Hardy–Littlewood–Sobolev inequality

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.