Stepanov, V. D.; Ushakova, E. P. Hardy-Steklov operators and duality principle in weighted Sobolev spaces of the first order. (English. Russian original) Zbl 1403.47004 Dokl. Math. 97, No. 3, 232-235 (2018); translation from Dokl. Akad. Nauk, Ross. Akad. Nauk 480, No. 2, 150-154 (2018). The paper is aimed to present a boundedness criteria for the Hardy-Steklov operator in Lebesgue spaces and its applications. The authors define the Hardy-Steklov operator from \(L^p(c,d)\) to \(L^q(a,b)\) as \[ \mathcal Hf(x):=w(x)\int_{\phi(x)}^{\psi(x)}f(y)v(y)\,dy,\;\;\;x\in I:=(a,b), \]\[ -\infty\leq c\leq\phi(x)\leq\psi(x)\leq d\leq\infty,\;\phi(a)=\psi(a)=c,\;\phi(b)=\psi(b)=d, \] \(\phi,\psi\) are differentiable and strictly increasing, and weight functions \(v\) and \(w\) are nonnegative and locally summable. First, the authors introduce several numerical characteristics of boundary functions \(\phi\) and \(\psi\) and relate their finiteness to the boundedness of \(\mathcal H\). Second, these results are used to establish the duality principle in the first-order weighted Sobolev space \(W^1_{p,s}(I)\) and its subspaces. This duality principle consists in proving two-side estimates for \(J_X(g)\) and \(\mathbf J_X(g)\) in terms of functionals independent of \(f\in X\), where \[ J_X(g):=\sup\left|\int_Ifg\right|/\|f\|_{W^1_{p,s}(I)},\;\;\;\mathbf J_X(g):=\sup\left(\int_I|fg|\right)/\|f\|_{W^1_{p,s}(I)}, \] and the supremum is taken over all \(f\in X\) with positive norm. Reviewer: Dmitri V. Prokhorov (Saratov) Cited in 1 Document MSC: 47G10 Integral operators 46A20 Duality theory for topological vector spaces 46E35 Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems Keywords:Hardy-Steklov operator; Sobolev space; duality principle; Lebesgue space PDFBibTeX XMLCite \textit{V. D. Stepanov} and \textit{E. P. Ushakova}, Dokl. Math. 97, No. 3, 232--235 (2018; Zbl 1403.47004); translation from Dokl. Akad. Nauk, Ross. Akad. Nauk 480, No. 2, 150--154 (2018) Full Text: DOI References: [1] K. T. Mynbaev and M. Otelbaev, Weighted Function Spaces and Spectra of Differential Operators (Nauka, Moscow, 1988) [in Russian]. · Zbl 0651.46037 [2] Maz’ya, V. G.; Verbitsky, I. E., “boundedness and compactness criteria for the one-dimensional Schrödinger operator,”, 369-382, (2000) · Zbl 1034.34101 [3] Prokhorov, D. V.; Stepanov, V. D., No article title, Sib. Math. J., 43, 694-707, (2002) [4] Maz’ya, V. G., No article title, J. Comput. Appl. Mat., 194, 94-114, (2006) · Zbl 1104.46020 [5] Nasyrova, M. G.; Ushakova, E. P., No article title, Proc. Steklov Inst. Math., 293, 228-254, (2016) · Zbl 1358.47033 [6] Oinarov, R., No article title, Complex Var. Elliptic Equations, 56, 1021-1038, (2011) · Zbl 1226.26013 [7] Oinarov, R., No article title, Izv. Math., 78, 836-853, (2014) · Zbl 1305.47032 [8] Oinarov, R., No article title, J. London Math. Soc., 48, 103-116, (1993) · Zbl 0811.26008 [9] Prokhorov, D. V.; Stepanov, V. D.; Ushakova, E. P., No article title, “Hardy-Steklov integral operators,” Modern Problems in Mathematics, 22, 3-185, (2016) [10] Prokhorov, D. V.; Stepanov, V. D.; Ushakova, E. P., No article title, Math. Nachr., 290, 890-912, (2017) · Zbl 1375.46029 [11] Prokhorov, D. V.; Stepanov, V. D.; Ushakova, E. P., No article title, Dokl. Math., 93, 78-81, (2016) · Zbl 1358.46032 [12] C. Bennett and R. Sharpley, Interpolation of Operators (Academic, Boston, MA, 1988). · Zbl 0647.46057 [13] Eveson, S. P.; Stepanov, V. D.; Ushakova, E. P., No article title, Math. Nachr., 288, 877-897, (2015) · Zbl 1345.46028 [14] Stepanov, V. D.; Ushakova, E. P., No article title, Proc. Steklov Inst. Math., 232, 290-309, (2001) [15] Stepanov, V. D.; Ushakova, E. P., No article title, Math. Inequal. Appl., 13, 449-510, (2010) 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.