One and two weight estimates for one-sided operators in $L^{p(\cdot)}$ spaces

Various type weighted norm estimates for one-sided maximal functions and potentials are established in variable exponent Lebesgue spaces $L^{p(\cdot)}$. In particular, sufficient conditions (in some cases necessary and sufficient conditions) governing one and two weight inequalities for these operators are derived. Among other results generalizations of the Hardy–Littlewood, Fefferman–Stein and trace inequalities are given in $L^{p(\cdot)}$ spaces.