Annales Academiæ Scientiarum Fennicæ
Mathematica
Volumen 39, 2014, 887-896

A SHARP INTEGRAL HARDY TYPE INEQUALITY AND APPLICATIONS TO MUCKENHOUPT WEIGHTS ON R

Eleftherios N. Nikolidakis

National and Kapodistrian University of Athens, Department of Mathematics
Panepistimioupolis, GR 157 84, Athens, Greece; lefteris 'at' math.uoc.gr

Abstract. We prove a generalization of a Hardy type inequality for negative exponents valid for non-negative functions defined on [0,1). As an application we find the exact best possible range of p such that 1 < p ≤ q such that any non-decreasing φ which satisfies the Muckenhoupt A_q condition with constant c upon all open subintervals of [0,1) should additionally satisfy the A_p condition for another possibly real constant c'. The result have been treated in [9] based on [1], but we give in this paper an alternative proof which relies on the above mentioned inequality.

2010 Mathematics Subject Classification: Primary 26D15; Secondary 42B25.

Key words: Hardy inequalities, Muckenhoupt weights.

Reference to this article: E.N. Nikolidakis: A sharp integral Hardy type inequality and applications to Muckenhoupt weights on R. Ann. Acad. Sci. Fenn. Math. 39 (2014), 887-896.

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doi:10.5186/aasfm.2014.3947