Archive of the Research Seminar

On the evolution of fluid flows
We consider systems of partial differential equations, modeling the dynamics of fluids on Riemannian manifolds. In particular, we will demonstrate, how the theory of evolution equations contributes to the analysis of those problems.

Fourth-order elliptic operators with Wentzell boundary conditions
This talk investigates elliptic operators of order four with Wentzell (or dynamic) boundary conditions, which desribe exchange of free energy between interior and boundary of a domain. In the simplest case the system is given by




We show that the operator governing this equation on a suitable product space generates a strongly continuous, real, and analytic semigroup . In order to achieve this we follow two different approaches:
In the case of a Hilbert space we use two nested quadratic forms and generalized Neumann traces. This method works even for Lipschitz domains and we obtain Hölder-regular solutions, exponential stablity and eventual positivity.
In the -setting we use parabolic theory, instead. However, we do not employ the classical trace spaces but the base space which leads to a theory of boundary value problems with very rough data; also the boundary regularity is finite. Key ideas, here, are spaces with different regularities in tangential and normal direction.

Well-posedness and maximal regularity for parabolic Cauchy problems on weighted tent spaces
Used by the work of Koch-Tataru on Navier-Stokes equations, the theory of tent spaces turns out to be useful to deal with evolution equations with very rough initial data. In this talk, we shall discuss the recent progress on studying linear parabolic equations with time-independent, uniformly elliptic, bounded measurable complex coefficients via tent spaces. The talk is based on a joint work with Pascal Auscher.

Hardy spaces on open sets
The classical Hardy spaces can be considered as spaces adapted to the negative Laplacian on the Euclidean space. Taking this as the point of departure, we motivate and introduce Hardy spaces that are adapted to some elliptic operator on a domain, subject to suitable boundary conditions. We are interested in deriving an atomic description for such spaces. We will review constructions in the case of pure Dirichlet or Neumann boundary conditions and show how the definition of an atom in these two cases is related to the underlying boundary conditions of the operator. We will then work towards generalisations to so-called mixed boundary conditions and other relaxations of geometric requirements. This is based on joint work with T. Böhnlein.

Analysis of Hibler’s Sea Ice Model and Time Periodic Quasilinear Evolution Equations
The first part of the talk is dedicated to the rigorous analysis of several models in the context of Hibler’s viscous-plastic sea ice model. The latter model was introduced in 1979 and represents a large-scale dynamic-thermodynamic model. Even though there is a plethora of literature on numerical analysis and modeling, a rigorous mathematical analysis for this model has only been developed quite recently.
First, we consider a fully parabolic variant of Hibler’s model and establish the local strong well-posedness and global strong well-posedness close to equilibria. Key steps here are the reformulation as a quasilinear abstract Cauchy problem as well as the study of the so-called Hibler operator arising from the internal ice stress.
Sea ice is subject to atmospheric wind and oceanic forces. We thus take into account a coupled atmosphere-sea ice-ocean model, where Hibler’s model is coupled to two viscous incompressible primitive equations via atmospheric drag force, shear stress and continuity of the ocean and ice velocity. A decoupling argument involving a hydrostatic Dirichlet and Dirichlet-to- Neumann operator allows us to deduce the bounded -calculus of the linearized operator matrix with non-diagonal domain. This paves the way for similar well-posedness results as for the fully parabolic Hibler model.
We also investigate a parabolic-hyperbolic variant of Hibler’s model. Employing Lagrangian coordinates, we are able to handle the hyperbolic effects in the balance laws and obtain maximal -regularity of the linearization and then also local strong well-posedness of the parabolic-hyperbolic problem.
In the second part of the talk, we present approaches to time periodic quasilinear equations by the Arendt-Bu theorem on maximal periodic -regularity and by a time periodic version of the classical Da Prato-Grisvard theorem. Finally, a previously developed framework is used to obtain the existence of time periodic strong solutions to Hibler’s fully parabolic model for small forces.
The talk is based on joint work with Tim Binz, Karoline Disser, Robert Haller and Matthias Hieber.

