Schauder estimates for equationswith fractional derivatives

The equation where $D^\alpha_t$ and $D^\beta_x$ are fractional derivatives of order $\alpha$ and $\beta$ is studied. It is shown that if $f=f(\underline{t}, \underline{x})$, $h_1=h_1(\underline{x})$, and $h_2=h_2(\underline{t})$ are Hölder-continuous and $f(0,0) =0$, then there is a solution such that $D^\alpha_t u$ and $D^\beta_x u$ are Hölder-continuous as well. This is proved by first considering an abstract fractional evolution equation and then applying the results obtained to ($*$). Finally the solution of ($*$) with $f=1$ is studied.