On the Fourier transform of function of two variables which depend only on the maximum of these variables

For functions $f(x_1,x_2)=f_0(\max \{\mathrm{\mid }{x}_1\mathrm{\mid },\mathrm{\mid }{x}_2\mathrm{\mid }\})$ from $L_1({\mathbb{R}}^2)$, sufficient and necessary conditions for the belonging of their Fourier transform $\widehat{f}$ to $L_1({\mathbb{R}}^2)$ as well as of a function $t\cdot \sup\limits_{y_{1}^{2}+y_{2}^{2}\geq t^{2}}\big|\widehat{f}(y_{1},y_{2})\big|$to$L_{1}($\mathbb{R}$^{1}_{+})$. As for the positivity of $\widehat{f}$ on ${\mathbb{R}}^2$, it is completely reduced to the same question on ${\mathbb{R}}^1$ for a function $f_1(x)=\mathrm{\mid }x\mathrm{\mid }{f}_0\left(\mathrm{\mid }x\mathrm{\mid }\right)+{\int }_{\mathrm{\mid }x\mathrm{\mid }}^{\mathrm{\infty }}{f}_0(t)dt$.