A maximal function characterization of the Hardy space for the Gauss measure

Journal article, 2013

An atomic Hardy space $ H^1(\gamma )$ associated to the Gauss measure $ \gamma $ in $ \mathbb{R}^n$ has been introduced by the first two authors. We first prove that it is equivalent to use $ (1,r)$- or $ (1,\infty )$-atoms to define this $ H^1(\gamma )$. For $ n=1$, a maximal function characterization of $ H^1(\gamma )$ is found. In arbitrary dimension, we give a description of the nonnegative functions in $ H^1(\gamma )$ and use it to prove that $ L^p(\gamma )\subset H^1(\gamma )$ for $ 1