By Urban Cegrell

The aim of this publication is to review plurisubharmonic and analytic capabilities in n utilizing capability conception. The case n=l has been studied for a very long time and is particularly good understood. the idea has been generalized to mn and the consequences are in lots of circumstances just like the placement in . despite the fact that, those effects aren't so good tailored to advanced research in different variables - they're extra concerning harmonic than plurihar monic features. Capacities should be regarded as a non-linear generali zation of measures; capacities are set services and lots of of the capacities thought of the following might be acquired as envelopes of measures. within the mn idea, the hyperlink among features and capa towns is usually the Laplace operator - the corresponding hyperlink within the n conception 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 capabilities is harmonic, however it can ensue that the sum of 2 plurisubharmonic capabilities has confident Monge-Ampere mass whereas all the capabilities has vanishing Monge-Ampere mass. to provide an instance of similarities and alterations, think of the next statements. think first that's an open subset VIII of n and that ok is a closed subset of Q. think about the next homes that ok mayor won't have.

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**Additional info for Capacities in Complex Analysis (Aspects of Mathematics) (German Edition)**

**Example text**

P ) have i s cont i nu ou s . By Theorem V : 6 we f MA ( u K , · . , u K ) < = { u <

H h ) =0 o n RB \k s by Theorem V : 7 . H e n c e , ks s by Coro l l ar y V : 2 H h � ( 1 - s1 ) cp o n RB s o we have t hat k s 1 « l - -1 ) h < h = H a . e . ( do ) . « l -) hk and so H h Hh s s k s- k s h k ks ks s s more , MA ( H h k I t f o l l ows that hk = 0 s wh i ch proves the c l a i m . a . e . ( do ) S ome consequences h. We have now seen that i n the p l u r i s uperharmon i c c a s e , F and c sat i s f i e s cond i t i o n s 1) and 5) r e spect i v e l y ; here we l i s t some conseque n c e s o f th i s .

H e n c e , ks s by Coro l l ar y V : 2 H h � ( 1 - s1 ) cp o n RB s o we have t hat k s 1 « l - -1 ) h < h = H a . e . ( do ) . « l -) hk and so H h Hh s s k s- k s h k ks ks s s more , MA ( H h k I t f o l l ows that hk = 0 s wh i ch proves the c l a i m . a . e . ( do ) S ome consequences h. We have now seen that i n the p l u r i s uperharmon i c c a s e , F and c sat i s f i e s cond i t i o n s 1) and 5) r e spect i v e l y ; here we l i s t some conseque n c e s o f th i s . Theorem V : 8 .