Hardy–Landau–Littlewood Inequalities for Fractional Derivatives in Weighted L p Spaces

The inequality of Hardy, Landau and Littlewood is established in weighted L p spaces, for weights satisfying Muckenhoupt's ( A p ) condition. That is, for f ∈ Dom ( D γ ), where D γ denotes the γth fractional derivative acting inL ω p (0, ∞),‖ D α f ‖ ω ⩽ K‖ f ‖ ω 1 − ( α / γ )‖ D γ f ‖ ω α / γ, for 0 < α < γ, and‖ f ‖ ω =(∫ 0 ∞| f ( x ) |p w ( x ) d x )1 / p. In addition, we show that the indefinite integral possesses a strongly continuous group of imaginary powers inL ω p (0, ∞).