Carleson measures on bounded \(\mathbb{C}\)-convex domains. (English) Zbl 07909842
Summary: In this paper, we completely characterize the Carleson and the vanishing Carleson measures on bounded \(\mathbb{C}\)-convex domains in terms of the Carathéodory (or the Kobayashi) geometry and Bergman geometry. As a corollary, we obtain a sufficient and necessary condition for the composition operators on \(\mathbb{C}\)-convex domains to be bounded on associated Bergman spaces.
MSC:
| 46E15 | Banach spaces of continuous, differentiable or analytic functions |
| 32A25 | Integral representations; canonical kernels (Szegő, Bergman, etc.) |
| 32A36 | Bergman spaces of functions in several complex variables |
References:
| [1] | Abate, M.; Raissy, J., Skew Carleson measures in strongly pseudoconvex domains, Complex Anal. Oper. Theory, 13, 2, 405-429, 2019 · Zbl 1426.32016 |
| [2] | Abate, M.; Raissy, J.; Saracco, A., Toeplitz operators and Carleson measures in strongly pseudoconvex domains, J. Funct. Anal., 263, 11, 3449-3491, 2012 · Zbl 1269.32003 |
| [3] | Abate, M.; Saracco, A., Carleson measures and uniformly discrete sequences in strongly pseudoconvex domains, J. Lond. Math. Soc., 83, 3, 587-605, 2011 · Zbl 1227.32008 |
| [4] | Andersson, M.; Passare, M.; Sigurdsson, R., Complex Convexity and Analytic Functionals, 2004, Birkhäuser: Birkhäuser Berlin · Zbl 1057.32001 |
| [5] | Balakumar, G.; Mahajan, P.; Verma, K., Bounds for invariant distances on pseudoconvex Levi corank one domains and applications, Ann. Fac. Sci. Toulouse Math. (6), 24, 2, 281-388, 2015 · Zbl 1326.32020 |
| [6] | Calzi, M.; Peloso, M., Carleson and reverse Carleson measures on homogeneous Siegel domains, Complex Anal. Oper. Theory, 16, 1, Article 4 pp., 2022 · Zbl 1484.32009 |
| [7] | Carleson, L., Interpolations by bounded analytic functions and the corona problem, Ann. Math., 76, 547-559, 1962 · Zbl 0112.29702 |
| [8] | Cima, J.; Mercer, P., Composition operators between Bergman spaces on convex domains in \(\mathbb{C}^n\), J. Oper. Theory, 33, 2, 363-369, 1995 · Zbl 0840.47025 |
| [9] | Cima, J.; Wogen, W., A Carleson measure theorem for the Bergman space on the ball, J. Oper. Theory, 7, 1, 157-165, 1982 · Zbl 0499.42011 |
| [10] | Conrad, M., Nicht isotrope Abschätzungen für lineal konvexe Gebiete endlichen Typs, 2002, Universität Wuppertal, Dissertation |
| [11] | Čučković, Z̆.; Şahutoğlu, S., Essential norm estimates for the \(\overline{\partial} \)-Neumann operator on convex domains and worm domains, Indiana Univ. Math. J., 67, 1, 267-292, 2018 · Zbl 1410.32015 |
| [12] | Hu, Z.; Lv, X.; Zhu, K., Carleson measures and balayage for Bergman spaces of strongly pseudoconvex domains, Math. Nachr., 289, 10, 1237-1254, 2016 · Zbl 1360.32007 |
| [13] | Jarnicki, M.; Pflug, P., Invariant Distances and Metrics in Complex Analysis, 2013, de Gruyter · Zbl 1273.32002 |
| [14] | Khanh, T.; Liu, J.; Thuc, P., Bergman-Toeplitz operators on weakly pseudoconvex domains, Math. Z., 291, 1-2, 591-607, 2019 · Zbl 1486.47057 |
| [15] | Koo, H.; Li, S., Composition operators on bounded convex domains in \(\mathbb{C}^n\), Integral Equ. Oper. Theory, 85, 4, 555-572, 2016 · Zbl 1362.47010 |
| [16] | Mahajan, P.; Verma, K., Some aspects of the Kobayashi and Carathéodory metrics on pseudoconvex domains, J. Geom. Anal., 22, 2, 491-560, 2012 · Zbl 1254.32016 |
| [17] | Nikolov, N.; Pflug, P.; Thomas, P.; Zwonek, W., On a local characterization of pseudoconvex domains, Indiana Univ. Math. J., 58, 6, 2661-2671, 2009 · Zbl 1202.32010 |
| [18] | Nikolov, N.; Pflug, P.; Zwonek, W., Estimates for invariant metrics on \(\mathbb{C} \)-convex domains, Trans. Am. Math. Soc., 363, 12, 6245-6256, 2011 · Zbl 1232.32005 |
| [19] | Nikolov, N.; Pflug, P.; Zwonek, W., An example of a bounded \(\mathbb{C} \)-convex domain which is not biholomorphic to a convex domain, Math. Scand., 102, 1, 149-155, 2008 · Zbl 1155.32009 |
| [20] | Nikolov, N.; Trybuła, M., The Kobayashi balls of \(( \mathbb{C} \)-)convex domains, Monatshefte Math., 177, 4, 627-635, 2015 · Zbl 1326.32021 |
| [21] | Su, G., Geometric properties of the pentablock, Complex Anal. Oper. Theory, 14, 4, Article 44 pp., 2020 · Zbl 1453.32015 |
| [22] | Thuc, P., A remark on Carleson measures of domains in \(\mathbb{C}^n\), Proc. Am. Math. Soc., 150, 6, 2579-2592, 2022 · Zbl 1495.32011 |
| [23] | Yeung, S.-T., Geometry of domains with the uniform squeezing property, Adv. Math., 221, 2, 547-569, 2009 · Zbl 1165.32004 |
| [24] | Zhang, S., Carleson measures on the generalized Hartogs triangles, J. Math. Anal. Appl., 510, 2, Article 126027 pp., 2022 · Zbl 1487.32029 |
| [25] | Zwonek, W., Geometric properties of the tetrablock, Arch. Math. (Basel), 100, 2, 159-165, 2013 · Zbl 1268.32002 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
