| Title:
|
On some new sharp embedding theorems in minimal and pseudoconvex domains (English) |
| Author:
|
Shamoyan, Romi F. |
| Author:
|
Mihić, Olivera R. |
| Language:
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English |
| Journal:
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Czechoslovak Mathematical Journal |
| ISSN:
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0011-4642 (print) |
| ISSN:
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1572-9141 (online) |
| Volume:
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66 |
| Issue:
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2 |
| Year:
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2016 |
| Pages:
|
527-546 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| Summary:
|
We present new sharp embedding theorems for mixed-norm analytic spaces in pseudoconvex domains with smooth boundary. New related sharp results in minimal bounded homogeneous domains in higher dimension are also provided. Last domains we consider are domains which are direct generalizations of the well-studied so-called bounded symmetric domains in $\mathbb {C}^{n}.$ Our results were known before only in the very particular case of domains of such type in the unit ball. As in the unit ball case, all our proofs are heavily based on nice properties of the $r$-lattice. Some results of this paper can be also obtained in some unbounded domains, namely tubular domains over symmetric cones. (English) |
| Keyword:
|
embedding theorem |
| Keyword:
|
minimal domain |
| Keyword:
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pseudoconvex domain |
| Keyword:
|
Bergman-type space |
| MSC:
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42B15 |
| MSC:
|
42B30 |
| idZBL:
|
Zbl 06604484 |
| idMR:
|
MR3519619 |
| DOI:
|
10.1007/s10587-016-0273-y |
| . |
| Date available:
|
2016-06-16T13:01:55Z |
| Last updated:
|
2020-07-03 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/145741 |
| . |
| Reference:
|
[1] Abate, M., Raissy, J., Saracco, A.: Toeplitz operators and Carleson measures in strongly pseudoconvex domains.J. Funct. Anal. 263 (2012), 3449-3491. Zbl 1269.32003, MR 2984073, 10.1016/j.jfa.2012.08.027 |
| Reference:
|
[2] Abate, M., Saracco, A.: Carleson measures and uniformly discrete sequences in strongly pseudoconvex domains.J. Lond. Math. Soc., Ser. (2) 83 (2011), 587-605. Zbl 1227.32008, MR 2802500, 10.1112/jlms/jdq092 |
| Reference:
|
[3] Arsenović, M., Shamoyan, R. F.: On distance estimates and atomic decompositions in spaces of analytic functions on strictly pseudoconvex domains.Bull. Korean Math. Soc. 52 (2015), 85-103. Zbl 1308.32037, MR 3313426, 10.4134/BKMS.2015.52.1.085 |
| Reference:
|
[4] F. Beatrous, Jr.: $L^p$-estimates for extensions of holomorphic functions.Mich. Math. J. 32 (1985), 361-380. MR 0803838, 10.1307/mmj/1029003244 |
| Reference:
|
[5] Carleson, L.: Interpolations by bounded analytic functions and the corona problem.Ann. Math. (2) 76 (1962), 547-559. Zbl 0112.29702, MR 0141789, 10.2307/1970375 |
| Reference:
|
[6] Cima, J. A., Mercer, P. R.: Composition operators between Bergman spaces on convex domains in $\mathbb C^n$.J. Oper. Theory 33 (1995), 363-369. MR 1354986 |
| Reference:
|
[7] Cima, J. A., Wogen, W. R.: A Carleson measure theorem for the Bergman space on the ball.J. Oper. Theory 7 (1982), 157-163. Zbl 0499.42011, MR 0650200 |
| Reference:
|
[8] Cohn, W. S.: Tangential characterizations of BMOA on strictly pseudoconvex domains.Math. Scand. 73 (1993), 259-273. Zbl 0815.32012, MR 1269263, 10.7146/math.scand.a-12470 |
| Reference:
|
[9] Djrbashian, A. E., Shamoyan, F. A.: Topics in the Theory of $A^p_\alpha$ Spaces.Teubner-Texte zur Mathematik 105 Teubner, Leipzig (1988). MR 1021691 |
| Reference:
|
[10] Engliš, M., Hänninen, T. T., Taskinen, J.: Minimal $L^\infty$-type spaces on strictly pseudoconvex domains on which the Bergman projection is continuous.Houston J. Math. 32 (2006), 253-275. Zbl 1113.46017, MR 2202364 |
| Reference:
|
[11] Hastings, W. W.: A Carleson measure theorem for Bergman spaces.Proc. Am. Math. Soc. 52 (1975), 237-241. Zbl 0296.31009, MR 0374886, 10.1090/S0002-9939-1975-0374886-9 |
| Reference:
|
[12] Jimbo, T., Sakai, A.: Interpolation manifolds for products of strictly pseudoconvex domains.Complex Variables, Theory Appl. 8 (1987), 333-341. Zbl 0587.32032, MR 0898073, 10.1080/17476938708814242 |
| Reference:
|