Global Solutions for the half-wave maps equation in three dimensions
This talk will concern the three dimensional half-wave maps equation (HWM), a nonlocal geometric equation with close links to the well-known wave maps equation. In high dimensions, n≥4, HWM is known to admit global solutions for suitably small initial data. The extension of these results to three dimensions presents significant new difficulties due to the loss of a key Strichartz estimate. In this talk I will introduce the half-wave maps equation and discuss a global wellposedness result for the three dimensional problem under an additional assumption of angular regularity on the initial data. The proof combines techniques from the study of wave maps with new microlocal arguments involving commuting vector fields and improved Strichartz estimates.

Bochner-Riesz profile of harmonic oscillator, anharmonic oscillator and Laguerre expansions
We start with discussion of spectral multipliers and Bochner-Riesz means corresponding to the Schrödinger operator with anharmonic potential . We show that the Bochner-Riesz profile of the operator completely coincides with such profile of the harmonic oscillator . Then we extend our discussion to include order Laguerre expansion corresponding to the operator , which can be interpreted as radial part of multidimensional harmonic oscillators. Based on joint a work with Peng Chen, and Waldemar Hebisch and a current project wit Himani Sharma and Sundaram Thangavelu.

Exponential Decay of the Linear Maxwell System due to Interior Conductivity
We study the anisotropic, linear Maxwell system on a bounded domain with an conductivity in a collar around the boundary and perfectly conductive boundary conditions. We show that solutions with divergence free initial values decay exponentially to . In our approach we split the solution via a Helmholtz decomposition and then show an observability estimate for the homogenous problem.

Tent space maximal regularity for the Stokes operator on the half-space
Introduced by Coifman, Meyer and Stein in 1984, tent spaces play an important role in harmonic analysis, see, e.g., 1. In their paper, Auscher and Frey provide an alternative proof to the famous result of Koch and Tataru on the Navier-Stokes equations, c.f., 2. It is then a natural and interesting question whether it is possible, using the ideas of Auscher and Frey, to establish the Koch-Tataru Theorem in the presence of a boundary, e.g., on the half-space. One step in that proof is the boundedness of the maximal regularity operator on the tent space . In this talk I will present an approach to this result in the half-space setting and extensions thereof to different (weighted) tent spaces. One challenge is to define the Stokes semigroup as the natural way via the Helmholtz projection is not useful in our setting. This is joint work with Patrick Tolksdorf.
1 P. Auscher and D. Frey. On the well-posedness of parabolic equations of Navier-Stokes type with data. J. Inst. Math. Jussieu 16 (2017), no. 5, 947–985.
2 H. Koch and D. Tataru. Well-posedness for the Navier-Stokes equations. Adv. Math. 157 (2001), no. 1, 22–35.

Jazz Session on Magnetic Schrödinger Operators and Related Fields

Magnetic Weyl Calculus and Phase Space Transforms
A gauage-covariant framework for pseudo-differential calculus with magnetic fields has been developed for about 20 years now by Iftimie, Mantoiu and Purice. In this talk we will have a look at how this magnetic formulation closely resembles the non-magnetic case, and how the techniques from pseudo-differential theory can be borrowed. In particular, Smith, Tataru and others have successfully applied phase space transforms to obtain Strichartz estimates for wave and Schrodinger equations with rough coefficients. Our goal will be to explore analogous transforms in the magnetic framework and adapt such powerful machinery to treat magnetic wave and Schrodinger equations with variable coefficients.

Some examples of "curved" multilinear operators invariant under modulation
We display some examples of modulation-invariant multi-linear operators which present certain curvature features: they either carry (maximal) oscillatory factors that are complex exponentials or they can be represented as more classical operators along curves. This is joint work with Bernicot, Lie, Vitturi, Bernicot, Lie and respectively with Oliveira.

Spectral Multiplier Theorems for Abstract Harmonic Oscillators on UMD lattices
This talk is based on the joint work with Jan van Neerven and Pierre Portal. Here we consider abstract harmonic oscillators of the form for tuples of operators and , where and are assumed to generate groups and to satisfy the canonical commutation relations. We prove functional calculus results for these abstract harmonic oscillators that match classical Hörmander spectral multiplier estimates for the harmonic oscillator on . This covers situations where the underlying metric measure space is not doubling and the use of function spaces that are not particularly well suited to extrapolation arguments. For instance, as an application, we treat the harmonic oscillator on mixed norm Bargmann-Fock spaces.

