The Cauchy problem for operators with injective symbol in the Lebesgue space $L^2$ in a domain

Let $D$ be a bounded domain in $\mathbb R^n$ ($n\ge2$) with infinitely smooth boundary $\partial D$. We give some necessary and sufficient conditions for the Cauchy problem to be solvable in the Lebesgue space $L^2(D)$ in $D$ for an arbitrary differential operator $A$ having an injective principal symbol. Furthermore, using bases with double orthogonality, we construct Carleman's formula that restores a (vector-)function in $L^2(D)$ from the Cauchy data given on a relatively open connected set $\Gamma\subset\partial D$ and the values $Au$ in $D$ whenever the data belong to $L^2(\Gamma)$ and $L^2(D)$ respectively.