

List of Titles  
G. ALEXOPOULOS  Heat kernels on semisimple Lie groups  
N. BOURNAVEAS  Low regularity solutions of the DiracKleinGordon equations  
G. DAVID  Recents results on analytic capacity  
T. De PAUW  Size minimizations and approximating problems  
L. ESCAURIAZA  Unique continuation and parabolic equations  
B. FRANCHI  Rectifiability and perimeter in step 2 Carnot groups  
P. GERARD  Nonlinear Schrödinger equations on compact manifolds  
W. HANSEN  Normalized solutions of Schrödinger operators with potentially bounded measures  
W. HEBISCH  Singular integrals on Iwasawa AN groups  
A. HULANICKI  Estimates for the Poisson kernel and the Harris measure for NA groups  
P.G. LEMARIERIEUSSET  Espaces de Lorentz et NavierStokes : le problème des solutions autosimilaires de Leray  
Y. MARTEL  Blowup phenomenon in the energy space for the critical generalized KdV equation  
S. MEDA  On Holomorphic spectral multipliers for the OrnsteinUhlenbeck operator  
V. MAZ'YA  The Schrödinger and the relativistic Schrödinger operators on the energy space: boundedness and compactness criteria  
Y. MEYER  Improved GagliardoNirenberg inequalities  
D. MÜLLER  Mehlertype kernels and local solvability  
D.ROBINSON  Analysis on Lie groups of polynomial growth  
P. SJÖGREN  Some maximal operators for the OrnsteinUhlenbeck semigroup with complex time parameter  
P. TCHAMITCHIAN  Hardy Sobolev spaces on strongly Lipschitz domains of R^{n} and square roots of elliptic operators  
A. VOLBERG  Bellman functions, heat extension and sharp (weighted) estimates of Riesz transforms and their combinations 
The Scientific Program will start on Monday, April 22 at 9 a.m. and will end on Friday, April at 5.30 p.m. All talks will take place in the Amphithéâtre Hermite at the Institut Henri Poincaré.
Monday will be devoted to talks by young researchers from the European Networks HA and HARP (See Below)
Preliminary Schedule of Invited Talks (Subject
to modifications):
Tuesday  Wednesday  Thursday  Friday  
9:30 10:00  Coffee  Coffee  Coffee  Coffee 
10:00 10:50  A. HULANICKI  G. ALEXOPOULOS  L. ESCAURIAZA  B. FRANCHI 
11:00  11:50  D. MÛLLER  D. ROBINSON  P. GERARD  T. De PAUW 
Lunch  Lunch  Lunch  Lunch  
14:10 15:00  P. SJÖGREN  Y. MEYER  W. HANSEN  P. TCHAMITCHIAN 
15:10 16:00  S. MEDA  V. MAZ'YA  Y. MARTEL  N. BOURNAVEAS 
16:00  16:30  Coffee Break  Coffee Break  Coffee Break  Coffee Break 
16:30 17:20  A. VOLBERG  P.G. LEMARIE  W. HEBISCH  G. DAVID 
The conferences will start at 10:00AM after registration and coffee.
A. ALFONSECA (Universidad Autonoma de Madrid)  An almostorthogonality principle for families of maximal operators along directions 
S. DUBOIS (Université de PicardieJules Verne)  Solutions to the NavierStokes equations in L^{3,\infty }and energy inequalities 
V. FISCHER (Université de ParisSud)  Maximal inequalities for homogeneous spheres on type h groups 
G. FURIOLI (Università di Genova)  On the Cauchy problem for the Schrödinger equation on the Heisenberg group 
G. GIGANTE (Università degli Studi di Bergamo)  Bessel functions and oscilatory integrals associated with the Schrödinger equation 
D. LEVIN (Université de CergyPontoise)  On isoperimetric dimensions of product spaces 
R. LILJENDAHL (Göteborg University)  The maximal operator for some nondoubling measures 
V. OLEVSKII (University of Edimburgh)  On estimates for Schur multipliers in S_{p} 
E. SASSO (Università di Genova)  Functional calculus for the Laguerre operator 
PLANNING OF MONDAY APRIL 22:
Monday  
9:3010:00  Coffee 
10:0010:30  G. FURIOLI 
10:4011:10  V. FISCHER 
11:2012:00  A. ALFONSECA 
12:1012:40  R. LILJENDAHL 
Lunch  
14:1014:40  G. GIGANTE 
14:5015:20  S. DUBOIS 
15:3016:00  E. SASSO 
16:0016:30  Coffee Break 
16:3017:00  D. LEVIN 
17:1017:40  V. OLEVSKII 
ABSTRACTS
A. ALFONSECA: This is a joint work with F. Soria and A. Vargas. Given a set of angles W, we study the relation between the maximal function defined on a basis of rectangles forming an angle belonging to W with the xaxis, and the maximal functions associated to subsets of W. We obtain an almostorthogonality result in L^{2}, and an extension involving a square function for L^{p} with p other than 2. We give applications to related problems, in particular we give a simple proof of a recent result by N. Katz.
