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Seminar
FIELD Math:Analysis
DATE December 26 (Thu), 2019
TIME 13:20-14:20
PLACE 1503
SPEAKER Ryu, Jaehyeon
HOST Jung, Joeun
INSTITUTE 서울대학교
TITLE $L^p$ eigenfunction bounds for the Hermite operator
ABSTRACT We introduce the Hermite operator $H$, which is defined by $H=-\Delta+x^2$. As an operator acting on $L^2(\mathbb{R}^d)$, this has some interesting properties. It has distinct positive eigenvalues, and corresponding eigenfunctions which form a basis in $L^2(\mathbb{R}^d)$. Then we direct our attention to the eigenfunction $(-\Delta+x^2)\phi_\lambda=\lambda^2\phi_\lambda$. These are concentrated inside the ball $\{|x|\le\lambda\}$, and have exponential Airy type decay beyond this boundary. We consider the problem of obtaining $L^p$ bounds of eigenfunction for $H$. This question has been considerable interest in the context of Riesz summability for the Hermite operator in the work of Thangavelu and Karadzhov. In this talk, we present the work of Koch and Tataru, which strengthen the results of aforementioned two authors and gives a complete picture.
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