H\"older regularity of the nonlinear stochastic time-fractional slow and fast diffusion equations on $\mathbb{R}^d$

In this paper, we use local fraction derivative to show the H\"older continuity of the solution to the following nonlinear time-fractional slow and fast diffusion equation: $$\left(\partial^\beta+\frac{\nu}{2}(-\Delta)^{\alpha/2}\right)u(t,x) = I_t^\gamma\left[\sigma\left(u(t,x)\right)\dot{W}(t,x)\right],\quad t>0,\: x\in\mathbb{R}^d,$$ where $\dot{W}$ is the space-time white noise, $\alpha\in(0,2]$, $\beta\in(0,2)$, $\gamma\ge 0$ and $\nu>0$, under the condition that $2(\beta+\gamma)-1-d\beta/\alpha>0$. The case when $\beta+\gamma\le 1$ has been obtained in \cite{ChHuNu19}. In this paper, we have removed this extra condition, which in particular includes all cases for $\beta\in(0,2)$.