A Weighted Variant of Riemann - Liouville Fractional Integrals on R n

and Applied Analysis 3 Moreover, ∥Hωf∥Lp Rn →Lp Rn A. 1.7 Remark 1.1. Notice that the condition 1.6 implies that ω is integrable on 0, 1 since ∫1 0 ω t dt ≤ ∫1 0 t −n/pω t dt. We naturally assumeω is integrable on 0, 1 throughout this paper. Obviously, Theorem A implies the celebrated result of Hardy et al. 6, Theorem 329 , namely, for all 0 < α < 1 and 1 < p < ∞, ‖Iα‖Lp dx →Lp x−pαdx Γ ( 1 − 1/p) Γ ( 1 α − 1/p) . 1.8 The constant A in 1.6 also seems to be of interest as it equals to p/ p − 1 if ω ≡ 1 and n 1. In this case, Hω is precisely reduced to the classical Hardy operator H defined by Hf x 1 x ∫x 0 f t dt, x > 0, 1.9 which is the most fundamental integral averaging operator in analysis. Also, a celebrated operator norm estimate due to Hardy et al. 6 , that is, ‖H‖Lp R →Lp R p p − 1 1.10 with 1 < p < ∞, can be deduced from Theorem A immediately. Recall that BMO R is defined to be the space of all b ∈ Lloc R such that ‖b‖BMO : sup B⊂Rn 1 |B| ∫ B |b x − bB| dx < ∞, 1.11 where bB 1/|B| ∫ B b and the supremum is taken over all balls B in R n with sides parallel to the axes. It is well known that L∞ R BMO R , since BMO R contains unbounded functions such as log |x|. Another interesting result of Xiao in 5 is that the weighted Hardy operator Hω is bounded on BMO R , if and only if ∫1 0 ω t dt < ∞. 1.12