Quantitative functional calculus in Sobolev spaces
In the frame work of Sobolev (Bessel potential) spaces Hn(Rd,R or C), we consider the nonlinear Nemytskij operator sending a function x∈Rd↦f(x) into a composite function x∈Rd↦G(f(x),x). Assuming sufficient smoothness for G, we give a “tame” bound on the Hn norm of this composite function in terms of...
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| Main Authors: | , |
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| Format: | Article |
| Language: | English |
| Published: |
Wiley
2004-01-01
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| Series: | Journal of Function Spaces and Applications |
| Online Access: | http://dx.doi.org/10.1155/2004/832750 |
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| Summary: | In the frame work of Sobolev (Bessel potential) spaces Hn(Rd,R or C), we consider the nonlinear Nemytskij operator sending a function x∈Rd↦f(x) into a composite function x∈Rd↦G(f(x),x). Assuming sufficient smoothness for G, we give a “tame” bound on the Hn norm of this composite function in terms of a linear function of the Hn norm of f, with a coefficient depending on G and on the Ha norm of f, for all integers n,a,d with a>d/2. In comparison with previous results on this subject, our bound is fully explicit, allowing to estimate quantitatively the Hn norm of the function x↦G(f(x),x). When applied to the case G(f(x),x)=f2(x), this bound agrees with a previous result of ours on the pointwise product of functions in Sobolev spaces. |
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| ISSN: | 0972-6802 |