Consider a vector $p$ in $\mathbb{R}^3$ such that $||p||=1$.

Let $f$ in $S(\mathbb{R}^3)$ (you can assume as smooth as you want if desire).

Consider the following operator defined in Fourier by :

$$\widehat{F(f)}(\xi) = \frac{(\xi \cdot p)^2}{|\xi|^2} \hat{f}(\xi).$$

I want to compute the fourier operator $F$ in the physical space. It is of degree zero in $\xi$ so I guess the operator should not involved any derivatives of $f$.

Any help is welcomed, as always.


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