[13] Kobayashi, S.: Hyperbolic Complex Spaces.Grundlehren der Mathematischen Wissenschaften 318 Springer, Berlin (1998). Zbl 0917.32019, MR 1635983, 10.1007/978-3-662-03582-5 |
| Reference:
|
[14] Krantz, S. G.: Function Theory of Several Complex Variables.Pure and Applied Mathematics A Wiley-Interscience Publication. John Wiley & Sons, New York (1982). Zbl 0471.32008, MR 0635928 |
| Reference:
|
[15] Li, H.: BMO, VMO and Hankel operators on the Bergman space of strongly pseudoconvex domains.J. Funct. Anal. 106 (1992), 375-408. Zbl 0793.47025, MR 1165861, 10.1016/0022-1236(92)90054-M |
| Reference:
|
[16] Li, S.-Y., Luo, W.: Analysis on Besov spaces I\kern-1ptI: Embedding and duality theorems.J. Math. Anal. Appl. 333 (2007), 1189-1202. MR 2331724, 10.1016/j.jmaa.2006.03.074 |
| Reference:
|
[17] Li, S., Shamoyan, R.: On some estimates and Carleson type measure for multifunctional holomorphic spaces in the unit ball.Bull. Sci. Math. 134 (2010), 144-154. Zbl 1187.32003, MR 2592966, 10.1016/j.bulsci.2008.04.003 |
| Reference:
|
[18] Li, S., Shamoyan, R.: On some properties of analytic spaces connected with Bergman metric ball.Bull. Iran. Math. Soc. 34 (2008), 121-139. Zbl 1182.32002, MR 2477998 |
| Reference:
|
[19] Luecking, D.: A technique for characterizing Carleson measures on Bergman spaces.Proc. Am. Math. Soc. 87 (1983), 656-660. Zbl 0521.32005, MR 0687635, 10.1090/S0002-9939-1983-0687635-6 |
| Reference:
|
[20] McNeal, J. D., Stein, E. M.: Mapping properties of the Bergman projection on convex domains of finite type.Duke Math. J. 73 (1994), 177-199. Zbl 0801.32008, MR 1257282 |
| Reference:
|
[21] Oleinik, V. L.: Embedding theorems for weighted classes of harmonic and analytic functions.J. Sov. Math. 9 (1978), 228-243. Zbl 0396.31001, 10.1007/BF01578546 |
| Reference:
|
[22] Oleinik, V. L., Pavlov, B. S.: Embedding theorems for weighted classes of harmonic and analytic functions.J. Sov. Math. 2 (1974), 135-142. MR 0318867, 10.1007/BF01099672 |
| Reference:
|
[23] Ortega, J. M., Fàbrega, J.: Mixed-norm spaces and interpolation.Stud. Math. 109 (1994), 233-254. Zbl 0826.32003, MR 1274011 |
| Reference:
|
[24] Range, R. M.: Holomorphic Functions and Integral Representations in Several Complex Variables.Graduate Texts in Mathematics 108 Springer, New York (1986). Zbl 0591.32002, MR 0847923, 10.1007/978-1-4757-1918-5 |
| Reference:
|
[25] Shamoyan, R. F., Kurilenko, S. M., Sergey, M.: On a new embedding theorem in analytic Bergman type spaces in bounded strictly pseudoconvex domains of $n$-dimensional complex space.Journal of Siberian Federal University 7 (2014), 383-388. |
| Reference:
|
[26] Shamoyan, R. F., Mihić, O. R.: On distance function in some new analytic Bergman type spaces in $\Bbb C^n$.J. Funct. Spaces (2014), Article ID 275416, 10 pages. MR 3208648 |
| Reference:
|
[27] Shamoyan, R. F., Mihić, O. R.: On new estimates for distances in analytic function spaces in higher dimension.Sib. Èlektron. Mat. Izv. (electronic only) 6 (2009), 514-517. Zbl 1299.30106, MR 2586703 |
| Reference:
|
[28] Shamoyan, R. F., Mihić, O. R.: On some properties of holomorphic spaces based on Bergman metric ball and Luzin area operator.J. Nonlinear Sci. Appl. 2 (2009), 183-194. Zbl 1181.32009, MR 2521196, 10.22436/jnsa.002.03.07 |
| Reference:
|
[29] Shamoyan, R., Povprits, E.: Sharp theorems on traces in analytic spaces in tube domains over symmetric cones.Journal of Siberian Federal University 6 (2013), 527-538. MR 3496285 |
| Reference:
|
[30] Shamoyan, R. F., Povprits, E. V.: Multifunctional analytic spaces on products of bounded strictly pseudoconvex domains and embedding theorems.Kragujevac J. Math. 37 (2013), 221-244. MR 3150861 |
| Reference:
|
[31] Shamoyan, R., Radnia, M.: On some new embedding theorems for some analytic classes in the unit ball.J. Nonlinear Sci. Appl. 2 (2009), 243-250. Zbl 1181.32010, MR 2562265, 10.22436/jnsa.002.04.06 |
| Reference:
|
[32] Yamaji, S.: Composition operators on the Bergman spaces of a minimal bounded homogeneous domain.Hiroshima Math. J. 43 (2013), 107-127. Zbl 1304.47034, MR 3066527, 10.32917/hmj/1368217952 |
| Reference:
|
[33] Yamaji, S.: Positive Toeplitz operators on weighted Bergman spaces of a minimal bounded homogeneous domain.J. Math. Soc. Japan 65 (2013), 1101-1115. Zbl 1284.47025, MR 3127818, 10.2969/jmsj/06541101 |
| Reference:
|
[34] Zhu, K.: Spaces of Holomorphic Functions in the Unit Ball.Graduate Texts in Mathematics 226 Springer, New York (2005). Zbl 1067.32005, MR 2115155 |
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