| Title:
|
Libera and Hilbert matrix operator on logarithmically weighted Bergman, Bloch and Hardy-Bloch spaces (English) |
| Author:
|
Karapetrović, Boban |
| Language:
|
English |
| Journal:
|
Czechoslovak Mathematical Journal |
| ISSN:
|
0011-4642 (print) |
| ISSN:
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1572-9141 (online) |
| Volume:
|
68 |
| Issue:
|
2 |
| Year:
|
2018 |
| Pages:
|
559-576 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| Summary:
|
We show that if $\alpha >1$, then the logarithmically weighted Bergman space $A_{\log ^{\alpha }}^2$ is mapped by the Libera operator $\mathcal {L}$ into the space $A_{\log ^{\alpha -1}}^2$, while if $\alpha >2$ and $0<\varepsilon \leq \alpha -2$, then the Hilbert matrix operator $H$ maps $A_{\log ^\alpha }^2$ into $A_{\log ^{\alpha -2-\varepsilon }}^2$.\newline We show that the Libera operator $\mathcal {L}$ maps the logarithmically weighted Bloch space $\mathcal {B}_{\log ^{\alpha }}$, $\alpha \in \mathbb {R}$, into itself, while $H$ maps $\mathcal {B}_{\log ^{\alpha }}$ into $\mathcal {B}_{\log ^{\alpha +1}}$.\newline In Pavlović's paper (2016) it is shown that $\mathcal {L}$ maps the logarithmically weighted Hardy-Bloch space $\mathcal {B}_{\log ^{\alpha }}^1$, $\alpha >0$, into $\mathcal {B}_{\log ^{\alpha -1}}^1$. We show that this result is sharp. We also show that $H$ maps $\mathcal {B}_{\log ^{\alpha }}^1$, $\alpha \geq {0}$, into $\mathcal {B}_{\log ^{\alpha -1}}^1$ and that this result is sharp also. (English) |
| Keyword:
|
Libera operator |
| Keyword:
|
Hilbert matrix operator |
| Keyword:
|
Hardy space |
| Keyword:
|
Bergman space |
| Keyword:
|
Bloch space |
| Keyword:
|
Hardy-Bloch space |
| MSC:
|
30H25 |
| MSC:
|
47B38 |
| MSC:
|
47G10 |
| idZBL:
|
Zbl 06890390 |
| idMR:
|
MR3819191 |
| DOI:
|
10.21136/CMJ.2018.0555-16 |
| . |
| Date available:
|
2018-06-11T10:58:15Z |
| Last updated:
|
2020-07-06 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/147236 |
| . |
| Reference:
|
[1] Jevtić, M., Karapetrović, B.: Hilbert matrix operator on Besov spaces.Publ. Math. 90 (2017), 359-371. MR 3666637, 10.5486/PMD.2017.7518 |
| Reference:
|
[2] Jevtić, M., Karapetrović, B.: Libera operator on mixed norm spaces $H_{\nu}^{p,q,\alpha}$ when $0<p<1$.Filomat 31 (2017), 4641-4650. MR 3730385, 10.2298/FIL1714641J |
| Reference:
|
[3] Jevtić, M., Vukotić, D., Arsenović, M.: Taylor Coefficients and Coefficient Multipliers of Hardy and Bergman-Type Spaces.RSME Springer Series 2, Springer, Cham (2016). Zbl 1368.30001, MR 3587910, 10.1007/978-3-319-45644-7 |
| Reference:
|
[4] Łanucha, B., Nowak, M., Pavlović, M.: Hilbert matrix operator on spaces of analytic functions.Ann. Acad. Sci. Fenn., Math. 37 (2012), 161-174. Zbl 1258.47047, MR 2920431, 10.5186/aasfm.2012.3715 |
| Reference:
|
[5] Mateljević, M., Pavlović, M.: $L^p$-behaviour of the integral means of analytic functions.Stud. Math. 77 (1984), 219-237. Zbl 1188.30004, MR 0745278, 10.4064/sm-77-3-219-237 |
| Reference:
|
[6] Pavlović, M.: Definition and properties of the libera operator on mixed norm spaces.The Scientific World Journal 2014 (2014), Article ID 590656, 15 pages. 10.1155/2014/590656 |
| Reference:
|
[7] Pavlović, M.: Function Classes on the Unit Disc. An introduction.De Gruyter Studies in Mathematics 52, De Gruyter, Berlin (2014). Zbl 1296.30002, MR 3154590, 10.1515/9783110281903 |
| Reference:
|
[8] Pavlović, M.: Logarithmic Bloch space and its predual.Publ. Inst. Math. (Beograd) (N.S.) 100(114) (2016), 1-16. Zbl 06749634, MR 3586678, 10.2298/PIM1614001P |
| Reference:
|
[9] Zhu, K.: Operator Theory in Function Spaces.Monographs and Textbooks in Pure and Applied Mathematics 139, Marcel Dekker, New York (1990). Zbl 0706.47019, MR 1074007 |
| Reference:
|
[10] Zygmund, A.: Trigonometric Series. Vol. I, II.Cambridge University Press, Cambridge (1959). Zbl 1084.42003, MR 1963498 |
| . |
