On fractional differentiation and integration on spaces of homogeneous type

Abstract

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 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 a, 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 D_\alpha and then we follow the method of her thesis [N1], [N2].

Fractional and Hypersingular Operators in Variable Exponent Spaces on Metric Measure Spaces

“…We prove two versions of Sobolev-type theorems with variable exponents. Various versions of such theorems for constant p were proved in [18], [15], [16], [17], [30], [31]. We also give boundedness statements for corresponding fractional maximal operators.…”

“…A detailed information about hypersingular integrals (of constant order) of functions defined in R m can be found in [47] and [49]; variable order hypersingular integrals were studied in [46]. Hypersingular integrals of constant order on metric measure spaces were considered in [16], [17] within the frameworks of Lipschitz (Hölder) function spaces.…”

“…the book [18] for I α , and the book [13] and papers [15], [16], [17] for I α n . In the next theorem, for functions on doubling measure spaces with upper bound (2.6), we deal with the "quasi-Sobolev" exponent q = q(n, N ) defined by 1…”

“…In the case of constant exponents, hypersingular integrals on metric spaces, jointly with potential operators, were considered in [16], [17]. This paper is structured as follows.…”

A commutator theorem for fractional integrals in spacesof homogeneous type

“…Let now (X,δ,µ) be a normal space of homogeneous type and of order ρ ∈ (0, 1), such that µ(X) = ∞ and µ({x}) = 0, for every x ∈ X. Gatto, Segovia, and Vagi [10] defined, for every 0 < α < 1, a function δ α on X × X as follows:…”