Superposition operators in the Lebesgue spaces and differentiability of quasiadditive set functions. (Russian) Zbl 1050.47031

Vodopyanov, S. K.; Ukhlov, A. D.

Vladikavkaz. Mat. Zh. 4, No. 1, 11-33 (2002).

Let \(D\) and \(\widetilde{D}\) be measurable sets on homogeneous metric spaces \(\mathbb X\) and \(\widetilde{\mathbb X}\), respectively. The authors describe bounded embedding operators \(\varphi^*\:L^p(\widetilde{D})\to L^q(D)\), where \(1\leq q\leq p<\infty\). In the case when \(q<p\), i.e., when \(\varphi^*\) lowers the degree of summability, quasiadditive set functions play an important role in the description of bounded \(\varphi^*\). An inequality is established for the integral of the upper derivative for \(q\)-quasiadditive set functions. As a corollary, the authors obtain a simple proof of the Lebesgue theorem on differentiability of the integral.

Reviewer: N. A. Kudryavtseva (Novosibirsk)

MSC:

47B38	Linear operators on function spaces (general)
46E35	Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems

Keywords:

homogeneous space; quasiadditive set function; embedding theorems