It is well know that there is a generalization of Lebesgue decomposition theorem in the following way:
Any non negative Radon measure can be decomposed uniquely into the sum of an absolutely continuous measure and a singular term with respect to the harmonic capacity. The absolute continuous term itself can be decomposed (not uniquely) to a function in $L^1$ and a function in $H^{-1}$ (dual of $H_0^1$).
I will be thankful to any one that will help me answering the following question:
do there exist such kind of decomposition involving the fractional harmonic capacity?
