I have recently started to study the book "Fourier integrals in classical analysis " by Sogge mainly oscillatory integral decay methods. I have a question from the chapters 4 and 5. Mainly chapter 4, Weyl's law has been discussed and chapter 5 deals with $L^p$ estimate of
I have recently begun studying "Fourier Integrals in Classical Analysis" by Sogge, with a focus on oscillatory integral decay methods. I have specific questions regarding the content in Chapters 4 and 5.
Chapter 4 discusses Weyl's law, while Chapter 5 focuses on $L^p$ estimates of eigenfunctions. I would like to clarify whether Theorem 5.1.1 and Lemma 5.1.3 are exclusively applicable to Laplace eigenfunctions. For instance, Corollary 5.1.2 pertains specifically to Laplace eigenfunctions. Are these results generally valid for other operators beyond the Laplacian, such as the $p$-Laplacian, fractional Laplacian, or higher-order Laplacians? If not, what is the reason that they fail to have such estimates?
Additionally, I have a similar inquiry regarding the sharp Weyl formula presented in Section 4.2 of Chapter 4 whether it only includes Laplace eigenfunction. Any idea or thought is greatly appreciated.