A solution to a problem of Lions, Temam and Wang
In their seminal articles 1,2 from 1993 Lions, Temam and Wang introduced the primitive equations of the coupled atmosphere-ocean. It describes the dynamics of the atmosphere and the ocean on a large scale. This model consists of two primitive equations coupled by non-linear wind-driven boundary conditions or non-linear traction conditions at the interface.
In this talk we show the existence and uniqueness of global strong solutions to this problem. Our proof rely on a new maximal -regularity result for the hydrostatic Stokes operator with inhomogeneous boundary conditions, a Kato-Ponce type para-product inequality in Triebel-Lizorkin spaces due to Chae, and the splitting into barotropic and baroclinic modes for primitive equations discovered by Cao and Titi, as well as a careful analysis of the boundary coupling terms in each step.
1 J.L. Lions, R. Temam, Sh. H. Wang, Mathematical theory for the coupled atmosphere-ocean models (CAO III). J. Math. Pures Appl. 74 (1995), 105–163
2 J.L. Lions, R. Temam, Sh. H. Wang, Models for the coupled atmosphere and ocean. (CAO I,II). Comput. Mech. Adv. 1 (1993), 3–119

Stability of abstract coupled systems
We present an abstract framework for studying the asymptotic behaviour of coupled linear systems. Our approach combines ideas from systems theory with results in the quantitative asymptotic theory of strongly continuous operator semigroups, and it allows us to study composite systems by looking separately at the (often much simpler) constituent components and the properties of a certain “transfer function”. We illustrate the power of our abstract results by using them to obtain (typically sharp) rates of energy decay in certain wave-heat systems and for a wave equation with an acoustic boundary condition. The talk is based on joint work with Lassi Paunonen and Serge Nicaise.

On off-diagonal behavior of the generalized Stokes operator II
Motivated by the theory of elliptic operators with bounded measurable coefficients, Tolksdorf first presented a non-local approach to study off-diagonal behavior of the generalized Stokes semigroup on generated by the generalized Stokes operator, formally given by . In this talk we will improve these off-diagonal decay estimates and show how they can be used to study operators such as the semigroup and the maximal regularity operator on tent spaces.

Lie splitting for semilinear wave equations at regularity
We consider the semilinear wave equation in with energy-(sub)critical power nonlinearity, and analyze a frequency-filtered Lie splitting scheme for the semidiscretization in time. For initial data in the energy space , we prove first-order convergence in . The error analysis relies on time-discrete Strichartz estimates for the wave propagator.

On well-posedness of parabolic equations of Navier-Stokes type with data
In this talk I will present the topic of my master thesis which is based on a paper by Auscher and Frey. They reproved the well-posedness result of Koch and Tataru of the incompressible Navier-Stokes equations with initial data in by tools coming from harmonic analysis. Moreover, they adapted their new proof to show even well-posedness of parabolic equations with similar quadratic nonlinearities but rougher second-order elliptic operators with bounded measurable coefficients for initial data in the operator-adapted space .

Oscillatory integral operators with homogeneous phase functions
We consider oscillatory integral operators with 1-homogeneous phase functions satisfying a convexity condition. These generalize the cone extension operator. We show --estimates via polynomial partitioning and decoupling estimates. The -estimates are sharp up to endpoints, which follows from examples of operators exhibiting Kakeya compression. We use the estimates to prove new local smoothing estimates for wave equations on Riemannian manifolds. The talk is based on arXiv:2109.14040 (accepted to Journal d'Analyse Mathématique).

The Finite Section Method and Periodic Schrödinger Operators
We study discrete Schrödinger operators on with periodic potentials as they are typically used to approximate aperiodic Schrödinger operators like the Fibonacci Hamiltonian. We prove an efficient test for applicability of the finite section method, a procedure that approximates by growing finite square submatrices . The study of the applicability of the finite section method also gives further insights on the location of Dirichlet eigenvalues of half-line Schrödinger operators on . This talk is based on the findings in arXiv:2110.09339 (to appear in Operator Theory: Advances and Applications) and the analysis code doi:10.15480/336.3828.

