Page 486 - 8th European Congress of Mathematics ∙ 20-26 June 2021 ∙ Portorož, Slovenia ∙ Book of Abstracts
P. 486
HARMONIC ANALYSIS AND PARTIAL DIFFERENTIAL EQUATIONS (MS-28)
Microlocal analysis of singular measures
Valeria Banica, Valeria.Banica@sorbonne-universite.fr
Sorbonne Université, France
In this talk I shall present a study of scalar and vectorial measures from a microlocal point
of view, by introducing a notion of L1-regularity wave front set. I shall give several results
including a full L1-elliptic regularity result, properties of the polarisation of vectorial measures
constrained by a PDE, as well as a propagation of singularities result. This is a joint work with
Nicolas Burq.
Pointwise convergence for the Schrödinger equation with orthonormal
initial data
Neal Bez, nealbez@mail.saitama-u.ac.jp
Saitama University, Japan
For the one-dimensional Schrödinger equation, we will explain how to obtain maximal-in-time
estimates for systems of orthonormal initial data and, as a result, certain pointwise convergence
results associated with systems of infinitely many fermions. Our argument proceeds by estab-
lishing a maximal-in-space estimate for the fractional Schrödinger equation which, at the same
time, addresses an endpoint problem raised by Rupert Frank and Julien Sabin. The talk is based
on joint work with Sanghyuk Lee and Shohei Nakamura.
p-ellipticity, generalized convexity and applications
Andrea Carbonaro, carbonaro@dima.unige.it
University of Genova, Italy
I will review some recent applications of the notions of p-ellipticity and generalized convexity
introduced by O. Dragicˇevic´ and myself [3]. M. Dindoš and J. Pipher [7], simultaneously
and independently of us, found that p-ellipticity is a critical tool in different elliptic regularity
problems they were studying. A condition weaker than p-ellipticity appeared in a different
formulation in the 2005 work by A. Cialdea and V. Maz’ya [6].
The applications I will discuss include: (i) optimal holomorphic functional calculus for gen-
erators of symmetric contraction semigroups [1] and for non-symmetric Ornstein–Uhlenbeck
operators [2] (ii) Lp-contractivity of the semigroups generated by divergence-form operators
with complex coefficients [3,4,8,6] and (iii) maximal parabolic regularity of the generators sub-
ject to mixed boundary conditions on generic open subsets of Rd [4] (iv) trilinear estimates and
Kato-Ponce-type inequalities [5].
References
[1] A. Carbonaro, O. Dragicˇevic´, Functional calculus for generators of symmetric contraction
semigroups, Duke Math. J. 166, 937–974 (2017).
[2] A. Carbonaro, O. Dragicˇevic´, Bounded holomorphic functional calculus for nonsymmetric
Ornstein-Uhlenbeck operators, Ann. Sc. Norm. Super. Pisa Cl. Sci. (5) 19 (2019), no. 4,
1497–1533.
484
Microlocal analysis of singular measures
Valeria Banica, Valeria.Banica@sorbonne-universite.fr
Sorbonne Université, France
In this talk I shall present a study of scalar and vectorial measures from a microlocal point
of view, by introducing a notion of L1-regularity wave front set. I shall give several results
including a full L1-elliptic regularity result, properties of the polarisation of vectorial measures
constrained by a PDE, as well as a propagation of singularities result. This is a joint work with
Nicolas Burq.
Pointwise convergence for the Schrödinger equation with orthonormal
initial data
Neal Bez, nealbez@mail.saitama-u.ac.jp
Saitama University, Japan
For the one-dimensional Schrödinger equation, we will explain how to obtain maximal-in-time
estimates for systems of orthonormal initial data and, as a result, certain pointwise convergence
results associated with systems of infinitely many fermions. Our argument proceeds by estab-
lishing a maximal-in-space estimate for the fractional Schrödinger equation which, at the same
time, addresses an endpoint problem raised by Rupert Frank and Julien Sabin. The talk is based
on joint work with Sanghyuk Lee and Shohei Nakamura.
p-ellipticity, generalized convexity and applications
Andrea Carbonaro, carbonaro@dima.unige.it
University of Genova, Italy
I will review some recent applications of the notions of p-ellipticity and generalized convexity
introduced by O. Dragicˇevic´ and myself [3]. M. Dindoš and J. Pipher [7], simultaneously
and independently of us, found that p-ellipticity is a critical tool in different elliptic regularity
problems they were studying. A condition weaker than p-ellipticity appeared in a different
formulation in the 2005 work by A. Cialdea and V. Maz’ya [6].
The applications I will discuss include: (i) optimal holomorphic functional calculus for gen-
erators of symmetric contraction semigroups [1] and for non-symmetric Ornstein–Uhlenbeck
operators [2] (ii) Lp-contractivity of the semigroups generated by divergence-form operators
with complex coefficients [3,4,8,6] and (iii) maximal parabolic regularity of the generators sub-
ject to mixed boundary conditions on generic open subsets of Rd [4] (iv) trilinear estimates and
Kato-Ponce-type inequalities [5].
References
[1] A. Carbonaro, O. Dragicˇevic´, Functional calculus for generators of symmetric contraction
semigroups, Duke Math. J. 166, 937–974 (2017).
[2] A. Carbonaro, O. Dragicˇevic´, Bounded holomorphic functional calculus for nonsymmetric
Ornstein-Uhlenbeck operators, Ann. Sc. Norm. Super. Pisa Cl. Sci. (5) 19 (2019), no. 4,
1497–1533.
484
