On fractional differentiation and integration on spaces of homogeneous type

• Autores: S. Vági, A. Gatto, Carlos Segovia Fernández
• Localización: Revista matemática iberoamericana, ISSN 0213-2230, Vol. 12, Nº 1, 1996, págs. 111-146
• Idioma: inglés
• DOI: 10.4171/rmi/196
• Títulos paralelos:
  • Diferenciación e integración fraccionales sobre espacios de tipo homogéneo.
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• Resumen

  • In this paper we define derivatives of fractional order on spaces of homogeneous type by generalizing a classical formula for the fractional powers of the Laplacean [S1], [S2], [SZ] and introducing suitable quasidistances related to an approximation of the identity. We define integration of fractional order as in [GV] but using quasidistances related to the approximation of the identity mentioned before.

    We show that these operators act on Lipschitz spaces as in the classical cases. We prove that the composition Ta of a fractional integral Ia and a fractional derivative Da of the same order and its transpose (a fractional derivative composed with a fractional integral of the same order) are Calderón-Zygmund operators. We also prove that for small order a, Ta is an invertible operator in L2. In order to prove that Ta is invertible we obtain Nahmod type representations for Ia and Da and then we follow the method of her thesis [N1], [N2].