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For $0<\lambda<d$ and suitable conjugate exponents, the Hardy–Littlewood–Sobolev inequality bounds convolution with the Riesz kernel $|x|^{-\lambda}$. One useful form is
$$
\iint_{\mathbb R^d\times\mathbb R^d}
\frac{f(x)g(y)}{|x-y|^\lambda}\,dx\,dy
\lesssim\|f\|_p\|g\|_q,
\qquad
\frac1p+\frac1q+\frac\lambda d=2.
$$

Hardy–Littlewood–Sobolev inequality

Global H1 solutions of the three-dimensional gravitational Hartree equation

Past exam of the mathematics course of the University of Cambridge / 2021 / iii / Paper 154 / 2 / 3 / Solution

Hardy–Littlewood–Sobolev inequality

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Hardy–Littlewood–Sobolev inequality

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