I'm working through the problems in the initial value formulation chapter in Wald's General Relativity.
A short summary of the problem. I have to show that $$\sup_{x\in A}|f(x)|\le C||f||_{A,k}$$ where $A\subset \mathbb{R}^n$ satisfies the uniform interior cone condition, $C$ is a constant, $k>n/2$ is an integer and $||.||_{A,k}$ is the Sobolev norm. Let $Q$ denote the solid closed cone in $\mathbb{R}^n$ of height $H$ and solid angle $\Omega$, with vertex at the origin. Let $\psi:\mathbb{R}\longrightarrow\mathbb{R}$ be a $C^\infty$ function with $\psi(r)=1$ for $r<H/3$ and $\psi(r)=0$ for $r >2H/3$. I have shown that $$\tag{1}f(0)=C_1\int_{Q\subset\mathbb{R}^n} r^{k-n}\frac{d^k}{dr^k}(\psi f)\,dx$$ The hint says to show $$\tag{2}|f(0)|\le C||f||_{Q,k}$$ using Cauchy-Schwarz. From this I can obtain the final result using the uniform interior cone condition.
Any help would be greatly appreciated.
EDIT: I should be using Cauchy-Schwarz with the $L^2$ inner product, not the Sobolev one. I can write $$f(0)=C_1\langle r^{k-n},\partial_r^k(\psi f)\rangle_{L^2}$$ and thus $$\tag{3}|f(0)|\le C_1 ||r^{k-n}||_{L^2}\cdot ||\partial^k_r(\psi f)||_{L^2}$$ For the first factor I get $$||r^{k-n}||_{L^2}=\int_Q |r^{k-n}|^2\,dx=\int_Q r^{2k-2n}\,dx=\int_Q r^{2k-2n}r^{n-1}\,drd\Omega=C_2$$ a constant, iff $k>n/2$. Putting this into (3), I get $$\tag{4}|f(0)|\le C_3||\partial^k_r(\psi f)||_{L^2}$$ I can obtain (2) by showing that $$||\partial^k_r(\psi f)||_{L^2}\le C_4 ||f||_{Q,k}$$ Intuitively, this is clear, because $||f||_{Q,k}$ contains "more terms" which are all positive. But my problem then lies with the equals in $\le$. I don't see how it could be anything other than strictly larger, but since I'm using my gut here, I could very well be wrong.
EDIT 2: I guess the equality could hold based on the precise form of $\psi$ perhaps, but I'm still not sure.