S. DUBOIS: We give simple conditions guaranteeing that solutions to the NavierStokes equations fulfil complete energy equalities. Combining this with a uniqueness and persistency result, we prove that solutions in C([0,T^{*}[,X), where X = L^{3,\infty}(R^{3}), with initial data in the closure in X of the Schwartz class and in L^{2}(R^{3}) fulfil complete energy inequalities with a loss of energy at the possible singularities.
V. FISCHER: fischer.tex On each group N = AÅ^{^}Z of Heisenberg type (where Z is the center), let n=(X,Z) ® n =( X ^{4}+ Z ^{2})^{[1/4]} denote the chosen homogeneous quasinorm, and let M .f (n) be the supremum of the averages of f taken over all homogeneous spheres (for the previous quasinorm) centered at n . We prove results on the L^{p}(N)boundedness of f® M .f.
G. FURIOLI: We present some recent results obtained in a joint work with A. Veneruso. We consider the Schrödinger operator related to the full Laplacian D_{F} on the Heisenberg group H_{n} and prove new Strichartz inequalities for the solution of the Cauchy problem u_{t }= iD_{F }u +f, f in L^{1}((0,T), L^{2}( H_{n}))) with initial data u(0,x)=u_{0} in L^{2}( H_{n}). A key point consists in estimating the decay in time of the L^{\infty} norm of the free solution; this requires a careful analysis due to the non homogeneous nature of the full Laplacian.
G.GIGANTE: (This is joint work with Fernando Soria, from Universidad Autónoma de Madrid (Spain).) In this talk I would like to discuss the magnitude of some oscillatory integrals involving Bessel functions. These integrals arise in the problem of determining the minimal Sobolev smoothness a required to bound a maximal square function for solutions to the Schrödinger equation in R^{n+1}_{+} with initial datum f in H^{a}(R^{n}).
D. LEVIN: It is wellknown that dimensions of Euclidean spaces add up, if one considers their product, \BbbR^{d}=\BbbR^{m} ?\BbbR^{n}, d=m+n. For Riemannian manifolds, the notion of dimension is more delicate, e.g. the topological dimension does not reflect their geometry at infinity. However, one may introduce an isoperimetric dimension through isoperimetric inequalities. The dimension introduced in this way is not a number but a family of functions indexed by a parameter p, 1 < p < ¥. Our main result generalizes the addition of dimensions in the euclidean case using the notion of the isoperimetric dimension.
R. LILJENDAHL: We consider the HardyLittlewood maximal operator with respect to noncentered balls and a nondoubling measure. We give some results on the boundedness of the maximal operator when the measure only depends on the first coordinate.
V. OLEVSKII: We give sharp estimates for the Schur multiplier norm of some classes of kernels, through their variation. We shall also discuss the relation with the commutative case (Rubio de Francia's inequality).
E. SASSO: In this talk we shall present results on the boundedness of spectral multipliers associated to the multidimensional Laguerre operator L_{a}. It is well known that, for special values of a, the analysis of the Laguerre operator can be interpreted as the analysis of the OrnsteinUhlenbeck operator acting on "polyradial" functions. Exploiting this relation, we prove that if M is a bounded holomorphic function on the sector S_{p}={z in C:  argz < arcsin2/p1}, for 1 < p < \infty, and satisfies suitable Hörmander type conditions on the boundary, then the spectral operator M(L_{a}) is bounded on L^{p} with respect to the Laguerre measure. We also obtain a weak type (1,1) estimate for multipliers of Laplace trasform type, as well as a necessary condition for multipliers whose norm is dilation invariant. Our results are the countepart, for the Laguerre operator, of results of GarciaCuerva, Mauceri, Meda, Sjógren, Torrea for the OrnsteinUhlenbeck operator.
