## Tuesday Seminar of Analysis

Seminar information archive ～09/18｜Next seminar｜Future seminars 09/19～

Date, time & place | Tuesday 16:00 - 17:30 156Room #156 (Graduate School of Math. Sci. Bldg.) |
---|---|

Organizer(s) | ISHIGE Kazuhiro, SAKAI Hidetaka, ITO Kenichi |

**Seminar information archive**

### 2013/12/17

16:30-18:00 Room #128 (Graduate School of Math. Sci. Bldg.)

Dispersion and observability for completely integrable Schrödinger flows (ENGLISH)

**Fabricio Macia**(Universidad Politécnica de Madrid)Dispersion and observability for completely integrable Schrödinger flows (ENGLISH)

[ Abstract ]

I will present some results on weak dispersion and unique continuation (observability) for linear Schrödinger

equations that are obtained as the quantization of a completely integrable Hamiltonian system.

The model case corresponds to the linear Schrödinger equation (with a potential) on the flat torus.

Our results are obtained through a detailed analysis of semiclassical measures corresponding to

sequences of solutions, which is performed using a two-microlocal approach.

This is a joint work with Nalini Anantharaman and Clotilde Fermanian-Kammerer.

I will present some results on weak dispersion and unique continuation (observability) for linear Schrödinger

equations that are obtained as the quantization of a completely integrable Hamiltonian system.

The model case corresponds to the linear Schrödinger equation (with a potential) on the flat torus.

Our results are obtained through a detailed analysis of semiclassical measures corresponding to

sequences of solutions, which is performed using a two-microlocal approach.

This is a joint work with Nalini Anantharaman and Clotilde Fermanian-Kammerer.

### 2013/12/10

16:30-18:00 Room #128 (Graduate School of Math. Sci. Bldg.)

Quantitative unique continuation principle, local behavior of solutions, and bounds on the density of states for Schr\\"odinger operators (ENGLISH)

**Abel Klein**(UC Irvine)Quantitative unique continuation principle, local behavior of solutions, and bounds on the density of states for Schr\\"odinger operators (ENGLISH)

[ Abstract ]

We establish bounds on the density of states measure for Schr\\"odinger operators. These are deterministic results that do not require the existence of the density of states measure, or, equivalently, of the integrated density of states. The results are stated in terms of a ``density of states outer-measure'' that always exists, and provides an upper bound for the density of states measure when it exists. We prove log-H\\"older continuity for this density of states outer-measure in one, two, and three dimensions for Schr\\"odinger operators, and in any dimension for discrete Schr\\"odinger operators. Our proofs use a quantitative unique continuation principle and the local behavior of approximate solutions of the stationary Schr\\"odinger equation.

(Joint work with Jean Bourgain.)

References: Jean Bourgain and Abel Klein: Bounds on the density of states for Schr\\"odinger operators. Invent. Math. 194, 41-72 (2013).

We establish bounds on the density of states measure for Schr\\"odinger operators. These are deterministic results that do not require the existence of the density of states measure, or, equivalently, of the integrated density of states. The results are stated in terms of a ``density of states outer-measure'' that always exists, and provides an upper bound for the density of states measure when it exists. We prove log-H\\"older continuity for this density of states outer-measure in one, two, and three dimensions for Schr\\"odinger operators, and in any dimension for discrete Schr\\"odinger operators. Our proofs use a quantitative unique continuation principle and the local behavior of approximate solutions of the stationary Schr\\"odinger equation.

(Joint work with Jean Bourgain.)

References: Jean Bourgain and Abel Klein: Bounds on the density of states for Schr\\"odinger operators. Invent. Math. 194, 41-72 (2013).

### 2013/11/26

16:30-18:00 Room #128 (Graduate School of Math. Sci. Bldg.)

