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An <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084810/s0848101.png" />-dimensional [[Differential form|differential form]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084810/s0848102.png" /> on an open subset <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084810/s0848103.png" /> such that the co-mass (cf. [[Mass and co-mass|Mass and co-mass]]) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084810/s0848104.png" /> and the co-mass Lipschitz constant
+
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<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084810/s0848105.png" /></td> </tr></table>
+
{{TEX|auto}}
 +
{{TEX|done}}
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084810/s0848106.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084810/s0848107.png" /> is the length of the vector <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084810/s0848108.png" />, are finite. The number
+
An  $  r $-
 +
dimensional [[Differential form|differential form]]  $  \omega $
 +
on an open subset  $  R \subset  E  ^ {n} $
 +
such that the co-mass (cf. [[Mass and co-mass|Mass and co-mass]])  $  | \omega | _ {0} $
 +
and the co-mass Lipschitz constant
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084810/s0848109.png" /></td> </tr></table>
+
$$
 +
{\mathcal L} _ {0} ( \omega )  = \sup 
 +
\frac{| \omega ( p) - \omega
 +
( q) | }{| p - q | }
 +
,
 +
$$
  
is known as the sharp norm of the form <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084810/s08481010.png" />.
+
where  $  p , q \in R $
 +
and  $  | p - q | $
 +
is the length of the vector  $  p - q $,
 +
are finite. The number
  
Whitney's theorem. To each <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084810/s08481011.png" />-dimensional sharp cochain <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084810/s08481012.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084810/s08481013.png" /> corresponds a unique <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084810/s08481014.png" />-dimensional sharp form <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084810/s08481015.png" /> for which
+
$$
 +
| \omega |  ^  \sharp   = \sup \{ | \omega | _ {0} ,\
 +
( r + 1 ) {\mathcal L} _ {0} ( \omega ) \}
 +
$$
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084810/s08481016.png" /></td> </tr></table>
+
is known as the sharp norm of the form  $  \omega $.
  
for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084810/s08481017.png" />-dimensional oriented simplices <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084810/s08481018.png" />; <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084810/s08481019.png" /> is defined by the formula
+
Whitney's theorem. To each  $  r $-
 +
dimensional sharp cochain  $  X $
 +
in  $  R $
 +
corresponds a unique  $  r $-
 +
dimensional sharp form  $  \omega _ {X} $
 +
for which
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084810/s08481020.png" /></td> </tr></table>
+
$$
 +
X \sigma  ^ {r}  = \int\limits _ {\sigma  ^ {r} } \omega _ {X}  $$
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084810/s08481021.png" /> is a sequence of simplices containing the point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084810/s08481022.png" />, with diameters tending to zero, and lying in the same plane. This correspondence is a one-to-one mapping of the space of cochains <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084810/s08481023.png" /> into the space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084810/s08481024.png" /> of sharp forms; moreover:
+
for all  $  r $-
 +
dimensional oriented simplices $  \sigma  ^ {r} $;
 +
$  \omega _ {X} ( p) $
 +
is defined by the formula
  
<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084810/s08481025.png" />, i.e. the co-mass of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084810/s08481026.png" />;
+
$$
 +
\omega _ {X} ( p)  = \lim\limits 
 +
\frac{X \sigma _ {i} }{| \sigma _ {i} | }
 +
,
 +
$$
  
<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084810/s08481027.png" />, i.e. the Lipschitz constant of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084810/s08481028.png" />;
+
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:
  
<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084810/s08481029.png" />, i.e. the [[Sharp norm|sharp norm]] of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084810/s08481030.png" />;
+
$  | \omega _ {X} | _ {0} = | X | $,  
 +
i.e. the co-mass of $  X $;
  
<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084810/s08481031.png" /> is a Banach space.
+
$  {\mathcal L} ( \omega _ {X} ) = {\mathcal L} ( X) $,
 +
i.e. the Lipschitz constant of  $  X $;
 +
 
 +
$  | \omega _ {X} |  ^  \sharp  = | X |  ^  \sharp  $,
 +
i.e. the [[Sharp norm|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).
 
