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We shall consider the truncated singular integral operatorsT   μ  ,  K   ε      f   (  x  )      =      ∫    ℝ  n   \  B   (   x  ,  ε   )K   (   x     −     y   )   f   (  y  )      d  μ  yand related maximal operatorsT   μ  ,  K   *      f   (  x  )      =       sup    ε  >  0        |    T   μ  ,  K   ε      f   (  x  )    |    . We shall prove for a large class of kernels  K  and measures  μ  and ν that if  μ  and ν are separated by a Lipschitz graph, thenT   ν  ,  K   *      :      L  p    (  ν  )      →      L  p    (  μ  )is bounded for 1 <  p  < ∞. We shall also show that the truncated operatorsT   μ  ,  K   εconverge weakly in some dense subspaces of  L  2 ( μ ) under mild assumptions for the measures and the kernels.

Boundedness and convergence for singular integrals of measures separated by Lipschitz graphs