| 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 |
| . |
| Category:
|
math |
| . |
| 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 |
| . |
| Date available:
|
2009-01-08T19:27:52Z |
| Last updated:
|
2012-04-30 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/119371 |
| . |
| 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 |
| . |
