Sharp Hardy’s inequalities in Hilbert spaces

Dimitar K. Dimitrov; Ivan Gadjev; Mourad E. H. Ismail

doi:10.4171/jst/520

JournalsjstVol. 14, No. 3pp. 1243–1256

Sharp Hardy’s inequalities in Hilbert spaces

Dimitar K. Dimitrov
Universidade Estadual Paulista, São José do Rio Preto, Brazil
Ivan Gadjev
University of Sofia, Sofia, Bulgaria
Mourad E. H. Ismail
University of Central Florida, Orlando, USA

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Abstract

We study the behavior of the smallest possible constants $d (a, b)$ and $d_{n}$ in Hardy’s inequalities

\int_{a}^{b} (\frac{1}{x} \int_{a}^{x} f (t) d t)^{2} d x \leq d (a, b) \int_{a}^{b} [f (x)]^{2} d x

and

k = 1 \sum n (\frac{1}{k} j = 1 \sum k a_{j})^{2} \leq d_{n} k = 1 \sum n a_{k}^{2} .

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.

Cite this article

Dimitar K. Dimitrov, Ivan Gadjev, Mourad E. H. Ismail, Sharp Hardy’s inequalities in Hilbert spaces. J. Spectr. Theory 14 (2024), no. 3, pp. 1243–1256

DOI 10.4171/JST/520