I have been trying to read Theorem 2.7.1 in the Book Topics in Functional Analysis and Applications by S. Kesavan. New Age International (formerly Wiley-Eastern), 1989. However, I am struggling to fill in some gaps in the proof. For instance:
(a) why do we have $$|v(x',0)|^2 = -\int_0^\infty \frac{\partial}{\partial x_n}(|v(x',x_n)|^2)dx_n?$$ Is it from the definitions of $L^p$ norms? Is it from the definition of functions in $C^\infty_0$?
(b) Why $$-\int_0^\infty \frac{\partial}{\partial x_n}(|v(x',x_n)|^2)dx_n=-2\int_0^\infty \frac{\partial}{\partial x_N}(x',x_N)v(x',x_N) dx_N?$$ I thought this step was due to integration by parts. But still it does not come out clearly.
(c) Which inequality related to integration yields this step? $$-2\int_0^\infty \frac{\partial}{\partial x_N}(x',x_N)v(x',x_N) dx_N\leq \int_0^\infty \left(\left|\frac{\partial}{\partial x_N}(x',x_N)\right|^2+ \left|v(x',x_N)\right|^2\right) dx_N.$$
I am a beginner in PDE, someone help me.