Global Strichartz estimates for Schr\\"odinger equations with long range metrics (JAPANESE)

**Haruya MIZUTANI**(Gakushuin University)Global Strichartz estimates for Schr\\"odinger equations with long range metrics (JAPANESE)

[ Abstract ]

We consider Schr\\"odinger equations on the asymptotically Euclidean space

with the long-range condition on the metric.

We show that if the high energy resolvent has at most polynomial growth with respect to the energy,

then global-in-time Strichartz estimates, outside a large compact set, hold.

Under the non-trapping condition we also discuss global-in-space Strichartz estimates.

This talk is based on a joint work with J.-M. Bouclet (Toulouse University).

We consider Schr\\"odinger equations on the asymptotically Euclidean space

with the long-range condition on the metric.

We show that if the high energy resolvent has at most polynomial growth with respect to the energy,

then global-in-time Strichartz estimates, outside a large compact set, hold.

Under the non-trapping condition we also discuss global-in-space Strichartz estimates.

This talk is based on a joint work with J.-M. Bouclet (Toulouse University).

### 2013/11/19

16:30-18:00 Room #128 (Graduate School of Math. Sci. Bldg.)

Inverse spectral problem for positive Hankel operators (ENGLISH)

**Alexander Pushnitski**(King's Colledge London)Inverse spectral problem for positive Hankel operators (ENGLISH)

[ Abstract ]

Hankel operators are given by (infinite) matrices with entries

$a_{n+m}$ in $\\ell^2$. We consider inverse spectral problem

for bounded self-adjoint positive Hankel operators.

A famous theorem due to Megretskii, Peller and Treil asserts

that such operators may have any continuous spectrum of

multiplicity one or two and any set of eigenvalues of multiplicity

one. However, more detailed questions of inverse spectral

problem, such as the description of isospectral sets, have never

been addressed. In this talk I will describe in detail the

direct and inverse spectral problem for a particular sub-class

of positive Hankel operators. The talk is based on joint work

with Patrick Gerard (Paris, Orsay).

Hankel operators are given by (infinite) matrices with entries

$a_{n+m}$ in $\\ell^2$. We consider inverse spectral problem

for bounded self-adjoint positive Hankel operators.

A famous theorem due to Megretskii, Peller and Treil asserts

that such operators may have any continuous spectrum of

multiplicity one or two and any set of eigenvalues of multiplicity

one. However, more detailed questions of inverse spectral

problem, such as the description of isospectral sets, have never

been addressed. In this talk I will describe in detail the

direct and inverse spectral problem for a particular sub-class

of positive Hankel operators. The talk is based on joint work

with Patrick Gerard (Paris, Orsay).

### 2013/07/09

16:30-18:00 Room #118 (Graduate School of Math. Sci. Bldg.)

A Trace Formula for Long-Range Perturbations of the Landau Hamiltonian

(Joint work with Georgi Raikov) (ENGLISH)

**Tom\'as Lungenstrass**(Pontificia Universidad Catolica de Chile)A Trace Formula for Long-Range Perturbations of the Landau Hamiltonian

(Joint work with Georgi Raikov) (ENGLISH)

[ Abstract ]

The Landau Hamiltonian describes the dynamics of a two-dimensional

charged particle subject to a constant magnetic field. Its spectrum

consists in eigenvalues of infinite multiplicity given by $B(2q+1)$, $q\\in Z_+$. We

consider perturbations of this operator by including a continuous

electric potential that decays slowly at infinity (as $|x|^{-\\rho}$, $0<\\rho<1$).

The spectrum of the perturbed operator consists of eigenvalue clusters

which accumulate to the Landau levels. We provide estimates for the

rate at which the clusters shrink as we move up the energy levels.

Further, we obtain an explicit description of the asymptotic density

of eigenvalues for asymptotically homogeneous long-range potentials in

terms of a mean-value transform of the associated homogeneous

function.

The Landau Hamiltonian describes the dynamics of a two-dimensional

charged particle subject to a constant magnetic field. Its spectrum

consists in eigenvalues of infinite multiplicity given by $B(2q+1)$, $q\\in Z_+$. We

consider perturbations of this operator by including a continuous

electric potential that decays slowly at infinity (as $|x|^{-\\rho}$, $0<\\rho<1$).

