RMC - Vol. 19, Num. 2 - Kruglyak et al.

Full paper in PDF:
$%N. Kruglyak, L. Maligranda, and L.-E. Persson, Structure of the Hardy operator related to Laguerre polynomials and the Euler differential equation, Rev. Mat. Complut. 19 (2006), no. 2, 467–476.%$

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 Luleå — Sweden

natan@sm.luth.se lech@sm.luth.se
larserik@sm.luth.se

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.