On a weighted inequality of Hardy type in spaces [formula omitted

Abstract

The boundedness of Hardy type operator H f ( x ) = ∫ { t ∈ R n : | t | ⩽ | x | } f ( t ) d t is studied in weighted variable exponent Lebesgue spaces L p ( ⋅ ) . The necessary and sufficient criterion established on the weight functions v ( x ) , ω ( x ) and exponents p ( x ) , q ( x ) for the Hardy operator to be bounded from L p ( ⋅ ) ( ω ) to L q ( ⋅ ) ( v ) . The exponents satisfy a modified logarithmic condition near zero and at infinity: ∃ δ > 0 , ∃ f ∞ , ∃ f ( 0 ) ∈ R sup x ∈ B ( 0 , δ ) | f ( x ) − f ( 0 ) | ln 1 W ( x ) < ∞ ; ∃ N > 1 sup x ∈ R n ∖ B ( 0 , N ) | f ( x ) − f ∞ | ln W ( x ) < ∞ , where W ( x ) = ∫ { t ∈ R n : | t | ⩽ | x | } ω − 1 / ( p ( t ) − 1 ) ( t ) d t .