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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

Complete characterization of functions which act, via superposition, on Sobolev spaces


Authors: Moshe Marcus and Victor J. Mizel
Journal: Trans. Amer. Math. Soc. 251 (1979), 187-218
MSC: Primary 46E35; Secondary 47H15
MathSciNet review: 531975
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Abstract: Given a domain $ \Omega \subset {R_N}$ and a Borel function $ h:\,{R_m} \to R$, conditions on h are sought ensuring that for every m-tuple of functions $ {u_i}$ belonging to the first order Sobolev space $ {W^{1,p}}(\Omega )$, the function $ h({u_1}( \cdot ), \ldots ,{u_m}( \cdot ))$ will belong to a first order Sobolev space $ {W^{1,r}}(\Omega )$, $ 1 \leqslant r \leqslant p < \infty $.In this paper conditions are found which are both necessary and sufficient in order that h have the above property. This result is based on a characterization obtained here for those Borel functions $ g:\,{R_m} \times {({R_N})_m} \to R$ satisfying the requirement that for every m-tuple of functions $ {u_i} \in {W^{1,p}}(\Omega )$ the function $ g({u_1}( \cdot ), \ldots ,{u_m}( \cdot ),\nabla {u_1}( \cdot ), \ldots ,\nabla {u_m}( \cdot ))$ belongs to $ {L^r}(\Omega )$. A needed result on the measurability of the set of $ {R_k}$-Lebesgue points of a function on $ {R_N}$ is presented in an appendix.


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Additional Information

DOI: http://dx.doi.org/10.1090/S0002-9947-1979-0531975-1
PII: S 0002-9947(1979)0531975-1
Keywords: Locally Lipschitz function, cone condition, k-dimensional Hausdorff measure, regular Lebesgue point
Article copyright: © Copyright 1979 American Mathematical Society