A “deformation estimate" for the Toeplitz operators on harmonic Bergman spaces

Abstract:

Let $B$ denote the open unit ball in $\mathbb R^n$ for $n\geq 2$ and $dx$ the Lebesgue volume measure on $\mathbb R^n$. For $\alpha >-1$, the (weighted) harmonic Bergman space $b^{2,\alpha }(B)$ is the space of all harmonic functions $u$ which are in $L^2(B,(1-|x|^2)^{\alpha }dx)$. For $f\in L^{\infty }(B)$, the Toeplitz operator $T_f^{(\alpha )}$ is defined on $b^{2,\alpha }(B)$ by $T_f^{(\alpha )}u = Q_{\alpha }[fu]$, where $Q_{\alpha }$ is the orthogonal projection of $L^2(B,(1-|x|^2)^{\alpha }dx)$ onto $b^{2,\alpha }(B)$. In this note, we prove that for $f\in C(B)\cap L^{\infty }(B)$ radial, $\lim _{\alpha \to \infty } \|T_f^{(\alpha )}\|=\|f\|_{\infty }$.