Partial Differential Equations – Book Recommendation on Fourier Analysis Technique

Question

I want a graduate-level textbook which discusses Fourier analysis techniques for solving PDE. To elaborate what I want to study, consider the Laplace operator $-\Delta$. Since the differential operator transforms as a multiplication operator, Fourier analysis techniques can be useful.
The following article by Terence Tao
https://terrytao.wordpress.com/tag/rage-theorem/?fbclid=IwAR3ovIoURjm8yisDgJp1-Evd1xXcj4Jmfwk1P9kLtVFI1i0THaBGzNuVqrQ
illustrates what I want to study. I am interested in "spectral multipliers", which I does not know the definition but it is like a function that is multiplied after Fourier transformation.

I am also interested in fundamental solution of Laplace equations, Helmholtz equations, etc.

Please suggest a book that deals with above topics. I have a strong background in Lebesgue integration and some knowledge in functional analysis.

Answer 1

I would suggest Michael Taylor's first book on PDEs ("Basic Theory"). In particular, the third chapter seems to contain everything that you're looking for.