Local solvability of some partial differential operators with polynomial coefficients

• Autores: Christopher J. Winfield
• Localización: Houston journal of mathematics, ISSN 0362-1588, Vol. 30, Nº 3, 2004, págs. 891-927
• Idioma: inglés
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• Resumen

  • We study the local solvability of partial differential operators which can be expressed as certain polynomials in the vector fields X=Dx and Y=Dy+xkDw for odd integers k greater than or equal to 1. More specifically, we study operators of the form P(X,Y), where P is a homogenous polynomial in X and Y of degree n greater than or equal to 2 where, for complex z, P(z,0) = zn and where P(iz,1) has distinct roots. We apply asymptotic estimates by F.M. Christ (1993) to form a constructive proof of local solvability; to do so, we assume the absence of non-trivial Schwartz-class functions in the kernels of the operators P(-iDx,+x)* and P(-iDx,-x))* (* denoting adjoint) and we assume a certain technical condition on the kernels of the operators P(-iDx,+(z-xk)) and P(-iDx,-(z-xk)) with parameter z>0. The novelty of our results lie in those for k greater than or equal to 3, odd. As a special case, these results prove the local solvability of the operators X2 + Y2 + ia[X,Y] for any complex constant, a, not an odd integer.