Hostname: page-component-848d4c4894-75dct Total loading time: 0 Render date: 2024-05-24T09:15:48.156Z Has data issue: false hasContentIssue false
  • English
  • Français

Two-weighted inequalities for the derivatives of holomorphic functions and Carleson measures on the unit ball

Published online by Cambridge University Press:  22 January 2016

Hyeonbae Kang  and
Hyungwoon Koo
Hyeonbae Kang
Affiliation:
Department of Mathematics, Seoul National University, Seoul, 151-742, Korea, hkang@math.snu.ac.kr
Hyungwoon Koo
Affiliation:
Department of Mathematics, Hankuk University of Foreign Studies, Yongin, Kyungki-Do, 449-791, Korea, koohw@maincc.hufs.ac.kr
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

We characterize those positive measure µ’s on the higher dimensional unit ball such that “two-weighted inequalities” hold for holomorphic functions and their derivatives. Characterizations are given in terms of the Carleson measure conditions. The results of this paper also distinguish between the fractional and the tangential derivatives.

Keywords

32A1032A3632A3526A33
Type
Research Article
Information
Nagoya Mathematical Journal , Volume 158 , December 2000 , pp. 107 - 131
Copyright
Copyright © Editorial Board of Nagoya Mathematical Journal 2000

References

[B] Beatrous, F., Estimates for derivatives of holomorphic functions in pseudoconvex domains, Math. Z., 191 (1986), 91–116.Google Scholar
[BB] Beatrous, F. and Burbea, J., Sobolev spaces of holomorphic functions in the unit ball, Dissertations Math., 276 (1989), 1–60.Google Scholar
[C] Carleson, L., Interpolation by bounded analytic functions and the corona problem, Ann. of Math., 76 (1962), 547–559.Google Scholar
[CKK] Choe, B., Kang, H. and Koo, H., preprint.Google Scholar
[D] Duren, P., Extention of a theorem of Carleson, Bull. Amer. Math. Soc, 75 (1969), 143–146.CrossRefGoogle Scholar
[GLS] Girela, D., Lorente, M. and Sarrion, M. D., Embedding derivatives of weighted Hardy spaces into Lebesgue spaces, Math. Proc. Camb. Phil. Soc., 116 (1994), 151-166.CrossRefGoogle Scholar
[G] Gu, D., Two-weight norm inequality and Carleson measure in weighted Hardy spaces, Canad. J. Math., 44 (6) (1992), 1206–1219.Google Scholar
[Gr] Grellier, S., Behaviour of holomorphic functions in complex tangential directions in a domain of finite type in n , Publications Matematique, 36 (1992), 251–292.CrossRefGoogle Scholar
[KK] Kang, H. and Koo, H., Carleson measure characterizations of BMOA on pseudo-convex domains, Pacific J. of Math., 178(2) (1997), 279–291.Google Scholar
[L1] Luecking, D., Forward and reverse Carleson inequalities for functions in the Bergman spaces and their derivatives, Amer. J. Math., 107 (1985), 85–111.Google Scholar
[L2] Luecking, D., Embedding derivatives of Hardy spaces into Lebesgue spaces, Proc. London. Math. Soc., 63(3) (1991), 595–619.Google Scholar
[LP] Littlewood, J. and Paley, R., Theorems on Fourier series and power series. II, Proc. London. Math. Soc., 42 (1936), 52–89.Google Scholar
[S1] Stein, E., Boundary behavior of holomorphic functions of several complex variables, Princeton University Press, Princeton, NJ, 1972.Google Scholar
[S2] Stein, E., Harmonic Analysis, real-variable methods, orthogonality, and oscillatory integrals, Princeton University Press, Princeton, NJ, 1993.Google Scholar
[Sh1] Shirokov, N. A., Some generalization of the Littlewood-Paley Theorem, Zap Nauch. Sem. LOMI, 39 (1974), 162–175; J. Soviet Math. 8 (1977) 119–129.Google Scholar
[Sh2] Shirokov, N. A., Some embedding theorems for spaces of harmonic functions, Zap Nauch. Sem. LOMI, 56 (1976), 191–194; J. Soviet Math. 14 (1980) 1173–1176.Google Scholar
[ST] Strömberg, J.-O. and Torchinsky, A., Weighted Hardy Spaces, Lecture Notes in Math. 1381, Springer-Verlag, 1989.Google Scholar