Minimal periods for semilinear parabolic equations
We show that, if generates a bounded holomorphic semigroup in a Banach space , , and satisfies , then a non-constant -periodic solution of the equation satisfies , where is a constant depending on and the semigroup. This extends results by Robinson and Vidal-Lopez, which have been shown for self-adjoint operators in Hilbert space. For the latter case, we obtain the optimal constant , which only depends on , and we also include the case . This is joint work with Gerd Herzog (Karlsruhe).

Optimal Sobolev Regularity for Degenerate Equations of Porous Media Type
We consider solutions to the nonlocal, nonlinear, degenerate equation
in ,
in
where and . This is a nonlocal variant of the porous medium equation (which corresponds to and for ), for which optimal space-time regularity has been established recently by B. Gess, the speaker and E. Tadmor. In this talk I explain how the method of kinetic formulation and averaging lemmas can be utilized to obtain (both in the local and nonlocal case) Sobolev regularity results that are in line with the optimal regularity suggested by scaling arguments and which are consistent with the limiting linear case .
The talk is based on a joint work with B. Gess and E. Tadmor and on ongoingwork with B. Gess. (B. Gess, J. Sauer and E. Tadmor, Optimal Regularity in Time and Space for the Porous Medium Equation, Anal. PDE 13(8):2441–2480, 2020)

The absolute functional calculus and regularity estimates for evolution equations
We recall some of the properties and examples of the absolute functionalcalculus introduced by N. Kalton and T. Kucherenko, and show how it can be used to prove regularity estimates for deterministic, and if time allows, also for stochastic evolution equations.

Vertical and horizontal Square and Maximal Functions on manifolds with ends
We consider connected sum of a finite number of N-dimensional manifolds of the form . We demonstrate that the vertical square function operator

is bounded on for and weak-type (1,1).
We also investigate family of vertical resolvent where . We show that the family is uniformly continuous on all for . We prove that the corresponding maximal function is bounded in the same range except that it is only weak-type (1,1) for . The Fefferman-Stein vector-valued maximal function is again of weak-type (1,1) but bounded if and only if , and not at .

Homogeneous function spaces on half-spaces and -maximal regularities
This presentation will mainly discuss the realization of homogeneous function spaces on half-spaces, which extends established approaches on the whole space. The construction we focus on is particularly well designed to deal with nonlinear problems and boundary value problems in Partial Differential Equations. We will specifically discuss their interpolation, trace results and the adapted operator theory to reach global-in-time -maximal regularity and many other variants in this setting, providing a natural extension of the results obtained by Danchin, Hieber, Mucha, and Tolksdorf. When considering the flat half-space, one is able to obtain a Hodge/Helmholtz decomposition for homogeneous Besov spaces with "high enough" regularity indices, which also allows us to recover various global-in-time -maximal regularity such has a -one. Finally, if we have enough time, we will discuss applications to certain nonlinear PDEs, such as the Hall-MagnetoHydroDynamic (Hall-MHD) system in arbitrary dimensions on the flat half-space, with minimal boundary conditions.

11:45-12:30	Noa Bihlmaier (Tübingen)	Das Zyklizitätsproblem
12:30-13:45		Lunch Break
13:45-14:30	Raphael Wagner (Ulm)	Vanishing long time average p-enstrophy dissipation rate in the 2D inviscid limit
14:45-15:30	Patrick Tolksdorf (Karlsruhe)	-extrapolation of non-local operators
15:30-16:15		Coffee Break
16:15-17:00	Franz Gmeineder (Konstanz)	KMS inequalities and limiting -estimates
		More information on TULKKA is here.

