The boundedness of classical operators on variable L^p spaces

We show that many classical operators in harmonic analysis ---such as maximal operators, singular integrals, commutators and fractional integrals--- are bounded on the variable Lebesgue space $L^{p(\cdot)}$ whenever the Hardy-Littlewood maximal operator is bounded on $L^{p(\cdot)}$. Further, we show that such operators satisfy vector-valued inequalities. We do so by applying the theory of weighted norm inequalities and extrapolation. As applications we prove the Calder\'on-Zygmund inequality for solutions of $\bigtriangleup u=f$ in variable Lebesgue spaces, and prove the Calder\'on extension theorem for variable Sobolev spaces.