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Weighted norm inequalities for positive operators restricted on the cone of λ-quasiconcave functions
- Part of
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- Journal:
- Proceedings of the Royal Society of Edinburgh. Section A: Mathematics / Volume 150 / Issue 1 / February 2020
- Published online by Cambridge University Press:
- 23 January 2019, pp. 17-39
- Print publication:
- February 2020
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- Article
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Let ρ be a monotone quasinorm defined on
${\rm {\frak M}}^ + $, the set of all non-negative measurable functions on [0, ∞). Let T be a monotone quasilinear operator on 
${\rm {\frak M}}^ + $. We show that the following inequality restricted on the cone of λ-quasiconcave functions
where
$$\rho (Tf) \les C_1\left( {\int_0^\infty {f^p} v} \right)^{1/p},$$
$1\les p\les \infty $ and v is a weighted function, is equivalent to slightly different inequalities considered for all non-negative measurable functions. The case 0 < p < 1 is also studied for quasinorms and operators with additional properties. These results in turn enable us to establish necessary and sufficient conditions on the weights (u, v, w) for which the three weighted Hardy-type inequality
holds for all λ-quasiconcave functions and all 0 < p, q ⩽ ∞.
$$\left( {\int_0^\infty {{\left( {\int_0^x f u} \right)}^q} w(x){\rm d}x} \right)^{1/q} \les C_1\left( {\int_0^\infty {f^p} v} \right)^{1/p},$$
