On the boundary behavior of functions in spaces of Hardy type

Let $X$ be a topological space with a measure $\mu$. In the product $\mathscr X=X\times (0,T]$ (or $\mathscr X=X\times [0,1)$) simple axioms are used to distinguish a family $\Gamma=\{\Gamma(x)\colon x\in X\}$ of domains for approaching the boundary of $\mathscr X$. Associated with the family $\Gamma$ is the maximal function
$$ \mathscr M_\Gamma u(x)=\sup\ \{|u(y,t)|\colon (y,t)\in\Gamma(x)\}. $$
The spaces $\mathscr H^p(\mathscr X,\Gamma,\mu)$ consisting of functions $u$ continuous on $\mathscr X$ with $\mathscr M_\Gamma u\in L^p$ are introduced, along with the subspaces of them consisting of the functions having a $\Gamma$-limit a.e. The properties of the spaces $\mathscr H^p$ and the action in them of operators of smoothing type are studied. The results are applied to Hardy spaces of harmonic or holomorphic functions.