# Characterisation of functions for which the Fourier transform commutes with a particular operator

Defining the operator $\phi$ by: $\phi(f(x))=\frac{1}{|x|} f(\frac{1}{x})$, and noting $\mathcal{F}$ the Fourier transform on the real line, can we characterize all the functions (with real variable and complex value) satisfying the following relation ?

$$\phi \circ \mathcal{F} (f(x))= c \; \mathcal{F} \circ \phi (f(x))$$

where $c$ is a complex constant.

All functions (tempered distribution to be precise) $f(x)=|x|^{\alpha}$ with $0>\Re (\alpha)>-1$, satisfy above equations but can we find other functions ?

# The Second question with same operator $\phi$:

Can we find a way to characterize all functions $f$ satisfying

$$\int_0^{\infty}\phi \circ \mathcal{F} (f(x)) dx = 0$$