Integral and derivative operators of functional order on generalized Besov and Triebel-Lizorkin spaces in the setting of spaces of homogeneous type

Hartzstein, Silvia I. ; Viviani, Beatriz E.

Commentationes Mathematicae Universitatis Carolinae, Tome 43 (2002) no. 4, pp. 723-754

Résumé

In the setting of spaces of homogeneous-type, we define the Integral, $I_{\phi}$, and Derivative, $D_{\phi}$, operators of order $\phi$, where $\phi$ is a function of positive lower type and upper type less than $1$, and show that $I_{\phi}$ and $D_{\phi}$ are bounded from Lipschitz spaces $\Lambda^{\xi}$ to $\Lambda^{\xi \phi}$ and $\Lambda^{\xi/\phi}$ respectively, with suitable restrictions on the quasi-increasing function $\xi$ in each case. We also prove that $I_{\phi}$ and $D_{\phi}$ are bounded from the generalized Besov $\dot{B}_{p}^{\psi, q}$, with $1 \leq p, q \infty $, and Triebel-Lizorkin spaces $\dot{F}_{p}^{\psi, q}$, with $1 p, q \infty $, of order $\psi$ to those of order $\phi \psi$ and $\psi/\phi$ respectively, where $\psi$ is the quotient of two quasi-increasing functions of adequate upper types.

Détail

Classification : 26A33, 42B35, 46E35, 47G10
Keywords: integral and derivative operators of functional order; fractional integral operator; fractional derivative operator; spaces of homogeneous type; Besov spaces; Triebel-Lizorkin spaces