By Borsuk M.V.
Read or Download A behavior of generalized solutions of the Dirichlet problem for quasilienar elliptic divergence equations of second order near a conical point PDF
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Extra info for A behavior of generalized solutions of the Dirichlet problem for quasilienar elliptic divergence equations of second order near a conical point
2. 1 equivalent : For a Banach space E the following assertions are : (i) E has (RNP) (ii) Every uniformly integrable martingale is 1-11-convergent (iii) Every uniformly bounded martingale is Proof : (i) - (ii) Let (Xn$Fn)n E III be the uniformly integrable martingale. Define, for each n E IN and each A E F : n Un(A) = f Xn dP A By the martingale property, lim pn(A) =: p(A) exists for each A E U Fn. n n-Ko Using uniform integrability we hence see that u(A) = lim u(A) n exists for each A CE 6(V Fn) _ n F Furthermore, using uniform integrability, u is countably additive.
A1 ; .. 'Anq } n qi q and A. ,n, we have qj n I' II S2 i=1 j=1 n E f XA q. E IIXTF - X7T ,III = f qi XA.. (P Al P(A. (W4 1 1J 1 dP(W) 13 XAiYI P(A. 2 XAl XAL XAi XAi P(Ai) P(Aij) P(Aij) SII P(Ai) 11 f f + Aij Ai \ Aij + II dP f 0 \ A. ) i P( A3 i Hence 2-2 n qi P(Aij)2 E E P(A) i=1 j=1 1 Hence, VTr E II, uTr' > Tr such that IIXTr - XTr11 > 1, a contradiction. 2 : Let < p < -. 1, (iii) - (i). (i) - (ii) implies uniform integrability. 1 it follows that there is a Xu E LE such that (X F 1-convergent to X is .
In E, it follows that , (x1-x2)X 1 [0' ) X E IN = (x 1 -x 2 -x )X 3 + (x1+x2)X 1 [2' 1) + (x -x +x )X 1 [0,1) 4 2 3 [11 + (x +x -x )X 2 1 3 [13 42 + (x1+x2+x3)X 3 24 ... t. Fn = o(X1.... ,Xn), the smallest o-algebra making X1,.... Xn measurable (take F = B[0,1), the Borel-o-algebra on [0,1)). 5 we suppose that F = FF for convenience. 2(1) converges to X in for each 1 < p < oo if X E L. P Proof : The martingale is certainly 11-11p-convergent in case X is a U Fn F stepfunction, since in this case (E nX,F ) nnEIN n is eventually constant.