CHAPTER 2 Preliminaries o n Capacitie s Let Lipl oc (Rn) b e th e se t o f al l comple x value d locall y Lipschit z function s in R n (wit h exponen t 1) . Th e fundamenta l solutio n $ fo r th e Laplac e equatio n Af = 0 in R n i s define d b y $(x) = i JtT-2 ? n 3 , wher e a n 0 is a constant . $ W = - ^ l o g R for n = 2 . We now introduce the class of admissible functions fo r the definition o f Lipschitz harmonic capacity . Fo r a compact se t E i n R n , se t L(E, 1 ) := { / Lipl c (Rn) supp(A/) c E, ||V/||o o 1 , V/(oo) = 0}, where supp(A/ ) i s the suppor t o f the distributio n A/ . We shal l conside r function s modul o constant s i n L(E, 1) , meanin g tha t w e shall writ e / = g fo r / , g G L(E, 1 ) i f / g i s constant . Not e tha t eac h functio n / G L(JE7, 1) is harmonic i n R n \ E an d / = $ * A/ - f constant . The Lipschit z harmoni c capacit y o f the se t E i s defined b y 7 ( £ ) : = s u p { | A / , l | : / L ( i 5 , l ) } , where (a s usual) (5 , /?) means th e actio n o f the distributio n wit h compac t suppor t on a smooth tes t function . Letting a(n 1) be the volum e o f the uni t bal l in R n _ 1 , w e define th e ( n 1)- dimensional Hausdorf f measur e fo r a subset E i n R n b y oo Hn-l{E) := liminf{V a(n- 1 ) T ? - 1 : E C 1 5(a i ,r i ), n 5} . i=l Then th e restrictio n o f 7Y n_1 t o sufficientl y regula r hypersurface s give s the surfac e measure. A se t E i n R n i s called removabl e fo r Lipschit z harmoni c function s i f fo r eac h domain D i n MJ 1 ever y locall y Lipschit z functio n / : D C whic h i s harmonic i n D \ E i s harmonic i n D. I t i s proved i n [35 ] tha t A set E i s removable fo r Lipschit z harmoni c function s i f and onl y i f y(E) = 0 . A set E i n M71 is called removable for subharmonic Lipschitz harmoni c function s if fo r eac h domai n D i n R n ever y locall y Lipschit z functio n / : D C whic h i s harmonic i n D \ E an d subharmoni c i n D i s harmonic i n D. Such set s als o ca n b e characterize d usin g a natura l capacity . Le t u s conside r 7 + ( £ ) : = sup{|(A/, 1)| :/ L(E, 1) , A/ = M e M + (E)} , where M+(E) i s the famil y o f positive Bore l measure s o n E. Similarl y t o [35] , one can se e tha t A set E i s removable fo r subharmoni c Lipschit z harmoni c function s if and onl y i f 7+(£) = 0 . 7 http://dx.doi.org/10.1090/cbms/100/02
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