The Fourier transform of bivariate functions that depend only on the maximum of the absolute values of their variables

Given an $L_1(\mathbb{R}^2)$-function $f(x_1,x_2)=f_0(\max\{|x_1|,|x_2|\})$, necessary conditions and sufficient conditions for its Fourier transform $\widehat{f}$ to lie in $L_1(\mathbb{R}^2)$ and for the function $t\mapsto t\sup_{y_1^2+y_2^2\geqslant t^2}|\widehat{f}(y_1,y_2)|$ to be in $L_1(\mathbb{R}_{+})$ are indicated. The problem of the positivity of $\widehat{f}$ on $\mathbb{R}^2$ is shown to be completely reducible to the same problem for the function $\displaystyle f_1(x)=|x|f_0(x)+\int_{|x|}^\infty f_0(t)\,dt$ in $\mathbb{R}$.
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