= Hardy–Littlewood–Sobolev inequality
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Let $0<\lambda<d$ and $1<p,q<\infty$ satisfy $1/p+1/q+\lambda/d=2$. The <Hardy–Littlewood–Sobolev inequality> states that
$$
\iint_{\mathbb R^d\times\mathbb R^d}\frac{|f(x)||g(y)|}{|x-y|^\lambda}\,dx\,dy
\le C_{d,\lambda,p}\|f\|_p\|g\|_q.
$$
Here 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.<Sharp constants in the Hardy-Littlewood-Sobolev and related inequalities by Elliott H. Lieb>[Introduction and Theorem 1.1, convolution and bilinear forms]{ref}
For $d=3$, $\lambda=1$ and $p=q=6/5$, applying it to $f=g=|u|^2$ gives
$$
\iint\frac{|u(x)|^2|u(y)|^2}{|x-y|}\,dx\,dy\le C\|u\|_{12/5}^4.
$$
This 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>.
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