# Integral Representations of Functions of Classes L_{p}^{l}(G) and Embedding Theorems

## Abstract

The principal objective of the present article is to obtain integral representations of functions adapted to any of the various normed spaces (classes) L _{p} ^{l} (G). Inasmuch as these classes coincide with the Sobolev classes W _{p} ^{l} (G) (see [1]) for integral value of the index l, the resulting representations may be regarded as generalizations in a certain direction of the well-known integral representations of functions of the classes W _{p} ^{l} (G). They enable one to investigate the indicated classes of functions in domains satisfying the so-called horn (cone) condition, i.e., in domains of the same type as those in which functions of the classes W _{p} ^{l} (G) and B _{p,θ} ^{l} (G) have been investigated (see [1, 2]). It is essential to point out that the admissibility of using such representations in the theory of classes L _{p} ^{l} (G) did not become a reality until Strichartz [3] came forth with a new norming of the spaces L _{p} ^{l} (in the case of noninteger-valued l) equivalent to the one used previously. In the discussion that follows we shall abide by the norming given in [3], accommodating it to the anisotropic case (vectorial l).

## Preview

Unable to display preview. Download preview PDF.

## Literature Cited

- 1.Sobolev, S. L., Some Applications of Functional Analysis in Mathematical Physics, Izd. LGU, Leningrad (1950).Google Scholar
- 2.Besov, O. V., and Il’in, V. P., A natural extension of the class of domains in embedding theorems, Matem. Sborn., 75(117) (4): 483–495 (1968).Google Scholar
- 3.Strichartz, R. S., Multipliers on fractional Sobolev spaces, J. Math. Mech., 16 (9): 1031–1060 (1967).Google Scholar
- 4.Lizorkin, P. I., Nonisotropic Bessel potentials; embedding theorems for Sobolev spaces with fractional derivatives, Dokl. Akad. Nauk SSSR, 170 (2): 508–511 (1966).Google Scholar
- 5.Lizorkin, P. I., Generalized Liouville differentiation and the method of multipliers in embedding theory, Trudy Matem. Inst. Steklov, 105 (1969).Google Scholar
- 6.Zygmund, A., Trigonometric Series, Vol. 2, Cambridge Univ. Press (1959).Google Scholar