The spectrum of the perturbed operator consists of eigenvalue clusters

which accumulate to the Landau levels. We provide estimates for the

rate at which the clusters shrink as we move up the energy levels.

Further, we obtain an explicit description of the asymptotic density

of eigenvalues for asymptotically homogeneous long-range potentials in

terms of a mean-value transform of the associated homogeneous

function.

### 2013/05/21

16:30-18:00 Room #128 (Graduate School of Math. Sci. Bldg.)

Homogenization in a Thin Layer with an Oscillating Interface and Highly Contrast Coefficients (JAPANESE)

**Masaaki Uesaka**(Graduate School of Mathematical Sciences, The University of Tokyo)Homogenization in a Thin Layer with an Oscillating Interface and Highly Contrast Coefficients (JAPANESE)

[ Abstract ]

We consider the homogenization problem of the elliptic boundary value problem in a thin domain which has a high and low conductivity zones. In our model, two media are separated by a highly oscillating interface. The asymptotic behavior is governed by the order of the thickness of the domain, oscillation period of the interface and contrast between two media. In this talk, we show that the limit problem is changed by these parameters. We also introduce the two-scale convergence result in a thin domain which is the key ingredient of the proof.

We consider the homogenization problem of the elliptic boundary value problem in a thin domain which has a high and low conductivity zones. In our model, two media are separated by a highly oscillating interface. The asymptotic behavior is governed by the order of the thickness of the domain, oscillation period of the interface and contrast between two media. In this talk, we show that the limit problem is changed by these parameters. We also introduce the two-scale convergence result in a thin domain which is the key ingredient of the proof.

### 2012/12/11

16:30-18:00 Room #128 (Graduate School of Math. Sci. Bldg.)

This talk was cancelled! (JAPANESE)

**Rafe Mazzeo**(Stanford University)This talk was cancelled! (JAPANESE)

### 2012/12/04

16:30-18:30 Room #128 (Graduate School of Math. Sci. Bldg.)

Evolution of smooth shapes and the KP hierarchy (ENGLISH)

Group of diffeomorphisms of the unit circle and sub-Riemannian geometry (ENGLISH)

**Alexander Vasiliev**(Department of Mathematics, University of Bergen, Norway) 16:30-17:30Evolution of smooth shapes and the KP hierarchy (ENGLISH)

[ Abstract ]

We consider a homotopic evolution in the space of smooth

shapes starting from the unit circle. Based on the Loewner-Kufarev

equation we give a Hamiltonian formulation of this evolution and

provide conservation laws. The symmetries of the evolution are given

by the Virasoro algebra. The 'positive' Virasoro generators span the

holomorphic part of the complexified vector bundle over the space of

conformal embeddings of the unit disk into the complex plane and

smooth on the boundary. In the covariant formulation they are

conserved along the Hamiltonian flow. The 'negative' Virasoro

generators can be recovered by an iterative method making use of the

canonical Poisson structure. We study an embedding of the

Loewner-Kufarev trajectories into the Segal-Wilson Grassmannian,

construct the tau-function, the Baker-Akhiezer function, and finally,

give a class of solutions to the KP hierarchy, which are invariant on

Loewner-Kufarev trajectories.

We consider a homotopic evolution in the space of smooth

shapes starting from the unit circle. Based on the Loewner-Kufarev

equation we give a Hamiltonian formulation of this evolution and

provide conservation laws. The symmetries of the evolution are given

by the Virasoro algebra. The 'positive' Virasoro generators span the

holomorphic part of the complexified vector bundle over the space of

conformal embeddings of the unit disk into the complex plane and

smooth on the boundary. In the covariant formulation they are

conserved along the Hamiltonian flow. The 'negative' Virasoro

generators can be recovered by an iterative method making use of the

canonical Poisson structure. We study an embedding of the

Loewner-Kufarev trajectories into the Segal-Wilson Grassmannian,

construct the tau-function, the Baker-Akhiezer function, and finally,

give a class of solutions to the KP hierarchy, which are invariant on

Loewner-Kufarev trajectories.