Über Operator adaptierte Hardy Räume
In 2005 haben Duong et al. zum ersten Mal Operator adaptierte Hardy Räume eingeführt, um die schönen Eigenschaften, die die klassischen Hardy Räume mit dem Laplace Operator verbinden, auf allgemeinere Operatoren zu übertragen. In diesem Vortrag betrachten wir sektorielle Operatoren auf , wobei ein Raum von homogenen Typ ist, die einen beschränkten holomorphen Funktionalkalkül auf haben und deren Wärmeleitungshalbgruppe, , Davies-Gaffney Abschätzungen erfüllt. Nach einem kurzen Ausflug in die von Coifman, Meyer und Stein eingeführten Tent spaces, werden wir uns den Operator adaptierten Hardy Räumen zuwenden. Für definieren wir diese dann auf verschiedene Weisen und werden sehen, dass die verschiedenen Definitionen äquivalent sind. Ähnlich zu den klassischen Hardy Räumen werden wir zuletzt zeigen, dass die zu assoziierte Riesz Transformation beschränkt von nach abbildet.

Quasilinear Maxwell Equations with Anisotropic Material Laws
We consider Maxwell equations with pointwise, fully anisotropic material laws. The characteristic surface is given by the Fresnel surface, which contains conical singularities. We prove Strichartz estimates for Hölder-continuous coefficients, which allows us to solve quasilinear Maxwell equations in the fully anisotropic case for rough initial data.

An invitation to the regularity of maximal functions
The Hardy–Littlewood maximal inequality bounds maximal functions on Lebesgue spaces, but it does not reveal much about the behaviour of their gradients. Following a 1997 paper of Kinnunen, an area of research has developed that is concerned with the behaviour of maximal functions on Sobolev spaces. In this talk, we give an overview of this theory and use plenty of examples to illustrate its key features. We also present some recent results and open questions.

Strichartz estimates for Wave equations with structured coefficients

11:30-12:15	Constantin Bilz (Karlsruhe)	Large sets without Fourier restriction theorems
12:15-14:00		Lunch
14:00-14:45	Alexandra Neamtu (Konstanz)	A semigroup approach to quasilinear rough PDEs
15:00-15:45	Lukas Niebel (Ulm)	Kinetic maximal -regularity
16:00-16:45		Coffee Break
16:45-17:30	Moritz Egert (Darmstadt)	Four Critical Numbers for Elliptic Systems with Block Structure
from 18:00		Dinner in the restaurant "Il Caminetto" (Kronenstr. 5)
		More information about Tulkka is here.

Functional calculi for Stokes operators with first order boundary conditions on unbounded domains
We study functional calculi in for Stokes operators with Hodge, Navier, and Robin type boundary conditions on uniform -domains . Our research complements recent results on the -theory of such operators and also sheds new light on the cases and .

Strichartz and dispersive estimates for equations with structured Lipschitz coefficients
We shall discuss Strichartz estimates for both Schrödinger and wave equations with structured Lipschitz coefficients. The arguments are based on Phillips calculus, which allows to deduce dispersive estimates from the constant coefficient case. For fixed time estimates we require a more refined wave packet analysis.

Local wellposedness of Maxwell systems with scalar-type retarded material laws
In the first part of the talk, local wellposedness of an abstract retarded evolution equation is studied using the concept of a mild solution and Banach's fixed point theorem. The second part is an application to Maxwell equations in the context of a model from nonlinear optics.

Strichartz estimates for Maxwell equations on domains with perfectly conducting boundary conditions
We consider Maxwell equations on a domain with perfectly conducting boundary conditions in isotropic media. In the charge-free case we recover Strichartz estimates due to Blair-Smith-Sogge for wave equations on domains. We shall also consider the quasilinear case of the Kerr nonlinearity, in which case we recover the Strichartz estimates and well-posedness results from Euclidean space. This is joint work with Nicolas Burq (Universite Paris-Sud).

Almost sure scattering for the energy-critical cubic nonlinear Schrödinger equation with supercritical data
The local and global wellposedness theory of nonlinear dispersive equations with randomized data has attracted a lot of interest over the last years. In particular in the scaling-supercritical regime, where a deterministic wellposedness theory fails, randomization has become an important tool to study the generic behaviour of solutions.
In this talk we study the energy-critical NLS on with supercritical initial data. We present a randomization based on a unit-scale decomposition in frequency space, a decomposition in the angular variable, and a unit-scale decomposition in physical space. We then discuss the resulting (almost surely) improved space-time estimates for solutions of the linear Schrödinger equation with randomized data and how these estimates yield almost sure scattering for the energy-critical cubic NLS.

