Local Spectral Formula for Integral Operators on \(L_{p}({\mathbb T})\)

Abstract

Let \(1\leq p\leq \infty \), \(f\in L_{p}({\mathbb T})\) and \(0 \notin \text {supp} \hat {f}\). Then, in this paper, we obtain the following local spectral formula for the integral operator I on \(L_{p}({\mathbb T})\), the space of 2π-periodic functions belonging to L p (−π,π): \( \lim _{n\rightarrow \infty } \|I^{n} f\|_{p,{\mathbb T}}^{1/n}= \sigma ^{-1}, \) where \(\sigma =\min \{ |k| : k \in \text {supp} \hat {f} \}, If(x)={{\int }_{0}^{x}} f(t) dt -c_{f}, x\in \mathbb {R}\) and the constant c f is chosen such that \({\int }_{0}^{2\pi } If (x) dx=0\). The local spectral formula for polynomial integral operators on \(L_{p}({\mathbb T})\) is also given.

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Acknowledgements

The research of V. N. Huy is funded by the Vietnam National University, Hanoi (VNU) under project number QG.16.08. A part of this work was done when V. N. Huy is working at the Vietnam Institute for Advanced Study in Mathematics (VIASM); the author would like to thank the VIASM for providing a fruitful research environment and working condition. The authors would like to thank the referees for useful remarks and comments.

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Correspondence to Ha Huy Bang.

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Bang, H.H., Huy, V.N. Local Spectral Formula for Integral Operators on \(L_{p}({\mathbb T})\) . Vietnam J. Math. 45, 737–746 (2017). https://doi.org/10.1007/s10013-017-0242-2

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Keywords

  • L p - spaces
  • Fourier transform
  • Generalized functions

Mathematics Subject Classification (2010)

  • Primary 26D10
  • Secondary 46E30