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On the Gleason problem
Lemmers, F.A.M.O.
Publication date
2002
Link to publication
Citation for published version (APA):
Lemmers, F. A. M. O. (2002). On the Gleason problem.
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Importantt notation
Heree we give some important notation. For other terms, we refer to the index. The bookk of Krantz ([36]) is also a good reference manual.
Inn this thesis, ft shall always be a domain in Cn, that is, a connected open set. Forr a set D in Rn (or Cn) we denote its boundary by dD, whereas its convex hull is denotedd by Co(D).
Wee will deal with several spaces of functions on ft (or ft), namely :
ƒƒ is k times continuously differentiable on ft}, ƒƒ is continuous on SI},
ƒƒ is bounded on ft}, ƒƒ is holomorphic on ft},
ƒƒ is bounded and holomorphic on ft}, ƒƒ is holomorphic on ft and continuous on ft}, {ƒƒ € H(ü) : Daƒ € C(ft) 0 < \a\ < m).
Lett U C Rn be an open set. Then C1+€(U) contains those ƒ € Cl(U) for which DDaaƒƒ{x{x + h)- Daf(x) CCkk(Q) (Q) C(ft) ) L°°(ft) ) if(ft) ) #°°(ft) ) A(ft) ) Am(ft) ) == {ƒ
==
if
==
if
==
{/
== {/
==
if
==
if
YlYl HA>/iiL-(tf)+ E
SUP
,, ,_1 x,x+h€U \a\a <1 << +CX)}. Aa(C^)) := {ƒ € C(tf) : s u p ^ ^ ^ \f(x + h) - f{x)\f\h\" + ||/||Lco( £ / ) < oo}.Withh d(-, ) we shall denote the Euclidean distance.
Thee open ball with center a and radius r is denoted by B(a,r). Thee supremum of | ƒ | on a set U is denoted by | | / | | L « = ( I / ) , or simply by iff it is clear over which set the supremum is being taken.
Forr a (p, g)-form A, d\ := £ "= 1 E|a|=P,|/3|=9 % f <% A <^a A < ^
-or r
Thee map L : (z, w) t—> (log jzj,log|w;|) maps a Reinhardt domain ft C C2 to its logarithmicc image UJ C M2. 5(ft) denotes the C2 strictly pseudoconvex boundary pointss of ft, whereas S{uS) denotes the C2 strictly convex boundary points of u.