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Title: On the composition of the integral and derivative operators of functional order (English)
Author: Hartzstein, Silvia I.
Author: Viviani, Beatriz E.
Language: English
Journal: Commentationes Mathematicae Universitatis Carolinae
ISSN: 0010-2628 (print)
ISSN: 1213-7243 (online)
Volume: 44
Issue: 1
Year: 2003
Pages: 99-120
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Category: math
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Summary: The Integral, $I_{\phi}$, and Derivative, $D_{\phi}$, operators of order $\phi$, with $\phi$ a function of positive lower type and upper type less than $1$, were defined in [HV2] in the setting of spaces of homogeneous-type. These definitions generalize those of the fractional integral and derivative operators of order $\alpha$, where $\phi(t)=t^{\alpha}$, given in [GSV]. In this work we show that the composition $T_{\phi}= D_{\phi}\circ I_{\phi}$ is a singular integral operator. This result in addition with the results obtained in [HV2] of boundedness of $I_{\phi}$ and $D_{\phi}$ or the $T1$-theorems proved in [HV1] yield the fact that $T_{\phi}$ is a Calder'on-Zygmund operator bounded on the generalized Besov, $\dot{B}_{p}^{\psi,q}$, $1 \le p,q < \infty$, and Triebel-Lizorkin spaces, $\dot{F}_{p}^{\psi,q}$, $1< p, q < \infty$, of order $\psi= \psi_1/\psi_2$, where $\psi_1$ and $\psi_2$ are two quasi-increasing functions of adequate upper types $s_1$ and $s_2$, respectively. (English)
Keyword: fractional integral operators
Keyword: fractional derivative operators
Keyword: spaces of homogeneous type
Keyword: Besov spaces
Keyword: Triebel-Lizorkin spaces
MSC: 26A33
MSC: 42B20
MSC: 46E35
MSC: 47B38
idZBL: Zbl 1127.42305
idMR: MR2045849
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Date available: 2009-01-08T19:27:52Z
Last updated: 2012-04-30
Stable URL: http://hdl.handle.net/10338.dmlcz/119371
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Reference: [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
Reference: [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.
Reference: [HV1] 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. Zbl 0995.42011, MR 1852730
Reference: [HV2] Hartzstein S.I., Viviani B.E.: Integral and derivative operators of functional order on generalized Besov and Triebel-Lizorkin spaces in the setting of spaces of homogeneous type.Comment. Math. Univ. Carolinae 43 (2002), 723-754. Zbl 1091.26002, MR 2046192
Reference: [MS] Macías R.A., Segovia C.: Lipschitz functions on spaces of homogeneous type.Adv. in Math. 33 (1979), 257-270. MR 0546295
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