The Brown analysis seminar typically meets on Mondays at 3pm in Kassar 105.

Upcoming talks, Spring 2024

Feb. 5 : Bingyang Hu,  Auburn University

Title: On the curved Trilinear Hilbert transform

Abstract: 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 Holder range \frac{1}{p_1}+\frac{1}{p_2}+\frac{1}{p_3}=\frac{1}{r} with 1<p_1, p_3<\infty, 1<p_2 \le \infty and 1 \le r <\infty. 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 approach -- referred to as Rank II LGC (which is also motivated 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 decomposition 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.

     Feb. 12 : Nathan Wagner,  Brown University

Title: Weighted Estimates for the Bergman Projection on Planar Domains

Abstract: Let $\Omega \subsetneq \mathbb{C}$ be a simply connected domain. A fundamental object of study in complex analysis is the Bergman projection $\Pi_\Omega$, which is the orthogonal projection from $L^2(\Omega)$  to the closed subspace of holomorphic functions, $A^2(\Omega)$. Unweighted and weighted $L^p$ estimates for $\Pi_\Omega$ have close connections to duality of Bergman spaces, Toeplitz and Hankel operators on Bergman spaces, conformal function theory, and complex partial differential equations. We generalize well-known weighted inequalities due to Békollè and Bonami by giving a sufficient condition on the domain $\Omega$ such that the Bergman projection $\Pi_\Omega$ is bounded on $L^p(\Omega,\sigma)$ for $1<p<\infty$ and $\sigma$ belonging to the Muckenhoupt class $B_p(\Omega)$ whose basis consists of the images, under the Riemann map, of Carleson boxes in $\mathbb{D}$. We also show that in the special case of bounded uniform domains, the same condition is also necessary for the full-range weighted estimates, and moreover provide an alternative characterization of the weight class that is intrinsic to $\Omega$. We in addition prove a weighted weak-type estimate for the endpoint $p=1.$ Our approach uses techniques from both conformal mapping and dyadic harmonic analysis. This talk is based on joint work with Walton Green at Washington University in St. Louis. 

    Feb. 26 : Zihui Zhao,  Johns Hopkins University 

Title:  Boundary unique continuation of harmonic functions

Abstract: Unique continuation property is a fundamental property for harmonic     functions, as well as a    large class of elliptic and parabolic PDEs. It says that if a harmonic function vanishes at a point to infinite order, it must vanish everywhere. In the same spirit, we are interested in quantitative unique continuation problems, where we use the local growth rate of a harmonic function to deduce some global estimates, such as estimating the size of its singular or critical set. In this talk, I will talk about some recent results together with C. Kenig on boundary unique continuation.

March 4: Brian Cole, Brown University

    Title:

March 11: Simon Bortz,  U. of Alabama

Title:  Parabolic Lipschitz Domains and Caloric Measure

Abstract: Since the pioneering work of Dahlberg, the study of quantitative “L^p” solvability of boundary value problems for elliptic and parabolic operators in non-smooth domains have been of considerable interest. (So much so that I won’t attempt to put sufficient history in this abstract!) Dahlberg’s fundamental contribution to the area was to show that in a Lipschitz domain the density of harmonic measure exists and it satisfies a certain scale invariant L^2 estimate (reverse Hölder inequality). This in turn gives L^2 control of a certain conical/non-tangential maximal function by the L^2 norm of the data, for the solution to the Dirichlet problem (harmonic function with prescribed boundary values).

For quite some time, less was known about the corresponding parabolic problem, that is, the Dirichlet problem for the heat equation with prescribed lateral boundary data. It was conjectured that if the lateral boundary is given by a Lip(1,1/2) graph (parabolic Lipschitz) then Dahlberg’s theorem should hold for the caloric measure. Kaufmann and Wu showed this to be false; however, it was shown by Lewis and Murray that if one imposes a mild additional (fractional) time-regularity on the graph, then the L^p Dirichlet problem is solvable. With Hofmann, Martell and Nyström, I have shown that this additional regularity is, in fact, necessary. This resolves a problem open for 30 years.

The talk is meant to be accessible, with a historical overview and the proof given at the level of ideas and pictures.

