Let $ u, v $ be two analytic functions on the open unit disk $ {\mathbb D} $ in the complex plane, $ \varphi $ an analytic self-map of $ {\mathbb D} $, and $ m, n $ nonnegative integers such that $ m < n $. In this paper, we consider the generalized Stević-Sharma type operator $ T_{u, v, \varphi}^{m, n}f(z) = u(z)f^{(m)}(\varphi(z))+v(z)f^{(n)}(\varphi(z)) $ acting from the derivative Hardy spaces into Zygmund-type spaces, and investigate its boundedness, essential norm and compactness.
Citation: Zhitao Guo, Jianyong Mu. Generalized Stević-Sharma type operators from derivative Hardy spaces into Zygmund-type spaces[J]. AIMS Mathematics, 2023, 8(2): 3920-3939. doi: 10.3934/math.2023196
Let $ u, v $ be two analytic functions on the open unit disk $ {\mathbb D} $ in the complex plane, $ \varphi $ an analytic self-map of $ {\mathbb D} $, and $ m, n $ nonnegative integers such that $ m < n $. In this paper, we consider the generalized Stević-Sharma type operator $ T_{u, v, \varphi}^{m, n}f(z) = u(z)f^{(m)}(\varphi(z))+v(z)f^{(n)}(\varphi(z)) $ acting from the derivative Hardy spaces into Zygmund-type spaces, and investigate its boundedness, essential norm and compactness.
| [1] | P. L. Duren, Theory of $H^{p}$ spaces, New York: Academic Press, 1970. |
| [2] | M. D. Contreras, A. G. Hernández-Díaz, Weighted composition operators on spaces with derivative in a Hardy space, J. Operator Theory, 52 (2004), 173–184. |
| [3] |
N. Hu, Weighted composition operators from derivative Hardy spaces into $n$-th weighted-type spaces, J. Math., 2021 (2021), 4398397. http://dx.doi.org/10.1155/2021/4398397 doi: 10.1155/2021/4398397
|
| [4] | B. D. MacCluer, Composition operators on $\mathcal{S}^{p}$, Houston J. Math., 13 (1987), 245–254. |
| [5] | R. C. Roan, Composition operators on the space of functions with $H^{p}$-derivative, Houston J. Math., 4 (1978), 423–438. |
| [6] |
E. Abbasi, X. Zhu, Product-type operators from the Bloch space into Zygmund-type spaces, Bull. Iran. Math. Soc., 48 (2022), 385–400. http://dx.doi.org/10.1007/s41980-020-00523-1 doi: 10.1007/s41980-020-00523-1
|
| [7] |
K. Esmaeili, M. Lindström, Weighted composition operators between Zygmund type spaces and their essential norms, Integr. Equ. Oper. Theory, 75 (2013), 473–490. http://dx.doi.org/10.1007/s00020-013-2038-4 doi: 10.1007/s00020-013-2038-4
|
| [8] |
Z. Guo, L. Liu, Product-type operators from Hardy spaces to Bloch-type spaces and Zygmund-type spaces, Numer. Funct. Anal. Optim., 43 (2022), 1240–1264. http://dx.doi.org/10.1080/01630563.2022.2097693 doi: 10.1080/01630563.2022.2097693
|
| [9] |
Z. Guo, L. Liu, Y. Shu, On Stević-Sharma operator from the mixed norm spaces to Zygmund-type spaces, Math. Inequal. Appl., 24 (2021), 445–461. http://dx.doi.org/10.7153/mia-2021-24-31 doi: 10.7153/mia-2021-24-31
|
| [10] |
S. Li, S. Stević, Volterra-type operators on Zygmund spaces, J. Inequal. Appl., 2007 (2007), 1–10. http://dx.doi.org/10.1155/2007/32124 doi: 10.1155/2007/32124
|
| [11] |
S. Li, S. Stević, Generalized composition operators on Zygmund spaces and Bloch type spaces, J. Math. Anal. Appl., 338 (2008), 1282–1295. http://dx.doi.org/10.1016/j.jmaa.2007.06.013 doi: 10.1016/j.jmaa.2007.06.013
|
| [12] |
