Abstract:
In this paper we study scalar pseudodifferential operators for which the gradient gradx,ξp0(x,ξ) of the principal part of the symbol does not vanish and is not proportional to a real vector at any characteristic point (x,ξ)∈Ω×{Rn∖0}. Such operators are called nondegenerate. It is assumed in addition that for each point of Ω×{Rn∖0} there exists an operator in the Lie algebra generated by the operators P and P∗ the principal part of the symbol of which does not vanish at this point. For these operators we present here hypoellipticity conditions, conditions for the local solvability of the equation Pu=f, a theorem on the smoothness of the solutions of this equation, and so on. All of the conditions obtained have a simple algebraic character and are exact, necessary and sufficient.
Bibliography: 13 titles.
This publication is cited in the following 9 articles:
M. I. Vishik, L. R. Volevich, A. M. Il'in, A. S. Kalashnikov, V. A. Kondrat'ev, O. A. Oleinik, “Yurii Vladimirovich Egorov (on his 60th birthday)”, Russian Math. Surveys, 54:2 (1999), 465–476
R. S. Zolotareva, V. P. Markov, V. A. Apanas'eva, S. S. Matveichuk, Z. S. Atroshenko, “Grinding slagsitall with free abrasives”, Glass Ceram, 33:3 (1976), 197
A. Menikoff, “Carleman estimates for partial differential operators with real coefficients”, Arch. Rational Mech. Anal., 54:2 (1974), 118
L. Khermander, “O suschestvovanii i regulyarnosti reshenii lineinykh psevdodifferentsialnykh
uravnenii”, UMN, 28:6(174) (1973), 109–164
V. V. Grushin, “Hypoelliptic differential equations and pseudodifferential operators with operator-valued symbols”, Math. USSR-Sb., 17:4 (1972), 497–514
Yu. V. Egorov, “On the solubility of differential equations with simple characteristics”, Russian Math. Surveys, 26:2 (1971), 113–130
V. V. Grushin, “On a class of hypoelliptic operators”, Math. USSR-Sb., 12:3 (1970), 458–476