Exponential Decay of the quasilinear Maxwell system due to surface conductivity
We consider a quasilinear Maxwell system on a bounded domain with absorbing boundary conditions and derive an estimate for the normal traces of the electric and magnetic field. This allows us to strengthen a result on exponential decay for small initial data.

A Fourier Integrator for the cubic Schrödinger equation at regularity
In this talk I will present the topic of my master thesis which is based on a paper by Ostermann, Rousset and Schratz. They introduced a new exponential-type time integration method for the cubic Schrödinger equation. While only assuming regularity of the solution, -Convergence with order strictly larger than 1/2 was shown. An important ingredient in this context are discrete-time Strichartz estimates.

Resolvent estimates for time-harmonic Maxwell's equations
We prove resolvent estimates for time-harmonic Maxwell's equations in -spaces with pointwise, spatially homogeneous, and possibly anisotropic material laws. These allow for the proofs of Limiting Absorption Principles and construction of solutions. In the fully anisotropic case, which is joint work with Rainer Mandel, the construction relies on new Bochner-Riesz estimates with negative index for non-elliptic surfaces.

References:

arXiv:2103.16951: Resolvent estimates for time-harmonic Maxwell's equations in the partially anisotropic case
arXiv:2103.17176: Time-harmonic solutions for Maxwell's equations in anisotropic media and Bochner-Riesz estimates with negative index for non-elliptic surfaces

Time integration of Maxwell equations with low regularity
The Peaceman-Rachford alternating direction implicit (ADI) scheme is very attractive for the time discretization of linear Maxwell equations on cuboids. Indeed, it is numerically stable and has optimal linear complexity. During the error analysis, it however turns out that the accuracy of the scheme heavily depends on the regularity of the solution of the Maxwell equations.
In the talk, we study linear time-dependent Maxwell equations on a heterogeneous cuboid, modelling a rectangular waveguide. The medium is assumed to consist of several small subcuboids with different material properties. Here the solution of the Maxwell equations is not -regular. We hence construct a new directional splitting scheme for the time discretization, and we provide a rigorous time-discrete error estimate for the scheme in .

Schrödinger operators with complex potentials
I will report on recent progress concerning eigenvalues of Schrödinger operators with complex potentials. We are interested in the magnitude and distribution of eigenvalues, and we seek bounds that only depend on an norm of the potential.
These questions are well understood for real potentials, but completely new phenomena arise for complex potentials. I will explain how techniques from harmonic analysis, particularly those related to Fourier restriction theory, can be used to prove upper and lower bounds. We will also discuss some open problems. The talk is based on recent joint work with Sabine Bögli (Durham).

Eigenvalue bounds and spectral stability of Lamé operator with complex potential
In this talk I will show how to get quantitative bounds on the location of eigenvalues, both discrete and embedded, of the non-self-adjoint Lamé operator of elasticity in terms of suitable norms of the potential. In order to do that we will use a nowadays well-oiled machinery based on the use of the Birman-Schwinger principle together with suitable uniform resolvent estimates. To emphasise the challenging feature of problems involving non-self-adjoint operators, we will show how, in the self-adjoint framework, such spectral enclosures are easily obtained as a consequence of the variational characterisation of the spectrum (no-longer available in a complex-valued context) and Sobolev inequalities.
The talk is based on the following two works:
L. Cossetti, Bounds on eigenvalues for perturbed Lamé operators with complex potentials, Math. Eng. 4 (2021), 5, 1-29
B. Cassano, L. Cossetti, L. Fanelli, Eigenvalue bounds and spectral stability of Lamé operator with complex potentials, Journal of Differential Equations 298 (2021), 528-559

Duality for outer spaces and relation to tent spaces
The theory of spaces for outer measures, or outer spaces, was introduced by Do and Thiele to encode the proof of boundedness of certain multilinear operators in a streamlined argument. Therefore, it was developed in the direction of the real interpolation features, e.g. Hölder's inequality and Marcinkiewicz interpolation, while other questions remained untouched. For example, whether the outer quasi-norms are equivalent to norms and satisfy any reasonable (Köthe) duality property.
In this talk, we will answer these questions, with a particular focus on two settings in the upper half plane and in the upper half space that are relevant in harmonic analysis. This allows us to clarify the relation between the outer spaces and the tent spaces introduced by Coifman, Meyer and Stein, and to get a glimpse at the use of the outer machinery in the proof of boundedness of multilinear operators mentioned above.

