Sharp constants for Moser‐Trudinger inequalities on spheres in complex space ℂn
Sharp constants for Moser‐Trudinger inequalities on spheres in complex space ℂn
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DOI：
10.1002/cpa.20043
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发表时间：
2004-11
影响因子：
3
通讯作者：
W. Cohn;G. Lu
中科院分区：
文献类型：
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作者：
W. Cohn;G. Lu
本文的主要结果涉及复空间中球面上的Moser - Trudinger不等式的尖锐常数。我们推导了光滑函数和具有不同锐常数的全纯函数的Moser - Trudinger不等式（见定理1.1）。考虑到尖锐的Moser - Trudinger不等式涉及函数的复杂切向梯度，因此我们在这里展示了CR设置下的这种不等式。虽然这里在复球上证明的不等式与Heisenberg群上证明的在任何有限域上具有紧支持的函数的不等式在精神上有密切的联系b[17]，但在复球上推导Moser - Trudinger不等式的尖锐常数比在Heisenberg群上推导更为复杂和难以获得。作为尖锐不等式的应用，在复球面上也给出了Moser - Onofri型不等式的变体（参见定理1.2和1.3）。推导主要定理的关键因素之一是用复切梯度表示复球面上的函数的清晰表示公式（见定理1.4）。©2004 Wiley期刊公司
The main results of this paper concern sharp constants for the Moser‐Trudinger inequalities on spheres in complex space ℂn. We derive Moser‐Trudinger inequalities for smooth functions and holomorphic functions with different sharp constants (see Theorem 1.1). The sharp Moser‐Trudinger inequalities under consideration involve the complex tangential gradients for the functions and thus we have shown here such inequalities in the CR setting. Though there is a close connection in spirit between inequalities proven here on complex spheres and those on the Heisenberg group for functions with compact support in any finite domain proven earlier by the same authors [17], derivation of the sharp constants for Moser‐Trudinger inequalities on complex spheres are more complicated and difficult to obtain than on the Heisenberg group. Variants of Moser‐Onofri‐type inequalities are also given on complex spheres as applications of our sharp inequalities (see Theorems 1.2 and 1.3). One of the key ingredients in deriving the main theorems is a sharp representation formula for functions on the complex spheres in terms of complex tangential gradients (see Theorem 1.4). © 2004 Wiley Periodicals, Inc.