A. Eduardo Gatto

On fractional differentiation and integration on spaces of homogeneous type

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 Calderon-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].

Boundedness on inhomogeneous Lipschitz spaces of fractional integrals singular integrals and hypersingular integrals associated to non-doubling measures

In the context of a finite measure metric space whose measure satisfies a growth condition, we prove “T1” type necessary and sufficient conditions for the boundedness of fractional integrals, singular integrals, and hypersingular integrals on inhomogeneous Lipschitz spaces. We also indicate how the results can be extended to the case of infinite measure. Finally we show applications to Real and Complex Analysis.

On Gaussian Lipschitz spaces and the boundedness of fractional integrals and fractional derivatives on them

Abstract The main purpose of this paper is to study the boundedness of Gaussian fractional integrals and derivatives associated to Hermite polynomial expansions on Gaussian Lipschitz spaces Lipα (γ). To get these results we introduce formulas for these operators in terms of the Hermite-Poisson semigroup as well as the Gaussian Lipschitz spaces. This approach was originally developed for the classical Poisson integral. These proofs can also be extended to the case of Laguerre and Jacobi expansions. In subsequent papers we will study the same operators on Gaussian Besov-Lipschitz and Triebel-Lizorkin spaces.

Riesz Potentials, Bessel Potentials, and Fractional Derivatives on Besov-Lipschitz Spaces for the Gaussian Measure

Gaussian Lipschitz spaces Lip α(γ d ) and the boundedness properties of Riesz potentials, Bessel potentials and fractional derivatives there were studied in detail in Gatto and Urbina (On Gaussian Lipschitz Spaces and the Boundedness of Fractional Integrals and Fractional Derivatives on them, 2009. Preprint. arXiv:0911.3962v2). In this chapter we will study the boundedness of those operators on Gaussian Besov-Lipschitz spaces B p, q α(γ d ). Also, these results can be extended to the case of Laguerre or Jacobi expansions and even further to the general framework of diffusions semigroups.