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The continuum $\varphi^4_2$ and $\varphi^4_3$ measures are shown to satisfy alog-Sobolev inequality uniformly in the lattice regularisation under theoptimal assumption that their susceptibility is bounded. In particular, thisapplies to all coupling constants in any finite volume, and uniformly in thevolume in the entire high temperature phases of the $\varphi^4_2$ and$\varphi^4_3$ models.  The proof uses a general criterion for the log-Sobolev inequality in terms ofthe Polchinski (renormalisation group) equation, a recently proved remarkablecorrelation inequality for Ising models with general external fields, thePerron--Frobenius theorem, and bounds on the susceptibilities of the$\varphi^4_2$ and $\varphi^4_3$ measures obtained using skeleton inequalities.

Log-Sobolev inequality for the $\varphi^4_2$ and $\varphi^4_3$ measures