By Michael Rockner
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Extra info for A Dirichlet Problem for Distributions and Specifications for Random Fields
Is linear on ft . (U) for every P €M . c x If be a second harmonic sheaf on D such that Brelot space. Assume that the associated harmonic measures U belong to and a AL . g. the case if (D,H) (D,H) y , is a x € U, for is a harmonic space COm- ing from a uniformly elliptic differential operator Lu » . I. a.. u + Z bL. u + c u ifj iJ iJ i with sufficiently smooth coefficients (cf. ). Then the harmonic measures u y , x € U, of (P,H) above may always be replaced by *m p , x € U . 4. THE PWB-SOLUTION FOR DISTRIBUTIONS GIVEN AN ARBITRARY OPEN SET In this section let exhaustion of U U be a fixed open subset of by compact sets.
U) and x -+ v°(£), x u U (£) x € U, s u U (n) is the classical PWB-solution. (iv) Let £ € P' on some open set for every x € U . (v) x € U For be such that V containing the function £i U, X n X 0 o(U ^ U n . (U) and a * 2 X y 6 L (P) u x for every P € W . 9. Remark, (i) Assume that 3U is compact. 8) by the same symbol £ € V* , x € U. 8) u U ( O , 5 € V, x,c x € U. 8). (ii) Let [11, P € M. _. n nfcJN (A ) £_, o :» A € € V , x € W, Prop. 2] n and Q. (W) and Then, since that for x € U . be an exhaustion of W := U U V .
Let n €U be such that K U supp y c U n k . , x - || w ° * ||E , x e u , sucn tnat DIRICHLET PROBLEM AND RANDOM FIELDS are y - i n t e g r a b l e . By 2 . 9 ( i i ) we have f o r 29 n > nQ K U n -»*"E -<2||^o|| E . 9 (iii) and the theorem of dominated convergence imply the assertion. a We see that the subsequence hence on K. 3 depends on is fixed we set from now on for simplicity (U ) ^_, := (U_ ). Cm . 6). Fix y , (U ) ^ Q (U,K) as before in I , it will satisfy the properties P € Bi . 8 that the maps x -• X M , n € K , and x -» X y 2 are continuous from Hence for n,k € U supp y to (L (P),il llp ~) • the Bochner integrals L 2 " f|X un - X n|dy(x) x x and L 2 " f|X u.