Jain, P.; Kanjilal, S.; Stepanov, V. D.; Ushakova, E. P. On bilinear Hardy-Steklov operators. (English. Russian original) Zbl 1480.47067 Dokl. Math. 98, No. 3, 634-637 (2018); translation from Dokl. Akad. Nauk, Ross. Akad. Nauk 483, No. 6, 596-599 (2018). The Hardy-Steklov operator on \(L_{p}\)-space on \({\mathbb R}_{+}=(0,\infty)\) is defined by \((Hf)(x)=\int_{a(x)}^{b(x)}f(y)v(y)\,dy\), \(x > 0\), where \(a\) and \(b\) are smooth strictly increasing functions on \({\mathbb R}_{+}\) such that \(0 < a(x) < b(x)\) for all \(x\in {\mathbb R}_{+}\), \(v\) is a measurable nonnegative function on \({\mathbb R}_{+}\). The authors analyse the inequality \(\left( \int_{0}^{\infty} (H_{1}f\cdot H_{2}g)^{q}w^{q}\right)^{1/q}\le C \Vert f \Vert_{p_{1}}\Vert g \Vert_{p_{2}}\), where \(H_{1}\) and \(H_{2}\) are operators of the type mentioned above, \(w\) is a measurable nonnegative function on \({\mathbb R}_{+}\), \(0 < q < \infty\), \(p_{1}\), \(p_{2} > 1\), \(f\in L_{p_{1}}\), \(g\in L_{p_{2}}\) are arbitrary nonnegative functions, and \(C\) is the best possible constant. Under several assumptions, they obtain sharp estimates of this constant. Reviewer: Vladimir S. Pilidi (Rostov-na-Donu) Cited in 5 Documents MSC: 47G10 Integral operators 47A07 Forms (bilinear, sesquilinear, multilinear) Keywords:Hardy-Steklov operator; bilinear form; sharp estimate PDFBibTeX XMLCite \textit{P. Jain} et al., Dokl. Math. 98, No. 3, 634--637 (2018; Zbl 1480.47067); translation from Dokl. Akad. Nauk, Ross. Akad. Nauk 483, No. 6, 596--599 (2018) Full Text: DOI References: [1] Canestro, M. I. A.; Salvador, P. O.; Torreblanca, C. R., No article title, J. Math. Anal. Appl., 387, 320-334 (2012) · Zbl 1244.26033 [2] Salvador, P. O., No article title, Proc. R. Soc. Edinburgh, Sect. A, 144, 819-829 (2014) · Zbl 1298.26080 [3] Krepela, M., No article title, Publ. Mat., 61, 3-50 (2017) · Zbl 1359.26020 [4] Krepela, M., No article title, Proc. Edinburgh Math. Soc., 60, 955-971 (2018) · Zbl 1377.26019 [5] Prokhorov, D. V., No article title, Proc. Steklov Inst. Math., 293, 272-287 (2016) · Zbl 1365.26021 [6] Stepanov, V. D.; Shambilova, G. E., No article title, Dokl. Math., 96, 631-635 (2017) · Zbl 1390.26046 [7] Stepanov, V. D.; Shambilova, G. E., No article title, Sib. Math. J., 59, 505-522 (2018) · Zbl 1473.26033 [8] D. V. Prokhorov, V. D. Stepanov, and E. P. Ushakova, “Hardy-Steklov integral operators,” Modern Problems in Mathematics (Mat. Inst. Ross. Akad. Nauk, Moscow, 2016), Vol. 22, pp. 3-185 [in Russian]. · Zbl 06908415 [9] Stepanov, V. D.; Ushakova, E. P., No article title, Math. Ineq. Appl., 13, 449-510 (2010) [10] Nasyrova, M. G.; Ushakova, E. P., No article title, Proc. Steklov Inst. Math., 293, 228-254 (2016) · Zbl 1358.47033 [11] Stepanov, V. D.; Ushakova, E. P., No article title, Proc. Steklov Inst. Math., 232, 290-309 (2001) [12] Stepanov, V. D.; Ushakova, E. P., No article title, J. Funct. Spaces Appl., 1, 1-15 (2003) · Zbl 1063.26020 [13] Ushakova, E. P., No article title, J. Funct. Spaces Appl., 9, 67-107 (2011) · Zbl 1250.47032 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.