Optimal local H？lder index for density states of superprocesses with $(1+β)$-branching mechanism

For $0<\alpha\leq2$, a super-$\alpha$-stable motion $X$ in $\mathsf{R}^d$ with branching of index $1+\beta\in(1,2)$ is considered. Fix arbitrary $t>0$. If $d<\alpha/\beta$, a dichotomy for the density function of the measure $X_t$ holds: the density function is locally H\"{o}lder continuous if $d=1$ and $\alpha>1+\beta$ but locally unbounded otherwise. Moreover, in the case of continuity, we determine the optimal local H\"{o}lder index.