April 12: Alex Iosevich, University of Rochester.

Title: "Restriction theory, uncertainty principle, and exact signal recovery" (joint work with Azita Mayeli, CUNY). 

Abstract: Click here for abstract.

April 15: Stefano Decio, University of Minnesota and IAS

    Title:  Robin harmonic measure in rough domains

Abstract: I will describe the construction of a harmonic measure that reproduces a harmonic function from its Robin boundary data, which is a combination of the value of the function and its normal derivative. I shall discuss the surprising fact that this measure turns out to be (quantitatively) mutually absolutely continuous with respect to surface measure on a wide class of domains that includes the complement of certain fractals. Based on joint work with Guy David, Max Engelstein, Svitlana Mayboroda and Marco Michetti.

April 29: Xiumin Du, Northwestern University

    Title: Weighted Fourier Extension Estimates

Abstract: In this talk, we will survey recent results on weighted Fourier extension estimates and its variants. Such estimates ask for $L^2$ bounds of the Fourier extension operator on $\alpha$-dimensional sets, and they have applications to several problems in PDEs and geometric measure theory, including size of divergence set of Schrodinger solutions, spherical average Fourier decay rates of fractal measures, and Falconer’s distance set problem.

May 6: Azita Mayeli, Graduate Center, CUNY

    Title: Eigenvalue distribution of spation-spectral limiting operators in higher dimenstions

Abstract:

Let $F$, $S$ be bounded measurable sets in $\R^d$. Let $P_F : L^2(\R^d) \rightarrow L^2(\R^d) $ be the orthogonal projection on the subspace of functions with compact support on $F$, and let $B_S : L^2(\R^d) \rightarrow L^2(\R^d)$ be the orthogonal projection  on the subspace of functions with Fourier transforms having compact support on $S$. In this talk, I will report on the distributional estimates on the eigenvalue sequence $1 \geq \lambda_1(F,S) \geq \lambda_2(F,S) \geq \cdots > 0$ of the \emph{spatio-spectral limiting operator} $B_S P_F B_S : L^2(\R^d) \rightarrow L^2(\R^d)$. The significance of such estimates lies in their diverse applications in wirless communications, medical imaging, signal processing, geophysics and astronomy.  More precisely, for suitable domains $F$ and $S$, we prove that for any $ \epsilon \in (0,1)$

$$ \# \{ k : \lambda_k(F,S) > \epsilon \}  = (2 \pi)^{-d} |F| \cdot |S| + \mathrm{Err}(F,S,\epsilon),$$

where the error term depends on $\cH_{d-1}(\partial F)$ and $\cH_{d-1}(\partial S)$, denoting the $(d-1)$-dimensional Hausdorff measures of the boundaries of $F$ and $S$, respectively, and the geometric constants related to an Ahlfors regularity condition on the domain boundaries. 

    Our proof is based on the dyadic decomposition of both spatial and frequency domains, the          decomposition techniques of operators,  and the application of the results in my previous joint paper with Arie Israel (ACHA 2024) on the eigenvalues of spatio-spectral limiting operators associated to cubical domains. \\

 This is joint work with Kevin Hughes (Edinburgh Napier University) and Arie Israel (UT at Austin). 

ARCHIVE of Fall 2023 Semester talks.

Oct. 9: No seminar (Indigeneous people's day)

Oct. 13: Jose Conde Alonso  (4:15 pm Kassar 205) **Time change!**

Title: Schur multipliers as singular integral operators

Abstract: A Schur multiplier is an operator associated to a matrix M that acts on other matrices by entry wise multiplication (that is, multiplying matrices as a bad student would). The boundedness of Schur multipliers in the Schatten-von Neumann classes is an interesting question in operator algebra. In this talk, we will show sufficient smoothness conditions on the symbol M that ensure said boundedness via a surprising connection with Fourier multipliers and noncommutative Calderón-Zygmund theory. Based on joint work with Adrián M. González Pérez, Javier Parcet and Eduardo Tablate.

Oct. 16: Eyvindur Palsson  (Virginia Tech)

Title: Distance problems and geometric averaging operators.

