Full paper in PDF:
$%N. Kruglyak, L. Maligranda, and L.-E. Persson, Structure of the Hardy operator related to Laguerre polynomials and the Euler differential equation, Rev. Mat. Complut. 19 (2006), no. 2, 467476.%$

Structure of the Hardy Operator Related to Laguerre Polynomials and the Euler Dierential Equation
Natan KRUGLYAK, Lech MALIGRANDA,
and Lars-Erik PERSSON
Department of Mathematics
Luleå University of Technology
SE-971 87 LuleåSweden

Received: February 1, 2006
Accepted: March 21, 2006

ABSTRACT

We present a direct proof of a known result that the Hardy operator Hf(x) = 1  integral xf(t)dt
        x 0  in the space L2 = L2(0, oo )  can be written as H = I- U  , where U  is a shift operator (Uen = en+1  , n  (- Z)  for some orthonormal basis {en} . The basis {en} is constructed by using classical Laguerre polynomials. We also explain connections with the Euler differential equation of the first order y'- 1xy = g  and point out some generalizations to the case with weighted L2w(a,b)  spaces.

Key words: Hardy inequality, Hardy operator, Laguerre polynomials, isometry, Lebesgue spaces, basis in L2  space, weighted L2w(a,b)  spaces.
2000 Mathematics Subject Classication:
47B38.