Given a BMO${}^1$ function f on $\mathbb{C}^n$,
we can take the Berezin transform (better known as the ``heat
transform") $\widetilde{f} ^{(t_0)}$ of $f$ with respect to the weighted
Segal-Bargmann space $H^2 (\mathbb{C}^n, d\mu_{t_0})$. In this talk,
we will discuss the question of whether $\widetilde{f} ^{(t_0)}$
vanishing at infinity for some $t_0 > 0$ implies that
$\widetilde{f} ^{(t)}$ vanishes at infinity for all $t > 0$, and we discuss
the same question in the context of the weighted Bergman space of
the unit ball. Moreover, we discuss what implications these results
have for the compactness of Toeplitz operators in both the weighted
Bergman and Segal-Bargmann space situations, and finally briefly
discuss recent work by H. Issa that extends of some of these results to
the weighted Bergman space of general bounded symmetric domains.
This is joint work with W. Bauer and L. Coburn.
수학과 세미나
제목 | Heat flow, BMO, and the compactness of Toeplitz operators |
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