; TeX output 2003.03.14:1021 7 ;_ 9K`y UU cmr10AnnalesAcademiScienĽtiarumFN8ennic MathematicaVN8olumen28,2003,55{68!D[:7"V cmbx10REGULAٚTEDDOMAINSANDiO BERGMANTYPEPROJECTIONS|s ˜"V cmbx10JariTT askinenaUnivĽersityofJo;Bensuu,DepartmenĽtofMathematics >PN8.O.BoĽx111,FIN-80101Jo;Bensuu,Finland;Jari.Taskinen@Jo;Bensuu.Fi|sD"V UU cmbx10DAbstract.VWN8eanalyzetherelationsofthegeometryofaregulatedcomplexdomain with theexistenceofBergmantĽyp;Beprovjectionsfrom<b> UU cmmi10LՍ=0er U cmmi7p!L( )ontotheBergmanspace5}>y UU rsfs10AՍp!( ) .[ThemaintecĽhnicaldeviceisaMuckenhoupttyp;Beweightcondition.LjInparticularwendb;BoundedBergmanGtĽyp;BeprovjectionsonGAp( )evenGinthecaseG hasGarbitraryinwardoroutwardcusps.Asaconsequence,AՍp!( )isisomorphicasaBanacĽhspacetol?po. 1.pIn9troQduction~э K`y cmr10Let !", cmsy10CbGeacomplexdomainboundedbyaJordancurve.&^W*ewanttondBergmantypGepro 8jectionsfromb> cmmi10L^ 0er cmmi7p!( )ontothecorrespGondingBergmanspace.KMoreover,ƶwe|wanttoanalyzetherelationsofthegeometryof| withtheKbGoundednessofvqariousBergmanpro 8jections..Here, thespaceKL^p!( )isKwithrespGecti}toaweightedi}2 -dimensionali}Lebesguemeasure,nwheretheweighti}!Visi}oftheUUsimplestpGossibletype:qitisapowerUUoftheboundarydistance,(1:1) L![٫(zp)=bu cmex10 \odist1(z;@ 8 )bWߴforDsomeD9v>/ 1 ..BytheBergmanspace2}>y rsfs10A^Jp!5( )wemeanthespaceofanalyticmappingsUUf:UP !CendowedUUwiththenorm(1:2)bkfk:=kfkp;!a:=^ #cZƟ yٓR cmr7 ŷjfjpR!dm^\tۮ1=p5<1:,|W*eUUonlydealwiththecaseUU1