Complete characterization of functions which act, via superposition, on Sobolev spaces
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- by Moshe Marcus and Victor J. Mizel PDF
- Trans. Amer. Math. Soc. 251 (1979), 187-218 Request permission
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.References
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Additional Information
- © Copyright 1979 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 251 (1979), 187-218
- MSC: Primary 46E35; Secondary 47H15
- DOI: https://doi.org/10.1090/S0002-9947-1979-0531975-1
- MathSciNet review: 531975