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C) is a consequence of b) and the standard regularity theorem of linear elliptic partial differential equations. The last half of b) follows from Proposition 1. Let h be a ranic r Hermitian matrix valued function defined on an n-dimensional Kahler manifold (Y, u) which belongs to H1 Assume that h and hr1 are uniformly bounded and it satisfies A^a/i/i" 1 ) = / (1) in a weak sense with a uniformly bounded function f, then h belongs to C}'a for anyO < a < 1 and admits an estimate depending only on ||/i||£,°°, | | ^ _ 1 | U ~ , | | / | | L ° ° and the geometry of (Y, UJ).

Il(THH,W)U + (Q(U(Q{U ,THH)U T"h. V ]^ T U",,T" ' >*. )) r . Now \T» Tl"\- dE* / 3 E " \ 5E" . -£*(3> [TH uH\ = tit ds »p(rj)a0 - ^(rf P )^] + 2L- ^ ^ j «P. 1 dl>dY, 3E" . / d l > \ . [W >,(4)^ - ^(r- jdj + ^ 5 ( ^ )fi,P^-Kw{*-«a*| ~~arH~arr/l tf*(D*-|*(f)* UH]-- [T",[/"] = A tit. ds dE" . (dY,"\ ~dT8*\-arj dE" . / d E " \ l . 1) yield \T", U") + [TH,U»] + [ r " , UH] + [W,W] H HH = ,U,W) ) = T(UHT(U ,W)H,TH)--T(T - T(T H + T(UH,TH)--T(T",U + T(U",W) T(T",W). ) Furthermore, if V e V we have (VVU»,T»)&.

1 . Let F: T1,0M - » 1 + be a strongly pseudoconvex Finsler metric on a complex manifold M. Then ((VwwQ)(H,K)x,x) {(V n)(H,K)X,x)-- =--(T»(H,0(K,W)),X) (TH(H,6(K,W)),X) In particular, forallWeV eH. for all W € V and H, H, K Ken. ((vwn)(H,x)x,x) = o for all W £ V and H € H. Proof: Since we are interested only in the horizontal part, we may replace fi by QH = Q% ® dz13 ® 6a. Since Vw dz0 = 0 and V - ^ 6 a = 0, we have Vwn" = (VWQ%) 9 dz0 ® Sa. Again, we only need the horizontal part, that is 1 v P jr(Vw«|) = R%^(W) dz» A dz".