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Equivalence of Hardy-type inequalities with general measures on the cones of non-negative respective non-increasing functions. (English) Zbl 1093.26023

The aim of the paper is to extend results for Hardy’s operators proved in [G.Sinnamon, Function Spaces, Differential Operators and Nonlinear Analysis (Eds.: P.Drábek and J.Rákosník), Math. Inst. Acad. Sciences of the Czech Republic, Prague, 292-310 (2005)] to more general integral operators. To be more specific, we mention one of the main results here:
Let \(\lambda\) and \(\mu\) be regular Borel measures on \(\mathbb R_+ : = [0,\infty)\), let \(k(x,y)\) be a non-negative function on \(\mathbb R_+ \times \mathbb R_+\) which is non-increasing in \(y\) for every \(x>0\) and let \[ (\mathbb Kf) (x) : = \int_{[0,x]} k(x,y) f(x) \,d\lambda (y),\quad x\in \mathbb R_+. \] Denote by \(E\) the set of all non-negative \(\lambda\)-measurable functions on \(\mathbb R_+\) and by \(E^{\downarrow}\) its subset consisting of those functions that are non-increasing on \(\mathbb R_+\). Let \(1\leq p <\infty\) and \(0<q<\infty\). Then the inequalities
\[ \| Kf\| _{L^q(\mu)} \leq C_1 (p,q)\, \| f\| _{L^p(\lambda)} \quad\text{for all} \;f\in E\tag{1} \]
and
\[ \| Kf\| _{L^q(\mu)} \leq C_2 (p,q)\, \| f\| _{L^p(\lambda)} \quad\text{for all} \;f\in E^{\downarrow} (2) \] are equivalent and the least constants \(C_1(p,q)\) and \(C_2(p,q)\) in (1) and (2) coincide.
In the case that \((\mathbb Kf)(x):=\int_{[0,x]} f\,d\lambda/\int_{[0,x]}\, d\lambda\), this result is essentially that of G. Sinnamon (loc. cit.). Moreover, if \(p>1\), then (1) holds if and only if \[ \| f\| _{L^q(\mu)} \leq C_3(p,q) \| f\| _{L^p(\lambda)} \quad\text{for all} \;f\in E^{\downarrow}. (3) \]

MSC:

26D15 Inequalities for sums, series and integrals
47B38 Linear operators on function spaces (general)
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References:

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