Some New Hardy Type Inequalities and their Limiting Inequalities

Volume 4, Issue 3 (GI8), 2003

Article 61

SOME NEW HARDY TYPE INEQUALITIES AND THEIR LIMITING INEQUALITIES

DEPARTMENT OF MATHEMATICS
LULEA UNIVERSITY
SE- 971 87 LULEA
SWEDEN.
E-Mail: annaw@sm.luth.se

Received 12 December, 2002; Accepted 08 September, 2003.
Communicated by: B. Opic

ABSTRACT. A new necessary and sufficient condition for the weighted Hardy inequality is proved for the case $1<p\leq q<\infty$ . The corresponding limiting Pólya-Knopp inequality is also proved for $0<p\leq q<\infty$ . Moreover, a corresponding limiting result in two dimensions is proved. This result may be regarded as an endpoint inequality of Sawyer's two-dimensional Hardy inequality. But here we need only one condition to characterize the inequality whereas in Sawyer's case three conditions are necessary.

Key words:
Inequalities, Weights, Hardy's inequality, Pólya-Knopp's inequality, Multidimensional inequalities, Sawyer's two-dimensional operator.

2000 Mathematics Subject Classification:
26D15.

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