International Journal of Mathematics and Mathematical Sciences
Volume 2008 (2008), Article ID 214030, 8 pages
doi:10.1155/2008/214030
Research Article
Some Estimates of Schrödinger-Type Operators with Certain Nonnegative Potentials
1Department of Mathematics and Mechanics, University of Science and Technology Beijing, Beijing 100083, China
2College of Science, Shandong Jianzhu University, Jinan 250101, Shandong Province, China
Received 4 September 2008; Revised 1 November 2008; Accepted 3 November 2008
Academic Editor: Jie Xiao
Copyright © 2008 Yu Liu and Youzheng Ding. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
Abstract
We consider the Schrödinger-type operator H=(−Δ)2+V2, where the nonnegative potential V belongs to the reverse Hölder class Bq1 for q1≥n/2, n≥5. The Lp estimates of the operator ∇4H−1 related to H are obtained when V∈Bq1 and 1<p≤q1/2. We also obtain the weak-type estimates of the operator ∇4H−1 under the same condition of V.