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 $$\left\lbrace \begin{array}{ll} -\mathcal{L}u + a(x) g(u) = f(x) \quad &\mbox{in} \;\; \Omega, u = 0 \quad &\mbox{on} \;\; \partial\Omega, \end{array} \right.$$ where $\Omega\subseteq\mathbb{R}^N$ is a bounded domain and $\mathcal{L}$ is an $X$-elliptic operator introduced by Lanconelli and Kogoj. If $f \in L^1(\Omega)$, 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 $a$.