Weak convergence of measures and weak type $(1,q)$ of maximal convolution operators

Abstract:

Let ${G^ * }$ be the maximal convolution operator associated with a sequence of ${L^1}$ kernels. We show that if ${G^ * }$ is of weak type $(1,q)$, $1 \leq q < \infty$, over a subset ${\mathcal N}$ of ${\mathcal M}$ (the space of all finite positive Borel measures on ${{\bf {R}}^h}$ endowed with the weak topology), then ${G^ * }$ is of weak type $(1,q)$ over the closed cone in ${\mathcal M}$ generated by ${\mathcal N}$. As a particular case we obtain a well-known result by de Guzman.