 Radial decreasing kernel domination by the maximal function

If $k(x)=\varphi (|x|)\ge 0$ is decreasing in radius and has integral one, then $|k_t\ast f(x)|\le Mf(x)$, 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.