Oinarov, Ryskul A weighted estimate for an intermediate operator on the cone of nonnegative functions. (Russian, English) Zbl 1043.47035 Sib. Mat. Zh. 43, No. 1, 161-173 (2002); translation in Sib. Math. J. 43, No. 1, 128-139 (2002). Summary: Consider the following integral operator \[ K_\beta f(x)=\int_{0}^{x} K^\beta (x,t) f(t)\, dt, \quad x>0, \;0\leq\beta\leq 1, \;K\equiv K_1. \] Under some restrictions on a positive continuous function \(K(x,s)\), we obtain necessary and sufficient conditions on weight functions \(u\), \(v\), and \(\rho\) that guarantee the inequality \(\| uK_\beta f\| _q \leq C(\| \rho f\| _p + \| vKf\| _r)\) for \(f\geq 0\), \(1<p,q,r<\infty\), and \(q\geq\max\{p,r\}\). MSC: 47G10 Integral operators 45P05 Integral operators 47B38 Linear operators on function spaces (general) 26D10 Inequalities involving derivatives and differential and integral operators Keywords:weighted norm inequalities for an integral operator; three-weight inequality; fractional integration; Riemann–Liouville operator; locally integrable function PDFBibTeX XMLCite \textit{R. Oinarov}, Sib. Mat. Zh. 43, No. 1, 161--173 (2002; Zbl 1043.47035); translation in Sib. Math. J. 43, No. 1, 128--139 (2002) Full Text: EuDML