I have to study the following spatial fractional differential equation $$ \dfrac{\partial x(z,\,t)}{\partial t}=\dfrac{\partial ^\alpha x(z,\,t)}{\partial z^\alpha} $$ with $1\leq \alpha<2$ and $0<z<l$.
I ask how to select the appropriate functional space for $x(z,\,t)$ ? Is $L^2(z,\,t)$ a best choice?