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

Abstract EN

The variable exponent Hardy inequality x β ( x ) - 1 ∫ 0 x f ( t ) d t L p ( .

) ( 0 , l ) ≤ C x β ( x ) f L p ( .

) ( 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.