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

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.

MSC:

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