A Necessary and Sufficient Condition for Hardy’s Operator in the Variable Lebesgue Space

The variable exponent Hardy inequalityxβ(x)-1∫0x‍f(t)dtLp(.)(0,l)≤Cxβ(x)fLp(.)(0,l),f≥0is proved assuming that the exponentsp:(0,l)→(1,∞),β:(0,l)→ℝnot rapidly oscilate near origin and1/p′(0)-β>0. The main result is a necessary and sufficient condition onp,βgeneralizing known results on this inequality.

Weak-type inequalities and convergence of the ergodic averages in variable Lebesgue spaces with weights

We characterize the weighted weak-type inequalities with variable exponents for the maximal operator associated with an ergodic, invertible, measure-preserving transformation and prove the almost everywhere convergence of the ergodic averages for all functions in a variable Lebesgue space with a weight verifying a suitable condition.