Interpolation properties of Besov spaces defined on metric spaces
Interpolation properties of Besov spaces defined on metric spaces
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DOI：
10.1002/mana.200810242
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发表时间：
2010-02
影响因子：
1
通讯作者：
Amiran Gogatishvili;P. Koskela;N. Shanmugalingam
中科院分区：
文献类型：
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作者：
Amiran Gogatishvili;P. Koskela;N. Shanmugalingam
设X =（X，d，μ）是一个双重度量测度空间.对于0 < α < 1，1 ≤p，q < ∞，我们定义半范数当q = ∞时，通常在定义中进行从积分到上确界的改变。Besov空间Bp，qα（X）是Llocp（X）中的函数f的集合，对于这些函数f，半范数B p，q α（X）是有限的。本文证明了：如果一个双重度量测度空间（X，d，μ）支持一个（1，p）-Poincaré不等式，则Besov空间Bp，qα（X）与真实的插值空间（Lp（X），KS 1，p（X））α，q重合，其中KS 1，p（X）是Korevaar和Schoen [15]定义的Sobolev空间.这导致（尖锐）嵌入定理。我们进一步证明了Besov空间的定义与Bourdon和Pajot [3]给出的定义是等价的，并建立了迹定理（© 2010 WILEY‐VCH Verlag GmbH & Co. KGaA，魏因海姆）
Let X = (X, d, μ)be a doubling metric measure space. For 0 < α < 1, 1 ≤p, q < ∞, we define semi‐norms When q = ∞ the usual change from integral to supremum is made in the definition. The Besov space Bp, qα (X) is the set of those functions f in Llocp(X) for which the semi‐norm ‖f ‖ B p, q α (X) is finite. We will show that if a doubling metric measure space (X, d, μ) supports a (1, p)‐Poincaré inequality, then the Besov space Bp, qα (X) coincides with the real interpolation space (Lp (X), KS1, p(X))α, q, where KS1, p(X) is the Sobolev space defined by Korevaar and Schoen [15]. This results in (sharp) imbedding theorems. We further show that our definition of a Besov space is equivalent with the definition given by Bourdon and Pajot [3], and establish a trace theorem (© 2010 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)