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

Abstract

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