司靈得 (Daniel Spector)

In this talk we will provide a brief introduction to the eigenfunctions of the Laplace-Beltrami operator. We will then focus on the so-called symmetry conjecture and a natural variant raised by Nadirashvili and Jakobson. In particular we will present certain results (positive and negative) obtained in collaboration with F. Torres de Lizaur, and extended by a group of students during last summer at the Fields Undergraduate Summer Research Program under our supervision. Time permitting we will comment on work in progress or open questions.

Differently from their integer versions, the fractional Sobolev spaces \( W^{\alpha,p}(\mathbb{R}^n) \) do not seem to have a clear distributional nature. By exploiting suitable notions of fractional gradient and divergence already existing in the literature, we introduce via a distributional approach the new spaces \( BV^{\alpha,p}(\mathbb{R}^n) \) of \( L^p \) functions with bounded \( \alpha \)-fractional variation in \( \mathbb{R}^n \), for \( \alpha \in (0,1) \) and \( p \in [1, \infty] \). In addition, we define in a similar way the distributional fractional Sobolev spaces \( S^{\alpha,p}(\mathbb{R}^n) \), which extend naturally \( W^{\alpha,p}(\mathbb{R}^n) \). The properties of these spaces have been analyzed in a series of papers in collaboration with Giorgio Stefani, Daniel Spector, Elia Bruè, and Mattia Calzi. In this talk, we shall focus ourselves on the absolute continuity property of the fractional variation with respect to a suitable Hausdorff measure and on the existence of precise representatives for \( S^{\alpha,p} \) and \( BV^{\alpha,p} \) functions. Subsequently, we shall derive fractional Leibniz rules and Gauss-Green formulas involving \( BV^{\alpha,p} \) and \( S^{\alpha,p} \) functions. As an application of these results, we will show the well-posedness of the boundary-value problem for a general \( 2\alpha \)-order fractional elliptic operator in divergence form.

I In this seminar, I shall talk about analogues of pseudo-differential operators (pseudo-multipliers) associated with the joint functional calculus of the Grushin operator. In particular, I shall discuss some sufficient conditions on a symbol function, those which imply $L^2$-boundedness of the associated Grushin pseudo-multiplier. This talk is based on a joint work (arXiv:2111.10098) with Sayan Bagchi. 

Consider families of spherical means where the radii are restricted to a given subset of a compact interval. One is interested in the Lp improving estimates for the associated maximal operators and related objects.  Results depend on several notions of fractal dimension of the dilation set, or subsets of it. There are some surprising  statements on the shape of the possible type sets.  Joint work with J. Roos and with T. Anderson, K. Hughes and J. Roos.

We consider integrals in the sense of Choquet with respect to the Hausdorff content. In particular, we discuss the validity of Poincare, Poincare-Sobolev, and Trudinger type inequalities for functions defined on John domains in this context. This is joint work with Petteri Harjulehto.

The study of a 2-dimensional wave profile of an inviscid, incompressible fluid acted upon by gravity traveling at constant speed goes back to 1847 and the work of Stokes, who conjectured that there should exists a one-parameter family of smooth wave profiles and an extreme wave of maximal height which had a peak with a 120 degree angle. It turns out that a wave profile in this setting is the free-boundary of a solution to a related free-boundary problem. The first half of this talk will serve as an incomplete and biased introduction to the study of this free boundary problem. Particularly, it will focus upon the ``density methods" developed by Varvaruca and Weiss. The second half of this talk will focus upon the issue of cusps. The aforementioned ``density methods" are unable to eliminate cusps, but recent work (joint with Lisa Naples) has recently found a rather simple approach to eliminating cusps.

In this talk I will discuss multilinear Kakeya estimates and the nonlinear Brascamp-Lieb inequality. Both may be viewed as perturbed versions of the classical version of the Brascamp-Lieb inequality and emerged around 2005 in connection with the multilinear restriction problem. Progress on these problems has relied heavily on induction-on-scales and I will try to give an overview of the main ideas.