Abstract: Two classic questions - the Erdos distinct distance problem, which asks about the least number of distinct distances determined by points in the plane, and its continuous analog, the Falconer distance problem - both focus on distance. Here, distance can be thought of as a simple two point configuration. Questions similar to the Erdos distinct distance problem and the Falconer distance problem can also be posed for more complicated patterns such as triangles, which can be viewed as three point configurations. In this talk I will go through some of the history of such point configuration questions, show how geometric averaging operators arise naturally and give some recent results.

Oct. 18: Benjamin Foster (Stanford): special analysis seminar 4:15pm in Kassar 105

Title:  Smallness sets for gradients of harmonic functions on Lipschitz surfaces

Abstract:  I’ll discuss some recent work on how understanding Almgren's frequency function gives us geometric information on how the gradients of harmonic functions (or more general solutions to elliptic equations) behave in two dimensions. In particular, harmonic functions have critical sets with size controlled by their degree (frequency) and if the gradient is small on a set with positive Hausdorff dimension, then the gradient can be quantitatively estimated to be "small" on a substantial fraction of the domain of interest. In studying the surface case, we are able to draw on a variety of rich tools from complex analysis.

Oct. 23: Alex Cohen (MIT)

Title: Higher dimensional fractal uncertainty

Abstract: The fractal uncertainty principle (FUP) roughly says that a function and its Fourier transform cannot both be concentrated on a fractal set. These were introduced to harmonic analysis in order to prove new results in quantum chaos: if eigenfunctions on hyperbolic manifolds concentrated in unexpected ways, that would contradict the FUP. Bourgain and Dyatlov proved FUP over the real numbers, and in this talk I will discuss an extension to higher dimensions. The bulk of the work is constructing certain plurisubharmonic functions on C^n. 

    Nov. 6: Dominique Maldague (MIT)

Title: Small cap decoupling for the moment curve in R^3

Abstract: One of the main motivations of decoupling theory is to study exponential sums over frequencies living in restricted sets. It is easier for a function whose frequencies lie in a line to stay large than for a function whose frequencies lie in a curve. Thus allows us to prove strong L^p bounds, called decoupling estimates, when the restricted sets satisfy some curvature assumptions. I will focus on small cap decoupling, which is a version of decoupling with applications in number theory. I will explain how the high-low method for decoupling, established for the parabola by Guth, Maldague, and Wang, may be generalized to the moment curve in R^3. This fully solves the 3-dimensional case of  Conjecture 2.5 from the original small cap decoupling paper of Demeter, Guth, and Wang. This is based on joint work with Larry Guth. 

Nov. 20: Hong Wang

Title: Incidence estimates for tubes

Abstract: Let P be a set of points and L be a set of lines in the plane, what can we say about the number of incidences between P and L,    I(P, L):= |\{ (p, l)\in P\times L, p\in L\}| ?

The problem changes drastically when we consider a thickening version, i.e. when P is a set of unit balls and L is a set of tubes of radius 1. Furstenberg set conjecture can be viewed as an incidence problem for tubes. It states that a set containing an s-dim subset of a line in every direction should have dimension at least  (3s+1)/2 when s>0. 

We will survey a sequence of results by Orponen, Shmerkin and a recent joint work with Ren that leads to the solution of this conjecture.

Dec. 4: Cody Stockdale

Title: On the theory of compact Calderón-Zygmund operators

Abstract: While the boundedness properties of Calderón-Zygmund singular integral operators are classical in harmonic analysis, a theory for compact CZ operators has more recently been established. We present new developments in the theory of compact CZ operators. In particular, we give a new formulation of the $T1$ theorem for compactness of CZ operators, which, compared to existing compactness criteria, more closely resembles David and Journé’s original $T1$ theorem for boundedness and follows from a simpler argument. Additionally, we discuss the extension of compact CZ theory to weighted Lebesgue spaces via sparse domination methods. This talk is based on joint works with Mishko Mitkovski, Paco Villarroya, and Brett Wick. 

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If you have questions or comment, please contact the organizers: Benoit Pausader, Jill Pipher and Nathan Wagner.