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Journal of Fourier Analysis and Applications

, Volume 25, Issue 1, pp 184–209 | Cite as

On Lebesgue Integrability of Fourier Transforms in Amalgam Spaces

  • Sekou Coulibaly
  • Ibrahim FofanaEmail author
Article
  • 67 Downloads

Abstract

Let f be an element of the subspace \((L^{q},l^{p})^{\alpha }({\mathbb {R}}^d)\) (\(1\le q \le \alpha \le p \le 2 \)) of the Wiener amalgam space \((L^{q},l^{p})({\mathbb {R}}^{d})\). It is well-known that the Fourier transform \(\widehat{f}\) of f belongs to \((L^{p'},l^{q'})^{\alpha '}({\mathbb {R}}^d)\) where \(p'\), \(q'\) and \(\alpha '\) are the conjugate exponents of p, q and \(\alpha \) respectively. We give sufficient conditions, in terms of a modulus of continuity of f, for \(\widehat{f}\) to be in a Lebesgue space \(L^{\beta }({\mathbb {R}}^{d})\) or in an amalgam space \((L^{p'},l^{s})^{\beta }({\mathbb {R}}^d)\) other than \((L^{p'},l^{q'})^{\alpha '}({\mathbb {R}}^d)\). As an application, we obtain a sufficient condition for the solvability in \([L^{2}({\mathbb {R}}^d)]^d\) of the equation \(\hbox {div}{F}=f\) .

Keywords

Fourier transform Modulus of continuity Amalgam spaces 

Mathematics Subject Classification

Primary 42B10 Secondary 42B35 

Notes

Acknowledgements

The authors thank the referee for valuable remarks that improved the presentation.

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Authors and Affiliations

  1. 1.D.E.R Mathématiques et InformatiqueUniversité de BamakoBamakoMali
  2. 2.Laboratoire de Mathématiques Fondamentales, UFR de Mathématiques et InformatiqueUniversité de CocodyAbidjan 22Côte d’Ivoire

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