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N. BOURNAVEAS: We shall discuss the KlainermanMachedon programme of low regularity local solutions of nonlinear wave equations and their relation to global solutions, with particular attention to the DiracKleinGordon equations. List of Titles
G. DAVID: Nous essaierons de décrire certains aspects des résultats de Xavier Tolsa sur la capacité analytique g. Si K est un ensemble compact du plan complexe, g(K) mesure combien de fonctions analytiques bornées on peut construire sur le complémentaire de K. Tolsa montre que g(K) est équivalente à la quantité g^{+}(K) où l'on ne considère que les fonctions analytiques bornées obtenues par convolution du noyau de Cauchy avec une mesure positive. Il en déduit une description géométrique des ensembles effaçables pour les fonctions analytiques bornées, et la semiadditivité de g. List of Titles
T. de PAUW: We will motivate and describe a common work with Robert Hardt. We consider Plateau type variational problems related to the size minimization of rectifiable currents. We realize the limit of a size minimizing sequence as a stationary varifold and a minimal set. Other examples of functionals to be minimized include the integral over the underlying carrying set of a power q of the multiplicity function, with 0 < q <1. Because minimizing sequences may have unbounded mass we make use of a more general object called a rectifiable scan for describing the limit. This concept is motivated by the possibility of recovering a flat chain from a sufficiently large collection of its slices. In case the given boundary is smooth and compact, the limiting scan has finite mass and corresponds to a rectifiable current. List of Titles
L. ESCAURIAZA: I will describe some results in collaboration with Luis Vega and Javier Fernández related to unique continuation properties for solutions to parabolic equations. The references are
1. L. Escauriaza. Carleman inequalities and the Heat operator. Duke Math. J.,104, n.1 (2000), 113127.
2. L. Escauriaza, L. Vega. Carleman inequalities and the heat operator II.Indiana Math. J. 50, n. 3 (2001), 11491169.
3. L. Escauriaza, J. Fernández. Unique continuation for parabolic operators. (to appear). List of Titles
B. FRANCHI: Carnot groups endowed with their CarnotCarathéodory distance provide the simplest example of metrics spaces that are not Euclidean even locally (i.e. they are not Riemannian manifolds), but nevertheless have a rich structure, since they have intrinsic translations and dilations. The Heisenberg group H^{n} is the simplest nontrivial example of Carnot group. It is possible to introduce a notion of perimeter in Carnot groups, and in addition to prove an intrinsic isoperimetric inequality in this setting. It is then natural to ask whether it is possible to develop a geometric measure theory for Carnot group; in particular we are interested to analyse the structure of sets of locally finite intrinsic perimeter in Carnot groups, extending the classical result due to De Giorgi in the Euclidean setting. A key point in this program is the correct definition of regular hypersurfaces, since, because of distortion of the geometry at any scale, Carnot groups do not contain surfaces meant as deformations of Euclidean spaces, but we need an utterly new notion. Other problems arise also from the lack of sophisticated covering lemmata that hold in the Euclidean setting (Besicovitch covering lemma). Nevertheless, it is possible to carry out the proof of a structure theorem that is the exact counterpart of De Giorgi's theorem.List of Titles
P. GERARD: (his is a jointwork with
Nicolas Burq and Nikolay Tzvetkov (Université de ParisSud)). I
shall report on some recent results on the Cauchy Problem for the nonlinear
Schrödinger equation (NLS) on compact manifolds. First we establish
Strichartztype estimates for the linear Schrödinger equation on arbitrary
compact manifolds, with some loss of derivatives which turns out to be
