Sharp form

An $ r $- dimensional differential form $ \omega $ on an open subset $ R \subset E ^ {n} $ such that the co-mass (cf. Mass and co-mass) $ | \omega | _ {0} $ and the co-mass Lipschitz constant

$$ {\mathcal L} _ {0} ( \omega ) = \sup \frac{| \omega ( p) - \omega ( q) | }{| p - q | } , $$

where $ p , q \in R $ and $ | p - q | $ is the length of the vector $ p - q $, are finite. The number

$$ | \omega | ^ \sharp = \sup \{ | \omega | _ {0} ,\ ( r + 1 ) {\mathcal L} _ {0} ( \omega ) \} $$

is known as the sharp norm of the form $ \omega $.

Whitney's theorem. To each $ r $- dimensional sharp cochain $ X $ in $ R $ corresponds a unique $ r $- dimensional sharp form $ \omega _ {X} $ for which

$$ X \sigma ^ {r} = \int\limits _ {\sigma ^ {r} } \omega _ {X} $$

for all $ r $- dimensional oriented simplices $ \sigma ^ {r} $; $ \omega _ {X} ( p) $ is defined by the formula

$$ \omega _ {X} ( p) = \lim\limits \frac{X \sigma _ {i} }{| \sigma _ {i} | } , $$

where $ \sigma _ {1} , \sigma _ {2} \dots $ is a sequence of simplices containing the point $ p $, with diameters tending to zero, and lying in the same plane. This correspondence is a one-to-one mapping of the space of cochains $ C ^ {\sharp r } ( R) $ into the space $ \Omega ^ {\sharp r } $ of sharp forms; moreover:

$ | \omega _ {X} | _ {0} = | X | $, i.e. the co-mass of $ X $;

$ {\mathcal L} ( \omega _ {X} ) = {\mathcal L} ( X) $, i.e. the Lipschitz constant of $ X $;

$ | \omega _ {X} | ^ \sharp = | X | ^ \sharp $, i.e. the sharp norm of $ X $;

$ \Omega ^ {\sharp r } $ is a Banach space.

In particular, there is a correspondence between zero-dimensional sharp cochains and sharp functions (bounded functions which satisfy a Lipschitz condition).

The space $ C _ {r} ^ \sharp ( R) $ of $ r $- dimensional sharp chains $ A $ of finite mass $ | A | $ with the sharp norm $ | A | ^ \sharp $ is isomorphic to the space $ \Gamma _ {r} ^ \sharp ( E ^ {n} ) $ of additive set functions whose values are $ r $- vectors $ \gamma $, provided with the sharp norm $ | \gamma | ^ \sharp $; this correspondence is defined by the formula:

$$ \tag{* } X A = \int\limits _ {E ^ {n} } \omega _ {X} d {\gamma _ {A} } = [ \omega \cdot \gamma ] ( E ^ {n} ) $$

for any cochain $ X $, where $ \omega _ {X} $ is the $ r $- dimensional sharp form corresponding to the cochain $ X $, and

$ \gamma _ {A} ( E ^ {n} ) = \{ A \} $, i.e. the covector of the chain $ A $;

$ | A | = | \gamma _ {A} | $, i.e. the complete variation of $ \gamma _ {A} $;

$ | \gamma _ {A} | ^ \sharp = | A | ^ \sharp $, i.e. the sharp norm of the chain $ A $.

Thus, (*) is a generalization of the ordinary Lebesgue–Stieltjes integral. In particular, the Lebesgue-measurable summable function $ \alpha ( p) $ associated with $ A $( cf. Flat form), i.e.

$$ X \cdot A = \int\limits _ {E ^ {n} } \omega _ {X} \cdot \alpha ( p) dp $$

for any cochain $ X $, exists for $ A $ if and only if $ \gamma _ {A} $ is absolutely continuous.

If $ \omega _ {A} $ is a regular form and $ X $ is a sharp cochain, then there exists a form $ \omega _ {dX } = d \omega _ {X} $, and Stokes' formula

$$ \int\limits _ {\partial \sigma } \omega _ {X} = \int\limits _ \sigma d \omega $$

applies. Other results established for regular forms can be generalized in an analogous manner.

For references see Sharp norm.