Abatract. Define the general extension operator

$$E_{\alpha,\beta}f(x',x_n)=\int_{\partial\mathbb{R}_+^n}\frac{x_n^\betaf(y)}{(|x'-y|^2+x_n^2)^{\frac{n-\alpha}{2}}} dy$$.

In this talk, I shall recall how we arrive at this general form in the study of the extension operators, what kind of sharp estimates we have obtained so far in the last ten years. I shall also report some recent results on the application of these estimates, in particular in the study of Carleman type inequalities in Hardy space.

Bio. 朱梅俊教授的研究领城为偏微分方程及其在整体几何和图像处理中的应用。近些年研究兴趣集中在非线性方程正解的分类及其爆破分析、最佳几何不等式、曲线流、图像处理中的数学理论等。在 Comm. Pure Appl. Math., Duke Math. J., GAFA, Adv. Math., J. Func. Anal., Int. Math. Res. Not. 等发表学术论文30多篇。2001年获美国数学会学会 (AMS) Centennial Fellowship奖。