Capacities In Complex Analysis(1st Edition)

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Book Price $0 : The Purpose Of This Book Is To Study Plurisubharmonic And Analytic Functions In N Using Capacity Theory. The Case N=l Has Been Studied For A Long Time And Is Very Well Understood. The Theory Has Been Generalized To Mn And The Results Are In Many Cases Similar To The Situation In . However, These Results Are Not So Well Adapted To Complex Analysis In Several Variables - They Are More Related To Harmonic Than Plurihar- Monic Functions. Capacities Can Be Thought Of As A Non-linear Generali- Zation Of Measures; Capacities Are Set Functions And Many Of The Capacities Considered Here Can Be Obtained As Envelopes Of Measures. In The Mn Theory, The Link Between Functions And Capa- Cities Is Often The Laplace Operator - The Corresponding Link In The N Theory Is The Complex Monge-Ampere Operator. This Operator Is Non-linear (it Is N-linear) While The Laplace Operator Is Linear. This Explains Why The Theories In Mn And N Differ Considerably. For Example, The Sum Of Two Harmonic Functions Is Harmonic, But It Can Happen That The Sum Of Two Plurisubharmonic Functions Has Positive Monge-Ampere Mass While Each Of The Two Functions Has Vanishing Monge-Ampere Mass. To Give An Example Of Similarities And Differences, Consider The Following Statements. Assume First That Is An Open Subset VIII Of N And That K Is A Closed Subset Of Q. Consider The Following Properties That K Mayor May Not Have. See more