Posted January 24, 2024
3:30 pm – 4:30 pm Lockett 232
Phuc Nguyen, Louisiana State University
Poincar\'e-Sobolev's inequalities for the class of $\mathcal{A}$-superharmonic functions
I will discuss about (weighted) Poincar\'e-Sobolev's inequalities for the class of $\mathcal{A}$-superharmonic functions which are solutions, possibly singular, to a class of quasi-linear elliptic equations with nonnegative measure data. A feature of these inequalities is that they hold for a wide range of exponents and a large class of weights over Boman/John domains. This talk is based on joint work with Seng-Kee Chua.
Posted February 3, 2024
3:30 pm – 4:30 pm Zoom
Cody Stockdale, Clemson University
On the Calderón-Zygmund theory of singular integrals
Calderón and Zygmund's seminal work on singular integral operators has greatly influenced modern harmonic analysis. We begin our discussion with some classical aspects of CZ theory, including examples and applications, and then focus on the crucial weak-type (1,1) estimate for CZ operators. We investigate techniques for obtaining weak-type inequalities that use the CZ decomposition and ideas inspired by Nazarov, Treil, and Volberg. We end with an application of these methods to the study of the Riesz transforms in high dimensions.
Posted February 26, 2024
3:30 pm – 4:30 pm Online Zoom
Yaghoub Rahimi, Georgia Institute of Technology
AVERAGES OVER THE GAUSSIAN PRIMES: GOLDBACH’S CONJECTURE AND IMPROVING ESTIMATES
In this discussion we will establish a density version of the strong Goldbach conjecture for Gaussian integers, restricted to sectors in the complex plane.
Posted January 24, 2024
Last modified March 13, 2024
Alan Chang, Washington University in St. Louis
Venetian blinds, digital sundials, and efficient coverings
Davies's efficient covering theorem states that we can cover any measurable set in the plane by lines without increasing the total measure. This result has a dual formulation, known as Falconer's digital sundial theorem, which states that we can construct a set in the plane to have any desired projections, up to null sets. The argument relies on a Venetian blind construction, a classical method in geometric measure theory. In joint work with Alex McDonald and Krystal Taylor, we study a variant of Davies's efficient covering theorem in which we replace lines with curves. This has a dual formulation in terms of nonlinear projections.