Summability of Fourier transforms in variable Hardy and Hardy-Lorentz spaces

• Ferenc Weisz ^[1]
  1. [1] Eötvös L. University
• Localización: Jaen journal on approximation, ISSN 1889-3066, ISSN-e 1989-7251, Vol. 10, Nº. 1-2, 2018, págs. 101-131
• Idioma: inglés
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• Resumen

  • Let p(·) : Rn → (0,∞) be a variable exponent function satisfying the globally log-H¨older condition and 0 < q ≤ ∞. We introduce the variable Hardy and HardyLorentz spaces Hp(·)(Rd) and Hp(·),q(Rd). A general summability method, the so called θ-summability is considered for multi-dimensional Fourier transforms. Under some conditions on θ, it is proved that the maximal operator of the θ-means is bounded from Hp(·)(Rd) to Lp(·)(Rd) and from Hp(·),q(Rd) to Lp(·),q(Rd). This implies some norm and almost everywhere convergence results for the θ-means, amongst others the generalization of the well known Lebesgue’s theorem. Some special cases of the θ-summation are considered, such as the Riesz, Bochner-Riesz, Weierstrass, Picard and Bessel summations.