18. Spectra of generalized Poisson integral operators on L<sup>p</sup>(ℝ<sup>+</sup>)

Miana, Pedro J.; Oliva-Maza, Jesús

doi:10.1515/9783110656756-018

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“…For real numbers, let for all . The spectral properties of its associated Hardy operator have been studied in [16]. Regarding the properties considered in the present paper, we have that, for and , if and only if , and in this case, .…”

Section: Examples and Applicationsmentioning

confidence: 86%

“…For α, β, μ real numbers, let P α,β, μ (r, s) = r α μ−β s β−1 (r α + s α ) −μ for all r, s > 0. The spectral properties of its associated Hardy operator have been studied in [16]. Regarding the properties considered in the present paper, we have that, for p ∈ [1, ∞) and α > 0, P α,β, μ ∈ H p if and only if 0 < β − 1/p < αμ, and in this case, P α,β, μ ∈ I p .…”

Section: Examples and Applicationsmentioning

confidence: 88%

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On Hardy kernels as reproducing kernels

Oliva-Maza

2022

Can. Math. Bull.

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Hardy kernels are a useful tool to define integral operators on Hilbertian spaces like $L^2(\mathbb R^+)$ or $H^2(\mathbb C^+)$ . These kernels entail an algebraic $L^1$ -structure which is used in this work to study the range spaces of those operators as reproducing kernel Hilbert spaces. We obtain their reproducing kernels, which in the $L^2(\mathbb R^+)$ case turn out to be Hardy kernels as well. In the $H^2(\mathbb C^+)$ scenario, the reproducing kernels are given by holomorphic extensions of Hardy kernels. Other results presented here are theorems of Paley–Wiener type, and a connection with one-sided Hilbert transforms.

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