Reference request: about inverse Laplacian operator

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I am currently studying some problems about inverse Laplacian and the Yosida approximation and wishing to learn more about it.

Here is a post about one of the problems that I am interested in.

I am looking for a book which has some referece about how to transform the inverse of Laplace operator into Fourier basis, in order to compute $L^2$ norm. i.e.,

$$ \|(I-t\Delta)^{-s}u\|_{L^2}^2=\sum_{k=0}?? $$ where $t$ and $s$ are real numbers.

If you know there are some reference which has similar computation. Please kindly direct me there.

Thank you!