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We show that the sharp constant in the Hardy-Littlewood-Sobolev inequalitycan be derived using the method that we employed earlier for a similarinequality on the Heisenberg group. The merit of this proof is that it does notrely on rearrangement inequalities; it is the first one to do so for the wholeparameter range.

A new, rearrangement-free proof of the sharp Hardy-Littlewood-Sobolev inequality