# Article

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Keywords:
integral and derivative operators of functional order; fractional integral operator; fractional derivative operator; spaces of homogeneous type; Besov spaces; Triebel-Lizorkin spaces
Summary:
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.
References:
[B] Blasco O.: Weighted Lipschitz spaces defined by a Banach space. García-Cuerva, J. et al., Fourier Analysis and Partial Differential Equations, CRC, 1995, Chapter 7, pp.131-140. MR 1330235 | Zbl 0870.46021
[FJW] Frazier M., Jawerth B., Weiss G.: Littlewood-Paley theory and the study of function spaces. CBMS, Regional Conference Series in Math., No. 79, 1991. MR 1107300 | Zbl 0757.42006
[GSV] Gatto A.E., Segovia C., Vági S.: On fractional differentiation and integration on spaces of homogeneous type. Rev. Mat. Iberoamericana 12 2 (1996), 111-145. MR 1387588
[GV] Gatto A.E., Vági S.: On Sobolev spaces of fractional order and $\epsilon$-families of operators on spaces of homogeneous type. Studia Math. 133.1 (1999), 19-27. MR 1671965
[H] Hartzstein S.I. Acotación de operadores de Calderón-Zygmund en espacios de Triebel-Lizorkin y de Besov generalizados sobre espacios de tipo homogéneo: Thesis, 2000, UNL, Santa Fe, Argentina.
[HV] Hartzstein S.I., Viviani B.E.: $T1$ theorems on generalized Besov and Triebel-Lizorkin spaces over spaces of homogeneous type. Revista de la Unión Matemática Argentina 42 1 (2000), 51-73. MR 1852730 | Zbl 0995.42011
[HS] Han Y.-S., Sawyer E.T.: Littlewood-Paley theory on spaces of homogeneous type and the classical function spaces. Memoirs Amer. Math. Soc., Vol. 110, No .530, 1994. MR 1214968 | Zbl 0806.42013
[I] Iaffei B.: Espacios Lipschitz generalizados y operadores invariantes por traslaciones. Thesis, UNL, 1996.
[J] Janson S.: Generalization on Lipschitz spaces and applications to Hardy spaces and bounded mean oscillation. Duke Math. J. 47 (1980), 959-982. MR 0596123
[MS] Macías R.A., Segovia C.: Lipschitz functions on spaces of homogeneous type. Adv. Math. 33 (1979), 257-270. MR 0546295

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