Fall 2024 Analysis Seminar: Difference between revisions
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Abstract: In this talk, we will discuss a dyadic variant of the Hilbert transform, which is a useful model of its continuous counterpart and the prototypical example of a so-called "Haar shift". After discussing some background and motivation in the Lebesgue measure case, we will turn to the situation where the L2 Haar functions are defined with respect to a locally finite Borel measure μ, which may not satisfy the dyadic doubling condition. In this more general setting, Lopez-Sanchez, Martell, and Parcet identified a weak regularity condition on the measure μ which characterizes weak-type and Lp estimates for this dyadic Hilbert transform. I then will discuss joint work with Jose Conde Alonso and Jill Pipher, where we obtain a domination of the dyadic Hilbert transform (and more generally, Haar shifts) by a modified sparse form. As an application, we characterize the class of weights where the dyadic Hilbert transform and related operators are bounded. A surprising novelty is that the usual (dyadic) Muckenhoupt A2 condition is necessary, but no longer sufficient in the non-doubling setting, and our modified weight condition reflects the "complexity" of the underlying Haar shift. Finally, we will examine a different dyadic Haar shift model of the Hilbert transform and its relationship to BMO (bounded mean oscillation) functions via commutators in the non-doubling setting (joint with Tainara Borges, Jose Conde Alonso, and Jill Pipher). | Abstract: In this talk, we will discuss a dyadic variant of the Hilbert transform, which is a useful model of its continuous counterpart and the prototypical example of a so-called "Haar shift". After discussing some background and motivation in the Lebesgue measure case, we will turn to the situation where the L2 Haar functions are defined with respect to a locally finite Borel measure μ, which may not satisfy the dyadic doubling condition. In this more general setting, Lopez-Sanchez, Martell, and Parcet identified a weak regularity condition on the measure μ which characterizes weak-type and Lp estimates for this dyadic Hilbert transform. I then will discuss joint work with Jose Conde Alonso and Jill Pipher, where we obtain a domination of the dyadic Hilbert transform (and more generally, Haar shifts) by a modified sparse form. As an application, we characterize the class of weights where the dyadic Hilbert transform and related operators are bounded. A surprising novelty is that the usual (dyadic) Muckenhoupt A2 condition is necessary, but no longer sufficient in the non-doubling setting, and our modified weight condition reflects the "complexity" of the underlying Haar shift. Finally, we will examine a different dyadic Haar shift model of the Hilbert transform and its relationship to BMO (bounded mean oscillation) functions via commutators in the non-doubling setting (joint with Tainara Borges, Jose Conde Alonso, and Jill Pipher). | ||
===[[Betsy Stovall|'''Betsy Stovall''']]=== | |||
Title: Incidences among flows | |||
Abstract: | |||
In this talk, we will discuss a family of combinatorial problems that may be viewed as discretized and (possibly) perturbed versions of various continuum incidence problems in harmonic analysis. Namely, given a finite collection L of integral curves of some “small” family of “nice” vector fields, how many incidences can be formed under various definitions of “small,” “nice,” and “incidence”? Well-studied special cases include the Joints Problem in R^n, the Szeméredi—Trotter Theorem for point-line intersections in the plane, and work of Magyar—Stein—Wainger, Pierce, and others, in which the discretization is to the integers. In this talk, we will discuss results and examples both old and new; the new results are joint with Huang and Tammen. | |||
=== [[Bingyuan Liu|'''Bingyuan Liu''']] === | === [[Bingyuan Liu|'''Bingyuan Liu''']] === | ||
Revision as of 17:21, 18 October 2024
Organizers: Shengwen Gan, Terry Harris and Andreas Seeger
Emails:
- Shengwen Gan: sgan7 at math dot wisc dot edu
- Terry Harris: tlharris4 at math dot wisc dot edu
- Andreas Seeger: seeger at math dot wisc dot edu
Time and Room: Wed 3:30--4:30 Van Vleck B119.
In some cases the seminar may be scheduled at different time to accommodate speakers.
If you would like to subscribe to the Analysis seminar list, send a blank email to analysis+subscribe (at) g-groups (dot) wisc (dot) edu
Link to Spring 2025 Analysis Seminar
| Date | Speaker | Institution | Title | Host(s) | Notes (e.g. unusual room/day/time) |
|---|---|---|---|---|---|
| We, 9-11 | Gevorg Mnatsakanyan | UW Madison | Almost everywhere convergence of the Malmquist Takenaka series | ||
| We, 9-18 | Lars Niedorf | UW Madison | Restriction type estimates and spectral multipliers on Métivier groups | ||
| Th, 9-26, 2:25-3:25, VV B215 | Niclas Technau | University of Bonn | Rational points on/near homogeneous hyper-surfaces | Andreas | Note changes of time/date/room, joint with Number Theory Seminar |
| We, 10-2 | Sergey Denisov | UW Madison | Applications of inverse spectral theory for canonical systems to NLS. | ||
| We, 10-9 | Shukun Wu | Indiana University | On almost everywhere convergence of planar Bochner Riesz means | Shengwen | |
| We, 10-16 | Nathan Wagner | Brown University | Dyadic shifts, sparse domination, and commutators in the non-doubling setting | Andreas | |
| We, 10-23 | Betsy Stovall | UW Madison | |||
| We, 10-30 | Burak Hatinoglu | Michigan State University | Alexei | ||
| We, 11-6 | Bingyuan Liu | University of Texas Rio Grande Valley | The Diederich--Fornaess index and the dbar-Neumann problem | Xianghong | |
| We, 11-13 | Maxim Yattselev | Indiana University (Indianapolis) | Sergey | ||
| We, 11-20 | Li Ji | Macquarie University | Brian | ||
| We, 11-27, no seminar | |||||
| We, 12-4 | Dallas Albritton | UW Madison | Andreas | ||
| We, 12-11 |
Abstracts
Gevorg Mnatsakanyan
Title: Almost everywhere convergence of the Malmquist Takenaka series
Link to Abstract: [1]
Lars Niedorf
Title: Restriction type estimates and spectral multipliers on Métivier groups
Abstract: We present a restriction type estimate for sub-Laplacians on arbitrary two-step stratified Lie groups. Although weaker than previously known estimates for the subclass of Heisenberg type groups, these estimates turn out to be sufficient to prove an Lp-spectral multiplier theorem with sharp regularity condition s > d|1/p-1/2| for sub-Laplacians on Métivier groups.
