## Almost everywhere convergence of inverse Fourier transforms

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- by Leonardo Colzani, Christopher Meaney and Elena Prestini
- Proc. Amer. Math. Soc.
**134**(2006), 1651-1660 - DOI: https://doi.org/10.1090/S0002-9939-05-08329-2
- Published electronically: October 18, 2005
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## Abstract:

We show that if $\log (2-\Delta )f\in L^2({\mathbb R}^d)$, then the inverse Fourier transform of $f$ converges almost everywhere. Here the partial integrals in the Fourier inversion formula come from dilates of a closed bounded neighbourhood of the origin which is star shaped with respect to $0$. Our proof is based on a simple application of the Rademacher-Menshov Theorem. In the special case of spherical partial integrals, the theorem was proved by Carbery and Soria. We obtain some partial results when $\sqrt {\log (2-\Delta )}f\in L^2({\mathbb R}^d)$ and $\log \log (4-\Delta )f\in L^2({\mathbb R}^d)$. We also consider sequential convergence for general elements of $L^2({\mathbb R}^d)$.## References

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## Bibliographic Information

**Leonardo Colzani**- Affiliation: Dipartimento di Matematica e Applicazioni, Università di Milano Bicocca, Edificio U5, via Cozzi 53, 20125 Milano, Italy
- MR Author ID: 50785
- Email: leonardo@matapp.unimib.it
**Christopher Meaney**- Affiliation: Department of Mathematics, Macquarie University, North Ryde NSW 2109, Australia
**Elena Prestini**- Affiliation: Dipartimento di Matematica, Università di Roma “Tor Vergata”, Via della Ricerca Scientifica, 00133 Roma, Italy
- Received by editor(s): December 27, 2004
- Published electronically: October 18, 2005
- Additional Notes: The second author was partially supported by Progetto cofinanziato MIUR “Analisi Armonica”. We are grateful to Fulvio Ricci and the Centro di Ricerca Matematica Ennio De Giorgi for their hospitality

The third author was partially supported by Progetto cofinanziato MIUR “Analisi Armonica”. - Communicated by: Andreas Seeger
- © Copyright 2005
American Mathematical Society

The copyright for this article reverts to public domain 28 years after publication. - Journal: Proc. Amer. Math. Soc.
**134**(2006), 1651-1660 - MSC (2000): Primary 42B10, 43A50; Secondary 42C15
- DOI: https://doi.org/10.1090/S0002-9939-05-08329-2
- MathSciNet review: 2204276