**Irina Markina**(Department of Mathematics, University of Bergen, Norway) 17:30-18:30Group of diffeomorphisms of the unit circle and sub-Riemannian geometry (ENGLISH)

[ Abstract ]

We consider the group of sense-preserving diffeomorphisms of the unit

circle and its central extension - the Virasoro-Bott group as

sub-Riemannian manifolds. Shortly, a sub-Riemannian manifold is a

smooth manifold M with a given sub-bundle D of the tangent bundle, and

with a metric defined on the sub-bundle D. The different sub-bundles

on considered groups are related to some spaces of normalized

univalent functions. We present formulas for geodesics for different

choices of metrics. The geodesic equations are generalizations of

Camassa-Holm, Huter-Saxton, KdV, and other known non-linear PDEs. We

show that any two points in these groups can be connected by a curve

tangent to the chosen sub-bundle. We also discuss the similarities and

peculiarities of the structure of sub-Riemannian geodesics on infinite

and finite dimensional manifolds.

We consider the group of sense-preserving diffeomorphisms of the unit

circle and its central extension - the Virasoro-Bott group as

sub-Riemannian manifolds. Shortly, a sub-Riemannian manifold is a

smooth manifold M with a given sub-bundle D of the tangent bundle, and

with a metric defined on the sub-bundle D. The different sub-bundles

on considered groups are related to some spaces of normalized

univalent functions. We present formulas for geodesics for different

choices of metrics. The geodesic equations are generalizations of

Camassa-Holm, Huter-Saxton, KdV, and other known non-linear PDEs. We

show that any two points in these groups can be connected by a curve

tangent to the chosen sub-bundle. We also discuss the similarities and

peculiarities of the structure of sub-Riemannian geodesics on infinite

and finite dimensional manifolds.

### 2012/11/06

16:30-18:00 Room #128 (Graduate School of Math. Sci. Bldg.)

Resonance free domains for homoclinic orbits (ENGLISH)

**Thierry Ramond**(Univ. Paris, Orsay)Resonance free domains for homoclinic orbits (ENGLISH)

### 2012/10/30

16:30-18:00 Room #128 (Graduate School of Math. Sci. Bldg.)

About the method of characteristics (ENGLISH)

**Francis Nier**(Univ. Rennes 1)About the method of characteristics (ENGLISH)

[ Abstract ]

While studying the mean field dynamics of a systems of bosons, one is led to solve a transport equation for a probability measure in an infinite dimensional phase-space. Since those probability measures are characterized after testing with cylindrical or polynomial observables, which make classes which are not invariant after composing with a nonlinear flow. Thus the standard method of characteristics for transport equations cannot be extended at once to the infinite dimensional case. A solution comes from techniques developed for optimal transport and a probabilistic interpretation of trajectories.

While studying the mean field dynamics of a systems of bosons, one is led to solve a transport equation for a probability measure in an infinite dimensional phase-space. Since those probability measures are characterized after testing with cylindrical or polynomial observables, which make classes which are not invariant after composing with a nonlinear flow. Thus the standard method of characteristics for transport equations cannot be extended at once to the infinite dimensional case. A solution comes from techniques developed for optimal transport and a probabilistic interpretation of trajectories.

### 2012/10/23

16:30-18:00 Room #002 (Graduate School of Math. Sci. Bldg.)