S. Li, S. Stević, Products of Volterra type operator and composition operator from $H^{\infty}$ and Bloch spaces to Zygmund spaces, J. Math. Anal. Appl., 345 (2008), 40–52. http://dx.doi.org/10.1016/j.jmaa.2008.03.063 doi: 10.1016/j.jmaa.2008.03.063
|
| [13] |
S. Li, S. Stević, Weighted composition operators from Zygmund spaces into Bloch spaces, Appl. Math. Comput., 206 (2008), 825–831. http://dx.doi.org/10.1016/j.amc.2008.10.006 doi: 10.1016/j.amc.2008.10.006
|
| [14] |
S. Li, S. Stević, Products of composition and differentiation operators from Zygmund spaces to Bloch spaces and Bers spaces, Appl. Math. Comput., 217 (2010), 3144–3154. http://dx.doi.org/10.1016/j.amc.2010.08.047 doi: 10.1016/j.amc.2010.08.047
|
| [15] |
F. Zhang, Y. Liu, On a Stević-Sharma operator from Hardy spaces to Zygmund-type spaces on the unit disk, Complex Anal. Oper. Theory., 12 (2018), 81–100. http://dx.doi.org/10.1007/s11785-016-0578-8 doi: 10.1007/s11785-016-0578-8
|
| [16] |
X. Zhu, E. Abbasi, A. Ebrahimi, Product-type operators on the Zygmund space, Iran. J. Sci. Technol. Trans. Sci., 45 (2021), 1689–1697. http://dx.doi.org/10.1007/s40995-021-01138-9 doi: 10.1007/s40995-021-01138-9
|
| [17] | C. C. Cowen, B. D. MacCluer, Composition operators on spaces of analytic functions, Boca Raton: CRC Press, 1995. |
| [18] |
R. A. Hibschweiler, N. Portnoy, Composition followed by differentiation between Bergman and Hardy spaces, Rocky Mountain J. Math., 35 (2005), 843–855. http://dx.doi.org/10.1216/rmjm/1181069709 doi: 10.1216/rmjm/1181069709
|
| [19] |
S. Li, S. Stević, Composition followed by differentiation from mixed-norm spaces to $\alpha$-Bloch spaces, Sbornik: Math., 199 (2008), 1847–1857. http://dx.doi.org/10.1070/SM2008v199n12ABEH003983 doi: 10.1070/SM2008v199n12ABEH003983
|
| [20] |
S. Ohno, Products of differentiation and composition on Bloch spaces, Bull. Korean Math. Soc., 46 (2009), 1135–1140. http://dx.doi.org/10.4134/BKMS.2009.46.6.1135 doi: 10.4134/BKMS.2009.46.6.1135
|
| [21] |
S. Stević, Norm and essential norm of composition followed by differentiation from $\alpha$-Bloch spaces to $H_{\mu}^{\infty}$, Appl. Math. Comput., 207 (2009), 225–229. http://dx.doi.org/10.1016/j.amc.2008.10.032 doi: 10.1016/j.amc.2008.10.032
|
| [22] |
S. Stević, A. K. Sharma, A. Bhat, Essential norm of products of multiplication composition and differentiation operators on weighted Bergman spaces, Appl. Math. Comput., 218 (2011), 2386–2397. http://dx.doi.org/10.1016/j.amc.2011.06.055 doi: 10.1016/j.amc.2011.06.055
|
| [23] |
S. Stević, A. K. Sharma, A. Bhat, Products of multiplication composition and differentiation operators on weighted Bergman space, Appl. Math. Comput., 217 (2011), 8115–8125. http://dx.doi.org/10.1016/j.amc.2011.03.014 doi: 10.1016/j.amc.2011.03.014
|
| [24] |
Y. Liu, Y. Yu, On a Stević-Sharma operator from Hardy spaces to the logarithmic Bloch spaces, J. Inequal. Appl., 22 (2015), 1–19. http://dx.doi.org/10.1186/s13660-015-0547-1 doi: 10.1186/s13660-015-0547-1
|
| [25] |
S. Wang, M. Wang, X. Guo, Differences of Stević-Sharma operators, Banach J. Math. Anal., 14 (2020), 1019–1054. http://dx.doi.org/10.1007/s43037-019-00051-z doi: 10.1007/s43037-019-00051-z
|
| [26] |
M. S. Al Ghafri, J. S. Manhas, On Stević-Sharma operators from weighted Bergman spaces to weighted-type spaces, Math. Inequal. Appl., 23 (2020), 1051–1077. http://dx.doi.org/10.7153/mia-2020-23-81 doi: 10.7153/mia-2020-23-81