optimal on spheres. Then, using these estimates, we prove global existence
of smooth solutions for defocusing NLS on arbitrary compact surfaces and
for defocusing cubic NLS on arbitrary compact threemanifolds, generalizing
a result due to Bourgain on standard tori. Finally, we show some new
instability
properties for NLS on spheres.List of Titles
W. HANSEN: Given a potentially bounded signed measure m on a Brelot space (X,H) with Green function G, it is well known that mharmonic functions (i. e., in the classical case, finely continuous versions of solutions to Du  um = 0) may be very discontinuous. In this paper it is shown that under very general assumptions on G (satisfied for large classes of elliptic second order linear differential operators) normalized perturbation, however, leads to a Brelot space (X,[(H)\tilde]^{m}) admitting a Green function T^{m}(G) which is locally comparable with G and has all properties required of G before. In particular, iterated perturbation is possible. Moreover, intrinsic Hölder continuity of quotients of harmonic functions with respect to the local quasimetric r:=(G^{1} +^{*} G^{1})/2 yields rHölder continuity for quotients of mharmonic functions as well.List of Titles
A. HULANICKI: hulanicki.tex´ We treat noncoercive invariant second order degenerate elliptic operators on simply connected homogeneous manifolds of negative curvature. J.Wolf [Wolf] and E.Heintze [Hei] proved that such a manifold is isometric with a solvable Lie group S=NA, being a semidirect product of a nilpotent Lie group N and A= R ^{+} and, moreover, for a H ÎA the Lie algebra of A the eigenvalues of Ad_{H}_{N} are all greater than 0. Conversely, every such a group equipped with a suitable leftinvariant metric becomes a Riemannian manifold with negative curvature.
We consider on S a second order leftinvariant operator


Let m_{t} be the semigroup of measures generated by L_{g}. If g? 0, then there is a unique (up to a constant) positive Radon measure n_{g} on N such that



P.G. LEMARIERIEUSSET: The role of Lorentz spaces in the NavierStokes problem has been underlined by Yves Meyer, who gave a simple proof for the uniqueness in L^{3} in the 3D case. We shall give in this lecture an elementary proof of the results of Necas and Tsai on the nonexistence of Leray's selfsimilar solutions with a profile in L^{3} (Necas  proof by L^{3} uniqueness) or fullfilling a local energy inequality in the neighbourhood of the blowup (Tsai  proof by Lorentz spaces) ; in both cases, we avoid using the regularity criterion of Caffarelli, Kohn and Nirenberg.List of Titles
Y. MARTEL: For the critical generalized KdV equation we present a series of papers proving the existence of a class of solutions which blow up in finite time in the energy space, and giving a precise qualitative study of the blow up phenomenon (blow up profile, blow up rate). List of Titles
V. MAZ'YA: mazya.tex (This is a joint work with I. Verbitsky.) We give a complete characterization of the class of functions (or, more generally, complexvalued distributions) Q such that the following inequality holds:

S. MEDA: meda.tex Let g be the Gauss measure on R^{d} and L the OrnsteinUhlenbeck operator

More specifically, for every p in (1,\infty) such that p is not equal to 2, set f_{p}^{*} = arcsin2/p1, and consider the sector S_{fp*} = {z in C: argz < f_{p}^{*}}. The first result is that if M is a bounded holomorphic function on S_{fp*}, whose boundary values on ¶S_{fp*} satisfy suitable Hörmander type conditions, then the spectral operator M(L) extends to a bounded operator on L^{p}(g) and hence on L^{q}(g) for all q such that 1/q1/2<1/p1/2. The result is sharp, in the sense that L does not admit a bounded holomorphic functional calculus in a sector smaller than S_{fp*}.
This is in striking contrast with some classical situations, such as that of spectral multipliers for the standard Laplacian on R^{d}, where a well known ``nonholomorphic functional calculus'' can be developed.
