Functions with Generalized Derivatives

In this chapter, the classes of functions W p l (U) introduced by Sobolev are studied. The investigation of these classes is based on certain integral representations of a function by its derivatives of order l. Deducing such representations, we use the properties of linear differential operators satisfying the complete integrability condition. In Section 1, integral representations of the Sobolev type are constructed for starlike domains. Section 2 studies integral representations of the same type for a wider class of domains. They are used to study the classes W p l (U). First, the representations of local character are constructed. The final form of the representations we are interested in is obtained in Section 4. At the same time, we establish in Section 2 some properties of the basic class of domains—the class J, for which the properties of the functional spaces of W p l (U) are investigated. In Section 3, estimates are established for integrals of the potential type, which are later used in Section 4 to establish the relations between the classes W p l (U) and the classes L q (U) and C(U). Section 4 studies the classes W p l (U). In Section 5 we give the general theorem about differentiability almost everywhere of functions of the classes W p l (U) and some of its corollaries. Besides, some theorems about the behaviour of functions of the classes W p l on almost all k-dimensional planes of the space R n for l ≤ k ≤ n are proved.