectors should be smooth functions and form an orthonormal basis of L 2 (H n). This approach leads to a differential operator on H n which is determined by the (stratified) Dynin-Folland Lie algebra. We provide an explicit expression for the operator as well as an asymptotic estimate for its eigenvalues. Résumé. Dans cette note, nous présentons une notion d'oscillateur harmonique sur le groupe de Heisenberg H n qui forme l'analogue naturel de l'oscillateur harmonique sur R n sous quelques hypothèses raisonnables : l'oscillateur harmonique sur H n devraît être une somme négative de carrés d'opérateurs liée au souslaplacien sur H n , être essentiellement auto-adjoint avec un spectre purement discret, et les vecteurs propres doivent former une base orthonormée de L 2 (H n). Cette approche conduit à un opérateur différentiel sur H n qui est déterminé par l'algèbre de Dynin-Folland de Lie (stratifiée). Nous fournissons une expression explicite pour l'opérateur ainsi qu'une estimation asymptotique pour ses valeurs propres.
Representation of Solutions and Regularity Properties for Weakly Hyperbolic Systems
In Boggiatto P and Rodino L and Toft J and Wong Mw Pseudo Differential Operators and Related Topics Birkhauser Verlag Ag, Mar 20, 2006
Abstract. Regularity properties of generic hyperbolic systems with diagonal-izable principal part... more Abstract. Regularity properties of generic hyperbolic systems with diagonal-izable principal part will be established in Lp and other function spaces. Sharp regularity of solutions will be discussed. Applications will be given to solutions of scalar weakly hyperbolic equations ...
Math Ann, 2006
In this paper a global smoothing property of Schrodinger equations is established in the critical... more 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.
Oscillatory integrals without convexity Theorem 4.3.1 requires the phase function to satisfy the ... more Oscillatory integrals without convexity Theorem 4.3.1 requires the phase function to satisfy the convexity condition of Definition 2.2.3; however, we will also investigate solutions to hyperbolic equations for which the characteristic roots do not necessarily satisfy such a condition. In this section we state and prove a theorem for this case. First, we give the key results that replaces Theorem 4.1.1 in the proof, the well-known van der Corput Lemma. We recall the standard van der Corput Lemma as given in, for example, [Sog93, Lemma 1.
Rendiconti del Seminario Matematico
The paper describes a new approach to global smoothing problems for inhomogeneous dispersive evol... more The paper describes a new approach to global smoothing problems for inhomogeneous dispersive evolution equations based on an idea of canonical transformation. In our previous papers, we introduced such a method to show global smoothing estimates for homogeneous dispersive equations. It is remarkable that this method allows us to carry out a global microlocal reduction of equations to some low dimensional model cases. The purpose of this paper is to pursue the same treatment for inhomogeneous equations. Especially, time-global smoothing estimates for the operator $a(D_x)$ with lower order terms are the benefit of our new method.
Journal of Functional Analysis, 2011
We establish the sharp Gårding inequality on compact Lie groups. The positivity condition is expr... more We establish the sharp Gårding inequality on compact Lie groups. The positivity condition is expressed in the non-commutative phase space in terms of the full matrix symbol, which is defined using the representations of the group. Applications are given to the L 2 and Sobolev ...
Army & Navy Medical Intelligence
The Lancet, 1846
Hokkaido Mathematical Journal, 1999
We will extend the sharpness results on L^{p_{-}} and L^{p}-L^{q_{-}} continuity of Fourier integ... more We will extend the sharpness results on L^{p_{-}} and L^{p}-L^{q_{-}} continuity of Fourier integral operators for an arbitrary rank of the canonical projection For the elliptic operators of small negative orders we will show that by a coordinate change they are equivalent to pseud0-differential operators.
Recent progress in the regularity theory of Fourier integrals with real and complex phases and solutions to partial differential equations
Banach Center Publications, 2003
Abstract. In this paper we will give a brief survey of recent regularity results for Fourier inte... more Abstract. In this paper we will give a brief survey of recent regularity results for Fourier integral operators with complex phases. This will include the case of real phase functions. Ap-plications include hyperbolic partial differential equations as well as non-hyperbolic classes of ...
Nonlinear W (infinity) algebra as asymptotic symmetry of three-dimensional higher spin Anti-de Sitter gravity
arXiv preprint arXiv:1008.4579
Comments: 29 pages, 57 figures. This paper is the fifth paper of the series of our works on highe... more Comments: 29 pages, 57 figures. This paper is the fifth paper of the series of our works on higher dimensional homology algebra. In our coming papers, we shall define $\ cE $ xt 2-functor and spectral sequence in an abelian 2-category, try to give the relation between $\ cE $ xt 2-functor and the extension of 2-modules
The domination properties of elliptic invariant di erential operators on symmetric spaces of nonc... more The domination properties of elliptic invariant di erential 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 di erential operator applied to a function can be uniformly estimated by function and an elliptic operator of higher order applied to the function in L p 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.
Geom Dedic, 1995
The domination properties of the Laplace operator on a class of symmetric spaces of noncompact ty... more 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 . W e will also discuss some relative aspects of the theory of convolutions.
Holomorphic factorization for the solution operators for hyperbolic equations
Results in Mathematics, 2002
In this paper we consider the Cauchy problem for a class of hyperbolic pseudo-differential operat... more In this paper we consider the Cauchy problem for a class of hyperbolic pseudo-differential operators. The considered class contains constant coefficient differential equations, also allowing the coefficients to depend on time. We establish sharp L p − L q , Lipschitz, and other estimates for their solutions. In particular, the ellipticity condition for the roots of the principal symbol is eliminated for certain dimensions. We discuss the situation with no loss of smoothness for solutions. In the space R 1+n with n ≤ 4 (total dimension ≤ 5), we give a complete list of L p − L q properties. In particular, this contains the very important case R 1+3 . 0 Mathematics Subject Classification (1991): 35A20, 35S30, 58G15, 32D20.
Proceedings of the American Mathematical Society, 2015
In this note we construct an integral boundary condition for the Kohn Laplacian in a given domain... more In this note we construct an integral boundary condition for the Kohn Laplacian in a given domain on the Heisenberg group extending to the setting of the Heisenberg group M. Kac's "principle of not feeling the boundary". This also amounts to finding the trace on smooth surfaces of the Newton potential associated to the Kohn Laplacian. We also obtain similar results for higher powers of the Kohn Laplacian.
PROCEEDINGS-AMERICAN MATHEMATICAL …, 2001
Abstract. In this paper we prove that if (X, d, µ) is a metric doubling space with segment proper... more Abstract. In this paper we prove that if (X, d, µ) is a metric doubling space with segment property, then inf r(E)/r(B) > 0 if and only if inf µ(E)/µ(B) > 0, where the infimum is taken over any collection C of balls E,B such that E ⊂ B ⊂ X. As a consequence we show that if X is a ...
Groups
Pseudo-Differential Operators and Symmetries, 2010
Elementary Functional Analysis
Pseudo-Differential Operators and Symmetries, 2010
Topological Groups
Pseudo-Differential Operators and Symmetries, 2010
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Papers by Michael Ruzhansky