Sharp norm

in the space of $ r $-dimensional polyhedral chains $ C _ {r} ( E ^ {n}) $

The largest semi-norm $ {| \cdot | } ^ \prime $ which, for any cell $ \sigma ^ {r} $ of volume $ | \sigma ^ {r} | $, satisfies the inequalities

$$ {| \sigma ^ {r} | } ^ \prime \leq | \sigma ^ {r} | , $$

$$ {| \partial \sigma ^ {r+1} | } ^ \prime \leq | \sigma ^ {r+1} | , $$

$$ {| T _ {v} \sigma ^ {r} - \sigma ^ {r} | } ^ \prime \leq \frac{ {| \sigma ^ {r} | } ^ \prime | v | }{r+1} , $$

where $ T _ {v} \sigma ^ {r} $ is the cell obtained by shifting by a vector $ v $ of length $ | v | $.

If $ A = \sum a _ {i} \sigma _ {i} ^ {r} $, the sharp norm $ | A | ^ \sharp $ is expressed as follows:

$$ | A | ^ \sharp = \inf \left \{ \frac{\sum | a _ {i} | | \sigma _ {i} ^ {r} | | v _ {i} | }{r+1} + \left | \sum a _ {i} T _ {v _ {i} } \sigma _ {i} ^ {r} \right | ^ \flat \right \} , $$

where $ | C | ^ \flat $ is the flat norm of the chain $ C $, and the infimum is taken over all shifts $ v $.

One has

$$ | a A | ^ \sharp = | a | | A | ^ \sharp , $$

$$ | A + B | ^ \sharp \leq | A | ^ \sharp + | B | ^ \sharp , $$

$$ | A | ^ \sharp = 0 \iff A = 0 , $$

$$ | A | ^ \sharp \leq | A | ^ \flat ; $$

if $ r = 0 $, then $ | A | ^ \sharp = | A | ^ \flat $.

The completion of the space $ C _ {r} ( E ^ {n} ) $ is the separable Banach space $ C _ {r} ^ \sharp ( E ^ {n} ) $, whose elements are known as $ r $-dimensional sharp chains. For any $ r $-dimensional polyhedral chain $ A $ and any vector $ v $,

$$ | T _ {v} A - A | ^ \sharp \leq \frac{| A | | v | }{r+1} , $$

where $ T _ {v} A $ is the chain obtained by shifting $ A $ by the vector $ v $ of length $ | v | $. A flat chain of finite mass is a sharp chain; in general, any flat chain may also be considered as a sharp chain in the following sense: If $ A = \lim\limits ^ \flat A _ {i} $, where $ A _ {i} $ are polyhedral chains, and $ \psi A = \lim\limits ^ \sharp A _ {i} $, where $ \psi $ is a linear bijective mapping from the space $ C _ {r} ^ \flat ( E ^ {n)} $ into the space $ C _ {r} ^ \sharp ( E ^ {n)} $, and $ \psi C _ {r} ^ \flat $ is dense in $ C _ {r} ^ \sharp $ in the sharp norm.

It is not possible to give a correct definition of the boundary $ \partial A $ of a sharp chain [1]; an $ r $-dimensional sharp chain $ X = XA $ is an element of the space $ C ^ {\sharp r } ( E ^ {n} ) $ dual to $ C _ {r} ^ \sharp ( E ^ {n} ) $; it is a flat cochain, and

$$ | X | \leq | X | ^ \flat \leq | X | ^ \sharp , $$

where $ | X | $ is the co-mass of $ X $, while the sharp co-norm $ | X | ^ \sharp $ is defined similarly to the flat norm $ | X | ^ \flat $. The co-boundary $ dX $ of a sharp cochain is not necessarily sharp [1], but

$$ | dX | \leq | X | ^ \flat \leq | X | ^ \sharp . $$

The Lipschitz constant $ {\mathcal L} ( X) $ of a cochain $ X $ is defined as follows:

$$ {\mathcal L} ( X) = \sup \left \{ \frac{| X \cdot ( T _ {v} A - A ) | }{| A | | v | } \right \} , $$

where the $ A $ are polyhedral chains. For sharp cochains this supremum is finite, and

$$ ( r + 1 ) {\mathcal L} ( X) \leq | X | ^ \sharp . $$

Any flat cochain with a finite Lipschitz constant is sharp, and

$$ | X | ^ \sharp = \sup \{ | X | ^ \flat ,\ ( r + 1 ) {\mathcal L} ( X) \} , $$

and also

$$ | dX | \leq ( r + 1 ) {\mathcal L} ( X) . $$

Similar concepts are introduced for $ r $-dimensional polyhedral chains in open subsets $ R \subset E ^ {n} $. See also Sharp form.

The sharp norm in the space of additive functions $ \gamma $ whose values are $ r $-vectors is the largest of the semi-norms $ | \cdot | ^ \prime $ which satisfy the conditions:

$ | \gamma | ^ \prime \leq | \gamma | $, where $ | \gamma | $ is the complete variation of $ \gamma $;

$$ | T _ {v} \gamma - \gamma | ^ \prime \leq \ \frac{| v | | \gamma | }{r+1} , $$

where $ T _ {v} \gamma ( Q) = \gamma T _ {-v} ( Q) $ is the shift of the function $ \gamma $ by the vector $ v $ of length $ | v | $:

$$ T _ {-v} ( Q) = \{ {q - v } : {q \in Q \subset E ^ {n} } \} ; $$

for each point $ p $ and an arbitrary $ \epsilon $ there exists an $ \eta > 0 $ such that $ | \gamma | ^ \prime \leq \epsilon | \gamma | $ if the support $ \supp t \gamma \subset U _ \eta ( p) $ and $ \gamma ( E ^ {n}) = 0 $.

The sharp norm $ | \gamma | ^ \sharp $ is represented as follows:

$$ | \gamma | ^ \sharp = \sup _ \omega \int\limits _ {E ^ {n} } \omega d \gamma , $$

where the $ \omega $ are $ r $-dimensional sharp forms for which $ | \omega | ^ \sharp \leq 1 $.

References

[1]	H. Whitney, "Geometric integration theory" , Princeton Univ. Press (1957)