Large sets without Fourier restriction theorems
Fourier restriction inequalities associated to submanifolds form a major area of research in harmonic analysis and are connected to problems in many other fields. Around 25 years ago, Mockenhaupt and Mitsis showed that the restriction of the Fourier transform to a fractal (instead of a submanifold) can also behave interestingly. But the class of fractals is vast and necessary conditions for restriction inequalities are not well understood. In this direction, I will show that Fourier restriction sets avoid a certain universal set of full Hausdorff dimension.

Hardy's inequality associated with orthogonal expansions
Let be a measure space, where and is doubling, and choose an orthonormal basis in . Hardy’s inequality on , , associated with has the following form

where , the symbol stands for the inner product in , and denotes an appropriate Hardy space. Here is a positive number referred to as the admissible exponent.
I will discuss a method of proving Hardy’s inequality for a large class of orthogonal bases, which I established in my PhD thesis. It relies on study of the corresponding semigroup (usually heat or Poisson semigroup). The strength of this method is evidenced by its generality and sharpness. This means that it can be applied in most (if not all) of the classical systems (e.q. Hermite, Laguerre, Jacobi) and it leads to the smallest possible admissible exponent.
If there is time left, I will mention my other projects, both finished and ongoing.

Continuity properties of semigroups in real interpolation spaces
Starting from a bi-continuous semigroup (which might actually be strongly continuous) in a Banach space we investigate continuity properties of the semigroup that is induced in real interpolation spaces between and the domain of the generator. Of particular interest is the case with second index infinity. We obtain topologies with respect to which the induced semigroup is bi-continuous, among them topologies induced by a variety of norms, and indicate applications.

On off-diagonal behavior of the generalized Stokes operator
Let denote a second-order elliptic operator in divergence form and let denote the corresponding strongly continuous heat semigroup on . If denote measurable sets with and if is supported in , then by the strong continuity of the semigroup, one finds that

An estimate that quantifies the convergence rate is often viewed as an off-diagonal estimate and it is well-known, that heat semigroups satisfy the following type of off-diagonal decay

In this talk, we study off-diagonal behavior of the generalized Stokes semigroup that is generated on by the generalized Stokes operator with bounded measurable coefficients , formally given by

In contrast to the elliptic operator , the operator exhibits a non-local behavior due to the presence of the pressure function . This non-locality affects the non-local behavior of the generalized Stokes semigroup and it is not clear how fast the support of a divergence free vector field that is supported in a set is smeared out. In this talk, first results in this direction are presented. We further discuss how possible optimal estimates could look like and try to pinpoint what has to be improved in the existing proof.

The talks are streamed via ZOOM.

The Control Problem for the heat equation on rectangular domains
We study the internal controllability of the heat equation on unbounded rectangular domains, for example the whole space, the half-space or the infinite strip, where the control set is a thick subset, meaning that it is well-distributed in the domain. We show that thickness is a sufficient condition for null-controllability of the system and that it is also possible to provide a sharp bound for the control cost in terms of the geometric parameters of the problem. We will also show that thickness is necessary for null-controllability.

Local Hardy spaces on measured metric spaces with local doubling property
After a short recap of the euclidean situation, we will introduce local Hardy spaces for and on more general measured metric spaces satisfying only some mild geometric assumptions. It will turn out, similar to the euclidean case, that the dual space of is the space of functions with (local) bounded mean oscillation to the parameters and , which we will denote by . We will then show, that both the local Hardy spaces and the spaces are independent of the parameters and , and will therefore all be denoted by and respectively. We will then conclude, that for the function space is a complex interpolation space between and . Therefore, we get by duality, that for the space is an interpolation space between and . With that in mind, we will finally give some criteria under which linear Operators are bounded from to .

You can find an overview of older talks here.