Sharp Hardy's Inequalities in Hilbert Spaces

2023/06/13 by Dimitrov, Dimitar K., Gadjev, Ivan, Ismail, Mourad E. H.
#26D15 #Classical Analysis and ODEs (math.CA) #FOS: Mathematics #Primary 26D10 #Secondary 33D45

paper · doi:10.48550/arxiv.2306.08172

Abstract

We study the behavior of the smallest possible constants d(a,b) and d_n in Hardy's inequalities ∫_a^b((1)/(x)∫_a^xf(t)dt)² dx≤ d(a,b) ∫_a^b [f(x)]² dx and ∑_k=1ⁿ((1)/(k)∑_j=1^ka_j)²≤ d_n ∑_k=1ⁿa_k². The exact constant d(a,b) and the precise rate of convergence of d_n are established and the extremal function and the ``almost extremal'' sequence are found.

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