Article

Full entry | Fulltext not available (moving wall 24 months)
Keywords:
Libera operator; Hilbert matrix operator; Hardy space; Bergman space; Bloch space; Hardy-Bloch space
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.
References:
[1] Jevtić, M., Karapetrović, B.: Hilbert matrix operator on Besov spaces. Publ. Math. 90 (2017), 359-371. DOI 10.5486/PMD.2017.7518 | MR 3666637
[2] Jevtić, M., Karapetrović, B.: Libera operator on mixed norm spaces $H_{\nu}^{p,q,\alpha}$ when $0. Filomat 31 (2017), 4641-4650. DOI 10.2298/FIL1714641J | MR 3730385 [3] Jevtić, M., Vukotić, D., Arsenović, M.: Taylor Coefficients and Coefficient Multipliers of Hardy and Bergman-Type Spaces. RSME Springer Series 2, Springer, Cham (2016). DOI 10.1007/978-3-319-45644-7 | MR 3587910 | Zbl 1368.30001 [4] Łanucha, B., Nowak, M., Pavlović, M.: Hilbert matrix operator on spaces of analytic functions. Ann. Acad. Sci. Fenn., Math. 37 (2012), 161-174. DOI 10.5186/aasfm.2012.3715 | MR 2920431 | Zbl 1258.47047 [5] Mateljević, M., Pavlović, M.:$L^p\$-behaviour of the integral means of analytic functions. Stud. Math. 77 (1984), 219-237. DOI 10.4064/sm-77-3-219-237 | MR 0745278 | Zbl 1188.30004
[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. DOI 10.1155/2014/590656
[7] Pavlović, M.: Function Classes on the Unit Disc. An introduction. De Gruyter Studies in Mathematics 52, De Gruyter, Berlin (2014). DOI 10.1515/9783110281903 | MR 3154590 | Zbl 1296.30002
[8] Pavlović, M.: Logarithmic Bloch space and its predual. Publ. Inst. Math. (Beograd) (N.S.) 100(114) (2016), 1-16. DOI 10.2298/PIM1614001P | MR 3586678 | Zbl 06749634
[9] Zhu, K.: Operator Theory in Function Spaces. Monographs and Textbooks in Pure and Applied Mathematics 139, Marcel Dekker, New York (1990). MR 1074007 | Zbl 0706.47019
[10] Zygmund, A.: Trigonometric Series. Vol. I, II. Cambridge University Press, Cambridge (1959). MR 1963498 | Zbl 1084.42003

Partner of