CR holomorphic functions
$begingroup$
Let $Omega subset mathbb{C}$ be a domain, $mathcal{O}(Omega)$ denote holomorphic functions on $Omega$ and $mathcal{C}^{infty}(overline{Omega})$ functions smooth up to the boundary.
I'm struggling with the following isomorphism:
$${mbox{CR hol. functions on } partialOmega =: M} cong mathcal{O}left(Omegaright) ;cap; mathcal{C}^{infty}(overline{Omega}).$$
I've noticed that one inlcusion follows from the Hartogs theorem, but I have no idea how to show the other part of the statement.
differential-geometry complex-geometry holomorphic-functions complex-manifolds
$endgroup$
add a comment |
$begingroup$
Let $Omega subset mathbb{C}$ be a domain, $mathcal{O}(Omega)$ denote holomorphic functions on $Omega$ and $mathcal{C}^{infty}(overline{Omega})$ functions smooth up to the boundary.
I'm struggling with the following isomorphism:
$${mbox{CR hol. functions on } partialOmega =: M} cong mathcal{O}left(Omegaright) ;cap; mathcal{C}^{infty}(overline{Omega}).$$
I've noticed that one inlcusion follows from the Hartogs theorem, but I have no idea how to show the other part of the statement.
differential-geometry complex-geometry holomorphic-functions complex-manifolds
$endgroup$
$begingroup$
What means ${mbox{CR hol. functions on } partialOmega =: M}$ ? With $D$ the unit disk then $mathcal{O}(D)cap C^infty(overline{D}) ={ sum_{n=0}^infty a_n z^n, forall k, sum_{n=0}^infty |a_n n^k| < infty}= { int_{partial D} frac{h(s)}{s-z}ds, hin C^infty(partial D)}$. Something similar holds for $partial Omega$ smooth and of finite length
$endgroup$
– reuns
Jan 8 at 8:50
add a comment |
$begingroup$
Let $Omega subset mathbb{C}$ be a domain, $mathcal{O}(Omega)$ denote holomorphic functions on $Omega$ and $mathcal{C}^{infty}(overline{Omega})$ functions smooth up to the boundary.
I'm struggling with the following isomorphism:
$${mbox{CR hol. functions on } partialOmega =: M} cong mathcal{O}left(Omegaright) ;cap; mathcal{C}^{infty}(overline{Omega}).$$
I've noticed that one inlcusion follows from the Hartogs theorem, but I have no idea how to show the other part of the statement.
differential-geometry complex-geometry holomorphic-functions complex-manifolds
$endgroup$
Let $Omega subset mathbb{C}$ be a domain, $mathcal{O}(Omega)$ denote holomorphic functions on $Omega$ and $mathcal{C}^{infty}(overline{Omega})$ functions smooth up to the boundary.
I'm struggling with the following isomorphism:
$${mbox{CR hol. functions on } partialOmega =: M} cong mathcal{O}left(Omegaright) ;cap; mathcal{C}^{infty}(overline{Omega}).$$
I've noticed that one inlcusion follows from the Hartogs theorem, but I have no idea how to show the other part of the statement.
differential-geometry complex-geometry holomorphic-functions complex-manifolds
differential-geometry complex-geometry holomorphic-functions complex-manifolds
asked Jan 8 at 4:57
user626292
$begingroup$
What means ${mbox{CR hol. functions on } partialOmega =: M}$ ? With $D$ the unit disk then $mathcal{O}(D)cap C^infty(overline{D}) ={ sum_{n=0}^infty a_n z^n, forall k, sum_{n=0}^infty |a_n n^k| < infty}= { int_{partial D} frac{h(s)}{s-z}ds, hin C^infty(partial D)}$. Something similar holds for $partial Omega$ smooth and of finite length
$endgroup$
– reuns
Jan 8 at 8:50
add a comment |
$begingroup$
What means ${mbox{CR hol. functions on } partialOmega =: M}$ ? With $D$ the unit disk then $mathcal{O}(D)cap C^infty(overline{D}) ={ sum_{n=0}^infty a_n z^n, forall k, sum_{n=0}^infty |a_n n^k| < infty}= { int_{partial D} frac{h(s)}{s-z}ds, hin C^infty(partial D)}$. Something similar holds for $partial Omega$ smooth and of finite length
$endgroup$
– reuns
Jan 8 at 8:50
$begingroup$
What means ${mbox{CR hol. functions on } partialOmega =: M}$ ? With $D$ the unit disk then $mathcal{O}(D)cap C^infty(overline{D}) ={ sum_{n=0}^infty a_n z^n, forall k, sum_{n=0}^infty |a_n n^k| < infty}= { int_{partial D} frac{h(s)}{s-z}ds, hin C^infty(partial D)}$. Something similar holds for $partial Omega$ smooth and of finite length
$endgroup$
– reuns
Jan 8 at 8:50
$begingroup$
What means ${mbox{CR hol. functions on } partialOmega =: M}$ ? With $D$ the unit disk then $mathcal{O}(D)cap C^infty(overline{D}) ={ sum_{n=0}^infty a_n z^n, forall k, sum_{n=0}^infty |a_n n^k| < infty}= { int_{partial D} frac{h(s)}{s-z}ds, hin C^infty(partial D)}$. Something similar holds for $partial Omega$ smooth and of finite length
$endgroup$
– reuns
Jan 8 at 8:50
add a comment |
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$begingroup$
What means ${mbox{CR hol. functions on } partialOmega =: M}$ ? With $D$ the unit disk then $mathcal{O}(D)cap C^infty(overline{D}) ={ sum_{n=0}^infty a_n z^n, forall k, sum_{n=0}^infty |a_n n^k| < infty}= { int_{partial D} frac{h(s)}{s-z}ds, hin C^infty(partial D)}$. Something similar holds for $partial Omega$ smooth and of finite length
$endgroup$
– reuns
Jan 8 at 8:50