Kruglyak, Natan; Maligranda, Lech; Persson, Lars-Erik Structure of the Hardy operator related to Laguerre polynomials and the Euler differential equation. (English) Zbl 1121.47025 Rev. Mat. Complut. 19, No. 2, 467-476 (2006). Let \(\omega \) be a positive locally integrable function on \((a,b)\), \(-\infty \leq a<b\leq + \infty \), such that \(\int_a^b \omega (t)\,dt =\infty \). Suppose that \(W(x)=\int_a^x \omega (t)\,dt < \infty \) for any \(x\in (a,b)\). Denote \(H_\omega f(x)=[1/W(x)] \int_a^x f(t)\,dt\), \(A_\omega f(x)= [\sqrt{\omega (x)}/W(x)]\int_a^x f(t)\sqrt{\omega (t)}\,dt\). It is shown that \(I-A_\omega \) is a shift isometry in \(L^2(a,b)\) and \(I-H_\omega \) is a shift isometry in \(L^2_\omega (a,b)\). The authors use this result to study properties of a solution of the Euler differential equation of the first order \(y'(x)-y(x)/x=g(x)\), \(y(0)=0\), \(x>0\). Reviewer: Dagmar Medková (Praha) Cited in 3 Documents MSC: 47B38 Linear operators on function spaces (general) 47N20 Applications of operator theory to differential and integral equations 42B35 Function spaces arising in harmonic analysis Keywords:Hardy inequalities; Hardy operator; Laguerre polynomials; basis PDFBibTeX XMLCite \textit{N. Kruglyak} et al., Rev. Mat. Complut. 19, No. 2, 467--476 (2006; Zbl 1121.47025) Full Text: DOI EuDML