Reverse Hardy-type inequalities for supremal operators with measures

In this paper we characterize the validity of the inequalities $\mathrm{\parallel }g{\mathrm{\parallel }}_{p,(a,b),\lambda }\le c\mathrm{\parallel }u(x)\mathrm{\parallel }g{\mathrm{\parallel }}_{\mathrm{\infty },(x,b),\mu }{\mathrm{\parallel }}_{q,(a,b),\nu }$ and $\label{eq.0.1.2} |g|{p,(a,b),\lambda} \le c |u(x) |g|{\infty,(a,x),\mu}|_{q,(a,b),\nu} $ for all non-negative Borel measurable functions $g$ on the interval $(a,b)$, where $0<p\le +\mathrm{\infty }$, $0 < q \le +\infty$,$\lambda$,$\mu$and$\nu$arenon-negativeBorelmeasureson$(a,b)$,and$u$isaweightfunctionon$(a,b)$.