|
| [27] |
Z. Guo, On Stević-Sharma operator from weighted Bergman-Orlicz spaces to Bloch-type spaces, Math. Inequal. Appl., 25 (2022), 91–107. http://dx.doi.org/10.7153/mia-2022-25-07 doi: 10.7153/mia-2022-25-07
|
| [28] |
Z. Guo, Y. Shu, On Stević-Sharma operators from Hardy spaces to Stević weighted spaces, Math. Inequal. Appl., 23 (2020), 217–229. http://dx.doi.org/10.7153/mia-2020-23-17 doi: 10.7153/mia-2020-23-17
|
| [29] |
X. Zhu, E. Abbasi, A. Ebrahimi, A class of operator-related composition operators from the Besov spaces into the Bloch space, Bull. Iran. Math. Soc., 47 (2021), 171–184. http://dx.doi.org/10.1007/s41980-020-00374-w doi: 10.1007/s41980-020-00374-w
|
| [30] |
E. Abbasi, Y. Liu, M. Hassanlou, Generalized Stević-Sharma type operators from Hardy spaces into $n$th weighted type spaces, Turkish J. Math., 45 (2021), 1543–1554. http://dx.doi.org/10.3906/mat-2011-67 doi: 10.3906/mat-2011-67
|
| [31] |
S. Stević, Composition operators from the weighted Bergman space to the $n$th weighted spaces on the unit disc, Discrete Dyn. Nat. Soc., 2009 (2009), 1–12. http://dx.doi.org/10.1155/2009/742019 doi: 10.1155/2009/742019
|
| [32] |
S. Stević, Weighted differentiation composition operators from the mixed-norm space to the $n$th weighted-type space on the unit disk, Abstr. Appl. Anal., 2010 (2010), 1–16. http://dx.doi.org/10.1155/2010/246287 doi: 10.1155/2010/246287
|
| [33] |
S. Stević, A. K. Sharma, R. Krishan, Boundedness and compactness of a new product-type operator from a general space to Bloch-type spaces, J. Inequal. Appl., 2016 (2016), 1–32. http://dx.doi.org/10.1186/s13660-016-1159-0 doi: 10.1186/s13660-016-1159-0
|
| [34] |
E. Abbasi, The product-type operators from Hardy spaces into $n$th weighted-type spaces, Abstr. Appl. Anal., 2021 (2021), 1–8. http://dx.doi.org/10.1155/2021/5556275 doi: 10.1155/2021/5556275
|
| [35] |
Z. Guo, X. Zhao, On a Stević-Sharma type operator from $Q_{k}(p, q)$ spaces to Bloch-type spaces, Oper. Matrices, 16 (2022), 563–580. http://dx.doi.org/10.7153/oam-2022-16-43 doi: 10.7153/oam-2022-16-43
|
| [36] |
S. Stević, Composition operators from the Hardy space to the $n$th weighted-type space on the unit disk and the half-plane, Appl. Math. Comput., 215 (2010), 3950–3955. http://dx.doi.org/10.1016/j.amc.2009.11.043 doi: 10.1016/j.amc.2009.11.043
|
| [37] |
F. Colonna, M. Tjani, Weighted composition operators from Banach spaces of analytic functions into Bloch-type spaces, Probl. Recent Methods Oper. Theory, 687 (2017), 75–95. http://dx.doi.org/10.1090/conm/687/13790 doi: 10.1090/conm/687/13790
|
| [38] |
P. Galindo, M. Lindström, S. Stević, Essential norm of operators into weighted-type spaces on the unit ball, Abstr. Appl. Anal., 2011 (2011), 1–14. http://dx.doi.org/10.1155/2011/939873 doi: 10.1155/2011/939873
|
| [39] |
S. Li, S. Stević, Generalized weighted composition operators from $\alpha$-Bloch spaces into weighted-type spaces, J. Inequal. Appl., 2015 (2015), 1–12. http://dx.doi.org/10.1186/s13660-015-0770-9 doi: 10.1186/s13660-015-0770-9
|
Zhitao Guo, Jianyong Mu. Generalized Stević-Sharma type operators from derivative Hardy spaces into Zygmund-type spaces[J]. AIMS Mathematics, 2023, 8(2): 3920-3939. doi: 10.3934/math.2023196
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