Hardy’s inequality with three measures

The Optimal Constant in Generalized Hardy's Inequality

“…According to [11, p.809], when p = q, the factor k q,p = p 1/p p * 1/p * , which is consistent with the result in [4,11,12,14,15,16,17].…”

“…(3) We can also present the sharp factor of the two-side estimate of the optimal constant in the Hardy's inequality with three measures just as in [17].…”

“…Hardy's inequality has been generalized in various direction. In [17], Prokhorov gave necessary and sufficient conditions for validity of the Hardy's inequality with three measures. He also claimed that the Hardy's inequality with three measures can be reduced to the following case with two measures.…”

Hardy-type inequalities on the cones of monotone functions

“…The case of (0.18) with arbitrary Borel measures was fractionally presented in [40], [39], for discrete inequalities in [5], [6], [7] and for three measures in [50]. The full characterization of the inequality (0.18) for the operators with the Oinarov kernels in the case 1 < p, q < ∞ was obtained by D.V.…”

“…The case u = v = w ≡ 1, where λ is a Lebesgue measure, was described in [39]. The more general cases for two (when λ = ν) and three measures for p > 1, q > 0 were in full characterized in [50]. The criterion for three measures is given in [50] in terms which include (ν a , ν s )− the Lebesgue decomposition of measure ν relative to λ, where dν a /dλ denotes the Radon-Nikodym derivative of ν a with respect to λ.…”