New generalized functions and multiplication of distributions.

*(English)*Zbl 0532.46019
North-Holland Mathematics Studies, 84. Amsterdam-New York-Oxford: North- Holland. XII, 375 p. $ 38.50; Dfl. 100.00 (1984).

In quantum field theories one faces the problem to consider products of distributions. Several attempts had been made to overcome the difficulties of L. Schwartz’ impossibility theorem. In the C.R.-article under review the author describes shortly a fundamental construction and basic results that give a manageable theory adaptable to many situations in QFT. In his book he also discusses this fundamental construction and numerous modifications and applications to QFT.

Let \(\Omega\) be an open subset of \({\mathbb{R}}^ n\) and \(X={\mathcal D}(\Omega)\) the space of compactly supported test functions. Then X is a locally convex topological vector space. Therefore it makes sense to talk about the algebra \({\mathcal E}(X)\) of differentiable functions on X, that contains the space \({\mathcal D}'\) of distributions. So there is a natural starting point for a definition of products of distributions. But when viewed as distributions the product of functions should equal the product of the corresponding distributions. This condition could not be satisfied for all locally integrable functions but only for differentiable functions (elements of \({\mathcal E}(\Omega))\). Let A and B be two algebras and T a not necessarily multiplicative linear map from A to B. Then there is a minimal ideal \({\mathfrak k}\) in B such that the map from A to B/\({\mathfrak k}\) associated to T becomes multiplicative.

Look at the natural mapping \({\mathcal E}(\Omega)\to {\mathcal E}(X)\) which is linear, but not multiplicative. There is a minimal ideal \({\mathfrak k}\) in \({\mathcal E}(X)\) such that \({\mathcal E}(\Omega)\to {\mathcal E}(X)/{\mathfrak k}\) becomes multiplicative. But now it is hard to see what \({\mathfrak k}\) looks like. The author goes another way that gives control on the ideal which ”generates multiplicativity”.

The construction, which opens the way for analytical reasonings, depends on a choice of a nested family \({\mathfrak A}=(A_ q)_{q\in {\mathbb{N}}}\) of sets in \({\mathcal D}(\Omega)\) subject to specific conditions (not repeated here). The author defines a subalgebra \({\mathcal E}_ M\) of \({\mathcal E}(X)\) and an ideal \({\mathcal N}\) in \({\mathcal E}_ M\) in terms of \({\mathfrak A}\) such that the image of \({\mathcal E}(\Omega)\) in \({\mathcal E}(X)\) is part of \({\mathcal E}_ M\) and such that the mapping of \({\mathcal E}(\Omega)\) into \({\mathcal E}_ M/{\mathcal N}\) is injective and multiplicative. Furthermore the mapping of \({\mathcal D}'\) into \({\mathcal E}_ M/{\mathcal N}\) is injective, and thus a product of distributions is defined. This fundamental construction is modified by using other algebras instead of \({\mathcal E}(X)\) and other nested families in place of \({\mathfrak A}.\)

The elements of the arising subquotient algebras \({\mathcal G}.(\Omega)\) are called (new) generalized functions. For classical features as point values, differentiation, integration, and Fourier transform, the possibilities to extend these notions are discussed, and applications to partial differential equations are given.

Let \(\Omega\) be an open subset of \({\mathbb{R}}^ n\) and \(X={\mathcal D}(\Omega)\) the space of compactly supported test functions. Then X is a locally convex topological vector space. Therefore it makes sense to talk about the algebra \({\mathcal E}(X)\) of differentiable functions on X, that contains the space \({\mathcal D}'\) of distributions. So there is a natural starting point for a definition of products of distributions. But when viewed as distributions the product of functions should equal the product of the corresponding distributions. This condition could not be satisfied for all locally integrable functions but only for differentiable functions (elements of \({\mathcal E}(\Omega))\). Let A and B be two algebras and T a not necessarily multiplicative linear map from A to B. Then there is a minimal ideal \({\mathfrak k}\) in B such that the map from A to B/\({\mathfrak k}\) associated to T becomes multiplicative.

Look at the natural mapping \({\mathcal E}(\Omega)\to {\mathcal E}(X)\) which is linear, but not multiplicative. There is a minimal ideal \({\mathfrak k}\) in \({\mathcal E}(X)\) such that \({\mathcal E}(\Omega)\to {\mathcal E}(X)/{\mathfrak k}\) becomes multiplicative. But now it is hard to see what \({\mathfrak k}\) looks like. The author goes another way that gives control on the ideal which ”generates multiplicativity”.

The construction, which opens the way for analytical reasonings, depends on a choice of a nested family \({\mathfrak A}=(A_ q)_{q\in {\mathbb{N}}}\) of sets in \({\mathcal D}(\Omega)\) subject to specific conditions (not repeated here). The author defines a subalgebra \({\mathcal E}_ M\) of \({\mathcal E}(X)\) and an ideal \({\mathcal N}\) in \({\mathcal E}_ M\) in terms of \({\mathfrak A}\) such that the image of \({\mathcal E}(\Omega)\) in \({\mathcal E}(X)\) is part of \({\mathcal E}_ M\) and such that the mapping of \({\mathcal E}(\Omega)\) into \({\mathcal E}_ M/{\mathcal N}\) is injective and multiplicative. Furthermore the mapping of \({\mathcal D}'\) into \({\mathcal E}_ M/{\mathcal N}\) is injective, and thus a product of distributions is defined. This fundamental construction is modified by using other algebras instead of \({\mathcal E}(X)\) and other nested families in place of \({\mathfrak A}.\)

The elements of the arising subquotient algebras \({\mathcal G}.(\Omega)\) are called (new) generalized functions. For classical features as point values, differentiation, integration, and Fourier transform, the possibilities to extend these notions are discussed, and applications to partial differential equations are given.

Reviewer: W.Kugler

##### MSC:

46F10 | Operations with distributions and generalized functions |

46G20 | Infinite-dimensional holomorphy |

46-02 | Research exposition (monographs, survey articles) pertaining to functional analysis |

81T05 | Axiomatic quantum field theory; operator algebras |