报告题目：The L² norm of the interior Cauchy transform beyond the first Dirichlet eigenvalue

报 告 人：David Kalaj教授（黑山共和国黑山大学）

报告时间：2026年4月26日  9:00-9:40

报告地点：数学与统计学院五楼数学研究中心528报告厅

报告摘要：

In this talk, we mainly discuss the interior Cauchy transform on bounded planar domains and its relationship with the first Dirichlet eigenvalue of the Laplacian. For the unit disk, the sharp L² operator norm is known to equal 2/√λ₁(D), which suggests a natural spectral conjecture for general domains. We show that this principle fails in general. In particular, the endpoint Fourier-weighted inequality underlying the conjectural argument is false even for the disk, and the corresponding sharp constant is instead governed by a potential operator $S_D$. We further prove a rigidity result: for bounded simply connected domains with C^1，αboundary, the identity||C_D||_L²_(D)→L²_(D)= 2/√λ₁(D) holds if and only if D is a disk. We also analyze annuli, where the norm is determined not by the first Dirichlet eigenvalue but by the first mixed Neumann--Dirichlet eigenvalue. These results show that the extremal behavior of the interior Cauchy transform goes beyond the first Dirichlet eigenvalue and is governed by more subtle geometric and spectral structure.

报告人简介：

David Kalaj是黑山共和国黑山大学数学系教授、博士生导师以及阿尔巴尼亚的外籍院士，主要研究方向为复分析与算子理论。2010-2013年，Kalaj教授主持黑山科技部的一项重大课题，并获得了2013年度黑山共和国最杰出科技奖。目前，Kalaj教授已在Duke Math. J.、Proc. London Math. Soc.、Adv. Math.、Math. Ann.、Trans. Amer. Math. Soc.、J. Funct. Anal.、IMRN、CVPDE、Math. Z.等国际知名期刊发表论文100余篇。