Let $ 0 < p, q < \infty $, $ \Phi $ be a generalized normal function and $ L_{p, q}(\Phi) $ the radial-angular mixed space. In this paper, we first generalize the classical Schur's test to radial-angular mixed spaces setting and then find the sufficient and necessary condition for the boundedness of integral operators from $ L_{p_1, p_2}(\Phi) $ to $ L_{q_1, q_2}(\Phi) $ for $ 1\leq p_i, q_i\leq \infty $ with $ i\in\{1, 2\} $. Moreover, we also establish the boundedness of Bergman-type operators $ P_{s, t} $, where $ s\in {\mathbb R} $ and $ t > 0 $, on holomorphic radial-angular mixed space $ H_{p, q}(\Phi) $ for all possible $ 0 < p, q < \infty $. As an application, we finally solve Gleason's problem on $ H_{p, q}(\Phi) $ for all possible $ 0 < p, q < \infty $.
Citation: Long Huang, Xiaofeng Wang. Schur's test, Bergman-type operators and Gleason's problem on radial-angular mixed spaces[J]. Electronic Research Archive, 2023, 31(10): 6027-6044. doi: 10.3934/era.2023307
Let $ 0 < p, q < \infty $, $ \Phi $ be a generalized normal function and $ L_{p, q}(\Phi) $ the radial-angular mixed space. In this paper, we first generalize the classical Schur's test to radial-angular mixed spaces setting and then find the sufficient and necessary condition for the boundedness of integral operators from $ L_{p_1, p_2}(\Phi) $ to $ L_{q_1, q_2}(\Phi) $ for $ 1\leq p_i, q_i\leq \infty $ with $ i\in\{1, 2\} $. Moreover, we also establish the boundedness of Bergman-type operators $ P_{s, t} $, where $ s\in {\mathbb R} $ and $ t > 0 $, on holomorphic radial-angular mixed space $ H_{p, q}(\Phi) $ for all possible $ 0 < p, q < \infty $. As an application, we finally solve Gleason's problem on $ H_{p, q}(\Phi) $ for all possible $ 0 < p, q < \infty $.
| [1] | K. Zhu, Operator Theory in Function Spaces, 2nd edition, American Mathematical Society, Providence, RI, 2007. https://doi.org/10.1090/surv/138 |
| [2] |
J. Schur, Bemerkungen zur Theorie der beschränkten Bilinearformen mit unendlich vielen Veränderlichen, J. Reine Angew. Math., 140 (1911), 1–28. https://doi.org/10.1515/crll.1911.140.1 doi: 10.1515/crll.1911.140.1
|
| [3] |
E. Gagliardo, On integral trasformations with positive kernel, Proc. Amer. Math. Soc., 16 (1965), 429–434. https://doi.org/10.2307/2034667 doi: 10.2307/2034667
|
| [4] |
R. Zhao, Generalization of Schur's test and its application to a class of integral operators on the unit ball of $\mathbb{C}^n$, Integr. Equations Oper. Theory, 82 (2015), 519–532. https://doi.org/10.1007/s00020-014-2215-0 doi: 10.1007/s00020-014-2215-0
|
| [5] |
G. Ren, U. Kähler, J. Shi, C. Liu, Hardy-Littlewood inequalities for fractional derivatives of invariant harmonic functions, Complex Anal. Oper. Theory, 6 (2012), 373–396. https://doi.org/10.1007/s11785-010-0123-0 doi: 10.1007/s11785-010-0123-0
|
| [6] | L. Huang, X. Wang, Z. Zeng, $L^{\vec{p}}-L^{\vec{q}}$ boundedness of multiparameter Forelli-Rudin type operators on the product of unit balls of $\mathbb{C}^n$, preprint, arXiv: 2304.04942. https://doi.org/10.48550/arXiv.2304.04942 |
| [7] | R. Zhao, L. Zhou, $L^p-L^q$ boundedness of Forelli-Rudin type operators on the unit ball of $\mathbb{C}^n$, J. Funct. Anal., 282 (2022) 109345. https://doi.org/10.1016/j.jfa.2021.109345 |
| [8] |
K. Zhu, Embedding and compact embedding of weighted Bergman spaces, Illinois J. Math., 66 (2022), 435–448. https://doi.org/10.1215/00192082-10120480 doi: 10.1215/00192082-10120480
|
| [9] | G. Ren, J. Shi, Bergman type operator on mixed norm spaces with applications, Chin. Ann. of Math., 18 (1997), 265–276. |
| [10] |
Y. Liu, Boundedness of the Bergman type operators on mixed norm spaces, Proc. Amer. Math. Soc., 130 (2002), 2363–2367. https://doi.org/10.1090/S0002-9939-02-06332-3 doi: 10.1090/S0002-9939-02-06332-3
|
| [11] |
Z. Lou, A note on the boundedness of Bergman-type operators on mixed norm spaces, J. Aust. Math. Soc., 82 (2007), 395–402. https://doi.org/10.1017/S1446788700036181 doi: 10.1017/S1446788700036181
|
| [12] | A. M. Gleason, Finitely generated ideals in Banach algebras, J. Math. Mech., 13 (1964), 125–132. |
| [13] | K. Zhu, The Bergman spaces, the Bloch space, and Gleason's problem, Trans. Amer. Math. Soc., 309 (1988), 253–268. https://doi.org/10.2307/2001168 |
| [14] | B. R. Choe, Projections, the weighted Bergman spaces, and the Bloch space, Proc. Amer. Math. Soc., 108 (1990), 127–136. https://doi.org/10.1090/S0002-9939-1990-0991692-0 |
| [15] |
Z. Hu, Gleason's problem for harmonic mixed norm and Bloch spaces in convex domains, Math. Nachr., 279 (2006), 164–178. https://doi.org/10.1002/mana.200310353 doi: 10.1002/mana.200310353
|
| [16] |
A. Benedek, R. Panzone, The space $L^p$, with mixed norm, Duke Math. J., 28 (1961), 301–324. https://doi.org/10.1215/S0012-7094-61-02828-9 doi: 10.1215/S0012-7094-61-02828-9
|
| [17] | K. Zhu, Spaces of Holomorphic Functions in the Unit ball, Springer-Verlag, New York, 1994. |
| [18] |
S. Li, A class of integral operators on mixed norm spaces in the unit ball, Czechoslovak Math. J., 57 (2007), 1013–1023. https://doi.org/10.1007/s10587-007-0091-3 doi: 10.1007/s10587-007-0091-3
|
| [19] |
A. L. Shields, D. L. Williams, Bonded projections, duality, and multipliers in spaces of analytic functions, Trans. Amer. Math. Soc., 162 (1971), 287–302. https://doi.org/10.2307/1995754 doi: 10.2307/1995754
|
Long Huang, Xiaofeng Wang. Schur's test, Bergman-type operators and Gleason's problem on radial-angular mixed spaces[J]. Electronic Research Archive, 2023, 31(10): 6027-6044. doi: 10.3934/era.2023307
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