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Symmetric Markov Processes, Time Change, and Boundary Theory (LMS-35) by Zhen-Qing Chen, Masatoshi Fukushima

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Appendix B

 

SOLUTIONS TO EXERCISES

image 1.1.10: Take approximating sequences {ƒn}, {gn} ⊂ images for ƒ, g. On account of the proof of Theorem 1.1.5(ii), we may assume that |ƒn| ≤ ||ƒ||, |gn| ≤ ||g||. Since

|ƒn(x)gn(x) – ƒn(y)gn(y)| ≤ ||g|||ƒn(x) – ƒn(y)| + ||ƒ|||gn(x) – gn(y)|,

and |ƒn(x)gn(x)| ≤ ||g|||ƒn(x)| + ||ƒ|||gn(x)|, we get from (1.1.13) that ATt(ƒngn)½≤ ||g||ATt(ƒn)½ + ||ƒ||ATt(gn)½. Dividing by t and letting t ↓ 0, we have ||ƒngn||E ≤ ||g|| ||ƒn||E + ||ƒ|| ||gn||E. Since ||ƒngn||E is uniformly bounded and ƒngn converges to fg as n → ∞, ...

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