The relationship between some special conditions of young functions and the validity of generalized $L^p$ estimate for the Poisson equations in a unit ball

In this paper, we consider the domain is in $B_1$, and we will show the relationship between the global ${\mathrm{\Delta }}_2$ and ${\mathrm{\nabla }}_2$ conditions of young functions and the validity of generalized $L^p$ estimate for the Poisson equations.