Topics in quantum entropy and entanglement (ENGLISH)

**Elliott Lieb**(Princeton Univ.)Topics in quantum entropy and entanglement (ENGLISH)

[ Abstract ]

Several recent results on quantum entropy and the uncertainty

principle will be discussed. This is partly joint work with Eric Carlen

on lower bounds for entanglement, which has no classical analog, in terms

of the negative of the conditional entropy, S1 - S12, whose negativity,

when it occurs, also has no classical analog. (see arXiv:1203.4719)

It is also partly joint work with Rupert Frank on the uncertaintly

principle for quantum entropy which compares the quantum von Neumann

entropy with the classical entropies with respect to two different

bases. We prove an extension to the product of two and three spaces, which

has applications in quantum information theory. (see arxiv:1204.0825)

Several recent results on quantum entropy and the uncertainty

principle will be discussed. This is partly joint work with Eric Carlen

on lower bounds for entanglement, which has no classical analog, in terms

of the negative of the conditional entropy, S1 - S12, whose negativity,

when it occurs, also has no classical analog. (see arXiv:1203.4719)

It is also partly joint work with Rupert Frank on the uncertaintly

principle for quantum entropy which compares the quantum von Neumann

entropy with the classical entropies with respect to two different

bases. We prove an extension to the product of two and three spaces, which

has applications in quantum information theory. (see arxiv:1204.0825)

### 2012/07/17

16:30-18:00 Room #128 (Graduate School of Math. Sci. Bldg.)

On non-radially symmetric solutions of the Liouville-Gel'fand equation on a two-dimensional annular domain (JAPANESE)

**Toru Kan**(Mathematical institute, Tohoku University)On non-radially symmetric solutions of the Liouville-Gel'fand equation on a two-dimensional annular domain (JAPANESE)

[ Abstract ]

指数関数を非線形項に持つ非線形楕円型方程式(Liouville-Gel'fand方程式)について考察する。特に2次元の円環領域では、この方程式の非球対称な解が球対称解から分岐する形で現れる。本講演では、この分岐解の分岐図上での大域的な構造に関して得られた結果を紹介する。

指数関数を非線形項に持つ非線形楕円型方程式(Liouville-Gel'fand方程式)について考察する。特に2次元の円環領域では、この方程式の非球対称な解が球対称解から分岐する形で現れる。本講演では、この分岐解の分岐図上での大域的な構造に関して得られた結果を紹介する。

### 2012/07/10

16:30-18:00 Room #128 (Graduate School of Math. Sci. Bldg.)

Hadamard variational formula for the Green function

of the Stokes equations with the boundary condition (JAPANESE)

**Erika Ushikoshi**(Mathematical Institute, Tohoku University)Hadamard variational formula for the Green function

of the Stokes equations with the boundary condition (JAPANESE)

### 2012/06/26

16:30-18:00 Room #128 (Graduate School of Math. Sci. Bldg.)

Absence of embedded eigenvalues for the Schr\\"odinger operator on manifold with ends (JAPANESE)

**Kenichi Ito**(Division of Mathematics, University of Tsukuba)Absence of embedded eigenvalues for the Schr\\"odinger operator on manifold with ends (JAPANESE)

[ Abstract ]

We consider a Riemannian manifold with, at least, one expanding end, and prove the absence of $L^2$-eigenvalues for the Schr\\"odinger operator above some critical value. The critical value is computed from the volume growth rate of the end and the potential behavior at infinity. The end structure is formulated abstractly in terms of some convex function, and the examples include asymptotically Euclidean and hyperbolic ends. The proof consists of a priori superexponential decay estimate for eigenfunctions and the absence of superexponentially decaying eigenfunctions, both of which employs the Mourre-type commutator argument. This talk is based on the recent joint work with E.Skibsted (Aarhus University).

We consider a Riemannian manifold with, at least, one expanding end, and prove the absence of $L^2$-eigenvalues for the Schr\\"odinger operator above some critical value. The critical value is computed from the volume growth rate of the end and the potential behavior at infinity. The end structure is formulated abstractly in terms of some convex function, and the examples include asymptotically Euclidean and hyperbolic ends. The proof consists of a priori superexponential decay estimate for eigenfunctions and the absence of superexponentially decaying eigenfunctions, both of which employs the Mourre-type commutator argument. This talk is based on the recent joint work with E.Skibsted (Aarhus University).

