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Published online by Cambridge University Press: 27 July 2019
Fractional calculus and the Riemann–Liouville integral are used to motivate the Hardy–Littlewood–Sobolev inequality. These are then generalised to the Riesz potential, and via heat kernels, toVaropoulos’s class of ultracontractive semigroups, which we show also satisfy this key inequality. As an application, we establish the Nash inequality for uniformly elliptic second-order diffusion operators and show that these are in the Varopoulos class.
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