**Exercise:*** Show that if is the Laplace operator on , then for , *

**Exercise:*** Let be an essentially self-adjoint diffusion operator on . Show that if the constant function and if , then *

**Exercise:*** Let be an essentially self-adjoint diffusion operator on . *

- Show that for every , the range of the operator is dense in .
- By using the spectral theorem, show that the following limit holds for the operator norm on ,

**Exercise:*** As usual, we denote by the Laplace operator on . The Mac-Donald’s function with index is defined for by .*

- Show that for and ,
- Show that for , .
- Show that
- Prove that for and , where (You may use Fourier transform to solve the partial differential equation ).

**Exercise:*** By using the previous exercise, prove that for , the limit being taken in . Conclude that almost everywhere, *

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