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 Mat. Sb. (N.S.), 1981, Volume 115(157), Number 4(8), Pages 614–631 (Mi msb2425)

On the functional dimension of the solution space of hypoelliptic equations

V. N. Margaryan, G. G. Kazaryan

Abstract: 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 $d_1<d_0$, $\lambda\in R_n$ and $\lambda_1\geqslant\lambda_2\geqslant…\geqslant\lambda_n=1$, then $P_0(0,…,0,\xi_j,0,…,0)\ne0$ for $\xi_j\ne0$ ($j=1,…,n$).
It is proved that $\operatorname{df}N=|\lambda|$ in case a), while in case b) $\displaystyle\operatorname{df}N=\frac1\Delta(\sum^{n-1}_{j=1}\lambda_j)+1$ 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.

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English version:
Mathematics of the USSR-Sbornik, 1982, 43:4, 547–562

Bibliographic databases:

UDC: 517.9
MSC: Primary 35E99, 35H05, 54F45; Secondary 26C10, 30C15, 32A99, 54C70

Citation: V. N. Margaryan, G. G. Kazaryan, “On the functional dimension of the solution space of hypoelliptic equations”, Mat. Sb. (N.S.), 115(157):4(8) (1981), 614–631; Math. USSR-Sb., 43:4 (1982), 547–562

Citation in format AMSBIB
\Bibitem{MarKaz81} \by V.~N.~Margaryan, G.~G.~Kazaryan \paper On the functional dimension of the solution space of hypoelliptic equations \jour Mat. Sb. (N.S.) \yr 1981 \vol 115(157) \issue 4(8) \pages 614--631 \mathnet{http://mi.mathnet.ru/msb2425} \mathscinet{http://www.ams.org/mathscinet-getitem?mr=629630} \zmath{https://zbmath.org/?q=an:0549.35027|0489.35031} \transl \jour Math. USSR-Sb. \yr 1982 \vol 43 \issue 4 \pages 547--562 \crossref{https://doi.org/10.1070/SM1982v043n04ABEH002580} 

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This publication is cited in the following articles:
1. G. G. Kazaryan, “On a functional index of hypoellipticity”, Math. USSR-Sb., 56:2 (1987), 333–347
2. Kazarian G., Markarian V., “On the Lower Bounds of Functional Dimension of Hypoelliptic Equation Solution Space”, 308, no. 1, 1989, 31–33
3. G. G. Kazaryan, V. N. Markaryan, “Lower bounds for the functional dimension of the solution space of hypoelliptic operators”, Math. USSR-Sb., 70:2 (1991), 341–353
4. G. G. Kazaryan, “Strictly hypoelliptic operators with constant coefficients”, Russian Acad. Sci. Sb. Math., 75:1 (1993), 265–276
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