In this paper, we find all Weierstrass points on the hyperelliptic modular curves $X_{0}(N)$ whose hyperelliptic involutions are non-exceptional, i.e., induced by matrices in $\mathrm{GL}_{2}(\mathbf{R})$.

The paper concerns the weighted Bergman spaces of the complex unit ball with exponential weights. We characterize the space with respect to Lipschitz type conditions using norm equivalence lemma.

Let $G$ be a real reductive Lie group and $H$ a closed subgroup. T. Kobayashi and T. Oshima established a finiteness criterion of multiplicities of irreducible $G$-modules occurring in the regular representation $C^{\infty}(G/H)$ by a geometric condition, referred to as \textit{real sphericity}, namely, $H$ has an open orbit on the real flag variety $G/P$. This note discusses a refinement of their theorem by replacing a minimal parabolic subgroup $P$ with a general parabolic subgroup $Q$ of $G$, where a careful analysis is required because the finiteness of the number of $H$-orbits on the partial flag variety $G/Q$ is not equivalent to the existence of $H$-open orbit on $G/Q$.

We correct and change Proposition 1 and the proof of Proposition 2 of the previous paper [7].