Integral Representations of Functions of Classes Lpl(G) and Embedding Theorems
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 ) 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  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 , accommodating it to the anisotropic case (vectorial l).
Unable to display preview. Download preview PDF.
- 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