Hardy’s inequalities, Hilbert inequalities and fractional integrals on function spaces of $q$-integral $p$-variation

We obtain Hardy’s inequalities and the Hilbert inequalities for function spaces of $q$-integral $p$-variation. We also establish the mapping properties of Riemann–Liouville integrals, Weyl integrals and Erdélyi–Kober fractional integrals on these function spaces.