Zur Darstellung von Lösungen einer Klasse linearer partieller Differentialgleichungen

Abstract

Questa nota riguarda la costruzione di operatori differenziali lineari \[ T=\overset{n}{\underset{i=0}{\sum}}\overset{m}{\underset{k=0}{\sum}}a_{ik}(z,\zeta)\frac{\partial^{i+k}}{\partial z^{i}\partial\zeta^{k}}, \] $(z,\zeta)\epsilon\mathbf{D\subset C^{\textrm{2}}}$che trasformano tutte le soluzioni u (z, $\zeta$) di equazioni u$_{z\zeta}$+ a($z,\zeta$)u$\zeta$+ b$(z,\zeta)$u$_{z}$= 0 in soluzioni $\tilde{u}$=Tu di equazioni $\tilde{u}_{z\zeta}$+$\tilde{a}$$(z,\zeta)$$\tilde{u_{\zeta}}$+$\tilde{b}$$(z,\zeta)\tilde{u_{\zeta}}$=0

\[ T=\overset{n}{\underset{i=0}{\sum}}\overset{m}{\underset{k=0}{\sum}}a_{ik}(z,\zeta)\frac{\partial^{i+k}}{\partial z^{i}\partial\zeta^{k}}, \] $(z,\zeta)\epsilon\mathbf{D\subset C^{\textrm{2}}}$which map all solutions u (z, $\zeta$) of equations u$_{z\zeta}$+ a($z,\zeta$)u$\zeta$+ b$(z,\zeta)$u$_{z}$= 0 into solutions $\tilde{u}$=Tu of equations $\tilde{u}_{z\zeta}$+$\tilde{a}$$(z,\zeta)$$\tilde{u_{\zeta}}$+$\tilde{b}$$(z,\zeta)\tilde{u_{\zeta}}$=0