 Hong-Wei ZHANG Institut Denis Poisson Université d'Orléans
Bâtiment de Mathématiques45067 Orléans cedex 2 France  Curriculum Vitae |
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Harmonic analysis and PDE on Riemannian
symmetric and locally symmetric spaces of non-compact
type |
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| [5] | Thesis: équation des ondes sur les espaces symétriques et localement symétriques de type non compact, (.pdf), (.slides) | |||
| By adding a detailed introduction
(in French), this thesis collects the research
papers [1]-[4] |
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| [4] |
Wave equation on general noncompact symmetric spaces, with J.-Ph. Anker, preprint, (.pdf) | |||
| [3] |
Wave equation on certain noncompact symmetric spaces, preprint, (.pdf) | |||
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The wave equation on
symmetric spaces has been considered for many
years, see for instance Semenov-Tian-Shansky,
Helgason,
Branson,
etc.. The interesting feature about non-compact
symmetric spaces is their rapid growth at
infinity. This means that properties of PDE such
as dispersion are more pronounced. This phenomenon
has been progressively understood on real
hyperbolic spaces since 2000, see for example Ionescu,
Tataru,
Metcalfe&Taylor,
Anker&Pierfelice,
etc., and these results can be extended without
difficulty to all non-compact symmetric spaces of
rank one. In these two papers, the wave
equation is studied in general rank. After
considering various special symmetric
spaces in [3], we give in [4] the definitive
solution to the problem on general symmetric
space of non-compact type. This is achieved by
overcoming a well-known difficulty in higher
rank analysis, namely the fact that the
Plancherel density is not a differential symbol
in general. We establish the time-sharp
pointwise kernel estimates and the dispersion
properties for the wave equation. As
consequence, we deduce the global-in-time
Strichartz inequality for a large family of
admissible pairs and prove global well-posedness
results for the corresponding semilinear
equation with low regularity data.
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| [2] |
Bottom of the L2 spectrum of the Laplacian on locally symmetric spaces, with J.-Ph. Anker, preprint, (.pdf) | |||
| In this paper, we estimate the
bottom of the L2 spectrum of the Laplacian on
locally symmetric spaces in terms of the critical
exponents of appropriate Poincaré series. Our main
result is the higher rank analog of a
characterization due to Elstrodt,
Patterson,
Sullivan
and Corlette
in rank one. It improves upon previous results
obtained by Leuzinger
and Weber
in higher rank. |
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| [1] | Wave and Klein-Gordon equations on certain locally symmetric spaces, J. Geom. Anal. 30 (2020), no. 4, 4386-4406, (.pdf) | |||
| This paper is devoted to study the
dispersion properties of the linear Klein-Gordon and
wave equations on a class of locally symmetric
spaces. We extend the previous results obtained on
non-compact symmetric spaces of rank one to more
general locally symmetric spaces, which are convex
cocompact and satisfy that the half sum of positive
roots is greater than the critical exponent of the
Poincaré series. This setting has been also
considered in Burq&Guillarmou&Hassell
and Fotiadis&Mandouvalos&Marias
for studying the Schrödinger equation. |
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Jan. 2020: | Wave equation on certain symmetric spaces. (.slides) |
| Introduction to spherical Fourier analysis on noncompact symmetric spaces. (.slides) | ||
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Workshop: Harmonic Analysis and Applications, TSIMF, Sanya, China | |
| Dec. 2018: | Wave and Klein-Gordon equations on certain
locally symmetric spaces. (.slides)
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| Conference: The Legacy of Joseph Fourier after 250 years, TSIMF, Sanya, China | ||
| Nov. 2018: | Équations des ondes sur certains espaces localement symétriques. (.slides) | |
| Seminar: Journée des doctorants, Orléans, France | ||