On fractional differentiation and integration on spaces of homogeneous type
Summary: 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 (cf. \textit{E. M. Stein} [Bull. Am. Math. Soc. 67, 102-104 (1961; Zbl 0127.32002); 68, 577-582 (1962; Zbl 0127.32002); Singular integrals and differentiability properties of functions (Princeton 1970; Zbl 0201.14702)]) and introducing suitable quasidistances related to an approximation of the identity. We define integration of fractional orders as in [\textit{A. E. Gatto} and \textit{S. Vági}, Analysis and partial differential equations, Notes Pure Appl. Math. 122, 171-216 (1990; Zbl 0695.43006)] but using quasidistances related to the approximation of the identity before. We show that these operators act on Lipschitz spaces as in the classical cases. We prove that the composition \(T_\alpha\) of a fractional integral \(I_\alpha\) and a fractional derivative \(D_\alpha\) 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 \(\alpha\), \(T_\alpha\) is an invertible operator in \(L^2\). In order to prove that \(T_\alpha\) is invertible we obtain Nahmod type representations for \(I_\alpha\) and then we follow the method of her thesis [\textit{A. R. Nahmod}, Geometry of operators and spectral analysis. Thesis. Yale University 1991; C. R. Acad. Sci., Paris, Sér. I 313, 721-725 (1991; Zbl 0739.47003)].
scientific article; zbMATH DE number 885336