In particular, there is a correspondence between zero-dimensional sharp cochains and sharp functions (bounded functions which satisfy a Lipschitz condition).
  
The space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084810/s08481032.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084810/s08481033.png" />-dimensional sharp chains <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084810/s08481034.png" /> of finite [[Mass|mass]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084810/s08481035.png" /> with the sharp norm <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084810/s08481036.png" /> is isomorphic to the space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084810/s08481037.png" /> of additive set functions whose values are <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084810/s08481039.png" />-vectors <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084810/s08481040.png" />, provided with the sharp norm <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084810/s08481041.png" />; this correspondence is defined by the formula:
+
The space $  C _ {r}  ^  \sharp  ( R) $
 +
of $  r $-
 +
dimensional sharp chains $  A $
 +
of finite [[Mass|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:
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084810/s08481042.png" /></td> <td valign="top" style="width:5%;text-align:right;">(*)</td></tr></table>
+
$$ \tag{* }
 +
X A  = \int\limits _ {E  ^ {n} } \omega _ {X}  d {\gamma _ {A} }
 +
= [ \omega \cdot \gamma ] ( E  ^ {n} )
 +
$$
  
for any cochain <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084810/s08481043.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084810/s08481044.png" /> is the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084810/s08481045.png" />-dimensional sharp form corresponding to the cochain <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084810/s08481046.png" />, and
+
for any cochain $  X $,  
 +
where $  \omega _ {X} $
 +
is the $  r $-
 +
dimensional sharp form corresponding to the cochain $  X $,  
 +
and
  
<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084810/s08481047.png" />, i.e. the covector of the chain <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084810/s08481048.png" />;
+
$  \gamma _ {A} ( E  ^ {n} ) = \{ A \} $,  
 +
i.e. the covector of the chain $  A $;
  
<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084810/s08481049.png" />, i.e. the complete variation of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084810/s08481050.png" />;
+
$  | A | = | \gamma _ {A} | $,  
 +
i.e. the complete variation of $  \gamma _ {A} $;
  
<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084810/s08481051.png" />, i.e. the sharp norm of the chain <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084810/s08481052.png" />.
+
$  | \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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084810/s08481053.png" /> associated with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084810/s08481054.png" /> (cf. [[Flat form|Flat form]]), i.e.
+
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|Flat form]]), i.e.
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084810/s08481055.png" /></td> </tr></table>
+
$$
 +
X \cdot A  = \int\limits _ {E  ^ {n} } \omega _ {X} \cdot \alpha ( p)  dp
 +
$$
  
for any cochain <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084810/s08481056.png" />, exists for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084810/s08481057.png" /> if and only if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084810/s08481058.png" /> is absolutely continuous.
+
for any cochain $  X $,  
 +
exists for $  A $
 +
if and only if $  \gamma _ {A} $
 +
is absolutely continuous.
  
If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084810/s08481059.png" /> is a regular form and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084810/s08481060.png" /> is a sharp cochain, then there exists a form <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084810/s08481061.png" />, and Stokes' formula
+
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
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084810/s08481062.png" /></td> </tr></table>
+
$$
 +
\int\limits _ {\partial  \sigma } \omega _ {X}  = \int\limits _  \sigma  d \omega
 +
$$
  
 
applies. Other results established for regular forms can be generalized in an analogous manner.
 
applies. Other results established for regular forms can be generalized in an analogous manner.
  
 
For references see [[Sharp norm|Sharp norm]].
 
For references see [[Sharp norm|Sharp norm]].

Latest revision as of 16:26, 22 February 2021


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.

How to Cite This Entry:
Sharp form. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Sharp_form&oldid=18828
This article was adapted from an original article by M.I. Voitsekhovskii (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article