Sharp Estimates of the Identity Minus Hardy Operator on the Cone of Decreasing Functions
2008 (English)In: Proceedings of the American Mathematical Society, ISSN 0002-9939, E-ISSN 1088-6826, Vol. 136, no 7, 2505-2513 p.Article in journal (Refereed) Published
It is shown that if we restrict the identity minus Hardy operator on the cone of nonnegative decreasing functions f in L-p, then we have the sharp estimate
parallel to(I - H)f parallel to(Lp) <= 1/(p - 1)(1/p) parallel to f parallel to(Lp) for p = 2, 3, 4, .... In other words,
parallel to f** - f*parallel to(Lp) <= 1/(p - 1)(1/p) parallel to f parallel to(Lp) for each f is an element of L-p and each integer p >= 2.
It is also shown, via a connection between the operator I - H and Laguerre functions, that
parallel to(1 - alpha)I + Phi(I - H)parallel to(L2 -> L2) = parallel to I - alpha H parallel to(L2 -> L2) = 1 for all a is an element of [ 0, 1].
Place, publisher, year, edition, pages
American Mathematical Society (AMS), 2008. Vol. 136, no 7, 2505-2513 p.
The Hardy operator, cone of decreasing functions, sharp estimates
IdentifiersURN: urn:nbn:se:liu:diva-90402DOI: 10.1090/S0002-9939-08-09200-9OAI: oai:DiVA.org:liu-90402DiVA: diva2:612938