Гладкие меры в бесконечномерных линейных пространствах
The branch of analysis, connected with the study of functions of an infinite-dimensional argument, has been recently intensively developing. This is due both to internal causes and to applications to theoretical and mathematical physics. It is well known which role in classical mathematical physics is played by distributions — linear continuous functionals on smooth basic functions. Therefore, it is natural to develop such a theory in infinite-dimensional spaces. In the infinite-dimensional case, in view of the absence of a standard measure of Lebesgue measure type, there are two ways of constructing a theory of generalized functions, each of which has its merits and demerits. One of them is that in an infinite-dimensional space a measure is fixed that has sufficiently good properties, by means of which a pairing between the basic and generalized functions is carried out. Very often, for these purposes, a Gaussian measure proves to be convenient. Such a theory of generalized functions was developed in the works of Yu. M. Berezansky, Yu. G. Kondratiev, Yu. S. Samoylenko. On the other hand, one can follow the path proposed by S. V. Fomin and developed in the works of V. I. Averbukh, O. G. Smolyanov, S. V. Fomin, Yu. L. Daletski˘i, and others. This way is that distributions are considered as generalized measures. In particular, the distributions include the usual countably additive measures, but do not include functions. Therefore, it becomes necessary to construct an analysis of measures parallel to the analysis of functions.
However, in all these papers the differentiation of measures with respect to constant directions was considered. In the transition to non-linear manifolds, the concept of a constant direction loses its meaning. Therefore, it is necessary to construct a theory of differentiating measures along vector fields. In this paper we introduce and study the notion of a derivative of a measure along a finite collection of vector fields. The basic properties of the derivative of a smooth measure are established, the form of the logarithmic derivative is specified, the law of transformation of these objects is found under a smooth invertible mapping.
Keywords: smooth measure, distribution, differentiation of a measure, derivative of a measure along vector field, logarithmic derivative of a measure.