Integrability of Functions in the Space $L^{1}_{p}(\varOmega )$

Abstract

Here we continue our study of integral inequalities for functions with unrestricted boundary values started in the previous chapter. For the embedding operator $L^{1}_{p}(\varOmega) \to L_{q}(\varOmega)$, p≥1, we find necessary and sufficient conditions on Ω ensuring the continuity of this operator (Sects. 6.2–6.4). To get criteria, analogous to those obtained in Sect. 2.2, for the space $L^{1}_{p}(\varOmega)$, we introduce classes of sets defined with the aid of the so-called p-conductivity, which plays the same role as p-capacity in Chap. 2. Geometrical conditions formulated in terms of the isoperimetric inequalities prove to be only sufficient if p>1.