Factorization of multilinear differential operators with constant coefficients (Q1190584)

The author considers the symmetric \(M\)-linear \((M\geq 3)\) differential operator \[ T(f_ 1,\dots,f_ M)=\sum_{n\in\mathbb{N}^ m}a_ n\prod^ M_{i=1}{d^{n_ i}f^ i\over dx^{n_ i}}, \] where \(n=(n_ 1,\dots,n_ m)\), \(a_ n\in\mathbb{C}\), \(a_ n\neq 0\) for only a finite number of multi-indices \(n\). He proves that there exist bounded linear operators \(A\in L({\mathcal A}_ N^{M'},H^{2M})\), \(B\in L(H^ 2,{\mathcal A}^ 2_{-M(M-1)})\), where \(1/M'+1/2M=1\), such that \(T\) has the factorization \(T(f_ 1,\dots,f_ M)=B\left(\prod^ M_{i=1}Af_ i\right)\), for \(f_ 1,\dots,f_ M\in{\mathcal A}_ N^{M'}\). In particular, \[ \sum_{n\in\mathbb{N}^ M}a_ n\prod^ M_{i=1}{d^{n_ i}f\over dx^{n_ i}}=B((Af)^ M) \] for \(f\in{\mathcal A}^{M'}_{N'}\) where \({\mathcal A}^ p_ N\) is the Banach space \[ {\mathcal A}^ p_ N=\{f:D\to\mathbb{C}\mid f(z)=\sum^ \infty_{n=0}a_ nz^ n,\;\| f\|_ p^{(N)}=\left(\sum^ \infty_{n=0}(n+1)^{p^ N}| a_ n|^ p\right)^{1/p}<\infty\}. \]