SOME NEW HARDY TYPE INEQUALITIES AND THEIR LIMITING INEQUALITIES

Abstract

ABSTRACT. A new necessary and sufficient condition for the weighted Hardy inequality is proved for the case 1 < p ≤ q < ∞. The corresponding limiting Pólya-Knopp inequality is also proved for 0 < p ≤ q < ∞. 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.