A Necessary and Sufficient Condition for Hardy’s Operator in the Variable Lebesgue Space - Farman Mamedov; Yusuf Zeren (Abstract and Applied Analysis)
The variable exponent Hardy inequality xβ(x)-1∫0x‍f(t)dtLp(.)(0,l)≤Cxβ(x)fLp(.)(0,l), f≥0 is proved assuming that the exponents p:(0,l)→(1,∞), β:(0,l)→ℝ not rapidly oscilate near origin and 1/p′(0)-β>0. The main result is a necessary and sufficient condition on p, β generalizing known results on this inequality.