Chain rule for pointwise Lipschitz mappings
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866909585873305600 |
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| author | Kolář, Jan Maleva, Olga |
| author_facet | Kolář, Jan Maleva, Olga |
| contents | The classical Chain Rule formula $(f\circ g)'(x;u)=f'(g(x);g'(x;u))$ gives the (partial, or directional) derivative of the composition of mappings $f$ and $g$. We show how to get rid of the unnecessarily strong assumption of differentiability at all of the relevant points: the mappings do not need to be defined on the whole space and it is enough for them to be pointwise Lipschitz. The price to pay is that the Chain Rule holds almost everywhere. We extend this construction to infinite-dimensional spaces with good properties (Banach, separable, Radon-Nikodým) with an appropriate notion of almost everywhere. Pointwise Lipschitzness is a local condition in contrast to the global Lipschitz property: we do not need the mappings to be defined on the whole space, or even locally in a neighbourhood, nor to know their behaviour far away from the points we consider. This distinguishes our results from recent research on the differentiation of the composition of Lipschitz mappings. The methods we develop for the purpose of proving the Chain Rule also allow us to strengthen the Rademacher-Stepanov type theorem on almost everywhere differentiability of a mapping. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2504_14385 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Chain rule for pointwise Lipschitz mappings Kolář, Jan Maleva, Olga Functional Analysis Primary 46G05, Secondary 26B30, 26B05, 26E15, 58C20 The classical Chain Rule formula $(f\circ g)'(x;u)=f'(g(x);g'(x;u))$ gives the (partial, or directional) derivative of the composition of mappings $f$ and $g$. We show how to get rid of the unnecessarily strong assumption of differentiability at all of the relevant points: the mappings do not need to be defined on the whole space and it is enough for them to be pointwise Lipschitz. The price to pay is that the Chain Rule holds almost everywhere. We extend this construction to infinite-dimensional spaces with good properties (Banach, separable, Radon-Nikodým) with an appropriate notion of almost everywhere. Pointwise Lipschitzness is a local condition in contrast to the global Lipschitz property: we do not need the mappings to be defined on the whole space, or even locally in a neighbourhood, nor to know their behaviour far away from the points we consider. This distinguishes our results from recent research on the differentiation of the composition of Lipschitz mappings. The methods we develop for the purpose of proving the Chain Rule also allow us to strengthen the Rademacher-Stepanov type theorem on almost everywhere differentiability of a mapping. |
| title | Chain rule for pointwise Lipschitz mappings |
| topic | Functional Analysis Primary 46G05, Secondary 26B30, 26B05, 26E15, 58C20 |
| url | https://arxiv.org/abs/2504.14385 |