TBA John Hoffman
- Series
- Analysis Seminar
- Time
- Wednesday, April 17, 2024 - 14:00 for 1 hour (actually 50 minutes)
- Location
- Skiles 005
- Speaker
- John Hoffman – Florida State University
We prove a wavelet T(1) theorem for compactness of multilinear Calderón -Zygmund (CZ) operators. Our approach characterizes compactness in terms of testing conditions and yields a representation theorem for compact CZ forms in terms of wavelet and paraproduct forms that reflect the compact nature of the operator. This talk is based on joint work with Walton Green and Brett Wick.
The goal of this talk is to discuss the Lp boundedness of the trilinear Hilbert transform along the moment curve. We show that it is bounded in the Banach range.
The main difficulty in approaching this problem(compared to the classical approach to the bilinear Hilbert transform) is the lack of absolute summability after we apply the time-frequency discretization(which is known as the LGC-methodology introduced by V. Lie in 2019). To overcome such a difficulty, we develop a new, versatile approch -- referred to as Rank II LGC (which is also motived by the study of the non-resonant bilinear Hilbert-Carleson operator by C. Benea, F. Bernicot, V. Lie, and V. Vitturi in 2022), whose control is achieved via the following interdependent elements:
1). a sparse-uniform deomposition of the input functions adapted to an appropriate time-frequency foliation of the phase-space;
2). a structural analysis of suitable maximal "joint Fourier coefficients";
3). a level set analysis with respect to the time-frequency correlation set.
This is a joint work with my postdoc advisor Victor Lie from Purdue.
This presentation is devoted to studying matrix solutions of the cubic Szegő equation, leading to the matrix Szegő equation on the 1-d torus and on the real line. The matrix Szegő equation enjoys a Lax pair structure, which is slightly different from the Lax pair structure of the cubic scalar Szegő equation introduced in Gérard-Grellier [arXiv:0906.4540]. We can establish an explicit formula for general solutions both on the torus and on the real line of the matrix Szegő equation. This presentation is based on the works Sun [arXiv:2309.12136, arXiv:2310.13693].
In this talk, I will present some joint works with Tom Courtade on characterizing probability measures that optimize the constant in a given functional inequalitiy via integration by parts formulas, and how Stein's method can be used to prove quantitative bounds on how close almost-optimal measures are to true optimizers. I will mostly discuss Poincaré inequalities and Gaussian optimizers, but also some other examples if time allows it.
Over the past decade, a rich theory of existence for the isoperimetric problem in spaces of nonnegative curvature has been established by multiple authors.
We will briefly review this theory, with a special focus on the reasons why one may expect the isoperimetric problem to have a solution in any nonnegatively curved space: it is true for large enough volumes, it is true if the ambient is 2-dimensional, and it is true under appropriate assumptions on the ambient space at infinity.
The main topic of the talk will be the presentation of a counterexample to this "intuition": a 3-dimensional manifold of positive sectional curvature without isoperimetric sets for small volumes.
This is a joint work with G. Antonelli.
The Poincaré constant of a body, or more generally a probability density, in $\mathbb R^n$ measures how "spread out" the body is - for instance, this constant controls how long it takes heat to flow from an arbitrary point in the body to any other. It's thus intuitively reasonable that convolving a "sufficiently nice" measure with a Gaussian, which tends to flatten and smooth out the measure, would increase its Poincaré constant ("spectral monotonicity"). We show that this is true if the original measure is log-concave, via two very different strategies - a dynamic variant of Bakry-Émery's $\Gamma$-calculus, and a mass-transportation argument. Moreover, we show that the dynamic $\Gamma$-calculus argument can also be extended to the discrete setting of measures on $\mathbb Z$, and that spectral monotonicity holds in this setting as well. Some results joint with B. Klartag.
The Poincaré constant of a body, or more generally a probability density, in $\mathbb R^n$ measures how "spread out" the body is - for instance, this constant controls how long it takes heat to flow from an arbitrary point in the body to any other. It's thus intuitively reasonable that convolving a "sufficiently nice" measure with a Gaussian, which tends to flatten and smooth out the measure, would increase its Poincaré constant ("spectral monotonicity"). We show that this is true if the original measure is log-concave, via two very different strategies - a dynamic variant of Bakry-Émery's $\Gamma$-calculus, and a mass-transportation argument. Moreover, we show that the dynamic $\Gamma$-calculus argument can also be extended to the discrete setting of measures on $\mathbb Z$, and that spectral monotonicity holds in this setting as well. Some results joint with B. Klartag.
The goal of this talk is to discuss the Lp boundedness of the trilinear Hilbert transform along the moment curve. More precisely, we show that the operator
$$
H_C(f_1, f_2, f_3)(x):=p.v. \int_{\mathbb R} f_1(x-t)f_2(x+t^2)f_3(x+t^3) \frac{dt}{t}, \quad x \in \mathbb R
$$
is bounded from $L^{p_1}(\mathbb R) \times L^{p_2}(\mathbb R) \times L^{p_3}(\mathbb R}$ into $L^r(\mathbb R)$ within the Banach H\"older range $\frac{1}{p_1}+\frac{1}{p_2}+\frac{1}{p_3}=\frac{1}{r}$ with $1
The main difficulty in approaching this problem(compared to the classical approach to the bilinear Hilbert transform) is the lack of absolute summability after we apply the time-frequency discretization(which is known as the LGC-methodology introduced by V. Lie in 2019). To overcome such a difficulty, we develop a new, versatile approch -- referred to as Rank II LGC (which is also motived by the study of the non-resonant bilinear Hilbert-Carleson operator by C. Benea, F. Bernicot, V. Lie, and V. Vitturi in 2022), whose control is achieved via the following interdependent elements:
1). a sparse-uniform deomposition of the input functions adapted to an appropriate time-frequency foliation of the phase-space;
2). a structural analysis of suitable maximal "joint Fourier coefficients";
3). a level set analysis with respect to the time-frequency correlation set.
This is a joint work with my postdoc advisor Victor Lie from Purdue.