I am interested in what are the largest families of initial conditions of the problem (in $\mathbb{R}^{4}$) \begin{equation}\label{kg} \left\lbrace \begin{array}{ll} (\square+m^2)F(x)=0\\ F(0,\vec{x})=g(\vec{x}) \\ \frac{\partial F}{\partial x^{0}}(0,\vec{x})=f(\vec{x}) \end{array} \right. \end{equation} I cut find something like "In order to this diferential equation be well posed, $g$ must live in $H^{2}(\mathbb{R}^{3})$ and $f\in H^{1}(\mathbb{R}^{3})$" on the web, but no for my equation.
So my question is ¿what are sufficent conditions over $f$ and $g$ in order to the probem be well posed?¿And is there any relation between this conditions and some Sobolev space?
I know probably the answer is in some classical book but I can find it.
Many thanks in advance.