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We give a representation of the fractional integral for symmetric Markovian semigroups as the projection of martingale transforms and prove the Hardy-Littlewood-Sobolev (HLS) inequality based on this representation. The proof rests on a new inequality for the fractional Littlewood-Paley –function.

Martingale Transforms and the Hardy-Littlewood-Sobolev Inequality for Semigroups