= Radial decreasing kernel domination by the maximal function
If $k(x)=\phi(|x|)\geq0$ is decreasing in radius and has integral one, then $|k_t*f(x)|\leq M f(x)$, where $k_t(x)=t^{-d}k(x/t)$ and $M$ is the centered <Hardy-Littlewood maximal function> of $|f|\,dx$. The <layer cake representation> expresses $k_t$ as a positive mixture of ball indicators; each corresponding integral is bounded by its ball volume times $Mf(x)$. Integrating these volumes gives exactly one. This applies to the <Poisson kernel for the upper half-space> and supplies a <weak type (1,1)> domination for its boundary convergence.
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