Fractional order inequalities of Hardy type (Q2708935)

The author considers a weighted generalization of the fractional Hardy type inequality in the form NEWLINE\[NEWLINE\left(\int_a^b\int_a^b\left|f(x)-f(y)\right|^q W(x,y) dx dy\right)^\frac{1}{q} \leq C \left(\int_a^b\left|f^\prime(x)\right|^p w(x) dx\right)^\frac{1}{p} \;.NEWLINE\]NEWLINE Given a non-negative function \(W(x,y)\), some sufficent conditions are proved which provide the construction of the weight \(w(x)\geq 0\) such that the above inequality is valid. The cases \(1<p\leq q <\infty\) and \(1<q<p <\infty\) are treated separately. Under some assumptions on \(W(x,y)\) there is also given a relation between \(W(x,y)\) and \(w(x)\) which is necessary and sufficient for the validity of this inequality in the case \(1<p\leq q <\infty\).NEWLINENEWLINEFor the entire collection see [Zbl 0947.00027].