On the boundedness of Toeplitz operators with radial symbols over weighted sup-norm spaces of holomorphic functions

We prove sufficient conditions for the boundedness and compactness of Toeplitz operators $T_a$ in weighted sup-normed Banach spaces $H_v^\infty$ of holomorphic functions defined on the open unit disc $\mathbb{D}$ of the complex plane; both the weights $v$ and symbols $a$ are assumed to be radial functions on $\mathbb{D}$. In an earlier work by the authors it was shown that there exists a bounded, harmonic (thus non-radial) symbol $a$ such that $T_a$ is not bounded in any space $H_v^\infty$ with an admissible weight $v$. Here, we show that a mild additional assumption on the logarithmic decay rate of a radial symbol $a$ at the boundary of $\mathbb{D} $ guarantees the boundedness of $T_a$. The sufficient conditions for the boundedness and compactness of $T_a$, in a number of variations, are derived from the general, abstract necessary and sufficient condition recently found by the authors. The results apply for a large class of weights satisfying the so called condition$(B)$, which includes in addition to standard weight classes also many rapidly decreasing weights. Our methods are also applied to study the boundedness of the Bergman projection in weighted spaces. We show that in the case of exponentially decreasing weights $v$, the Bergman (orthogonal) projection of the weighted space $L_w^2$ is bounded in the space $L_v^\infty$ for $w = v^{1+ \delta}$ with any $\delta > (1- \sqrt{2})^2/2 \approx 0.0858$. This should be compared with the well-known result of Dostanic that for such weights, the Bergman projection of $L_v^2$ is never bounded in any space $L_v^p$ with $1 \leq p \leq \infty$, $p \not=2$.