Niclas Technau
Title: Rational points on/near homogeneous hyper-surfaces
Abstract: How many rational points are on/near a compact hyper-surface? This question is related to Serre's Dimension Growth Conjecture. We survey the state of the art, and explain a standard random model. Furthermore, we report on recent joint work with Rajula Srivastava (Uni/MPIM Bonn). Our arguments are rooted in Fourier analysis and, in particular, clarify the role of curvature in the random model.
Sergey Denisov
Title: Applications of inverse spectral theory for canonical systems to NLS
Abstract: For nice initial data, NLS can be integrated using the inverse scattering theory for the Dirac equation on the line. We will discuss the connection of the Dirac equation to canonical systems and use the recent characterization of the Szegő class of measures on the real line to obtain a new semi-conserved quantity for the NLS. The bounds for the negative Sobolev norms will be presented as an application for the L2 NLS solutions (based on joint work with Roman Bessonov).
Shukun Wu
Title: On almost everywhere convergence of planar Bochner Riesz means
Abstract: We prove that the planar Bochner Riesz mean converges almost everywhere for any L^p function in the optimal range, for 5/3<p<2. Our approach is based on a weighted L^2 estimate, which may be of independent interest. For example, up to an epsilon loss, we can reprove Cordoba's L^4 orthogonality by solely considering L^2 space and using L^2 orthogonality. This is a joint work with Xiaochun Li.
Nathan Wagner
Title: Dyadic shifts, sparse domination, and commutators in the non-doubling setting
Abstract: In this talk, we will discuss a dyadic variant of the Hilbert transform, which is a useful model of its continuous counterpart and the prototypical example of a so-called "Haar shift". After discussing some background and motivation in the Lebesgue measure case, we will turn to the situation where the L2 Haar functions are defined with respect to a locally finite Borel measure μ, which may not satisfy the dyadic doubling condition. In this more general setting, Lopez-Sanchez, Martell, and Parcet identified a weak regularity condition on the measure μ which characterizes weak-type and Lp estimates for this dyadic Hilbert transform. I then will discuss joint work with Jose Conde Alonso and Jill Pipher, where we obtain a domination of the dyadic Hilbert transform (and more generally, Haar shifts) by a modified sparse form. As an application, we characterize the class of weights where the dyadic Hilbert transform and related operators are bounded. A surprising novelty is that the usual (dyadic) Muckenhoupt A2 condition is necessary, but no longer sufficient in the non-doubling setting, and our modified weight condition reflects the "complexity" of the underlying Haar shift. Finally, we will examine a different dyadic Haar shift model of the Hilbert transform and its relationship to BMO (bounded mean oscillation) functions via commutators in the non-doubling setting (joint with Tainara Borges, Jose Conde Alonso, and Jill Pipher).
Betsy Stovall
Title: Incidences among flows
Abstract: In this talk, we will discuss a family of combinatorial problems that may be viewed as discretized and (possibly) perturbed versions of various continuum incidence problems in harmonic analysis. Namely, given a finite collection L of integral curves of some “small” family of “nice” vector fields, how many incidences can be formed under various definitions of “small,” “nice,” and “incidence”? Well-studied special cases include the Joints Problem in R^n, the Szeméredi—Trotter Theorem for point-line intersections in the plane, and work of Magyar—Stein—Wainger, Pierce, and others, in which the discretization is to the integers. In this talk, we will discuss results and examples both old and new; the new results are joint with Huang and Tammen.
Bingyuan Liu
Title: The Diederich--Fornaess index and the dbar-Neumann problem
Abstract: Introduced in 1977, the Diederich–Fornaess index was developed to help construct bounded plurisubharmonic functions on bounded pseudoconvex domains. For the past three decades, it is believed that the Diederich–Fornaess index is linked with the global regularity of the dbar-Neumann operator. A longstanding open question has been whether a Diederich–Fornaess index of 1 implies this global regularity. In this talk, we will overview the background and present and sketch a proof of a recent theorem, jointly proved by Emil Straube and me. This theorem answers this open question affirmatively for (0, n-1) forms.