Structure of the Hardy operator related to Laguerre polynomials and the Euler differential equation (Q875700)

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 \(y'(x)-y(x)/x=g(x)\), \(y(0)=0\), \(x>0\).