The following is an exercise in Knapp's book, representation theory of Semisimple Groups.
For $f\in C_c^\infty(\mathbb R^2-\{0\})$ with norm $$\|f\|^2=\int_{-\infty}^\infty\int_{-\infty}^\infty |f(x,y)|^2 dx dy,$$ define $E_v:f\mapsto F_v$ by $$ F_v(x,y)=\int_{0}^\infty f(tx,ty)t^{iv} d t. $$ The image is a dense subspace of $$ H_v^+\oplus H_v^-=\{F:\mathbb R^2\rightarrow \mathbb C, F(tx,ty)=t^{-1-iv}F(x,y),\forall t>0\}$$ with norm $$ \|F\|_v^2=\frac{1}{2\pi}\int_{0}^{2\pi} |F(\cos\theta,\sin\theta)|^2 d\theta. $$
Let $h(s)=e^sf(e^sx,e^sy)$, $-\infty<s<\infty$. By Fourier inversion formula and Plancherel formula, establish $$ f(x,y)=\frac{1}{2\pi}\int_\infty^\infty F_v(x,y)dv $$ and $$ \|f\|^2=\int_{-\infty}^\infty\|F_v\|_v^2 d v $$ for $f\in C_c^\infty(\mathbb R^2-\{0\})$.
The first one is easy to obtained by $$ h(0)=\int_{-\infty}^\infty \widehat{h}(\xi)d\xi. $$ How to establish the second formula?