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.


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