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

• DOI: 10.1007/s11118-016-9571-0
• Publication Date: 2016-06-02T05:10:26Z
• Authors: Daesung Kim
• Keywords: Sobolev inequality; Potential theory

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 a fractional Littlewood-Paley $g$-function.<br/>13 pages<br/>