Discussion On The Existence Of Continuous Linear Right Inverses For Partial Differential Operators

In the early fifties of last century, L.Schwartz posed the problem of determining when a linear differential operator P(D) has a (continuous linear)right inverse; that is,when does there exist a cintinuous linear map R such thatP(D)R(f) = f, for all f ( ) or all f D' ( ) . In 1973, the Phragmen-Lindelof condition was introduced by Hormander[1] to characterize the surjectivity of operators P(D) on the space A( ) of real-analytic functions on a covex open set in Rn. In 1990,R.Meise,B. A.Taylor and D.Vogt gave a fairly complete solution of Schwartz's problem in [2]. They showed that for an open set in Rn and for P C[z1,z2,... ,zn] the differential operator P(D) has a right inverse on ( ) or D'( ) if and only if the algebraic variety V(P) = {z Cn : P(-z) = 0} satisfies a Phragmen-Lindelof condition. Recently,the problem is extended to the space of -ultradistributions and the space of -ultradifferentiable ([3]-[13]).LetP(D) = (Pm + Q)(D),where Pm is the principal part of P. If V(P) satisfies the P-L condition ,then V(Pm) satisfies the P-L condition-Therefore it is reasonable to treat general differential operators P(D) as perturbations of thir principal partPm.Braun,Meise and Vogt [8] have considered the operators S(D) = P(D)- . Inthis paper,we consider the operatorsand derive a necessary condition in terms of Pm for V(S) to have PL (Rn+1, log):Let P C[z1,Z2,... ,zn] have degree m 3 and principal part Pm R[Z1, z2, ..., zn}. Define S C[Z1,,z2,... ,zn+1] by S(z',Zn+1) = P(z')-z2n+1.If V(S)satisfiesPL(Rn+1,log), then for each V(Pm) En, | | = 1, Pm,the localization (Pm) of Pm in is square-free. This extends a result of [8].In[2],R.Meise and Vogt characterized the existence of the right inverse for the differential operator P(D) by an equivalent fact that satisfies a very strong form of P-convexity,which they called P- convexity with bounds.They also showed that a bounded set with C1- boundary is P-convex with bounds if and only if P is hyperbolic with respect to each non-characteristic direction.Roughly speaking, it is rather technical and difficult to check. Concerning this,we use the notation of a point of inner support to derive a method to see when is not P-convex with bounds for each non-constsnt polynomial P on Cn:Let 0 be an open subset of Rn for which Rn\ has a component Q with C1 -boundary and positive normal. QN = {Nx : x }, S = {x : |x| = l,x Rn}, B = S\QN, where Nx denotes the outer unit normal to Q at x. If for any N B, Pm(N) 0, then is not P-convex with bounds for each non-constsnt polynomial P on Cn.So,for some ,we can judge whether the linear partial differential operators P(D) admit a right inverse according to the boundary condition.