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A sufficient condition for solvability. (English) Zbl 0947.58019

Let \(\mathcal M\) be a \(C^{\infty}\) manifold, \(P\) be a classical pseudo-differential operator of order \(m\) defined on \(\mathcal M\). Let \(p\) be the principal symbol of \(P\) which is assumed to be a principal type.
In the paper the problem of local solvability of pseudo-differential equations \(Pu=f\) is studied. Earlier by N. Lerner it was proved that the pseudo-differential equation \(Pu=f\) is locally solvable if \[ |dp \wedge d \overline{p}|^2 \leq C \text{Im} \{p,\overline{p} \},\text{ when } p=0, \] where \(|dp \wedge d \overline{p}|\) stands for the norm of two-forms and \(\{p,\overline{p} \}\) is the Poisson bracket. The author suggests the new condition, namely \[ |H_p |dp \wedge d \overline{p}|^2|\leq C (\text{Im} \{p,\overline{p} \}+|dp \wedge d \overline{p}|^2),\text{ when } p=0, \] under which the equation \(Pu=f\) is \(L_2\) locally solvable.

MSC:

58J40 Pseudodifferential and Fourier integral operators on manifolds
35S99 Pseudodifferential operators and other generalizations of partial differential operators
53C21 Methods of global Riemannian geometry, including PDE methods; curvature restrictions
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