We characterize the power weights ? for which the fractional type operator ??,? is bounded from ??(??) into ??(??) for 1/(?−(?+?)) and 1/?=1/?−(?−(?+?))/? . If ?/(?−(?+?))≤?/(?−(?+?)−1)+ we prove that ??,? is bounded from a weighted weak ?? space into a suitable weighted ???? space for weights satisfying a doubling condition and a reverse Hölder condition. Also, we prove the boundedness of ??,? from a weighted local space ????0 into a weighted ???? space, for weights satisfying a doubling condition.
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