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We prove sufficient conditions for the boundedness and compactness ofToeplitz operators $T_a$ in weighted sup-normed Banach spaces $H_v^\infty$ ofholomorphic functions defined on the open unit disc $\mathbb{D}$ of the complexplane; both the weights $v$ and symbols $a$ are assumed to be radial functionson $\mathbb{D}$. In an earlier work by the authors it was shown that thereexists a bounded, harmonic (thus non-radial) symbol $a$ such that $T_a$ is notbounded in any space $H_v^\infty$ with an admissible weight $v$. Here, we showthat a mild additional assumption on the logarithmic decay rate of a radialsymbol $a$ at the boundary of $\mathbb{D} $ guarantees the boundedness of$T_a$.  The sufficient conditions for the boundedness and compactness of $T_a$, in anumber of variations, are derived from the general, abstract necessary andsufficient condition recently found by the authors. The results apply for alarge class of weights satisfying the so called condition$(B)$, which includesin addition to standard weight classes also many rapidly decreasing weights.

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