Capacities in Complex Analysis (Aspects of Mathematics) by Urban Cegrell

By Urban Cegrell

The aim of this booklet is to review plurisubharmonic and analytic services in n utilizing skill conception. The case n=l has been studied for a very long time and is especially good understood. the idea has been generalized to mn and the consequences are in lots of instances just like the location in . even if, those effects usually are not so good tailored to complicated research in different variables - they're extra relating to harmonic than plurihar monic features. Capacities may be considered a non-linear generali zation of measures; capacities are set services and lots of of the capacities thought of right here should be bought as envelopes of measures. within the mn concept, the hyperlink among features and capa towns is usually the Laplace operator - the corresponding hyperlink within the n idea is the advanced Monge-Ampere operator. This operator is non-linear (it is n-linear) whereas the Laplace operator is linear. This explains why the theories in mn and n vary significantly. for instance, the sum of 2 harmonic services is harmonic, however it can take place that the sum of 2 plurisubharmonic capabilities has confident Monge-Ampere mass whereas all of the services has vanishing Monge-Ampere mass. to offer an instance of similarities and adjustments, give some thought to the next statements. think first that's an open subset VIII of n and that okay is a closed subset of Q. examine the next houses that ok mayor would possibly not have.

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Example text

Thi s is purely lo c al so we can assume that U==RB where R) l , K=B , v=v J, outside B · Ta k e uEPSHnC 2 x { RB ) , u= c onst . near B and Then , for k�j , Stokes � formula g ives 1 u 1 co I , � == - fd { v j -v k ) A d ( u-u ) /\ ( dd c u ) n- l c � sin c e fd { v j -vk ) /\ d u A ( dd c u ) n - l = 0 . s ince { dd cu ) n- l i s a positive ( n- l , n- l ) -form we get I fd ( V j -V k ) /\ d C ( u-u } /\ { dd c u } n- l I � 1 2 nC A ( dd u ) l ] < [ f d ( v j -v k ) A d ( v j -v k ) 1 2 nc l ] .

1\ l im J � A /\ J 0 The r e f o r e so i f J f n v dd C v 1 A < o J nv . s i nce f n v °. dd c v 1. The r e f o r e C n dd v a nd t h e y have t h e s ame ma s s s o they have to b e equa l . Now , if i v . J r e g u l a r i z ed f u n c t i on s Then , if PSHnL are i n 00 ( eJ ) J. = 1 i v . J , 1S E: 00 l oc E PSHnC 00 (8) o n l y , w e c on s i de r t h e i n a sma l l e r ba l l . a n y s eq u e n c e d e c r e a s i ng t o z e r o w e < - 37 - get f rom above that o v 1. , ( . dd c v 1l.

Propo s i t i oD III: 1. Proof . 2) � 3}) or c l osed . S o i f = 2) h 9 f �. 1 . == 9 If h9 = H h 8 Hh is = g we g e t �. c. H lD . f i EN �. 1 w i th so a . e . ( do ) . 1 i n f � 1. do l

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