### 2012/05/22

16:30-18:00 Room #118 (Graduate School of Math. Sci. Bldg.)

Viscosity solutions for nonlinear elliptic-parabolic problems (ENGLISH)

**Norbert Pozar**(Graduate School of Mathematical Sciences, The University of Tokyo)Viscosity solutions for nonlinear elliptic-parabolic problems (ENGLISH)

[ Abstract ]

We introduce a notion of viscosity solutions for a general class of

elliptic-parabolic phase transition problems. These include the

Richards equation, which is a classical model in filtration theory.

Existence and uniqueness results are proved via the comparison

principle. In particular, we show existence and stability properties

of maximal and minimal viscosity solutions for a general class of

initial data. These results are new even in the linear case, where we

also show that viscosity solutions coincide with the regular weak

solutions introduced in [Alt&Luckhaus 1983]. This talk is based on a

recent work with Inwon Kim.

We introduce a notion of viscosity solutions for a general class of

elliptic-parabolic phase transition problems. These include the

Richards equation, which is a classical model in filtration theory.

Existence and uniqueness results are proved via the comparison

principle. In particular, we show existence and stability properties

of maximal and minimal viscosity solutions for a general class of

initial data. These results are new even in the linear case, where we

also show that viscosity solutions coincide with the regular weak

solutions introduced in [Alt&Luckhaus 1983]. This talk is based on a

recent work with Inwon Kim.

### 2012/05/15

16:30-18:00 Room #128 (Graduate School of Math. Sci. Bldg.)

Strichartz estimates for Schr\\"odinger equations with variable coefficients and unbounded electromagnetic potentials (JAPANESE)

**MIZUTANI, Haruya**(Research Institute for Mathematical Sciences, Kyoto University)Strichartz estimates for Schr\\"odinger equations with variable coefficients and unbounded electromagnetic potentials (JAPANESE)

[ Abstract ]

In this talk we consider the Cauchy problem for Schr\\"odinger equations with variable coefficients and unbounded potentials. Under the assumption that the Hamiltonian is a long-range perturbation of the free Schr\\"odinger operator, we construct an outgoing parametrix for the propagator near infinity, and give applications to sharp Strichartz estimates. The basic idea is to combine the standard approximation by using a time dependent modifier, which is not in the semiclassical regime, with the semiclassical approximation of Isozaki-Kitada type. We also show near sharp Strichartz estimates without asymptotic conditions by using local smoothing effects.

In this talk we consider the Cauchy problem for Schr\\"odinger equations with variable coefficients and unbounded potentials. Under the assumption that the Hamiltonian is a long-range perturbation of the free Schr\\"odinger operator, we construct an outgoing parametrix for the propagator near infinity, and give applications to sharp Strichartz estimates. The basic idea is to combine the standard approximation by using a time dependent modifier, which is not in the semiclassical regime, with the semiclassical approximation of Isozaki-Kitada type. We also show near sharp Strichartz estimates without asymptotic conditions by using local smoothing effects.

### 2012/02/14

16:30-18:00 Room #128 (Graduate School of Math. Sci. Bldg.)

Symmetry results for Caffarelli-Kohn-Nirenberg inequalities (ENGLISH)

**Michael Loss**(Georgia Institute of Technology)Symmetry results for Caffarelli-Kohn-Nirenberg inequalities (ENGLISH)

### 2012/01/31

16:30-18:00 Room #128 (Graduate School of Math. Sci. Bldg.)

Obstacle problems in unbounded domains (ENGLISH)

**Michel Chipot**(University of Zurich)Obstacle problems in unbounded domains (ENGLISH)

[ Abstract ]

We will present a formulation of obstacle problems in unbounded

domains when the energy method does not work, i.e. whenthe force does not belong to H^{-1}.

We will present a formulation of obstacle problems in unbounded

domains when the energy method does not work, i.e. whenthe force does not belong to H^{-1}.

