高次R-L分数积分的加权不等式

Weighted Lorentz Norm Inequalities for Riemann-Liouville Fractional Integrals of Order Oneand Greater

本文给出了R_+上的权函数(非负可测函数)v(x)、w(x)的一个充分必要条件,使得‖T_wf(x)‖L^(p,q(w))≤c‖f‖L^r(v),0<p<∞,1<q,r<∞. 这里T_,f(x)=(?),函数甲:(0,1)→(0,∞)非增且满足:甲(ab)≤D[0(甲(a)十甲(b)],0〈a,b〈1.T_。的特殊情形即为Riemann-Liouville分数积分T_。,(a≥1).

Necessary and sufficient conditions on locally finite positive Borel measures α and ω are obtained in order that the Lebesgue and Lorentz norm inequality of the form holds for 0<p<∞, 1<q,r<∞, where T(?) is a generalized Hardy operator T(?)(fσ)(x)= (t/x)f(t)dσ(t) and (?):(0,1)→(0,∞) is nonincreasing and sa- tisfies the condition (?)(ab)≤D[(?)(a) + (?)(b)] for 0<a,b<1. The Riemann-Liouville fractional integrals Tα with α≥1 are the special cases of T(?).