Full Length ArticleOn the optimal relationships between -norms for the Hardy operator and its dual for decreasing functions☆
Section snippets
Introduction and main results
Denote by the class of all nonnegative measurable functions on . Let . Set and These equalities define the classical Hardy operator and its dual operator . By Hardy’s inequalities [4, Ch. 9], these operators are bounded in for any . Furthermore, it is easy to show that for any and any (as usual, .
However, the constants in (1.1) are not optimal. Sharp constants are
Proof of Theorem 1.3
Let . Taking into account (1.1), we may assume that and belong to . Denote Since , we have Thus, integrating by parts, we obtain
Further, set and . Since , we have Thus, by Fubini’s theorem, Set
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