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Notes on integral transformations. (English) Zbl 0558.44001

The contents of this interesting work is organized as follows: Section 1 is an introduction to the topic of study. Section 2 is devoted to preliminaries and to some results about spaces of measurable functions, needed in the sequel. In Section 3 the author discusses the proper (or natural) domain \(D_ k\) of an integral transformation K with its natural topology. It is perhaps of interest that the graph topology of K on \(D_ k\) is not a suitable one, however, the graph topology of the sublinear transformation \(u\to \int | k(x,y)|\) \(| u(y)| dy\) seems to be appropriate to study questions of continuity. In section 4 the author considers various aspects of K as an unbounded transformation from \(L^ 0(Y)\) into \(L^ 0(X)\). In section 5 the author deals with extensions by continuity of integral transformations to topological spaces of measurable functions. An example of the situation considered here is the Fourier transform in \(L^ 2\). In this context the category of solid spaces seems to be particularly suitable - in this category there exists for every integral transformation K with a nontrivial domain, a maximal (in a suitable sense) solid space to which K can be extended by continuity. It is impossible to abandon entirely the hypothesis of solidity without imposing unreasonable restrictions on the kernel k. In section 6 the author presents some mostly known results about compactness of integral transformations; some effort is made here to avoid imposing restrictions on the absolute values of the kernel k. In section 7 the author collects various results that did not fit into the preceding sections yet seemed necessary to give a rounded up description of the theory. Section 8 is devoted to bibliographical remarks.
One of the conclusions that could be drawn from the present study is that it is expedient to study integral transformations in the context of spaces which are a priori not assumed to be locally convex (for instance \(L^ 0(X)\) is not locally convex unless X is purely atomic). Indeed, it is not known if the natural domain \(D_ k\) and the extended domain \(D_ k\) (see Section 5) are locally convex in general, even though this is the case in many concrete examples. Another conclusion is that even though the theory of integral transformations is relatively painless in the category of solid spaces, it would be of considerable interest to relax the latter hypothesis perhaps at the expense of imposition of some reasonable restrictions of K. Very little is known about unbounded integral transformations (e.g. in \(L^ 2)\) and part of the difficulties lies in the fact that their domains are not in general solid.
Reviewer: K.C.Gupta

MSC:

44-02 Research exposition (monographs, survey articles) pertaining to integral transforms
44A05 General integral transforms
42A38 Fourier and Fourier-Stieltjes transforms and other transforms of Fourier type