Reverse Hardy-type inequalities for supremal operators with measures

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In this paper we characterize the validity of the inequalities gₚ,₍ₐ,b₎,λ ≤ c u(x) g∞,₍ₓ,b₎,μq,₍ₐ,b₎,ν and eq.0.1.2 gₚ,₍ₐ,b₎,λ ≤ c u(x) g∞,₍ₐ,ₓ₎,μq,₍ₐ,b₎,ν for all non-negative Borel measurable functions g on the interval (a,b), where 0 < p ≤ +∞, 0 < q ≤ +∞, λ, μ and ν are non-negative Borel measures on (a,b), and u is a weight function on (a,b).