Abstract
In this paper, we consider functions \({u\in W^{m,1}(0,1)}\) where m ≥ 2 and u(0) = Du(0) = · · · = D m-1 u(0) = 0. Although it is not true in general that \({\frac{D^ju(x)}{x^{m-j}} \in L^1(0,1)}\) for \({j\in \{0,1,\ldots,m-1\}}\), we prove that \({\frac{D^ju(x)}{x^{m-j-k}} \in W^{k,1}(0,1)}\) if k ≥ 1 and 1 ≤ j + k ≤ m, with j, k integers. Furthermore, we have the following Hardy type inequality,
where the constant is optimal.
Similar content being viewed by others
References
Hardy, G.H., Littlewood, J.E., Pólya, G.: Inequalities, 2nd edn. Cambridge University Press, Cambridge (1952)
Brezis, H.: Functional Analysis, Sobolev Spaces and Partial Differential Equations. Springer, New York (2010)
Author information
Authors and Affiliations
Corresponding author
Additional information
Communicated by H. Brezis.
Rights and permissions
About this article
Cite this article
Castro, H., Wang, H. A Hardy type inequality for W m,1(0, 1) functions. Calc. Var. 39, 525–531 (2010). https://doi.org/10.1007/s00526-010-0322-6
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00526-010-0322-6