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Integral conditions for Hardy-type operators involving suprema

Research paper by Martin Křepela

Indexed on: 15 Aug '16Published on: 23 Apr '16Published in: Collectanea Mathematica



Abstract

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.AbstractWe 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 \(0<q<\infty \) . $$\begin{aligned} \left( \int _0^\infty \Big [ \sup _{s\in [t,\infty )} u(s) \int _s^\infty g(x)\,\mathrm {d}x\Big ]^q w(t)\,\mathrm {d}t\right) ^\frac{1}{q} \le C \left( \int _0^\infty g^p(t) v(t)\,\mathrm {d}t\right) ^\frac{1}{p} \end{aligned}$$ $$\begin{aligned} \left( \int _0^\infty \Big [ \sup _{s\in [t,\infty )} u(s) \int _s^\infty g(x)\,\mathrm {d}x\Big ]^q w(t)\,\mathrm {d}t\right) ^\frac{1}{q} \le C \left( \int _0^\infty g^p(t) v(t)\,\mathrm {d}t\right) ^\frac{1}{p} \end{aligned}$$g \((0,\infty )\) \((0,\infty )\) \(1\le p<\infty \) \(1\le p<\infty \) \(0<q<\infty \) \(0<q<\infty \)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. $$\begin{aligned} \left( \int _0^\infty \Big [ \sup _{s\in [t,\infty )} u(s) f(s) \Big ]^q w(t)\,\mathrm {d}t\right) ^\frac{1}{q} \le C \left( \int _0^\infty f^p(t) v(t)\,\mathrm {d}t\right) ^\frac{1}{p} \end{aligned}$$ $$\begin{aligned} \left( \int _0^\infty \Big [ \sup _{s\in [t,\infty )} u(s) f(s) \Big ]^q w(t)\,\mathrm {d}t\right) ^\frac{1}{q} \le C \left( \int _0^\infty f^p(t) v(t)\,\mathrm {d}t\right) ^\frac{1}{p} \end{aligned}$$f \((0,\infty )\) \((0,\infty )\) \(0<q<p<\infty \) \(0<q<p<\infty \)