Dirichlet Integrals on Harmonic Spaces by Fumi-Yuki Maeda (auth.)

By Fumi-Yuki Maeda (auth.)

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12) m ~j~(gj)l ~ ~(v 2) - ~(v I) ~ 3cma(s) j=l on W x. The first and the last terms are independent of W x. 50 Hence m Io(f) l on V. ~jo(gj) I ~; 3ema(s) j=l m Now, letting hence Theorem e -- O, we see t h a t a(f) Let f l , . . o i,j =i h j=l in ~k f = (fl'" "''fk )" containing and the f o l l o w i n g ~(U). equations If ~ 2 ( ~ ) , hold: ~) i { kz = i on V, and j=l on U. 3. is) such that k h > 0 on U; 8~ = j:l ] for any gE ~ ( U ) . 16) Proof. 15) i ~e C'(~) j for which ~oF e~(U) and hold. (I) i ~ and x .

3. x~W, h(x) Since choose we can on W. Rf V'% so that at Xo, Rf(x o) < +=. we have + 2e. semicontinuous, on W c U (W% ~ X ) . x. Since (Rf) V % ~ ( U ) , is h a r m o n i c it f o l l o w s For f ~ Rf, it f o l l o w s on V. Since each f ~ x%W, (Rf) V that x%W is on W. ~X such and f < Rf on W ~ that f is c o n t i n u o u s find f < ~h < Rf on V for Rf o f is c o n t i n u o u s = i. S i n c e continuous, that Hence . containing is h a r m o n i c For +,, + 2s ~ Rf(x o) Rf is l o w e r at x V~ ~ r c ( W ) suppose each ( R f ) v ( X o) and Therefore, Rf ( R f ) v ( X o) < f is s u b h a r m o n i c Finally, with ~ is c o n t i n u o u s suppose we c a n ue£ ~(U).

14. harmonic and space a base Bauer's Proposition Corollary harmonic U, then satisfies i-3). 10 Axiom , it f o l l o w s (S) a harmonic sets a Brelot's is B a u e r ' s harmonic for B a u e r ' s space having harmonic space is a space. If U is a P-set, holds: Proposition If u~ ~ ( U ) then the following and u ~ 0 o u t s i d e u ~ 0 on U. minimum principle a compact set in 26 Proof. 11 , there Since choose neighborhood in U such Vx£ ~rc(U) exists B Vx Px both set that which W x of Px x are such such continuous x.

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