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Structure of the Hardy operator related to Laguerre polynomials and the Euler differential equation. (English) Zbl 1121.47025
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 $$y'(x)-y(x)/x=g(x)$$, $$y(0)=0$$, $$x>0$$.

##### 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
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