Regularizing effect of the interplay between coefficients in linear and semilinear $X$-elliptic equations

We study the regularizing effect arising from the interaction between the coefficient $a$ of the zero order term and the datum $f$ in the problem $$\{\begin{array}{lll}-\mathcal{L}u+a(x)g(u)=f(x)& in\mathrm{\Omega },u=0& on\mathrm{\partial }\mathrm{\Omega },\end{array}$$ where $\mathrm{\Omega }\subseteq {\mathbb{R}}^N$ is a bounded domain and $\mathcal{L}$ is an $X$-elliptic operator introduced by Lanconelli and Kogoj. If $f\in L^1(\mathrm{\Omega })$, we prove that the $Q$-condition introduced by Arcoya and Boccardo is sufficient to ensure the existence and boundedness of solutions in the framework of $X$-elliptic operators as well. Finally, we prove the existence of a bounded solution for linear problems under a more general condition between $f$ and $a$.