| ZHONG JIAQING.[J].数学年刊A辑,1980,1(3-4):359~374 |
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| ON PRIME IDEALS OF THE RING OFDIFFERENTIAL OPERATORS |
| Received:December 13, 1978 |
| DOI： |
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| For given linear differential operators \[{P_1}(\frac{\partial }{{\partial {x_1}}},...,\frac{\partial }{{\partial {x_n}}}),...,{P_n}(\frac{\partial }{{\partial {x_1}}},...,\frac{\partial }{{\partial {x_n}}})\], let \[({P_1},...,{P_m})\] be the left ideal generated by \[{P_1},...,{P_m}\], and F be the space of the common solutions of the operators \[{P_i}(i = 1,...,m),i,e.{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} F = \{ f|{P_i}f = 0,i = 1,...,m\} \].Inspired by the Hilbert's Nullstailerisatz, we introduce another ideal
\[H({P_1},...,{P_m})\] related to \[{P_1},...,{P_m}\], \[H({P_1},...,{P_m}) = \{ P|Pf = 0,\forall f \in F\} \],
Obviously, \[({P_1},...,{P_m}) \subseteq H({P_1},...,{P_m})\].
Definition The ideal \[({P_1},...,{P_m})\] is prime if \[H({P_1},...,{P_m}) = ({P_1},...,{P_m})\].
The aim of this paper is to stndy under what conditions the ideal \[({P_1},...,{P_m})\] is
prime. We have the following results：
Theorem 1 \[(\Delta )\]is prime, where \[\Delta \] is the Laplace operator of \[{{\cal R}^n},i,e\],
\[\Delta = \sum\limits_{i = 1}^n {\frac{{{\partial ^2}}}{{\partial x_i^2}}} \]
Theorem 2 \[{\Delta _S}\] is prime, where \[\Delta \] is the Laplace-Beltrami operator of
\[B = \{ ({Z_1},...,{Z_n}) \in {{\cal L}^n}|\sum\limits_{i = 1}^n {|{z_i}} {|^2} < 1\} \]
it is well hmwn that
\[{\Delta _S} = \sum\limits_{i,j = 1}^n {({\delta _{ij}} - {z_i}{{\bar z}_j})\frac{{{\partial ^2}}}{{\partial {z_i}\partial {{\bar z}_j}}}} \]
Theorem 3
Suppose \[{T_1}(\frac{\partial }{{\partial {x_1}}},...,\frac{\partial }{{\partial {x_n}}}),...,{T_m}(\frac{\partial }{{\partial {x_1}}},...,\frac{\partial }{{\partial {x_n}}})\] are differential operators with constant coefficients, then \[({T_1},...,{T_m})\] is prime if the corresponding
polynomial ideal \[({T_1}({X_1},...,{X_n}),...,{T_m}({X_1},...,{X_n}))\] is prime in the usual sense.
For general linear differential operators with variable coeffcients, we have given
some sufficient conditions for which \[({P_1},...,{P_m})\] is prime (see Th. 4). The two
important ones among these sufficient conditions are that \[{P_1},...,{P_m}\] possess a Poisson
kernel and they satisfy group invariance in some sense.
Finally, as an example, we study in detail the case of the classical domain
\[R(2) = \{ Z = \left( {\begin{array}{*{20}{c}}
{{z_1}}&{{z_2}}\{{z_3}}&{{z_4}}
\end{array}} \right)|I - Z{Z^'} > 0\} \]
We show that \[H({\Delta _{R(2)}}),i,e,({\Delta _{R(2)}})\] is not prime, and we also give the basis
of the ideal \[H({\Delta _{R(2)}})\], where \[H({\Delta _{R(2)}})\] is the Laplaoe-Beltrami operator of R(2). |
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