We consider a function $f$ in $\mathcal{B}_1$ whose partial derivatives $\frac{\partial f}{\partial x}$ and $\frac{\partial f}{\partial y}$ are in $\mathcal{B}_2$. We set:
$$\forall q \in \mathbb{R}^+,\quad F(q) = \frac{1}{2\pi}\int_0^{2\pi} \hat{f}(q,\theta)\,\mathrm{d}\theta = 2\int_q^{+\infty} \frac{r\bar{f}(r)}{\sqrt{r^2-q^2}}\,\mathrm{d}r$$
Prove: $\forall \varepsilon > 0,\ \int_\varepsilon^{+\infty} \frac{F^{\prime}(q)}{q}\,\mathrm{d}q = -\frac{F(\varepsilon)}{\varepsilon} + 2\int_\varepsilon^{+\infty} \frac{1}{q^2}\left(\int_q^{+\infty} \frac{r\bar{f}(r)}{\sqrt{r^2-q^2}}\,\mathrm{d}r\right)\mathrm{d}q$.