Weighted norm inequalities in the variableLlebesgue spaces for the Bergman projector on the unit ball of ${\mathbb{c}}^n$

In this work, we extend the theory of Békollè-Bonami $B_p$ weights. Here we replace the constant $p$ by a non-negative measurable function $p(\cdot ),$ which is log-Hölder continuous function with lower bound $1$. We show that the Bergman projector on the unit ball of ${\mathbb{C}}^n$ is continuous on the weighted variable Lebesgue spaces $L^{p(\cdot )}(w)$ if and only if $w$ belongs to the generalised Békollè-Bonami class $B_{p(\cdot )}$. To achieve this, we define a maximal function and show that it is bounded on $L^{p(\cdot )}(w)$ if $w\in B_{p(\cdot )}$. We next state and prove a weighted extrapolation theorem that allows us to conclude.