| Title:
|
On some problems connected with diagonal map in some spaces of analytic functions (English) |
| Author:
|
Shamoyan, Romi |
| Language:
|
English |
| Journal:
|
Mathematica Bohemica |
| ISSN:
|
0862-7959 (print) |
| ISSN:
|
2464-7136 (online) |
| Volume:
|
133 |
| Issue:
|
4 |
| Year:
|
2008 |
| Pages:
|
351-366 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| Summary:
|
For any holomorphic function $f$ on the unit polydisk $\mathbb D ^n$ we consider its restriction to the diagonal, i.e., the function in the unit disc $\mathbb D \subset \mathbb C $ defined by $\mathop{\rm Diag} f(z)=f(z,\ldots ,z)$, and prove that the diagonal map ${\rm Diag}$ maps the space $Q_{p,q,s}(\mathbb D ^n)$ of the polydisk onto the space $\widehat Q^q_{p,s,n}(\mathbb D )$ of the unit disk. (English) |
| Keyword:
|
diagonal map |
| Keyword:
|
holomorphic function |
| Keyword:
|
Bergman space |
| Keyword:
|
polydisk |
| MSC:
|
30H05 |
| MSC:
|
47B35 |
| idZBL:
|
Zbl 1199.47126 |
| idMR:
|
MR2472484 |
| DOI:
|
10.21136/MB.2008.140625 |
| . |
| Date available:
|
2010-07-20T17:37:10Z |
| Last updated:
|
2020-07-29 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/140625 |
| . |
| Reference:
|
[1] Duren, P. L., Shields, A. L.: Restriction of $H^p$ functions on the diagonal of the polydiscs.Duke Math. J. 42 (1975), 751-753. MR 0402101, 10.1215/S0012-7094-75-04262-3 |
| Reference:
|
[2] Essen, M., Wulan, H., Xiao, J.: Several function theoretic characterizations of Möbius invariant $Q_k$ spaces.J. Funct. Anal. 230 (2006), 78-115. MR 2184185, 10.1016/j.jfa.2005.07.004 |
| Reference:
|
[3] Luecking, D.: A new proof of an inequality of Littlewood and Paley.Proc. Am. Math. Soc. 103 (1988), 887-893. Zbl 0665.30035, MR 0947675, 10.1090/S0002-9939-1988-0947675-0 |
| Reference:
|
[4] Lusin, N.: Sur une propriete des fonctions a carre sommable.Bulletin Calcutta M. S. 20 (1930), 139-154. |
| Reference:
|
[5] Ortega, J., Fabrega, J.: Hardy's inequality and embeddings in holomorphic Lizorkin-Triebel spaces.Ill. J. Math. 43 (1993), 733-751. 10.1215/ijm/1256060689 |
| Reference:
|
[6] Ortega, J., Fabrega, J.: Pointwise multipliers and decomposition theorems for $F^{\infty, q}_s$.Math. Ann. 329 (2004), 247-277. MR 2060362, 10.1007/s00208-003-0461-6 |
| Reference:
|
[7] Piranian, G., Rudin, W.: Lusin's theorem on areas of conformal maps.Mich. Math. J. 3 (1956), 191-199. Zbl 0074.05602, MR 0083553 |
| Reference:
|
[8] Ren, G., Huai, S. J.: The diagonal mapping in mixed norm spaces.Stud. Math. 163 (2004), 103-117. MR 2047374, 10.4064/sm163-2-1 |
| Reference:
|
[9] Rudin, W.: Function Theory in Polydiscs.Benjamin, New York (1969). Zbl 0177.34101, MR 0255841 |
| Reference:
|
[10] Shamoyan, R. F.: Generalized Hardy transformation and Toeplitz operators in BMOA-type spaces.Ukr. Math. J. (2001), 53 1519-1534. Zbl 1010.32005, MR 1900044, 10.1023/A:1014322910001 |
| Reference:
|
[11] Shamoyan, F. A., Djrbashian, A. E.: Topics in the Theory of ${\cal A}^p_\alpha$ Spaces.Teubner Texte zur Math. Leipzig (1988). |
| Reference:
|
[12] Shi, J. H.: Inequalities for the integral means of holomorphic functions and their derivatives in the unit ball of $\mathbb C^n$.Trans. Am. Math. Soc. 328 (1991), 619-637. MR 1016807, 10.1090/S0002-9947-1991-1016807-5 |
| Reference:
|
[13] c, S. Stevi': On an area inequality and weighted integrals of analytic functions.Result. Math. 41 (2002), 386-393. MR 1915936, 10.1007/BF03322780 |
| Reference:
|
[14] c, S. Stevi': On harmonic Hardy and Bergman spaces.J. Math. Soc. Japan 54 (2002), 983-996. MR 1921096, 10.2969/jmsj/1191592001 |
| Reference:
|
[15] c, S. Stevi': Weighted integrals of holomorphic functions on the unit polydisk II.Z. Anal. Anwend. 23 (2004), 775-782. MR 2110405 |
| Reference:
|
[16] Yamashita, S.: Criteria for functions to be of Hardy class $H^p$.Proc. Am. Math. Soc. 75 (1979), 69-72. MR 0529215 |
| Reference:
|
[17] Zhu, K.: The Bergman spaces, the Bloch spaces, and Gleason's problem.Trans. Am. Math. Soc. 309 (1988), 253-268. MR 0931533 |
| Reference:
|
[18] Zhu, K.: Duality and Hankel operators on the Bergman spaces of bounded symmetric domains.J. Funct. Anal. 81 (1988), 260-278. Zbl 0669.47019, MR 0971880, 10.1016/0022-1236(88)90100-0 |
| . |
