A New, Rearrangement-free Proof of the Sharp Hardy–Littlewood–Sobolev Inequality

By Rupert L. Frank, Elliott H. Lieb. 2010

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We show that the sharp constant in the Hardy–Littlewood–Sobolev inequality can be derived using the method that we employed earlier for a similar inequality on the Heisenberg group. The merit of this proof is that it does not rely on rearrangement inequalities; it is the first one to do so for the whole parameter range.

Subfields: Applied Mathematics

View Paper: https://www.semanticscholar.org/paper/f9ce2942e0fc59162d214e2618f18ab1efe66df2

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In Applied Mathematics

Citations in the corpus

Nondegeneracy of the bubble in the critical case for nonlocal equations (2013, background): “[10] Rupert L. Frank and Elliott H. Lieb, Inversion positivity and the sharp Hardy-LittlewoodSobolev inequality, Calc.”
Hardy–Littlewood–Sobolev and Stein–Weiss inequalities and integral systems on the Heisenberg group (2012, unknown): “Lieb used rearrangement method in [5], and recently, Frank and Lieb developed a new method to prove the sharp version of the same special case of (1.1) (r = s = 2N/(2N − λ)) without using rearrangement, see [7]. that the sharp constant and extremal problem for general 1 ≤ p < Q is still widely open.”