Weighted inequalities for Hardy-type operators involving suprema

Let $u$ and $b$ be two weight functions on $(0,\infty)$. Assume that u is continuous on $(0,\infty)$ and that $b$ is such that the function $B(t)=\int_0^t b(s)\,ds$ satisfies $0<B(t)<\infty$ for every $t\in (0,\infty)$. Let the operator $T_{u,b}$ be given at a measurable non-negative function $g$ on $(0,\infty)$ by $$ (T_{u,b}g)(t)=\sup_{t\leq \tau<\infty}\frac{u(\tau)}{B(\tau)}\int_0^\tau g(s)b(s)\,ds. $$ We give necessary and sufficient conditions on weights $v,w$ on $(0,\infty)$ for which there exists a positive constant $C$ such that the inequality $$\left(\int_0^\infty [(T_{u,b}g)(t)]^qw(t)\,dt\right)^{1/q}\leq C\left(\int_0^\infty [g(t)]^p v(t)\,dt\right)^{1/p}$$ holds for every measurable non-negative function $g$ on $(0,\infty)$, where $p, q\in (0,\infty)$. satisfy certain restrictions. We also characterize weights $v, w\in (0,\infty)$ for which there exists a positive constant $C$ such that the inequality $$\left(\int_0^\infty [(T_{u,b}\varphi)(t)]^qw(t)\,dt\right)^{1/q}\leq C\left(\int_0^\infty [\varphi(t)]^p v(t)\,dt\right)^{1/p}$$