Shift-Invariant Measures on Hilbert and Related Function Spaces
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Journal of Mathematical Sciences, Vol. 249, No. 6, September, 2020
SHIFT-INVARIANT MEASURES ON HILBERT AND RELATED FUNCTION SPACES V. M. Busovikov Moscow Institute of Physics and Technology 9, Institutskiy per., Dolgoprudny 141701, Moscow Region, Russia [email protected]
V. Zh. Sakbaev ∗ Moscow Institute of Physics and Technology 9, Institutskiy per., Dolgoprudny 141701, Moscow Region, Russia Institute of Mathematics, UFRC RAS 112, Chernyshevsky St., Ufa 450008, Russia Intitute of Information Technologies, Mathematics, and Mechanics Lobachevsky State University of Nizhny Novgorod 23, Gagarina Pr., Nizhny Novgorod 603950m Russia
UDC 517.98
[email protected]
We consider function spaces by analogy to Sobolev spaces and spaces of smooth functions on a finite-dimensional Euclidean space. For a real separable Hilbert space E we introduce the Hilbert space H of complex-valued functions on E that are square integrable with respect to some measure λ invariant under shifts and orthogonal transformations of E. For one-parameter semigroups of selfadjoint contractions in H we obtain the strong continuity criterion and study properties of their generators. For counterparts of Sobolev spaces and spaces of smooth functions we find necessary and sufficient embedding conditions and obtain the existence conditions for traces of functions in the Sobolev spaces on hyperplanes of codimension 1 in the space E. Bibliography: 13 titles.
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Introduction
To study shift-invariant measures in an infinite-dimensional real Hilbert space E, a nonnegative normalized finitely additive measure λ that is shift-invariant, but not countably additive on E was defined and studied in [1]. The theory of shift-invariant measures on infinite-dimensional linear spaces faces the nonexistence of a Lebesgue measure on an infinite-dimensional topological vector space in view of the Weyl theorem, i.e., on the σ-ring of Borel subsets of an infinite-dimensional topological vector ∗
To whom the correspondence should be addressed.
Translated from Problemy Matematicheskogo Analiza 103, 2020, pp. 43-62. c 2020 Springer Science+Business Media, LLC 1072-3374/20/2496-0864
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space, there exists no nonzero countably additive σ-finite locally finite measure that is invariant under shifts by vectors of this space (cf., for example, [2]). In connection with this fact, the question arises about the existence of measures defined on infinite-dimensional topological vector spaces and invariant under shifts by vectors of some maximal admissible subspace [3], measures that are invariant, but not σ-finite [4], and measures that are not countably additive [1]. In [3], a vector space and a Borel σ-finite measure on this space are defined in such a way that the measure is invariant under shifts by any elements of some infinite-dimensional subspace of the second category in itself. In [4], based on the Lebesgue–Carath´eodory extension procedure, the countably additive measure on R∞ is constructed from a set function defined on measurable blocks in R∞ . On the topological vector space R∞ equip
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