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Partial differential operators depending analytically on a parameter. (English) Zbl 0729.35011
Let \(P(\lambda,D)=\sum_{| \alpha | \leq m}a_{\alpha}(\lambda)D^{\alpha}\) be a differential operator with constant coefficients \(a_{\alpha}\) depending analytically on a parameter \(\lambda\). Assume that the family \(\{\) P(\(\lambda\),D)\(\}\) is of constant strength. We investigate the equation \(P(\lambda,D)f_{\lambda}\equiv g_{\lambda}\) where \(g_{\lambda}\) is a given analytic function of \(\lambda\) with values in some space of distributions and the solution \(f_{\lambda}\) is required to depend analytically on \(\lambda\), too. As a special case we obtain a regular fundamental solution of P(\(\lambda\),D) which depends analytically on \(\lambda\). This result answers a question of L. Hörmander.

MSC:
35B30 Dependence of solutions to PDEs on initial and/or boundary data and/or on parameters of PDEs
35E05 Fundamental solutions to PDEs and systems of PDEs with constant coefficients
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