Then, we will discuss a recent result concerning the problem of finding necessary conditions on M that imply that M(L) is bounded on L^{p}(g) for some p not equal to 2.List of Titles
Y. MEYER: We aim at discussing the new proof by Michel Ledoux of the improved GagliardoNirenberg inequalities. These inequalities allow to interpolate between the space BV of functions with bounded variation and the Besov space B_{\infty}^{1,\infty}. This new proof is no longer based on some wavelet analysis of the space BV of functions with bounded variations. It is instead very much in the spirit of the LittelwoodPaley theory `à la Stein'. This new proof yields better constants as the dimension tends to infinity. List of Titles
D. MÜLLER: mûll er.tex Let V_{j} be leftinvariant vector fields on the Heisenberg group H_{n}, such that [V_{j},V_{j+n}]=U, 1 < j < n) are the only nontrivial commutation relations, and consider the operator

D. ROBINSON: In this talk we give a brief review of estimates on heat kernels of complex, subelliptic, secondorder, right invariant operators acting on Lie groups of polynomial growth. The emphasis is on asymptotic estimates and global properties. The main features of the analysis are structure theory of the Lie groups, homogenization theory and De Giorgi estimates. List of Titles
P. SJÖGREN: sjögren. tex Let g be the Gaussian measure in R^{d}. The OrnsteinUhlenbeck semigroup H_{t}, t > 0, defines a bounded operator H_{z} on L^{p}(g) when t is replaced by a complex variable z staying in a closed set E_{p}. We examine the maximal operator H^{*}_{p}f=sup{ H_{z}f: z in E_{p}}. For p < 2 it is known that H^{*}_{p} is not of strong type nor of weak type p,p; however, it will be of weak type if a neighbourhood of a certain critical point is deleted from E_{p} in the definition of H^{*}_{p}. We shall consider instead the case p > 2, and prove that H^{*}_{p} is of weak type but not of strong type p,p. But if a neighbourhood of the origin is deleted from E_{p} in the definition of H^{*}_{p}, the resulting operator turns out to be of strong type. List of Titles
P. TCHAMITCHIAN: (joint work with P. Auscher and E. Russ) Let D be a strongly Lipschitz domain of R^{n}. Hardy spaces on D have been introduced by Miyachi, by Jonsson, Sjögren and Wallin and by Chang, Krantz and Stein. We investigate spaces of functions on D whose gradients belong to these Hardy spaces on D. In particular, we characterize these spaces via suitable maximal functions. Then, using this characterization, if L=divAGrad is an elliptic second order divergence operator on D subject to the Dirichlet or the Neumann boundary condition, we compare the norms of L^{1/2}f and Gradf in suitable Hardy spaces on D, depending on the boundary condition, under some technical assumptions on L. This solves the square root problem for L is this context. List of Titles
A. VOLBERG: volberg.tex Quasiregular map is a W_{1}^{2} solution of the Beltrami equation f_{\bar z}mf_{z} = 0, where the function m (called Beltrami coefficient) satisfies m_{\infty} = k < 1. These maps selfimprove, becoming of class W_{1}^{p}(k) with p=p(k) > 2. In particular, they are continuous, discrete, open,...
There were two questions (going sometimes by the name of problems of Gehring and Iwaniec):
Namely, we will be using the harmonic analysis Bellman function and
the heat extension to explain how the Stochastic Optimal Control approach
can be useful in estimating the AhlforsBeuling transform and many othe
types of Riesz transforms. List of Titles
Registration is mandatory and free. This will give you access to computer
facilities and to the library of the Institute.
To register for the conference, get back
to the page on the ihp site and click on registration form.
The Conference will take place at the Institut
Henri Poincaré: this is located
11, rue Pierre et Marie Curie
75231 Paris Cedex 05
tel: 33 1 44 27 67 89
fax: 33 1 43 25 40 67
How to get to the Conference Center and a list of hotels: here
Money exchange: French official currency is now the Euro, which is worth approximately 0.9 US Dollars.
Local transportation: Enjoy the many possibilities of Paris
transportation
: metro, RER, bus. The best if weather is appropriate
is your feet.
How to telephone in France: There are very few public phones that operate with cards or Visa. It is more convenient and also cheaper to use telephone cards. You can find them at most places selling newspapers or cigarettes.
Meals: Finding food will be your responsability. You'll find many restaurants (typical or exotic), bars, fast foods around the Institute.
Computer facilities: Registered participants to the conference can use machines (10 Mac and 5 PC), in common rooms. Some are equipped with Linux.
Library: Registered participants to the conference have access to the library of the Institute.
Weather: A French saying: "en Avril, ne te découvre
pas d'un fil." It means, be careful, the temperature could be colder than
you think. But one can expect nice, sunny and warm days.