$L^{p}$-APPROXIMATION OF HOLOMORPHIC FUNCTIONS ON A CLASS OF CONVEX DOMAINS. Issue 3 (23rd April 2018)

• Record Type: Journal Article
• Authors: HA, LY KIM
• Is Part Of: Bulletin of the Australian Mathematical Society. Volume 97:Issue 3(2018)
• Page Start: 446
• Page End: 452
• Publication Date: 2018-04-23
• Subjects: primary 32E30, -- secondary 32T25, -- 32W05, -- 41A63, -- 41A65; Lp approximation, -- holomorphic functions, -- Henkin solutions, -- the additive Cousin problem, -- finite/infinite type; Mathematics -- Societies, etc; Mathematics -- Periodicals; 510.5
• DOI: 10.1017/S0004972718000114
• Languages: English
• ISSNs: 0004-9727

Abstract

Abstract : Let $\unicode[STIX]{x1D6FA}$ be a member of a certain class of convex ellipsoids of finite/infinite type in $\mathbb{C}^{2}$ . In this paper, we prove that every holomorphic function in $L^{p}(\unicode[STIX]{x1D6FA})$ can be approximated by holomorphic functions on $\bar{\unicode[STIX]{x1D6FA}}$ in $L^{p}(\unicode[STIX]{x1D6FA})$ -norm, for $1\leq p<\infty$ . For the case $p=\infty$, the continuity up to the boundary is additionally required. The proof is based on $L^{p}$ bounds in the additive Cousin problem.