ABSTRACT
We present a direct proof of a known result that the Hardy operator in the space can be written as , where is a shift operator , for some orthonormal basis . The basis is constructed by using classical Laguerre polynomials. We also explain connections with the Euler differential equation of the first order and point out some generalizations to the case with weighted spaces.
Key words: Hardy inequality, Hardy operator, Laguerre polynomials, isometry,
Lebesgue spaces, basis in
space, weighted
spaces.
2000 Mathematics Subject Classication: 47B38.