Gogatishvili, A.; Stepanov, V. D. Integral operators on cones of monotone functions. (English. Russian original) Zbl 1263.47057 Dokl. Math. 86, No. 2, 650-653 (2012); translation from Dokl. Akad. Nauk. 446, No. 4, 367-370 (2012). Let \(\mathfrak{M}^{\downarrow}\) be the set of all nonnegative nonincreasing functions on \(\mathbb{R}_+=[0,\infty)\). The authors consider the following inequality \[ \left(\int_0^\infty\left(\int_x^\infty k(t,x) f(t) u(t)\, dt\right)^{q}w(x)\, dx\right)^{\frac{1}{q}}\leq C\left(\int_0^\infty (f(t))^p v(t)\,dt\right)^{\frac{1}{p}}, \;\;f\in \mathfrak{M}^{\downarrow}, \] where the kernel \(k(x,t)\geq 0\) satisfies the Oinarov condition: there exists a constant \(D\geq 1\) independent of \(x, y, z\) such that \(D^{-1}(k(x,z)+k(z,y))\leq k(x,y)\leq D(k(x,z)+k(z,y))\) for all \(x\geq z\geq y\geq 0\). They obtain necessary and sufficient conditions for the above inequality, which shown to hold in the following cases: (i) \(0<p\leq 1,\;p\leq q<\infty\); (ii) \(0<q<p\leq 1\); (iii) \(1<p\leq q<\infty\); (iv) \(1<q<p<\infty\); (v) \(q=1\leq p<\infty\); (vi) \(0<q<1<p<\infty\). Reviewer: Fayou Zhao (Shanghai) Cited in 3 Documents MSC: 47G10 Integral operators 26D10 Inequalities involving derivatives and differential and integral operators 26D15 Inequalities for sums, series and integrals 26D07 Inequalities involving other types of functions Keywords:integral inequalities; monotone functions; Oinarov’s condition PDFBibTeX XMLCite \textit{A. Gogatishvili} and \textit{V. D. Stepanov}, Dokl. Math. 86, No. 2, 650--653 (2012; Zbl 1263.47057); translation from Dokl. Akad. Nauk. 446, No. 4, 367--370 (2012) Full Text: DOI References: [1] A. Gogatishvili and V. D. Stepanov, Dokl. Math. 86, 562–565 (2012). · Zbl 1281.26014 [2] R. Oinarov, Proc. Steklov Inst. Math. 204, 205–214 (1994). [3] E. A. Myasnikov, L.-E. Persson, and V. D. Stepanov, Acta Sci. Math. Szeged 59, 613–624 (1994). [4] O. V. Popova, Sib. Math. J. 53, 152–167 (2012). · Zbl 1257.26025 [5] E. Sawyer, Stud. Math. 96, 145–158 (1990). · Zbl 0705.42014 [6] Ì. L. Goldman, Eurasian Math. J. 2(3), 143–146 (2011). [7] G. Sinnamon and V. D. Stepanov, J. London Math. Soc. 54, 89–101 (1996). · Zbl 0856.26012 [8] Q. Lai, Proc. London Math. Soc. 79, 649–672 (1999). · Zbl 1030.46030 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.