Boundedness and approximation of Hilbert-type operators in the Triebel–Lizorkin spaces: Applications to non-smooth volatility dynamics

Philip Ajibola Bankole; Mohsin Nasir; Manuel De la Sen; Sina Etemad; Philip Ajibola Bankole; Mohsin Nasir; Manuel De la Sen; Sina Etemad

doi:10.3934/math.2025969

AIMS Mathematics

2025, Volume 10, Issue 9: 21794-21819. doi: 10.3934/math.2025969

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Boundedness and approximation of Hilbert-type operators in the Triebel–Lizorkin spaces: Applications to non-smooth volatility dynamics

1.
Lagos State University of Education, Department of Mathematics, Oto/Ijanikin, Nigeria; Email: bankolepa@lasued.edu.ng (P.A.B)
2.
Abdus Salam School of Mathematical Sciences, Government College University, Lahore 54600, Pakistan; Email: mohsinnasir_22@sms.edu.pk (M.N.)
3.
Department of Mathematics, University of Management and Technology, Lahore 54770, Pakistan
4.
Department of Electricity and Electronics, Institute of Research and Development of Processes, Faculty of Science and Technology, University of the Basque Country, 48940-Leioa, Bizkaia, Spain
5.
Department of Mathematics, Azarbaijan Shahid Madani University, Tabriz 53751, Iran
6.
Jadara Research Center, Jadara University, Irbid 21110, Jordan

Received: 15 July 2025 Revised: 26 August 2025 Accepted: 08 September 2025 Published: 18 September 2025
MSC : 42B20, 46E35, 42B25, 47G10, 91G80

This paper investigates the boundedness and approximation properties of Hilbert-type singular integral operators within the framework of Triebel–Lizorkin spaces $ F^{s}_{p, q}(\mathbb{R}) $, a refined class of function spaces central to microlocal and harmonic analysis. We introduced a regularized Hilbert-type operator and established its strong convergence in Triebel–Lizorkin norms. Using Fourier analytic and interpolation techniques, we rigorously proved new boundedness results under optimal smoothness and integrability conditions. Furthermore, we demonstrated how this functional analytic framework enables robust modeling of non-smooth volatility structures in financial derivatives pricing, particularly under stochastic volatility regimes and market turbulence. This approach unifies singular operator theory and financial mathematics, offering a novel path for analyzing irregular price dynamics through spectral and geometric regularity tools.
- Hilbert-type operators,
- Triebel–Lizorkin spaces,
- singular integral approximation,
- stochastic volatility modeling,
- microlocal harmonic analysis
Citation: Philip Ajibola Bankole, Mohsin Nasir, Manuel De la Sen, Sina Etemad. Boundedness and approximation of Hilbert-type operators in the Triebel–Lizorkin spaces: Applications to non-smooth volatility dynamics[J]. AIMS Mathematics, 2025, 10(9): 21794-21819. doi: 10.3934/math.2025969

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Abstract

This paper investigates the boundedness and approximation properties of Hilbert-type singular integral operators within the framework of Triebel–Lizorkin spaces $ F^{s}_{p, q}(\mathbb{R}) $, a refined class of function spaces central to microlocal and harmonic analysis. We introduced a regularized Hilbert-type operator and established its strong convergence in Triebel–Lizorkin norms. Using Fourier analytic and interpolation techniques, we rigorously proved new boundedness results under optimal smoothness and integrability conditions. Furthermore, we demonstrated how this functional analytic framework enables robust modeling of non-smooth volatility structures in financial derivatives pricing, particularly under stochastic volatility regimes and market turbulence. This approach unifies singular operator theory and financial mathematics, offering a novel path for analyzing irregular price dynamics through spectral and geometric regularity tools.

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