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ABSTRACT.In this paper we develop some aspect of a general theory parallel to the Calderón-Zygmund theory for operator valued kernels, where the operators considered map functions defined on Rn into functions defined on Rn+1 = Rn x [0)OO).In particular, we apply the obtained results to get vector-valued inequalities for the Poisson integral and fractional integrals.Some weighted norm inequalities are also considered for fractional integrals.

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