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

By Urban Cegrell

The aim of this booklet is to check plurisubharmonic and analytic capabilities in n utilizing potential idea. The case n=l has been studied for a very long time and is particularly good understood. the speculation has been generalized to mn and the implications are in lots of situations just like the location in . even if, those effects aren't so good tailored to advanced research in numerous variables - they're extra regarding harmonic than plurihar monic features. Capacities will be considered a non-linear generali zation of measures; capacities are set features and plenty of of the capacities thought of the following might be got as envelopes of measures. within the mn thought, the hyperlink among capabilities and capa towns is frequently the Laplace operator - the corresponding hyperlink within the n concept is the complicated 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 fluctuate significantly. for instance, the sum of 2 harmonic services is harmonic, however it can occur that the sum of 2 plurisubharmonic features has optimistic Monge-Ampere mass whereas all of the services has vanishing Monge-Ampere mass. to offer an instance of similarities and transformations, think about the next statements. think first that's an open subset VIII of n and that okay is a closed subset of Q. give some thought to the subsequent homes that ok mayor won't have.

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1 6 ( 1 9 7 8 ) that n �EPSH ( � ) . F o r a s tocha s t i c po i n t o f v i ew s e e M. Fukush ima and M . Okada , On D i r i ch l e t f o r ms f o r p l u r i s u bharmo n i c f u n c t i on s . Acta Math . 1 59 : 3-4 , 1 987 .

J. d� = �EM j-++CXl �EM j-+ oo = sup �im I ( CP6 i nf( � cp� , 1 ) ) d� � l im sup I CP 6d� + E . But s=1 �EM J-++oo j-++oo � EM since all functions ( CP Oj ) j=1 and are lower semicontinuous we have by i i ) - + + 00 sup ICP�d� = sup inf I CP�d� � inf sup I CP�d� � G ( E J. ) . K �EM �EM �EM K which proves the theorem . Hence G( E ) -< l- im G ( E J. ) j ++oo + E If R and M satisfies i ) -v ) then M can be replaced, by its weak*-closure . Remark. Notes and references Example 111 : 1 i s due to B .

Out s i d e a f i xe d c ompa c t s e t v . E PSHnC ( B ) . J O < i

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