Abstract: The Cauchy-Riemann problem, also known as the $\overline\partial$-problem, is a central problem in several complex variables. It concerns the regularity estimates to the equation $\overline\partial u=f$ on forms in a bounded domain $\Omega\subset\mathbb C^n$. We will talk about some background of the $\overline\partial$-regularity theory, and the obstructions on solving the $\overline\partial$ equation when f is a generic distributions, and our recent works using new technique from extension operators. We use the so-called Rychkov's extension operator, which extends functions on a bounded Lipschitz domain and has boundedness on all Besov spaces and Triebel-Lizorkin spaces. This is based on multiple works which are in part joint with Ziming Shi and Yuan Zhang. 

Location and Zoom link: SCEN 322 and at https://uark.zoom.us/j/87503165233?pwd=S7QdLHdFaJYEkZL3at0OV4aE2rRPwh.1

Abstract: An important problem in several complex variables is to classify all proper holomorphic maps that send the unit ball in C^n to the unit ball in C^N. When such mappings extend continuously to the boundary, they necessarily send the unit sphere to the unit sphere and are called sphere maps. Many open questions remain, even for monomial sphere maps. In this talk, we focus on group-invariant monomial sphere maps, which satisfy some extra symmetry properties, and seek to identify the target ranks for such maps. Rather than give a detailed proof of one specific theorem, we give an overview of the recent results of several undergraduate and graduate student researchers at BYU.  

Location and Zoom link: SCEN 322 and at https://uark.zoom.us/j/87503165233?pwd=S7QdLHdFaJYEkZL3at0OV4aE2rRPwh.1

Abstract: 

The sequence  Besov space $b_p$, $p>1$, is the space of analytic functions $f(z)=\sum_{n=0}^{\infty} a_n\, z^n$ on the open unit disk 

$\mathbb D$ with $$\sum_{n=0}^{\infty} n^{p-1}\, |a_n|^p<\infty\, .$$

We will give a brief introduction to the space $b_p$, as well as explore how certain operators behave on the space. For example, let $\varphi$ be an analytic self-map of $\mathbb D$. We study the multiplication operator $M_\varphi$ on $b_p$ and look at examples including when $\varphi$ is a polynomial. We also study the composition operator $C_\varphi$ on $b_p$, and in more depth on $b_2$ the Dirichlet space. The culmination will be an analogue of the Littlewood Theorem on Hardy spaces for $b_2$.

Location and Zoom link: SCEN 322 and at https://uark.zoom.us/j/87503165233?pwd=S7QdLHdFaJYEkZL3at0OV4aE2rRPwh.1

Abstract: The Hilbert transform maps L¹ functions into weak-L¹ ones. In fact, this estimate holds true for any operator T(m) defined by a bounded Fourier multiplier m with singularity only in the origin. Tao and Wright identified the space replacing L¹ in the endpoint estimate for T(m) when m has singularities in a lacunary set of frequencies, in the sense of the Hörmander-Mihlin condition.

In this talk we will quantify how the endpoint estimate for T(m) for any arbitrary m is characterized by the lack of additivity of its set of singularities . This property of the set of singularities of m is expressed in terms of a Zygmund-type inequality. The main ingredient in the proof of the estimate is a multi-frequency projection lemma based on Gabor expansion playing the role of Calderón-Zygmund decomposition.

The talk is based on joint work with Bakas, Ciccone, Di Plinio, Parissis, and Vitturi.

Location and Zoom link: SCEN 322 and at https://uark.zoom.us/j/87503165233?pwd=S7QdLHdFaJYEkZL3at0OV4aE2rRPwh.1

Abstract:  We present a unified framework for the local-in-time existence of suitable weak solutions to the three-dimensional incompressible Navier-Stokes equations with non decaying initial data. The analysis is carried out in a broad family of local energy spaces defined through general coverings of $\mathbb{R}^3$ by balls of varying size, allowing for configurations in which spatial scales may grow or shrink at different rates and in different directions. This setting encompasses the classical $L^2_{uloc}$ framework as well as previously studied dyadic constructions.

A key feature of the approach is a new construction of weak solutions based on a linearized system that avoids divergence-free approximations at intermediate stages. An essential step in the analysis is the derivation of uniform pressure estimates that remain valid despite the presence of multiple, non-uniform spatial scales induced by the underlying ball covering. These estimates allow one to control the pressure consistently across regions with differing geometric behavior and are central to establishing the local energy bounds required for existence. The resulting framework provides a flexible and robust existence theory for Navier-Stokes flows with large-scale, non-decaying structure.

Location and Zoom link: SCEN 322 and at https://uark.zoom.us/j/87503165233?pwd=S7QdLHdFaJYEkZL3at0OV4aE2rRPwh.1