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Elliott H. Lieb proved sharp forms of the Hardy–Littlewood–Sobolev inequality and related inequalities. The work determines optimal constants in bounds for singular integral expressions. Sharp constants distinguish an estimate that merely controls an integral from one that identifies its exact limiting strength, with consequences for analysis and mathematical physics.^[ref]

Let $0<\lambda <d$ and $1<p,q<\infty$ satisfy $1/p+1/q+\lambda /d=2$. The Hardy–Littlewood–Sobolev inequality states thatHere the right side uses Lp norms, and the constant is independent of $f,g$. The exponents are linked by the displayed dimensional relation; they are not a conjugate exponents pair. This is a bound for a singular convolution rather than a pointwise bound on the kernel.^[ref]

For $d=3$, $\lambda =1$ and $p=q=6/5$, applying it to $f=g=|u{|}^2$ givesThis is the potential-energy estimate used for the Gravitational Hartree equation. A Gagliardo-Nirenberg interpolation inequality then relates that intermediate Lp norm to mass and the gradient L2 norm.