Integral conditions for Hardy-type operators involving suprema

Abstract

We characterize the validity of the weighted inequality

$$\begin{aligned} \left( \int _0^\infty \Big [ \sup _{s\in [t,\infty )} u(s) \int _s^\infty g(x)\,\mathrm {d}x\Big ]^q w(t)\,\mathrm {d}t\right) ^\frac{1}{q} \le C \left( \int _0^\infty g^p(t) v(t)\,\mathrm {d}t\right) ^\frac{1}{p} \end{aligned}$$

for all nonnegative functions g on $(0,\infty )$, with exponents in the range $1\le p<\infty$ and $0<q<\infty$.

Moreover, we give an integral characterization of the inequality

$$\begin{aligned} \left( \int _0^\infty \Big [ \sup _{s\in [t,\infty )} u(s) f(s) \Big ]^q w(t)\,\mathrm {d}t\right) ^\frac{1}{q} \le C \left( \int _0^\infty f^p(t) v(t)\,\mathrm {d}t\right) ^\frac{1}{p} \end{aligned}$$

being satisfied for all nonnegative nonincreasing functions f on $(0,\infty )$ in the case $0<q<p<\infty$, for which an integral condition was previously unknown.