司靈得 (Daniel Spector)

Weighted Carleson embedding (weighted paraproduct estimates in another language) lies in the core of various harmonic analysis and PDE results. Not much is known about it in multi-parameter situation, while one parameter is completely understood. I will formulate several new results on weighted multi-parameter Carleson embedding on multi-trees and their corollaries as embeddings of Hilbert spaces of analytic functions on poly-discs. I will also formulate corresponding Poincar\'e inequalities on multi-trees and poly-discs. Some of those results are final, but even embedding of Hardy space on bi-disc is not completely described. My presentation is based on joint works with N. Arcozzi, I. Holmes, P. Mozolyako, P. Zorin-Kranich.

We prove the global existence of the non-negative unique mild solution for the Cauchy problem of the cutoff Boltzmann equation for soft potential model −1<=γ<0 with the small initial data in three dimensional space. Thus our result fixes the gap for the case γ=−1 in three dimensional space in the authors' previous work where the estimate for the loss term was improperly used. The other gap there for the case γ=0 in two dimensional space is recently fixed by Chen, Denlinger and Pavlović. The initial data f0 is non-negative, small in weighted L3_{x,v} and finite in weighted L15/8_{x,v}. We also show that the solution scatters with respect to the kinetic transport operator. The novel contribution of this work lies in the exploration of the symmetric property of the gain term in terms of weighted estimate. It is the key ingredient for solving the model −1<γ<0 when applying the Strichartz estimates.

We discuss two ways that analysis and number theory have recently teamed up, using a back and forth interplay to make progress on two different types of counting problems. First we will count equilateral triangles in Euclidean space. Second we will determine how often a random polynomial fails to have "full" Galois group. Though easy to state, these questions have generated a lot of interesting techniques through the years, which we will glimpse during this talk.

Nonlocal variational problems arise in various applications, such as continuum mechanics, the theory of phase transitions, or image processing. Naturally, the presence of nonlocalities leads to new effects, and the standard methods in the calculus of variations, which tend to rely intrinsically on localization arguments, do not apply. In this talk, we address questions arising from the existence theory for three different classes of variational functionals: integrals depending on Riesz fractional gradients, double integrals, and double supremals - and find qualitatively very different results. Regarding the characterization of weak lower semicontinuity, it may be surprising that quasiconvexity, which is well-known from the classical local setting, also provides the correct convexity notion for the fractional integrals. Our proof relies on a translation mechanism that allows switching between classical and fractional gradients. In the case of double supremals, we show that the natural guess of separate level convexity fails in the vectorial case, and introduce the new Cartesian level convexity. As for relaxation, we discuss the central issue of why one cannot expect these nonlocal functionals, in contrast to their local counterparts, to be structure-preserving. This is based on joint work with Antonella Ritorto, Hidde Schönberger (both KU Eichstätt-Ingolstadt), and Elvira Zappale (Sapienza University of Rome).

We prove a general principle, called the principal alternative, which yields an easily verifiable necessary and sufficient condition for the existence or the non-existence of an optimal Orlicz space in a wide variety of specific tasks including boundedness of operators. We show that the key relation is the positioning of certain rearrangement-invariant space, characteristic for the task in question, to its fundamental Orlicz space. The main motivation stems from the imbalance between the expressivity, meaning the richness and versatility, of certain class of function spaces, and its accessibility, i.e., its complexity and technical difficulty. More precisely, while an optimal rearrangement-invariant space in a given task often exists, it might be too complicated or too implicit to be of any practical value. Optimal Orlicz spaces, on the other hand, are simpler and more manageable for applications, but they tend not to exist at all. We apply the general abstract result to several specific tasks including continuity of Sobolev embeddings or boundedness of integral operators such as the Hardy-Littlewood maximal operator and the Laplace transform. The proof of the principal alternative is based on relations of endpoint Lorentz spaces to unions or intersections of Orlicz spaces. This is a joint work with Vít Musil (Brno) and Jakub Takáč (Warwick).

In this talk we will describe several qualitative and quantitative unique continuation properties for the fractional discrete Laplacian. We will show that, in contrast to the fractional continuous Laplacian, global unique continuation fails to hold in general for fractional discrete elliptic equations. We will also discuss quantitative versions of unique continuation which illustrate how the properties in the continuous setting may be recovered if exponentially small (in terms of the lattice size) correction factors are added. Joint work with Aingeru Fernández-Bertolin and Angkana Rüland.

The aim of these lectures is to give a gentle introduction to Strichartz estimates, with an emphasis on particular cases such as the linear Schr\"odinger and wave equations. The associated dispersive estimates play a highly important role in the theory of Strichartz estimates so I will begin in Lecture 1 by proving the required dispersive estimates.

Next, in Lecture 2, I will prove the homogeneous Strichartz estimates in all admissible cases, including the so-called Keel--Tao endpoint case. Building on the content of the first two lectures, in Lecture 3, I will discuss the situation regarding inhomogeneous Strichartz estimates.

You can find the note of this lecture here!

The aim of these lectures is to give a gentle introduction to Strichartz estimates, with an emphasis on particular cases such as the linear Schr\"odinger and wave equations. The associated dispersive estimates play a highly important role in the theory of Strichartz estimates so I will begin in Lecture 1 by proving the required dispersive estimates.

Next, in Lecture 2, I will prove the homogeneous Strichartz estimates in all admissible cases, including the so-called Keel--Tao endpoint case. Building on the content of the first two lectures, in Lecture 3, I will discuss the situation regarding inhomogeneous Strichartz estimates.

You can find the note of this lecture here!

The aim of these lectures is to give a gentle introduction to Strichartz estimates, with an emphasis on particular cases such as the linear Schr\"odinger and wave equations. The associated dispersive estimates play a highly important role in the theory of Strichartz estimates so I will begin in Lecture 1 by proving the required dispersive estimates.

Next, in Lecture 2, I will prove the homogeneous Strichartz estimates in all admissible cases, including the so-called Keel--Tao endpoint case. Building on the content of the first two lectures, in Lecture 3, I will discuss the situation regarding inhomogeneous Strichartz estimates.

You can find the note of this lecture here!