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Titchmarsh theorems on Damek-Ricci spaces via moduli of continuity of higher order

Manoj Kumar, Vishvesh Kumar (UGent) and Michael Ruzhansky (UGent)
(2025) INTERNATIONAL JOURNAL OF MATHEMATICS. 36(6).
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Abstract
A classical theorem of Titchmarsh relates the 𝐿2-Lipschitz functions and decay of the Fourier transform of the functions. In this note, we prove the Titchmarsh theorem for Damek–Ricci space (also known as harmonic 𝑁⁢𝐴 groups) via moduli of continuity of higher orders. We also prove an analogue of another Titchmarsh theorem which provides integrability properties of the Fourier transform for functions in the Hölder Lipschitz spaces.
Keywords
FOURIER-TRANSFORMS, GROWTH-PROPERTIES, EXTENSIONS, Titchmarsh theorems, Damek-Ricci spaces, harmonic NA groups, Helgason transform, moduli of continuity, generalized Lipschitz class

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Citation

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MLA
Kumar, Manoj, et al. “Titchmarsh Theorems on Damek-Ricci Spaces via Moduli of Continuity of Higher Order.” INTERNATIONAL JOURNAL OF MATHEMATICS, vol. 36, no. 6, 2025, doi:10.1142/S0129167X25500016.
APA
Kumar, M., Kumar, V., & Ruzhansky, M. (2025). Titchmarsh theorems on Damek-Ricci spaces via moduli of continuity of higher order. INTERNATIONAL JOURNAL OF MATHEMATICS, 36(6). https://doi.org/10.1142/S0129167X25500016
Chicago author-date
Kumar, Manoj, Vishvesh Kumar, and Michael Ruzhansky. 2025. “Titchmarsh Theorems on Damek-Ricci Spaces via Moduli of Continuity of Higher Order.” INTERNATIONAL JOURNAL OF MATHEMATICS 36 (6). https://doi.org/10.1142/S0129167X25500016.
Chicago author-date (all authors)
Kumar, Manoj, Vishvesh Kumar, and Michael Ruzhansky. 2025. “Titchmarsh Theorems on Damek-Ricci Spaces via Moduli of Continuity of Higher Order.” INTERNATIONAL JOURNAL OF MATHEMATICS 36 (6). doi:10.1142/S0129167X25500016.
Vancouver
1.
Kumar M, Kumar V, Ruzhansky M. Titchmarsh theorems on Damek-Ricci spaces via moduli of continuity of higher order. INTERNATIONAL JOURNAL OF MATHEMATICS. 2025;36(6).
IEEE
[1]
M. Kumar, V. Kumar, and M. Ruzhansky, “Titchmarsh theorems on Damek-Ricci spaces via moduli of continuity of higher order,” INTERNATIONAL JOURNAL OF MATHEMATICS, vol. 36, no. 6, 2025.
@article{01JPAADAE2HJWEA3VJ4ZXX3K5Y,
  abstract     = {{A classical theorem of Titchmarsh relates the 𝐿2-Lipschitz functions and decay of the Fourier transform of the functions. In this note, we prove the Titchmarsh theorem for Damek–Ricci space (also known as harmonic 𝑁⁢𝐴 groups) via moduli of continuity of higher orders. We also prove an analogue of another Titchmarsh theorem which provides integrability properties of the Fourier transform for functions in the Hölder Lipschitz spaces.}},
  articleno    = {{S0129167X25500016}},
  author       = {{Kumar, Manoj and Kumar, Vishvesh and Ruzhansky, Michael}},
  issn         = {{0129-167X}},
  journal      = {{INTERNATIONAL JOURNAL OF MATHEMATICS}},
  keywords     = {{FOURIER-TRANSFORMS,GROWTH-PROPERTIES,EXTENSIONS,Titchmarsh theorems,Damek-Ricci spaces,harmonic NA groups,Helgason transform,moduli of continuity,generalized Lipschitz class}},
  language     = {{eng}},
  number       = {{6}},
  pages        = {{15}},
  title        = {{Titchmarsh theorems on Damek-Ricci spaces via moduli of continuity of higher order}},
  url          = {{http://doi.org/10.1142/S0129167X25500016}},
  volume       = {{36}},
  year         = {{2025}},
}
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