Previous |  Up |  Next

Article

Keywords:
diagonal map; holomorphic function; Bergman space; polydisk
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.
References:
[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
[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
[3] Luecking, D.: A new proof of an inequality of Littlewood and Paley. Proc. Am. Math. Soc. 103 (1988), 887-893. MR 0947675 | Zbl 0665.30035
[4] Lusin, N.: Sur une propriete des fonctions a carre sommable. Bulletin Calcutta M. S. 20 (1930), 139-154.
[5] Ortega, J., Fabrega, J.: Hardy's inequality and embeddings in holomorphic Lizorkin-Triebel spaces. Ill. J. Math. 43 (1993), 733-751.
[6] Ortega, J., Fabrega, J.: Pointwise multipliers and decomposition theorems for $F^{\infty, q}_s$. Math. Ann. 329 (2004), 247-277. MR 2060362
[7] Piranian, G., Rudin, W.: Lusin's theorem on areas of conformal maps. Mich. Math. J. 3 (1956), 191-199. MR 0083553 | Zbl 0074.05602
[8] Ren, G., Huai, S. J.: The diagonal mapping in mixed norm spaces. Stud. Math. 163 (2004), 103-117. MR 2047374
[9] Rudin, W.: Function Theory in Polydiscs. Benjamin, New York (1969). MR 0255841 | Zbl 0177.34101
[10] Shamoyan, R. F.: Generalized Hardy transformation and Toeplitz operators in BMOA-type spaces. Ukr. Math. J. (2001), 53 1519-1534. MR 1900044 | Zbl 1010.32005
[11] Shamoyan, F. A., Djrbashian, A. E.: Topics in the Theory of ${\cal A}^p_\alpha$ Spaces. Teubner Texte zur Math. Leipzig (1988).
[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
[13] c, S. Stevi': On an area inequality and weighted integrals of analytic functions. Result. Math. 41 (2002), 386-393. MR 1915936
[14] c, S. Stevi': On harmonic Hardy and Bergman spaces. J. Math. Soc. Japan 54 (2002), 983-996. MR 1921096
[15] c, S. Stevi': Weighted integrals of holomorphic functions on the unit polydisk II. Z. Anal. Anwend. 23 (2004), 775-782. MR 2110405
[16] Yamashita, S.: Criteria for functions to be of Hardy class $H^p$. Proc. Am. Math. Soc. 75 (1979), 69-72. MR 0529215
[17] Zhu, K.: The Bergman spaces, the Bloch spaces, and Gleason's problem. Trans. Am. Math. Soc. 309 (1988), 253-268. MR 0931533
[18] Zhu, K.: Duality and Hankel operators on the Bergman spaces of bounded symmetric domains. J. Funct. Anal. 81 (1988), 260-278. MR 0971880 | Zbl 0669.47019
Partner of
EuDML logo