Integral Representations of Functions of Classes L ^l_p (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 ^l_p (G). Inasmuch as these classes coincide with the Sobolev classes W ^l_p (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 ^l_p (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 ^l_p (G) and B ^l_p,θ (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 ^l_p (G) did not become a reality until Strichartz [3] came forth with a new norming of the spaces L ^l_p (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).