The behavior of the laplacian on weighted sobolev spaces
The behavior of the laplacian on weighted sobolev spaces
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DOI：
10.1002/cpa.3160320604
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发表时间：
1979-11
影响因子：
3
通讯作者：
R. McOwen
中科院分区：
文献类型：
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作者：
R. McOwen
注意，条件（ii）消除了该结果对R3或R4中A的Lz理论的适用性。这是有原因的：如果E和F是Banach空间，使得E c L2（Rn），FI G（R”）和A：E+ F是索引为0的Fredholm映射，则n 2 5（见[3]）。因此，如果我们想研究（*）的Fredholm性质，p= 2，n= 3，4，我们必须准备允许A4&$L2（即，6< 0）或允许（*）的索引为非零。我们将看到这两种选择都是可能的。事实上，下面的定理完全描述了（*）对所有p和6值的行为。(Let N表示非负整数，Xi表示j次齐次的调和多项式。）
Note that condition (ii) eliminates the applicability of this result to Lz-theory for A in R3 or R4. There is a reason for this: if E and F are Banach spaces such that E c L2 (Rn), FI G (R") and A: E+ F is a Fredholm map of index zero, then n 2 5 (see [3]). Thus if we want to study the Fredholm properties of (*) for p= 2 and n= 3, 4, we must be prepared to allow A4& $ L2 (ie, 6< 0) or allow the index of (*) to be nonzero. We shall see that both these alternatives are possible. In fact the following theorem completely describes the behavior of (*) for all values of p and 6.(Let N denote the non-negative integers, and Xi the harmonic polynomials which are homogeneous of degree j.)