Change from Fourier Space to Real Space

Question

I have a function in 3D fourier Space $$g(\textbf {k})=\frac{\hat{k}_i}{\hat{k_j}}f(\textbf {k}),$$ where $\hat{\alpha}$ is a fixed vector and $i$ and $j$ are the components of the relevant vector, and the function $f(\textbf {k})$ is given by $$f(r,\hat{r}.\hat{\alpha})=\int \mathop{\mathrm{d}^3k}e^{i\mathbf{k}.\mathbf{r}}f(\textbf {k})$$ My question is how do express the above function, which is fourier space, to a function in real space. That is, I want $g(\textbf {k})$ in real space, $g(r)$, in terms of derivatives of $f(r)$. I know that $i\hat{k}_i\rightarrow \partial/\partial r_i,$ and so on. But, I have no idea how to deal with the denominator.