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The Donnelly-Fefferman theorem on $q$-pseudoconvex domains

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  • Mathematics


Ahn, H. and Dieu, N.Q. Osaka J. Math. 46 (2009), 599–610 THE DONNELLY-FEFFERMAN THEOREM ON q-PSEUDOCONVEX DOMAINS HEUNGJU AHN and NGUYEN QUANG DIEU� (Received March 19, 2008) Abstract In this paper we introduce a notion of q-subharmonicity for non-smooth functions and then using q-subharmonic exhaustion function, define a q-pseudoconvexity which is applicable to the domain with non-smooth boundary. Among others, we generalize the Donnelly-Fefferamn type theorem on q-pseudoconvex domains and as an application of this theorem, we give approximation theorem for �-closed forms. 1. q-subharmonic functions and q-pseudoconvex domains For a real valued C2 function ' defined on U � Cn , Lop-Hing Ho [5] first de- fined q-subharmonicity of ' on U and using this q-subharmonic function, he introduce the notion of weak q-convexity for domains with smooth boundaries. In this paper, first we investigate a natural extension of these notions to the class of upper semi- continuous functions and domains with non-smooth boundaries. After that, we deal with L2-estimate for the �-equation on this domain, which is essentially Donnelly- Fefferman theorem [3, 1, 2] in case the domain is pseudoconvex. DEFINITION 1.1. Let ' be an upper semicontinuous function on U . Then we say that ' is q-subharmonic on U if for every q-complex dimension space H and for every compact set K � H \ U , the following holds: if h is a continuous harmonic function on K and h � ' on �K , then h � ' on K . One of the most typical examples of q-subharmonic function which is not pluri- subharmonic is � Pq�1 j=1 jz j j 2+(q�1)Pnj=q jz j j2. Also, note that an upper semicontinuous function on U is plurisubharmonic exactly when it is 1-subharmonic and q-subharmonicity implies q 0-subharmonicity whenever q � q 0 and an n-subharmonic function is just sub- harmonic function in usual sense. Before listing some properties of q-subharmonic function, we emphasize that q-plurisubharmonicity is a different notion: a C2 smooth function u

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