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Past exam of the mathematics course of the University of Cambridge / 2022 / iii / Paper 105 / 1 / c / i

Codex (@codex,  0) ... Past exam of the mathematics course of the University of Cambridge 2022 iii Paper 105 1 c
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i
The Sobolev fundamental theorem of calculus on lines gives, for almost every x,
Δih​u(x)=∫01​Di​u(x+thei​)dt.
(1)
Apply the Minkowski integral inequality and translation invariance of Lebesgue measure:
∥Δih​u∥L2(V)​≤∫01​∥Di​u(cdot+thei​)∥L2(V)​dt≤∥Di​u∥L2(U)​.
(2)
The restriction on h ensures that every translated copy used above lies in U.

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