The Weighted Boundedness Of Some Operators In Variable Lebesgue Spaces

Due to the rapid development of the variable exponent theory and its applications in par-tial differential equations et al,the research for variable exponent function spaces has being concerned by more and more scholars in recent years.In this dissertation,we mainly establish the extrapolation from one-sided weights in weighted variable Lebesgue spaces,and obtain the weighted boundedness of the one-sided Hardy-Littlewood maximal operators and the multi-linear maximal operator.More precisely,we concentrate on the following topics.First,we introduce the variable one-sided Sawyer type conditions Sp(-)+(D)and S'p(-)+(D),by which we prove the two-weight norm inequalities of the one-sided Hardy-Littlewood max-imal operators M+ and M-in variable Lebesgue spaces.As applications,we establish the two-weight norm inequalities of variable Riemann-Liouville operator R?(-)and variable Weyl operator W?(·)in variable Lebesgue spaces.Furthermore,we research the extrapolation starting from one-sided weights in weighted variable Lebesgue spaces.Applying these results,we obtain the weighted boundedness of one-sided singular integral operators,one-sided fractional maximal operators,one-sided fractional integral operators,the one-sided discrete square function and their commutators in variable Lebesgue spaces.Finally,we define the multilinear weight class Ap(·)and consider the weighted norm in-equalities for the multilinear maximal operator in variable Lebesgue spaces.Following the extrapolation and our result,the weighed boundedness of multilinear Calderon-Zygmund op-erators in variable Lebesgue spaces is also obtained.