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

This article is cited in 4 scientific papers (total in 4 papers)

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
Received: 23.05.1980

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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    Citing articles on Google Scholar: Russian citations, English citations
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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  mathnet  crossref  mathscinet  zmath
    2. Kazarian G., Markarian V., “On the Lower Bounds of Functional Dimension of Hypoelliptic Equation Solution Space”, 308, no. 1, 1989, 31–33  isi
    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  mathnet  crossref  mathscinet  zmath  adsnasa  isi
    4. G. G. Kazaryan, “Strictly hypoelliptic operators with constant coefficients”, Russian Acad. Sci. Sb. Math., 75:1 (1993), 265–276  mathnet  crossref  mathscinet  zmath  adsnasa  isi
  • Математический сборник (новая серия) - 1964–1988 Sbornik: Mathematics (from 1967)
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