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

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 $L^2(B,(1-\vert x\vert^2)^{\alpha}dx)$. 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 $L^2(B,(1-\vert x\vert^2)^{\alpha}dx)$ onto $b^{2,\alpha}(B)$. In this note, we prove that for $f\in C(B)\cap L^{\infty}(B)$ radial, $\lim_{\alpha\to\infty} \Vert T_f^{(\alpha)}\Vert=\Vert f\Vert _{\infty}$.