An Analysis Upon Surjectivity of Partial Differential Operators: Fundamental Solution to the Division Problem

An important milestone in the general theory of partial differential equations is the solution to the division problem: Let D be a nonzero partial differential operator with constant coefficients and T be a distribution; can one find a distribution S such that DS = T? That this is always possible was established by Ehrenpreis (1956). See also Peter Wagner(2009) for the solution of various avatars of the division problem for different spaces of distributions. In this article, we study the division problem in spaces of distributions on(where we think of there beingspace directions and one time direction) that are periodic in the spatial directions. The study of such solution spaces arises naturally in control theory when one considers the so-called "spatially invariant systems". In the "behavioral approach" to control theory for such spatially invariant systems, a fundamental question is whether this class of distributions is an injective module over the ring of partial differential operators with constant coefficients. In light of this, one can first ask what happens with the division problem. Thus besides being a purely mathematical question that fits in the classical theme mentioned in the previous paragraph, there is also a behavioral control theoretic motivation for studying the division problem for distributions that are periodic in the spatial directions. Upon taking Fourier transform with respect to the spatial variables, the problem amounts to the following. Problem: For which is surjective, where (denotes the 1-norm in.) An obvious necessary condition is that for all However, that this condition is not sufficient is demonstrated by considering the following example. Example 1. Let(a "Liouville number"). With,, Indeed, otherwise there would exist an m such that and in particular, with, and, a contradiction. We consider a simpler situation and set Our main result is the following:

(The rootsare arbitrarily arranged.)

the second condition is not superfluous either. Example 2. Take with the same c as in Example 1. and. Then

There holds that (For the first isomorphism we use the Closed Graph Theorem, while the second isomorphism follows, that is,is the dual of a Frechet space and there holds that for allthere existssuch that for all, Hence it follows that in Y: for allthere existssuch that for alland all, In particular forand, We will also need the following lemma. Lemma 1. Consider a monic polynomial

Proof. Let. Then by comparing coefficients of the powers of, we obtain We first show that. Suppose this is not true. Then there exists a sequence such that andfor all. But then from the first equation in the above equation array, it follows that Then from the second equation in the above equation array, we also obtain that Proceeding in this manner, we get eventually that a contradiction. We will be done once we show thatall belong toby an such that. But then by a similar argument as above, it follows from that(since we have already established that. Proceeding in this manner, we eventually obtain that, which clearly contradicts the last equation in the above equation array, namely that.

In this paper, we continue the study of the basic question when is surjective. (1) (1.1) Here P( D) is a partial differential operator with constant coefficients,is an open set andis the space of real analytic functions

011.

Solutions to this problem have been given mainly by two methods: Hormander (1973) has characterized (1) for convex open setsby means of a Phragmen-Lindelof condition valid on the complex variety of. His method has been adapted by several authors for further studies on this problem. Homiander s criterion is restricted to convex setsby the use of Fourier theory. On the other hand, Kawai (1972) used so-called "good elementary solutions" for P(D) to prove (1) for locally hyperbolic operators on special, not necessarily convex bounded open sets for the case of unbounded open sets and further results in the spirit of Kawai's work, and Andersson (1974) for). In Langenbruch (2004) we recently clarified the role of fundamental solutions for our problem, and we gave several characterizations of (1) by means of elementary solutions and also by conditions of type. In the present paper, a quantitative version of the latter characterization will be proved. Using this new condition and a result of Homiander (1973), we will show that P(D) is surjective on For convex, this is one of the main results of Homiander (1973). while the question had been open for general. Thus, surjectivity of P(D) onis a general necessary condition for (1). We also show that surjectivity of partial differential operators on real analytic functions is inherited

P(D) is surjective on for anythen P(D) is surjective onfor and for any but finitely many}. Also, if P(D) is surjective on, then P(D) is surjective 011for any where Next, we obtain the following result of Andreotti and Nacinovich (1980) and Zampieri (1984): For

A(H) for any tangent halfspace H of. We finally show an extension of this result for homogeneous operators P( D ) and open setswith-boundary: In this case. P(D) is surjective on A(conv()) and on A(H) for any tangent halfspace ofif (1) holds.

First we need a topological Paley-Wiener Theorem and in order to do so, we have to topologize the space. For this, introduce the spaceto be the space consisting of holomorphic mapssatisfying constant CN, and we topologize by this family of seminorms. This turns it into a Frechet space. The Weyl invariance still makes sense in this generalized setting, and thus we defineto be the subset of Weyl invariant elements. This is a closed subspace and hence a Frechet space. We have an obvious inclusionand this inclusion turns out to be surjective:

Proof. Fordefine Obviously,is a smooth function. By examining the proof of bijectivity of, it is seen thatis supported in the closed R-ball and thatalmost everywhere, and thusis a smooth representative of. Furthermoresatisfies the stronger growth condition and hence. Nowinherits the topology from(given by the seminorms ), and hence it becomes a Frechet space. Furthermore we define and give it the inductive limit topology.

stated and proved. Now we considerfor a given R. Forit is straightforward to check the inequality for each N: where D is the invariant differential operator (of order 2N) on X corresponding to the invariant polynomial and where C is a constant depending on N and R. Since and hence we see that Hence we get, whereis one of the standard semi norms on, i.e.is continuous. Thus the Fourier transform is a homeomorphism since these spaces are Frechet. Hence it is also a homeomorphism when defined on.

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