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DIVISION PROBLEM IN GENERALIZED GROWTH SPACES ON THE UNIT BALL IN ℂn

  • Cho, Hong Rae (Department of Mathematics, Pusan National University) ;
  • Lee, Han-Wool (Department of Mathematics, Pusan National University) ;
  • Park, Soohyun (Department of Mathematics, Pusan National University)
  • Received : 2014.10.22
  • Accepted : 2014.12.09
  • Published : 2015.01.31

Abstract

Let $\mathbb{B}$ be the unit ball in $\mathbb{C}^n$. For a weight function ${\omega}$, we define the generalized growth space $A^{\omega}(\mathbb{B})$ by the space of holomorphic functions f on $\mathbb{B}$ such that $${\mid}f(z){\mid}{\leq}C{\omega}({\mid}{\rho}(z){\mid},\;z{\in}\mathbb{B}$$. Our main purpose in this note is to get the corona type decomposition in generalized growth spaces on $\mathbb{B}$.

Keywords

References

  1. E. Amar, On the corona problem, J. Geom. Anal. 1 (1991), no. 4, 291-305. https://doi.org/10.1007/BF02921307
  2. M. Andersson, The $H^{p}$ corona problem in weakly pseudoconvex domains, Trans. Amer. Math. Soc. ,342 (1994), 241-255.
  3. M. Andersson and H. Carlsson, Wolff-type estimates for ${\overline{\partial}}_b$ and the $H^{p}$-corona problem in strictly pseudoconvex domains, Ark. Mat. 32 (1994) 255-276. https://doi.org/10.1007/BF02559572
  4. M. Andersson and H. Carlsson, $H^{p}$-estimates of holomorphic division formulas, Pacific J. Math. 173 (1996), 307-335. https://doi.org/10.2140/pjm.1996.173.307
  5. M. Andersson and H. Carlsson, Estimates of solutions of the $H^{p}$ and BMOA corona problem, Math. Ann. 316 (2000), 83-102. https://doi.org/10.1007/s002080050005
  6. B. Berndtsson and M. Andersson, Henkin-Ramirez formulas with weight factors, Ann. Inst. Fourier 32(1982), 91-110.
  7. J. Byun, H. R. Cho, and J.-D. Park, Weighted Lipschitz estimates for ${\overline{\partial}}$ on convex domains of finite type, J. Math. Anal. Appl. 368 (2010) 190210
  8. L. Carleson, Interpolation of bounded analytic functions and the corona problem, Ann. of Math. 76(1961), 547-559.
  9. H. R. Cho and E. G. Kwon, Embedding of Hardy spaces into weighted Bergman spaces in bounded domains with $C^2$ boundary, Illinois J. Math. 48-3(2004), 747-757.
  10. D. Girela, M. Nowak, and P. Waniurski, On the zeros of Bloch functions, Math. Proc. Camb. Phil. Soc. 129(2000), 117-128. https://doi.org/10.1017/S0305004199004338
  11. P. Koosis, Introduction to $H^{p}$-spaces. With an appendex on Wolff's proof of the corona theorem, Lond. Math. Soc. Lecture Note Series, 40(1980).
  12. S. G. Krantz and S.-Y. Li, Some remarks on the corona problem on strongly pseu-doconvex domains in ${\mathbb{C}}^n$, Illinois J. Math. 39-1(1995), 323-349.
  13. K. Lin, The ${\overline{\partial}}_u$ = f corona theorem for the polydisc, Trans. Amer. Math. Soc. 341(1994), 371-375.
  14. J. M. Ortega, J. Fabrega, Corona type decomposition in some Besov spaces, Math. Scand. 78(1996), 93-111. https://doi.org/10.7146/math.scand.a-12576
  15. J. M. Ortega, J. Fabrega, Pointwise multipliers and corona type decomposition in BMOA, Ann. Inst. Fourier 46(1996), 111-137. https://doi.org/10.5802/aif.1509
  16. J. M. Ortega, J. Fabrega, Pointwise multipliers and decomposition theorems in analytic Besov spaces, Math. Z. 235(2000), 53-81. https://doi.org/10.1007/s002090000123
  17. J. M. Ortega, J. Fabrega, Pointwise multipliers and decomposition theorems in $F_s^{{\infty},q}$, Math. Ann. 329(2004), 247-277 https://doi.org/10.1007/s00208-003-0461-6
  18. N. Varopoulos, BMO functions and the ${\overline{\partial}}$-equation, Pacific J. Math. 71(1977), 221-273. https://doi.org/10.2140/pjm.1977.71.221