On the functional dimension of the solution space of hypoelliptic equations

V. N. Margaryan ; G. G. Kazaryan

Sbornik. Mathematics, Tome 43 (1982) no. 4, pp. 547-562

Résumé

Let $P(D)$ be a linear differential operator with constant coefficients, and let $N=\{u;\ u\in C(E_n),\ P(D)u=0\}$. Exact formulas are established for the functional dimensional $\operatorname{df}N$ of $N$ when $P(D)$ is a) semielliptic or b) hypoelliptic, where if $P(D)$ is represented in the form $$ P(D)=\sum_{(\lambda,\alpha)=d_0}\gamma_\alpha D^\alpha+\sum_{(\lambda,\alpha)\leqslant d_1}\gamma_\alpha D^\alpha\equiv P_0(D)+P_1(D), $$ with $d_1$, $\lambda\in R_n$ and $\lambda_1\geqslant\lambda_2\geqslant\dots\geqslant\lambda_n=1$, then $P_0(0,\dots,0,\xi_j,0,\dots,0)\ne0$ for $\xi_j\ne0$ ($j=1,\dots,n$). It is proved that $\operatorname{df}N=|\lambda|$ in case a), while in case b) $\displaystyle\operatorname{df}N=\frac1\Delta\biggl(\sum^{n-1}_{j=1}\lambda_j\biggr)+1$ under certain restrictions on $P(D)$, where $$ \Delta=\inf(d_1-d_0+l(\tau))/l(\tau),\qquad\tau\in\Sigma(P_0), $$ with $\Sigma(P_0)=\{\xi\in R_n,\,|\xi|=1,\,P_0(\xi)=0\}$ and $l(\tau)$ the order of the zero $\tau\in\Sigma(P_0)$ of the polynomial $P_0(\xi)$. Bibliography: 19 titles.