I recently came across the following theorem of Martineau (and others). Let $P$ be a polynomial of $n$ variables with complex coefficients. If we identify naturally powers of partial derivatives $\frac{\partial}{\partial z_{i}}$ with the action of the appropiate differetial operators then $P(\frac{\partial}{\partial z_{1}},\ldots,\frac{\partial}{\partial z_{n}})$ becomes a well-defined differential operator with constant coefficients. The theorem says that the differential equation $P(\frac{\partial}{\partial z_{1}},\ldots,\frac{\partial}{\partial z_{n}})u=f$ has an analytic(holomorphic) solution for any analytic $f$ and any polynomial $P$ if and only if the domain on which we look for the solution is so-called $\mathbb C$-convex. The proofs that I found were in the books of H\"ormander "Notions of Convexity" and Andersson, Passare, Sigurdsson "Complex convexity and analytic functionals". Unlike the proof of a corresponding result for the operator $P(\frac{\partial}{\partial \bar z_{1}},\ldots,\frac{\partial}{\partial \bar z_{n}})$, which uses $L^2$ theoretic methods and is to be found far more frequently in the literature (see e.g., the aforementioned book of H\"ormander), the proof of Martineau theorem is based on completely different methods involving integral transforms and some properties of supports of analytic functionals. Now my question is: knowing that one can solve $P(\frac{\partial}{\partial z_{1}},\ldots,\frac{\partial}{\partial z_{n}})u=f$, can a solution be found with growth control? For example can the $L^2$ norm of a solution be controlled by the $L^2$ norm of $f$ as is in the case of $P(\frac{\partial}{\partial \bar z_{1}},\ldots,\frac{\partial}{\partial \bar z_{n}})$ (well-known)?

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