积分算子的奇异数和本征值

设Ω=0,1]×0,1]是单位正方形,W~(12)(Ω)表示由所有这样的K(x,y)∈L~2(Ω)构成的空间:它对每个y关于x绝对连续,对每个x关于y绝对连续,而且偏导数((?)/(?)x)K(x,y)((?)/(?)x)K(x,y)都在L~2(Ω)中。最近Reade证明,任何K(x,y)=K(Y,X)∈W~(12)(Ω)的本征值,满足。本文说明,任何K(x,y)∈W~(12)(Ω)的奇异数满足特别如K(x,y)∈W~(12)(Ω)还假定是对称的,那末Reade的结果可改进。

Singular Numbers and Eigenvalues of Integral Operators

Let Ω = 0,l]× 0,1] be the unit square. Let W12(Ω) denote the space of all K(x,y)∈ L2(Ω) which arc absolutely continuous in x for each t and absolutely continuous iny for each x, and the partial derivatives (?) /(?)xK (x,y),(?) /(?)yK(x,y) arc both in L2(Ω). RecentlyRcadc 〔4〕 has proved that the eigenvalues of any K(x,y) = K(y,x)∈ W12(Ω)satisfyn2Kn2(K)<∞. In this paper we show that the singular numbers of any K(x,y)e W12(Ω) satisfyParticularly, if K(x,y)e Wl2(Ω) is also assumed to be symmetric, then Rcadc's result can be improved to