### 2011/12/20

16:30-18:00 Room #128 (Graduate School of Math. Sci. Bldg.)

A trace formula for the perturbed Landau Hamiltonian (ENGLISH)

**Gueorgui Raykov**(Catholic University of Chile)A trace formula for the perturbed Landau Hamiltonian (ENGLISH)

[ Abstract ]

The talk will be based on a joint work with A. Pushnitski

and C. Villegas-Blas, the preprint is available here:

http://arxiv.org/abs/1110.3098 .

The talk will be based on a joint work with A. Pushnitski

and C. Villegas-Blas, the preprint is available here:

http://arxiv.org/abs/1110.3098 .

### 2011/12/13

16:30-18:00 Room #128 (Graduate School of Math. Sci. Bldg.)

Trivializable subriemannian structures and spectral analysis of associated operators (ENGLISH)

**Wolfram Bauer**(Mathematisches Institut, Georg-August-Universität)Trivializable subriemannian structures and spectral analysis of associated operators (ENGLISH)

### 2011/11/08

16:30-18:00 Room #128 (Graduate School of Math. Sci. Bldg.)

The University of Tokyo (JSPS Research Fellow))

Stochastic Power-Law Fluid equations: Existence and Uniqueness of weak solutions (joint work with Nobuo Yoshida) (JAPANESE)

**Yutaka Terasawa**(Graduate School of Mathematical Sciences,The University of Tokyo (JSPS Research Fellow))

Stochastic Power-Law Fluid equations: Existence and Uniqueness of weak solutions (joint work with Nobuo Yoshida) (JAPANESE)

### 2011/10/11

16:30-18:00 Room #128 (Graduate School of Math. Sci. Bldg.)

On the best constant of the weighted Trudinger-Moser

type inequality (JAPANESE)

**Hidemitsu Wadade**(Waseda University (JSPS-PD))On the best constant of the weighted Trudinger-Moser

type inequality (JAPANESE)

### 2011/07/12

16:30-18:00 Room #128 (Graduate School of Math. Sci. Bldg.)

The inclusion relation between Sobolev and modulation spaces (JAPANESE)

**Masaharau Kobayashi**(Tokyo University of Science)The inclusion relation between Sobolev and modulation spaces (JAPANESE)

[ Abstract ]

In this talk, we consider the inclusion relations between the $L^p$-Sobolev spaces and the modulation spaces. As an application, we give mapping properties of unimodular Fourier multiplier $e^{i|D|^\\alpha}$ between $L^p$-Sobolev spaces and modulation spaces.

Joint work with Mitsuru Sugimoto (Nagoya University).

In this talk, we consider the inclusion relations between the $L^p$-Sobolev spaces and the modulation spaces. As an application, we give mapping properties of unimodular Fourier multiplier $e^{i|D|^\\alpha}$ between $L^p$-Sobolev spaces and modulation spaces.

Joint work with Mitsuru Sugimoto (Nagoya University).

### 2011/04/26

16:30-18:00 Room #128 (Graduate School of Math. Sci. Bldg.)

On the system of fifth-order differential equations which describes surfaces containing six continuous families of circles (JAPANESE)

**Kiyoomi KATAOKA**(Graduate School of Mathematical Sciences, the University of Tokyo)On the system of fifth-order differential equations which describes surfaces containing six continuous families of circles (JAPANESE)

### 2010/11/16

16:00-18:30 Room #123 (Graduate School of Math. Sci. Bldg.)

Hyperfunctions and vortex sheets (ENGLISH)

Residual trace and equivariant asymptotic trace of Toeplitz operators (ENGLISH)

**Keisuke Uchikoshi**(National Defense Academy of Japan) 16:00-16:45Hyperfunctions and vortex sheets (ENGLISH)

**L. Boutet de Monvel**(University of Paris 6) 17:00-18:30Residual trace and equivariant asymptotic trace of Toeplitz operators (ENGLISH)