The Euler equation for the hardy operator

Yoshimi Saito UAB, USA

We define the Euler equation for the (generalized) Hardy operator

T f(x) = v(x) ∫_{a}^{x} u(t)f(t) dt

from L^{p}(I) into L^{q}(I), where I = (a, b), v ∈ L^{q}(I) and u ∈ L^{p′}(I) (1/p + 1/p′ = 1). We discuss the relationship between the ”eigenvalues” of the Euler equation and the approximation numbers a_{n}(T), n = 1, 2, · · ·. In the case p = q, it can be shown that ”the eigenvalue = the approximation numbers”.